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\begin{document}
\author{Volodymyr Derkach} \address{
Institut f\"ur Mathematik,
Technische Universit\"{a}t Ilmenau,
Germany} \address{
Department of Mathematics,
Vasyl Stus Donetsk National University,
Vinnytsya, Ukraine} \email{[email protected]}
\author{Dmytro Strelnikov} \address{
Institut f\"ur Mathematik,
Technische Universit\"{a}t Ilmenau,
Germany} \email{[email protected]}
\author{Henrik Winkler} \address{
Institut f\"ur Mathematik,
Technische Universit\"{a}t Ilmenau,
Germany} \email{[email protected]}
\thanks{The research of the first author was supported by a grant of the German Research Foundation (DFG, grant TR 903/22-1) and a grant of the Volkswagen Foundation.}
\begin{abstract} We study spectral problems for two--dimensional integral system with two given non-decreasing functions $R_1$, $R_2$ on an interval $[0,b)$ which is a generalization of the Krein string. Associated to this system are the maximal linear relation $T_{\max}$ and the minimal linear relation $T_{\min}$ in the space $L^2(R_2)$ which are connected by $T_{\max}=T_{\min}^*$. It is shown that the limit point condition at $b$ for this system is equivalent to the strong limit point condition for the linear relation $T_{\max}$. In the limit circle case the strong limit point condition fails to hold on $T_{\max}$ but it is still satisfied on a subspace $T_N^*$ of $T_{\max}$ characterized by the Neumann boundary condition at $b$. The notion of the principal Titchmarsh-Weyl coefficient of this integral system is introduced both in the limit point case and in the limit circle case. Boundary triples for the linear relation $T_{\max}$ in the limit point case (and for $T_{N}^*$ in the limit circle case) are constructed and it is shown that the corresponding Weyl function coincides with the principal Titchmarsh-Weyl coefficient of the integral system. The notion of the dual integral system is introduced by reversing the order of $R_1$ and $R_2$. It is shown that the principal Titchmarsh-Weyl coefficients $q$ and $\widehat q$ of the direct and the dual integral systems are related by the equality $\lambda \widehat q(\lambda) = -1/q(\lambda)$ both in the regular and the singular case.
\end{abstract}
\subjclass{Primary 34B24; Secondary 34L05, 47A06, 47A57, 47B25, 47E05}
\date{\today}
\keywords{integral systems, Kre\u{\i}n strings, dual systems, principal Titchmarsh-Weyl coefficient, boundary triples, symmetric linear relations}
\title{On a class of integral systems}
\section{Introduction} \label{sec:intro}
In this paper spectral problems for integral systems, associated dual systems and, in particular, Krein strings are investigated. We consider an integral system of the form \begin{equation}\label{eq:IntSys}
u(x,\lambda) = u(0,\lambda) -J \int_0^x
\begin{bmatrix} \lambda dR_2(t) & 0 \\ 0 & dR_1(t)\end{bmatrix} u(t,\lambda),\quad J=\begin{bmatrix} 0 & -1 \\ 1 & 0\end{bmatrix},\quad \end{equation} where $u = [u_1\ u_2]^T$, with some spectral parameter $\lambda\in\dC$ and measures $dR_1$ and $dR_2$ associated with non-decreasing functions $R_1(x)$ and $R_2(x)$ on an interval $[0,b)$, see~\cite{Ben89}. If $R_1(x)\equiv x$ then $u_2=u_1'$ and system \eqref{eq:IntSys} is reduced to the equation of a vibrating string in the sense of M.\,G.~Krein \begin{equation}\label{eq:Krein_S}
u_1(x,\lambda) = u_1(0,\lambda) +x u_1'(0,\lambda) - \lambda \int_0^x (x-t) u_1(t,s)\, dR_2(t),
\quad x\in[0,b). \end{equation} Integral systems \eqref{eq:IntSys} arise in the theory of diffusion processes with two measures \cite{Man68,LaSc90}. In the theory of stochastic processes the equation~\eqref{eq:Krein_S} describes generalized diffusion processes which includes both diffusion processes and birth and death processes ~\cite{Fel57,Fel59,IMcK65,Kas75}. In mechanics the equation~\eqref{eq:Krein_S} describes small transverse oscillations of the string with the mass distribution function $R_2(x)$,~\cite{KacK68}. Relations for the Titchmarsh-Weyl coefficients of Krein strings and their associated dual strings were studied in~\cite{KacK68, KWW07}.
Let $c(\cdot,\lambda)$ and $s(\cdot,\lambda)$ be the unique solutions of~\eqref{eq:Krein_S} satisfying the initial conditions \begin{equation}
c(0,\lambda) = 1,\ c'(0,\lambda) = 0,
\quad \text{and} \quad
s(0,\lambda) = 0,\ s'(0,\lambda) = 1. \end{equation} The function \begin{equation}\label{eq:m_NK}
q_S(\lambda) \coloneqq \lim_{x\to b}\frac{s(x,\lambda)}{c(x,\lambda)} \end{equation} is called \emph{the principal Titchmarsh-Weyl coefficient of the string} \cite{KWW07} or \emph{the dynamic compliance coefficient} in the terminology of I.\,S.~Kac and M.\,G.~Krein \cite{KacK68} and describes the spectral properties of the string. The principal Titchmarsh-Weyl coefficient $q(\lambda)$ is a Stieltjes function and the measure $d\sigma$ from its integral representation \begin{equation}\label{eq:Int_Rep_mS}
q_S(\lambda)=a+\int_0^\infty\frac{d\sigma(t)}{t-\lambda}, \quad a\ge 0 \end{equation} is the spectral measure of the string $S_1[b,R_2]$, which in the limit point case is given by the boundary condition $u'(0)=0$.
Denote the integral system~\eqref{eq:IntSys} by $S[R_1,R_2]$. In the present paper we define \emph{the principal Titchmarsh-Weyl coefficient} $q$ of the integral system $S[R_1,R_2]$ by \begin{equation}\label{eq:m_TW}
q(\lambda) \coloneqq \lim_{x\to b}\frac{s_1(x,\lambda)}{c_1(x,\lambda)}, \end{equation} where $c_1(\cdot, \lambda)$, $c_2(\cdot, \lambda)$ and $s_1(\cdot, \lambda)$, $s_2(\cdot, \lambda)$ are pairs of the unique solutions of~\eqref{eq:IntSys} satisfying the initial conditions \begin{equation} \label{eq:csK}
c_1(0,\lambda) = 1,\ c_2(0,\lambda) = 0,
\quad \text{and} \quad
s_1(0,\lambda) = 0,\ s_2(0,\lambda) = 1. \end{equation} Formula~\eqref{eq:m_TW} requires justification. For this purpose we use the operator approach to the integral system $S[R_1,R_2]$ developed in~\cite{Str19}, the boundary triples technique from~\cite{Koc75,GG91} and the theory of associated Weyl functions as introduced in ~\cite{DM91,DM95}. The maximal linear relation $T_{\max}$ is defined as the set of pairs $\bm{u} = [u_1\ f]^T$ such that $u_1,f\in L^2(R_2)$ and the equation~\eqref{eq:sys_lin_rel} is satisfied for some $u_2\in BV_{\loc}[0,b)$, see Definition~\ref{def:Tmax}. The closure of its restriction to the set of compactly supported functions is called the minimal linear relation $T_{\min}$.
In~\cite{Str19} it is shown that $T_{\min}$ is symmetric in $L^2(R_2)$, $T_{\max}=T_{\min}^*$
and boundary triples for the linear relation $T_{\min}$ were constructed both in the limit point and in the limit circle case.
In Theorem~\ref{thm:SLP} we show that the system $S[R_1,R_2]$ is in the limit point case at $b$ if and only if it satisfies \emph{the strong limit point condition} at $b$, see~\cite{Ever76}, which in our case is of the form \begin{equation}\label{eq:SLP}
\lim_{x\to b}u_1(x)u_2(x) = 0
\quad \text{for all} \quad \bm{u} \in T_{\max}. \end{equation} As a consequence of~\eqref{eq:SLP} we conclude that in the limit point case the linear relation $T_{\min}$ and its von Neumann extension $A_N$, characterized by the boundary condition $u_2(0)=0$, are nonnegative, the corresponding Weyl function is a Stieltjes function and coincides with the principal Titchmarsh-Weyl coefficient of the system $S[R_1,R_2]$. The strong limit point condition for second order differential operators was introduced by W. Everitt~\cite{Ever76}.
In the limit circle case the linear relation $T_{\min}$ has defect numbers $(2,2)$, in this case an intermediate symmetric extension $T_N$ with defect numbers $(1,1)$ of $T_{\min}$ is considered as the restriction of $T_{\max}$ to the set of elements $\bm{u}\in T_{\max}$ such that $u_1(0)=u_2(0)=u_2(b)=0$. In this case we show in Lemma~\ref{lem:psi} that the strong limit point condition \eqref{eq:SLP} fails to hold, but still the limit in~\eqref{eq:SLP} is vanishing on the subspace $T_{N}^*$ of $T_{\max}$, i.e. the following \emph{Evans--Everitt condition} holds, cf.~\cite{EvEver}: \begin{equation}\label{eq:Ever}
\lim_{x\to b} u_1(x)u_2(x) = 0
\quad \text{for all} \quad \bm{u}\in T_{N}^*. \end{equation} This result implies the nonnegativity of the linear relation $T_{N}$ and its selfadjoint extension \begin{equation*}
A_N = \{\bm{u} \in T_{\max} \colon u_2(0) = u_2(b) = 0\}. \end{equation*} In~\cite{Kost13} another analytical object --- \emph{the Neumann $m$-function} of the system $S[R_1,R_2]$ was introduced by the equality \begin{equation}\label{eq:m_N}
m_N(\lambda) \coloneqq \lim_{x\to b} \frac{s_2(x,\lambda)}{c_2(x,\lambda)}, \end{equation} which is a special case of a more general definition of the Neumann $m$-function presented in~\cite{Ben89}. In Proposition~\ref{prop:BT_IS_N} it is shown that the Neumann $m$-function $m_N(\lambda)$ is a Stieltjes function and it coincides with the principal Titchmarsh-Weyl coefficient of the integral system $S[R_1,R_2]$.
The system $S[R_1,R_2]$ is called \emph{regular} if $R_1(b)+R_2(b)<\infty$ and \emph{singular} otherwise. In the regular case we construct \emph{the canonical singular extension} $S[\widetilde R_1,\widetilde R_2]$ of the system $S[R_1,R_2]$ with $R_1, R_2$ extended to non-decreasing functions $\widetilde R_1, \widetilde R_2$ on the interval $(0,\infty)$, so that the principal Titchmarsh-Weyl coefficients of both systems coincide.
\emph{The dual system} $\widehat S[R_1,R_2]$ of the integral system $S[R_1,R_2]$ in the singular case is obtained by changing the roles of $R_1$ and $R_2$. In the regular case the dual system of the integral system $S[R_1,R_2]$ is defined as the dual of the canonical singular extension $S[\widetilde R_1,\widetilde R_2]$ of the system $S[R_1,R_2]$. The main result of the paper is Theorem~\ref{thm:2} where it is shown that the principal Titchmarsh-Weyl coefficient $\widehat q$ of the dual system is related to the principal Titchmarsh-Weyl coefficient $q$ of the system $S[R_1,R_2]$ by the equality \begin{equation}\label{eq:mS_whmS}
\widehat q(\lambda) = -\frac{1}{\lambda q(\lambda)}. \end{equation} both in the regular and the singular case.
In the case of a string ($R_1(x)=x$) the notion of the dual string and the formula~\eqref{eq:mS_whmS} connecting the principal Titchmarsh-Weyl coefficients of the direct and the dual string in the singular case was presented in~\cite{KacKr58}. In~\cite{KWW07} some further relations between strings, dual strings and canonical systems of differential equations were studied. Analogues of these relations between integral systems and canonical systems can also be established and will be presented in a forthcoming paper.
\section{Preliminaries} \subsection{Linear relations} \label{subsec:pre:lr} Let $\sH$ be a Hilbert space.
A linear relation $T$ in $\sH$ is a linear subspace of $\sH \times \sH$. Let us recall some basic definitions and properties associated with linear relations in~\cite{Arens,Ben72}.
The \emph{domain}, the \emph{range}, the \emph{kernel}, and the \emph{multivalued part} of a linear relation $T$ are defined as follows: \begin{align}
\dom{T} &\coloneqq \left\{ f \colon \begin{bmatrix} f \\ g \end{bmatrix} \in T \right\}, &
\ran{T} &\coloneqq \left\{ g \colon \begin{bmatrix} f \\ g \end{bmatrix} \in T \right\},\\
\ker{T} &\coloneqq \left\{ f \colon \begin{bmatrix} f \\ 0 \end{bmatrix} \in T \right\}, &
\mul{T} &\coloneqq \left\{ g \colon \begin{bmatrix} 0 \\ g \end{bmatrix} \in T \right\}. \end{align} The \emph{adjoint} linear relation $T^*$ is defined by \begin{equation}
T^* \coloneqq \left\{
\begin{bmatrix} u \\ f \end{bmatrix}
\in \sH \times \sH \colon \langle f,v\rangle_{\sH} = \langle u,g\rangle_{\sH}\ \text{for any}\
\begin{bmatrix} v \\ g \end{bmatrix}
\in T \right\}. \end{equation} A linear relation $T$ in $\sH$ is called \emph{closed} if $T$ is closed as a subspace of $\sH \times \sH$. The set of all closed linear operators (relations) is denoted by $\mathcal{C}(\sH)$ ($\widetilde{\mathcal{C}}(\sH)$). Identifying a linear operator $T \in \mathcal{C}(\sH)$ with its graph one can consider $\mathcal{C}(\sH)$ as a part of $\widetilde{\mathcal{C}}(\sH)$.
Let $T$ be a closed linear relation, $\lambda \in \mathbb{C}$, then
\begin{equation}
T - \lambda I \coloneqq \left\{ \begin{bmatrix} f \\ g-\lambda f \end{bmatrix} \colon
\begin{bmatrix} f \\ g \end{bmatrix}
\in T \right\}.
\end{equation}
A point $\lambda \in \mathbb{C}$ such that $\ker{\left( T - \lambda I \right)} = \{0\}$ and $\ran{\left( T - \lambda I \right)} = \sH$ is called a \emph{regular point} of the linear relation $T$. Let $\rho(T)$ be the set of regular points.
The \emph{point spectrum} $\sigma_p(T)$ of the linear relation $T$
is defined by
\begin{equation}\label{eq:Point_s}
\sigma_p(T) \coloneqq
\{\lambda\in\mathbb{C} \colon \ker(T-\lambda I)\ne\{0\}\},
\end{equation}
A linear relation $T$ is called \emph{symmetric} if $T \subseteq T^*$. A point $\lambda \in \mathbb{C}$ is called a \emph{point of regular type} (and is written as $\lambda \in \widehat{\rho}(T)$) for a closed symmetric linear relation $T$, if $\lambda \notin \sigma_p(T)$ and the subspace $\ran(T-\lambda I)$ is closed in $H$. For $\lambda \in \widehat{\rho}(T)$ let us set $\sN_{\lambda}(T^*) \coloneqq \ker (T^* -\lambda I)$ and \begin{equation}
\widehat{\sN}_{\lambda}(T^*) \coloneqq
\left\{
\bm{u}_{\lambda} =
\begin{bmatrix} u_{\lambda} \\ \lambda u_{\lambda} \end{bmatrix} \colon u_{\lambda} \in \sN_{\lambda}(T^*)
\right\}. \end{equation} The \emph{deficiency indices} of a symmetric linear relation $T$ are defined as \begin{equation}
n_{\pm}(T) \coloneqq \dim \ker (T^* \mp iI). \end{equation}
\subsection{Boundary triples and Weyl functions} \label{subsec:pre:triples} Let $T$ be a symmetric linear relation with deficiency indices $(1,1)$. In the case of a densely defined operator the notion of the boundary triple was introduced in \cite{Koc75,GG91}. Following the papers \cite{M92,DM95} we shall give a definition of a boundary triple for the linear relation $T^*$.
\begin{definition} \label{def:btriple}
A tuple $\Pi = (\dC,\Gamma_0,\Gamma_1)$, where $\Gamma_0$ and $\Gamma_1$ are linear mappings from $T^*$ to $\dC$, is called a \emph{boundary triple} for the linear relation $T^*$, if:
\begin{enumerate}
\item [(i)]
for all
$\bm{u} = \begin{bmatrix} u \\ f \end{bmatrix}$,
$\bm{v} = \begin{bmatrix} v \\ g \end{bmatrix} \in T^*$
the following generalized Green's identity holds
\begin{equation} \label{eq:1.9}
\langle f,v \rangle_{\sH} - \langle u,g \rangle_{\sH} =\Gamma_1 \bm{u} \conj{\Gamma_0\bm{v}} -\Gamma_0 \bm{u} \conj{\Gamma_1\bm{v}};
\end{equation}
\item [(ii)]
the mapping
$\Gamma=\begin{bmatrix}\Gamma_0 \\ \Gamma_1\end{bmatrix} \colon T^* \rightarrow \dC^2$
is surjective.
\end{enumerate} \end{definition}
Notice, that in contrast to~\cite{M92} the linear relation $T$ is not supposed to be single-valued. The following linear relations \begin{equation} \label{e q:A0A1}
A_0 \coloneqq \ker \Gamma_0, \qquad A_1 \coloneqq \ker \Gamma_1 \end{equation} are selfadjoint extensions of the symmetric linear relation $T$.
\begin{definition}[\cite{DM91,DM95}] \label{def:M_gamma}
Let $\Pi = (\dC,\Gamma_0,\Gamma_1)$ be a boundary triple for the linear relation $T^*$.
The scalar function $m(\cdot)$ and the vector valued function $\gamma(\cdot)$ defined by
\begin{equation}\label{def:Weyl}
m(\lambda) \Gamma_0 \bm{u}_\lambda = \Gamma_1 \bm{u}_\lambda, \quad
\gamma(\lambda) \Gamma_0 \bm{u}_\lambda = u_{\lambda}, \quad
\bm{u}_\lambda=
\begin{bmatrix} u_{\lambda} \\ \lambda u_{\lambda} \end{bmatrix} \in \widehat{\sN}_{\lambda}(T^*), \quad
\lambda \in \rho(A_0)
\end{equation}
are called \emph{the Weyl function} and \emph{the $\gamma$-field} of the symmetric linear relation $T$ corresponding to the boundary triple $\Pi$. \end{definition}
The Weyl function and the $\gamma$-field are connected via the next identity (see \cite{DM95}) \begin{equation} \label{eq:weyl-id}
m(\lambda) - m(\zeta)^* =
(\lambda - \conj{\zeta}) \gamma(\zeta)^* \gamma(\lambda),
\quad
\lambda,\zeta \in \rho(A_0). \end{equation}
\begin{definition}[\cite{KaKr74}]
A function
$m \colon \mathbb{C}\setminus\dR\to \cB ({\mathcal H})$ is said
to be a \emph{Herglotz-Nevanlinna function}, if the following conditions hold:
\begin{enumerate}
\item [(i)]
$m$ is holomorphic in $\mathbb{C}\setminus\dR$;
\item [(ii)]
$\operatorname{Im} m(\lambda) \geq 0$ as $\lambda \in \mathbb{C}_+:=\{\lambda\in\dC: \text{Im }\lambda>0\}$;
\item [(iii)]
$m(\conj{\lambda}) = m^*(\lambda)$ for $\lambda \in \mathbb{C}\setminus\dR$.
\end{enumerate} \end{definition}
It follows from~\eqref{eq:weyl-id} that the Weyl function $m(\cdot)$ is a Herglotz-Nevanlinna function. A Herglotz-Nevanlinna function $m$ which admits a holomorphic continuation to $\dR_-$ and takes nonnegative values for all $\lambda\in\dR_-$ is called a \emph{Stieltjes function}. Every Stieltjes function $m$ admits an integral representation \eqref{eq:Int_Rep_mS} with a non-decreasing function $\sigma(t)$ such that $\int_{\dR_+}(1+t)^{-1}d\sigma(t)<\infty$.
\subsection{Minimal and maximal relations associated with the integral system $S[R_1,R_2]$ }
Let $I=[0,b)$ be an interval with $b\le\infty$, let $R(x)$ be a non-decreasing left-continuous function on $I$ such that $R(0)=0$, let $dR$ be the corresponding Lebesgue-Stieltjes measure, and let ${\mathcal L}^2(R,I)$ be an inner product space which consists of complex valued functions $f$ such that \begin{equation}
\int_I |f(t)|^2\, dR(t) < \infty \end{equation} with inner product defined by \begin{equation}
\langle f,g \rangle_{R} = \int_I f(t) \conj{g(t)} dR(t). \end{equation} ${\mathcal L}^2_{\comp}(R,I)$ denotes the subspace consisting of those $f\in {\mathcal L}^2(R,I)$ with compact support in $I$, $BV[0,b)$ denotes the set of functions of bounded variation on $[0,b)$ and $BV_{\loc}[0,b)$ is the set of functions $f$ such that $f\in BV[0,b')$ for every $b'<b$. Denote by $L^2(R,I)$ the corresponding quotient space for ${\mathcal L}^2(R,I)$, which consists of equivalence classes w.r.t. $dR$ and denote by $\pi$ the corresponding quotient map, i.e. $\pi \colon {\mathcal L}^2(R,I) \to L^2(R,I)$. Further we omit $I$ in the notation if it coincides with $[0,b)$.
From now on the following convention is used for the integration limits for any measure $d\sigma$ on an interval: \begin{equation}
\int_a^x f\, d\sigma \coloneqq \int_{[a,x)} f\, d\sigma. \end{equation} Thus, an integral as a function of its upper limit is always left-continuous. With every function of bounded variation $f$ we associate the left-continuous and the right-continuous functions $f_-$ and $f_+$ defined by \begin{equation}
f_-(x) \coloneqq \lim_{t\uparrow x} f(t), \quad
f_+(x) \coloneqq \lim_{t\downarrow x} f(t). \end{equation}
Let $u$ and $v$ be left-continuous functions of bounded variation, $du$ and $dv$ be the corresponding Lebesgue-Stieltjes measures. The following integration-by-parts formula for the Lebesgue-Stieltjes integral (see e.g. \cite{Hew60}) is used throughout the paper \begin{equation} \label{eq:in-by-parts}
\int_a^x u\, dv + \int_a^x v_+\, du = u(x)v(x) - u(a)v(a). \end{equation}
If $u$ and $u_+$ have no zeros then it follows with $v=1/u$ from~\eqref{eq:in-by-parts} \begin{equation*}
d(1) = d \left( \frac{u}{u} \right) =
u\, d\left( \frac{1}{u} \right) + \frac{1}{u_+}\, du = 0. \end{equation*} This leads to the quotient-rule formula \begin{equation} \label{eq:quotient-rule}
d\left( \frac{1}{u} \right) = -\frac{du}{u u_+}. \end{equation}
The following existence and uniqueness theorem for integral systems was proven in \cite[Theorem 1.1]{Ben89}.
\begin{theorem}\label{thm:Ex_Uniq} Let $dS$ be a complex $n\times n$ matrix-valued measure. For every left continuous (either $n\times n$ or $n\times1$ matrix valued) function $A(x)$ in $BV_{\loc}[0,b)$ there is a unique function $U$ such that the equality
\begin{equation}\label{eq:sys_SolvTH}
U(x) = A(x) + \int_0^x dS \cdot U
\end{equation}
holds for every point $x\in[0,b)$. \end{theorem} \begin{remark}\label{rem:2.6}
Due to the properties of the Lebesgue-Stieltjes integral and the used convention, any solution $U$ to~\eqref{eq:sys_SolvTH} is left continuous and belongs to $BV_{\loc}[0,b)$, componentwise. \end{remark}
Now we focus on integral systems $S[R_1,R_2]$ of the form~\eqref{eq:IntSys}, where $R_1(x)$ and $R_2(x)$ are nondecreasing and left-continuous real-valued functions on the interval $I=[0,b)$ such that $R_1(0)=R_2(0)=0$. We define the corresponding inhomogeneous system.
\begin{definition} \label{def:A_triple}
Let $f \in {\mathcal L}^2(R_2)$ and $[u_1\ u_2]^T$ be a vector-valued function such that the following equation
\begin{equation} \label{eq:sys_lin_rel}
\begin{bmatrix} u_1\\u_2 \end{bmatrix} (x)
=
\begin{bmatrix} u_1\\u_2 \end{bmatrix} (0)
- J \int_0^x
\begin{bmatrix}
dR_2 & 0 \\
0 & dR_1
\end{bmatrix}
\begin{bmatrix} f\\u_2 \end{bmatrix}
\end{equation}
holds for every point $x\in[0,b)$.
The triple $(u_1,u_2,f)$ is said to belong to the set $\cT$
if $u_1 \in {\mathcal L}^2(R_2)$. \end{definition} Due to Remark~\ref{rem:2.6} for every $(u_1,u_2,f)\in\cT$ both functions $u_1$ and $u_2$ belong to $BV_{\loc}[0,b)$. Theorem~\ref{thm:Ex_Uniq} implies that for every $f \in {\mathcal L}^2(R_2)$ the vector-valued function $[u_1\ u_2]^T$ satisfying~\eqref{eq:sys_lin_rel} is uniquely determined by its initial values at zero, however $u_1 \in {\mathcal L}^2(R_2)$ is not guaranteed for an arbitrary $f$.
\begin{definition}\label{def:Tmax}
We define the maximal and the pre-minimal relations $T_{\max}$, $T'\subset L^2(R_2) \times L^2(R_2)$ by
\begin{equation} \label{eq:Pi_map}
T_{\max} \coloneqq \left\{
\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \colon
(u_1, u_2, f) \in \cT \right\}.
\end{equation}
\begin{equation} \label{eq:Pi_min}
T' \coloneqq \left\{
\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix}
\in T_{\max} \colon
(u_1, u_2, f) \in \cT, \ u_1, f\in L^2_{\comp}(R_2,I)
\right\}.
\end{equation} \end{definition}
Everywhere in the paper, except Remark~\ref{rem:MP}, we suppose that the following two natural assumptions hold. \begin{assumption} \label{assum:R1R2}
The functions $R_1$ and $R_2$ have no common points of discontinuity. \end{assumption}
\begin{assumption} \label{assum:surj}
There exists an interval $[0,b_0) \subseteq [0,b)$ such that
\begin{equation} \label{eq:assum_cond}
\dim\spn\{\pi 1,\pi R_1\} = 2
\end{equation}
where $\pi \colon {\mathcal L}^2(R_2,[0,b_0)) \to L^2(R_2,[0,b_0))$ is the corresponding quotient map. \end{assumption}
Assumption~\ref{assum:R1R2} has the important consequence that the first component of a solution has no discontinuity in common with the second component of any solution $(u_1, u_2, f) \in \cT$. Assumption~\ref{assum:surj} makes it possible to assign correctly the values $u_1(x)$ and $u_2(x)$ for every $\bm{u} \in T_{\max}$. In case of absolutely continuous functions $R_1$ and $R_2$ the equivalent to $S[R_1,R_2]$ differential system is \emph{definite} in the sense of \cite[Definition 2.14]{LM03} if and only if Assumption~\ref{assum:surj} holds. \begin{definition}
Let $(u_1,u_2,f) \in \cT$ and $ \bm{u} \in T_{\max}$ be its image under the mapping \begin{equation}\label{eq:Pi_map1}
\cT \ni (u_1,u_2,f) \mapsto
\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix}
\in T_{\max}. \end{equation}
The mappings $\phi_{1,2}[x] \colon T_{\max} \to {\mathbb C}$ are defined by
\begin{equation}
\phi_i [x] \bm{u} \coloneqq u_i (x), \quad i \in \{1,2\},\quad x\in[0,b).
\end{equation} \end{definition} The following Proposition provides an analog of \cite[Proposition 2.15]{LM03} for integral system $S[R_1,R_2]$. \begin{proposition} If Assumptions~\ref{assum:R1R2} and~\ref{assum:surj} hold then
the mappings $\phi_{1,2}[x]$ are well-defined. \end{proposition} \begin{proof}
In general, the mapping defined by~\eqref{eq:Pi_map1} is not invertible.
Suppose that $(u_1,u_2,f)$ and $(\widetilde{u}_1,\widetilde{u}_2,\widetilde{f})$ are two pre-images of
$\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \in T_{\max}$
as it is shown on the following diagram.
\begin{center}
\begin{tikzcd}
(u_1,u_2,f) \arrow{d} \arrow{r} &
\bm{u} \arrow[dashed]{d} &
\arrow{l} (\widetilde{u}_1,\widetilde{u}_2,\widetilde{f}) \arrow{d} \\
u_{1,2}(x) \arrow{r} &
\phi_{1,2}[x] \bm{u} &
\arrow{l} \widetilde{u}_{1,2}(x)
\end{tikzcd}
\end{center}
Let us show that due to Assumption~\ref{assum:surj}
\begin{equation}
u_1(x) = \widetilde{u}_1(x), \quad u_2(x) = \widetilde{u}_2(x), \quad \text{for}\ t \in [0,b).
\end{equation}
Clearly, $(u_1-\widetilde{u}_1, u_2-\widetilde{u}_2, Y-\widetilde{Y}) \in \cT$. Taking into account $\pi u_1 = \pi \widetilde{u}_1$, $\pi f = \pi \widetilde{f}$, it follows from~\eqref{eq:sys_lin_rel} that
\begin{equation} \label{eq:y1y2_recovery}
\begin{bmatrix}
u_1(x) - \widetilde{u}_1(x)\\ u_2(x) - \widetilde{u}_2(x)
\end{bmatrix}
=
\begin{bmatrix}
(u_1(0) - \widetilde{u}_1(0)) + (u_2(0) - \widetilde{u}_2(0))R_1(x)\\
u_2(0) - \widetilde{u}_2(0)
\end{bmatrix}.
\end{equation}
The mapping $\pi$ to applied the first line of~\eqref{eq:y1y2_recovery} gives
\begin{equation}
0 = (u_1(0) - \widetilde{u}_1(0)) \cdot \pi 1 + (u_2(0) - \widetilde{u}_2(0)) \cdot \pi R_1.
\end{equation}
Now it follows from~\eqref{eq:assum_cond} that $u_1(0) = \widetilde{u}_1(0)$, $u_2(0) = \widetilde{u}_2(0)$, which together with~\eqref{eq:y1y2_recovery} completes the proof. \end{proof}
Further in the text we will simply write $u_{1,2}(x)$ instead of $\phi_{1,2}[x]\bm{u}$ unless this can lead to confusion. For a pair of vector-valued functions $u = \begin{bmatrix} u_1 & u_2 \end{bmatrix}^T$, $v = \begin{bmatrix} v_1 & v_2 \end{bmatrix}^T$ we define the generalized Wronskian by \begin{equation}
[u,v](x) \coloneqq u_1(x) v_2(x) - u_2(x) v_1(x). \end{equation}
\begin{proposition} \label{prop:green_formulas} If $(u_1,u_2,f)$ and $(v_1,v_2,g)$ belong to $\cT$ then the following generalized first and second Green's identities hold
\begin{equation}\label{eq:G1}
\int_0^x f v_1\, dR_2 = \int_0^x u_2 v_2\, dR_1 - u_2(x) v_1(x) + u_2(0) v_1(0),
\end{equation}
\begin{equation}\label{eq:G2}
\int_0^x (f v_1 - u_1 g) \, dR_2 = [u,v](x) - [u,v](0).
\end{equation}
for an arbitrary interval $[0,x) \subset [0,b)$. \end{proposition} \begin{proof}
We recall that due to Assumption~\ref{assum:R1R2} the functions $R_1$ and $R_2$ do not have common points of discontinuity, so neither do the functions $v_1$ and $u_2$.
By virtue of~\eqref{eq:sys_lin_rel} we get
\begin{equation}
dv_1 = v_2\, dR_1, \quad du_2 = - f\, dR_2.
\end{equation}
and hence, using the integration-by-parts formula~\eqref{eq:in-by-parts},
\begin{equation} \label{eq:2.7}
d(u_2 v_1) =
v_1\, du_2 + u_2\, dv_1 = u_2 v_2\, dR_1 - f v_1\, dR_2.
\end{equation}
Integrating \eqref{eq:2.7} over $[0,x)$ provides~\eqref{eq:G1}.
Swapping the tuples $(u_1,u_2,f)$ and $(v_1,v_2,g)$ in~\eqref{eq:2.7} and subtracting the obtained expression from~\eqref{eq:2.7} proves~\eqref{eq:G2}. \end{proof}
Theorem~\ref{thm:Ex_Uniq} provides that system $S[R_1,R_2]$ has a unique solution for every choice of initial values. Let $c(\cdot,\lambda) = [c_1(\cdot,\lambda)\ c_2(\cdot,\lambda)]^T$ and $s(\cdot,\lambda) = [s_1(\cdot,\lambda)\ s_2(\cdot,\lambda)]^T$ be its unique solutions satisfying the initial conditions~\eqref{eq:csK}. \begin{corollary}
For every $\lambda\in\dC$ and $x\in [0,b)$ the following formulas hold:
\begin{equation} \label{eq:Liouv}
[c(\cdot,\lambda),s(\cdot,\lambda)](x) = c_1(x,\lambda) s_2(x,\lambda) - c_2(x,\lambda) s_1(x,\lambda) = 1,
\end{equation}
\begin{equation} \label{eq:Liouv+}
c_{1+}(x,\lambda) s_2(x,\lambda) - c_2(x,\lambda) s_{1+}(x,\lambda) = 1,
\end{equation}
\begin{equation} \label{eq:Liouv++}
c_1(x,\lambda) s_{2+}(x,\lambda) - c_{2+}(x,\lambda) s_1(x,\lambda) = 1.
\end{equation}
\end{corollary} \begin{proof}
Equality~\eqref{eq:Liouv} follows immediately from either~\eqref{eq:Kernel_U} or \eqref{eq:G2}. Further we subtract the left-hand side of \eqref{eq:Liouv} from the left-hand side of \eqref{eq:Liouv+}:
\begin{multline} \label{eq:cor_Liouv}
(c_{1+}(x,\lambda) s_2(x,\lambda) - c_2(x,\lambda) s_{1+}(x,\lambda)) -
(c_1(x,\lambda) s_2(x,\lambda) - c_2(x,\lambda) s_1(x,\lambda)) =\\
(c_{1+}(x,\lambda)-c_1(x,\lambda)) s_2(x,\lambda) -
c_2(x,\lambda) (s_{1+}(x,\lambda)- s_1(x,\lambda))
\end{multline}
One can immediately see that the expression~\eqref{eq:cor_Liouv} is equal to zero at every point of continuity of $R_1$. Let $x_0$ be a point of discontinuity of $R_1$. From~\eqref{eq:sys_lin_rel} one can see that
\begin{align}
c_{1+}(x_0,\lambda)-c_1(x_0,\lambda) = c_2(x_0, \lambda)\, dR_1(\{x_0\}),\\
s_{1+}(x_0,\lambda)-s_1(x_0,\lambda) = s_2(x_0, \lambda)\, dR_1(\{x_0\})
\end{align}
and hence
\begin{multline}
(c_{1+}(x_0,\lambda)-c_1(x_0,\lambda)) s_2(x_0,\lambda) -
c_2(x_0,\lambda) (s_{1+}(x_0,\lambda)- s_1(x_0,\lambda)) = \\
c_2(x_0,\lambda) s_2(x_0,\lambda)\, dR_1(\{x_0\}) -
s_2(x_0,\lambda) c_2(x_0,\lambda)\, dR_1(\{x_0\}) = 0.
\end{multline}
The proof of~\eqref{eq:Liouv++} is similar. \end{proof}
It follows from~\eqref{eq:G2} that the pre-minimal relation $T'$ is symmetric in $L^2(R_2)$. \begin{definition} \label{def:Amin} The minimal relation $T_{\min}$ is defined as the closure of the pre-minimal linear relation $T'$: $T_{\min} = \clos{T'}$. \end{definition} As was shown in~\cite{Str19} the linear relation $T_{\min}$ is also symmetric, $T_{\min}^* = T_{\max}$ and
\begin{equation} \label{eq:def-Amin}
T_{\min} \coloneqq
\left\{ \bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \in T_{\max} \colon
u_1(0) = u_2(0) = [u,v]_b = 0\ \text{for all}\
\bm{v} = \begin{bmatrix} \pi v_1 \\ \pi g \end{bmatrix} \in T_{\max} \right\}.
\end{equation}
\begin{lemma} \label{lem:Weyl_disks} Let $l<b$, $h\in\clos{{\mathbb C}_+}\cup\{\infty\}$, and let $m(\lambda,l,h)$ be some coefficient such that the function \begin{equation} \label{eTWeyl}
\psi(t,\lambda) \coloneqq s(t,\lambda)-m(\lambda,l,h)\, c(t,\lambda) \end{equation} satisfies the condition $\psi_1(l,\lambda) + h \psi_2(l,\lambda) = 0$. Then: \begin{enumerate}
\item [(i)] The coefficient $m$ is well-defined and can be calculated as \begin{equation} \label{eq:mlhl}
m(\lambda,l,h) =
\frac{s_1(l,\lambda) + h s_2(l,\lambda)}{c_1(l,\lambda) + h c_2(l,\lambda)}. \end{equation}
\item [(ii)] For every $\lambda \in {\mathbb C}_+$ the set
$D_l(\lambda) \coloneqq \{m(\lambda,l,h) \colon h\in\clos{\dC_+}\cup\{\infty\}\}$ is a disk in ${\mathbb C}_+$ such that $\omega \in D_l(\lambda)$ if and only if \begin{equation} \label{eq:WeylD}
\int_0^l |s_1(t,\lambda) - \omega c_1(t,\lambda)|^2 dR_2(t) \le
\frac{\operatorname{Im}\omega}{\operatorname{Im}\lambda}, \end{equation} and its radius can be calculated as \begin{equation}\label{Abh.13.10}
r_l(\lambda) = \left(2 \operatorname{Im} \lambda\int_0^l |s_1(t,\lambda)|^2 dR_2(t)\right)^{-1}. \end{equation}
\item [(iii)] The Weyl discs $D_l(\lambda)$ are nested, i.e. $D_{l_2} \subseteq D_{l_1}$ provided $l_1 < l_2 < b$, and the function $s_1(\cdot,\lambda) - \omega c_1(\cdot,\lambda)$ belongs to ${\mathcal L}^2(R_2)$ provided $\omega\in\cap_{l<b} D_l(\lambda)$. \end{enumerate} \end{lemma} \begin{proof}
(i) From~\eqref{eTWeyl} and the condition $\psi_1(l,\lambda) + h \psi_2(l,\lambda) = 0$ we get
\begin{equation}
\psi_1(l,\lambda) + h \psi_2(l,\lambda) =
(s_1(l,\lambda) + h s_2(l,\lambda)) - m(\lambda,l,h) (c_1(l,\lambda) + h c_2(l,\lambda)) = 0
\end{equation}
which results as~\eqref{eq:mlhl}.
(ii) It is clear from formula~\eqref{eq:mlhl} that the function $m(\lambda,l,\cdot)$ maps $\dR_+\cup\{\infty\}$ into a circle.
Let $h\in\clos{\dC_+}\cup\{\infty\}$ and $\omega \in D_l(\lambda)$.
Applying the second Green's identity~\eqref{eq:G2} to the tuples $\{\psi_1,\psi_2,\lambda\psi_1\}$ and $\{\psi_1^*,\psi_2^*,\conj{\lambda}\psi_1^*\}$ provides
\begin{equation}
(\lambda - \conj{\lambda}) \int_0^l |\psi_1(t,\lambda)|^2 dR_2(t) =
(\omega - \conj{\omega}) - (h - \conj{h}) |\psi_2(l)|^2
\end{equation}
and hence
\begin{equation} \label{eq:wde}
\int_0^l |s_1(t,\lambda) - \omega c_1(t,\lambda)|^2\, dR_2(t)
=
\frac{\operatorname{Im}\omega}{\operatorname{Im}\lambda} - \frac{\operatorname{Im} h}{\operatorname{Im}\lambda} |\psi_2(l)|^2.
\end{equation}
Since $\operatorname{Im} h > 0$, \eqref{eq:WeylD} follows now from \eqref{eq:wde}.
(iii) Let $l_1 < l_2 < d$ and let $\omega \in D_{l_2}$. Then
\begin{equation}
\int_0^{l_1} |s_1(t,\lambda) - \omega c_1(t,\lambda)|^2\, dR_2(t) \le
\int_0^{l_2} |s_1(t,\lambda) - \omega c_1(t,\lambda)|^2\, dR_2(t) \le
\frac{\operatorname{Im}\omega}{\operatorname{Im}\lambda}
\end{equation}
and therefore $D_{l_2} \subseteq D_{l_1}$. Assume now $\omega\in\cap_{l<b}D_l(\lambda)$. Passing to the limit as $l\to b$ in~\eqref{eq:WeylD}, one gets
\begin{equation}
\int_0^b |s_1(t,\lambda) - \omega c_1(t,\lambda)|^2\, dR_2(t) \le
\frac{\operatorname{Im}\omega}{\operatorname{Im}\lambda},
\end{equation}
which proves that $s_1(t,\lambda) - \omega c_1(t,\lambda) \in {\mathcal L}^2(R_2)$. \end{proof}
Assume that the point $b$ is singular for the system~\eqref{eq:IntSys}. Then the following alternative holds: \begin{enumerate}
\item [(i)]
either the discs $D_l(\lambda)$ shrink to a limit point as $l\to b$ and then $\dim\sN_{\lambda} (T_{\max}) = 1$;
\item [(ii)]
or the discs $D_l(\lambda)$ converge to a limit disc as $l\to b$ and then $\dim\sN_{\lambda} (T_{\max}) = 2$. \end{enumerate}
\begin{definition} \label{def:lc_lp_cases}
The system $S[R_1,R_2]$ is called \emph{limit point} at $b$ if $\dim\sN_{\lambda} (T_{\max}) = 1$, or \emph{limit circle} at $b$ if $\dim\sN_{\lambda} (T_{\max}) = 2$. \end{definition} \begin{remark} A matrix version of integral equation equivalent to the integral system $S[R_1,R_2]$ with $R_1(x)\equiv x$ and $R_2(x)$ continuous was considered in~\cite{ArDy12}. Such equation can be reduced to a canonical differential system, see~\cite[Section 2.2]{ArDy12}. Condition of definiteness of general matrix canonical differential system was found in \cite{LM03}. In the scalar case this condition coincides with Assumption~\ref{assum:surj}. \end{remark}
\section{Integral systems in the limit circle case} \subsection{The fundamental matrix of the system $S[R_1,R_2]$} We will start with some general properties of the fundamental matrix of the system $S[R_1,R_2]$. \begin{lemma}\label{lem:2.4} Let $U(x,\lambda)$ be the fundamental matrix function of the system $S[R_1,R_2]$ \begin{equation}\label{eq:Fund2}
U(x,\lambda) = \begin{bmatrix} c_1(x,\lambda) & s_1(x,\lambda) \\ c_2(x,\lambda) & s_2(x,\lambda) \end{bmatrix}.
\end{equation} Then: \begin{enumerate}
\item [(i)] The following identity holds
\begin{equation}\label{eq:Kernel_U}
J-U(x,\mu)^*JU(x,\lambda) = -(\lambda-\conj{\mu})\int_0^x
\begin{bmatrix} c_1(t,\conj{\mu}) \\ s_1(t,\conj{\mu})\end{bmatrix}
\begin{bmatrix} c_1(t,\lambda) & s_1(t,\lambda) \end{bmatrix} dR_2(t).
\end{equation}
\item [(ii)] For every $x\in [0,b)$, $U(x,\lambda)$ is entire in $\lambda$.
\item [(iii)] The entries of $U(x,\lambda)$ are nonnegative for $x\in [0,b)$, $\lambda\in\dR_-$. If, in addition, the interval $(0,x)$ contains growth points of $R_1$ and $R_2$, and
\begin{equation}\label{eq:supp}
a=\inf \supp dR_2, \quad
a_1=\inf (\supp dR_1\cap(a,b)), \end{equation} then \begin{equation}\label{eq:c_12}
\lim_{\lambda\to-\infty} c_1(x,\lambda) = +\infty, \quad x\in(a_1,b);\quad
\lim_{\lambda\to-\infty} c_2(x,\lambda) = +\infty, \quad x\in(a,b); \end{equation} \begin{equation}\label{eq:s_12}
\lim_{\lambda\to-\infty} s_1(x,\lambda) = +\infty, \quad x\in(a_1,b); \quad
\lim_{\lambda\to-\infty} s_2(x,\lambda) = +\infty, \quad x\in(a,b). \end{equation}
\item [(iv)] If $\lambda\in\dR_-$ then \begin{equation}\label{eq:sc1}
\frac{s_1(x,\lambda)}{c_1(x,\lambda)}< \frac{s_2(x,\lambda)}{c_2(x,\lambda)},\quad x\in (a,b), \end{equation} the function $\frac{s_1(x,\lambda)}{c_1(x,\lambda)}$ is increasing on $[0,b)$ and the function $\frac{s_2(x,\lambda)}{c_2(x,\lambda)}$ is decreasing on $(a,b)$. \end{enumerate} \end{lemma} \begin{proof} \textbf{1.} By \eqref{eq:G1}, for the triples $(c_1(\cdot,\lambda), c_2(\cdot,\lambda), \lambda c_1(\cdot,\lambda))$ and $(c_1(\cdot,\mu), c_2(\cdot,\mu), \mu c_1(\cdot,\mu))$ belonging to $\cT$ one obtains \begin{equation}\label{eq:V11}
(\lambda-\conj{\mu}) \int_0^x c_1(t,\lambda) c_1(t,\conj{\mu})\, dR_2 =
c_1(x,\lambda) c_2(x,\conj{\mu}) - c_2(x,\lambda) c_1(x,\conj{\mu}). \end{equation} Similarly, for $(s_1(\cdot,\lambda),s_2(\cdot,\lambda), \lambda s_1(\cdot,\lambda))$ and $(s_1(\cdot,\mu),s_2(\cdot,\mu), \mu s_1(\cdot,\mu))$ one obtains \begin{equation}\label{eq:V22}
(\lambda-\conj{\mu}) \int_0^x s_1(t,\lambda) s_1(t,\conj{\mu})\, dR_2 =
s_1(x,\lambda) s_2(x,\conj{\mu}) - s_2(x,\lambda) s_1(x,\conj{\mu}). \end{equation} And finally for $(c_1(\cdot,\lambda),c_2(\cdot,\lambda), \lambda c_1(\cdot,\lambda))$ and $(s_1(\cdot,\mu),s_2(\cdot,\mu), \mu s_1(\cdot,\mu))$ one obtains \begin{equation}\label{eq:V12}
(\lambda-\conj{\mu}) \int_0^x c_1(t,\lambda) s_1(t,\conj{\mu})\, dR_2 =
c_1(x,\lambda) s_2(x,\conj{\mu}) - c_2(x,\lambda) s_1(x,\conj{\mu}) - 1. \end{equation} The statement (i) is implied by~\eqref{eq:V11}--\eqref{eq:V12}.
\textbf{2.} It follows from~\eqref{eq:Kernel_U} that \begin{equation*}
U(x,\mu)^* = JU(x,\conj{\mu})^{-1} J^T. \end{equation*}
Therefore, \begin{equation*}
\frac{U(x,\lambda) - U(x,\conj{\mu})}{\lambda-\conj{\mu}}=
U(x,\conj{\mu}) J^T\int_0^x
\begin{bmatrix} c_1(t,\conj{\mu}) \\ s_1(t,\conj{\mu})\end{bmatrix}
\begin{bmatrix} c_1(t,\lambda) & s_1(t,\lambda) \end{bmatrix} dR_2(t), \end{equation*} hence $U(x,\lambda)$ is holomorphic on $\dC$which shows (ii).
\textbf{3.} To show (iii), expanding $c_1(x,\lambda)$ and $c_2(x,\lambda)$ in series in $\lambda$ \begin{equation*} c_1(x,\lambda)=1-\lambda\varphi_1(x)+\lambda^2\varphi_2(x)+\dots,\quad c_2(x,\lambda)=-\lambda\psi_1(x)+\lambda^2\psi_2(x)+\dots \end{equation*} one obtains from~\eqref{eq:IntSys} that \begin{equation}\label{eq:phi_1}
\psi_1(x)=R_2(x), \quad \varphi_1(x)=\int_0^xR_2(t)\,dR_1(t) \end{equation} \begin{equation}\label{eq:phi_n}
\psi_n(x)=\int_0^x\varphi_{n-1}(t)\,dR_2(t), \quad \varphi_n(x)=\int_0^xdR_1(t)\int_0^t\varphi_{n-1}(s)\,dR_2(s),\quad n\in\dN,\,n\ge 2. \end{equation} This implies that $\varphi_n(x)\ge 0$, $\psi_n(x)\ge 0$ for $n\in \dN$ and hence \begin{equation*}
c_1(x,\lambda)\ge 0, \quad c_2(x,\lambda)\ge 0
\quad \text{for} \quad x\in [0,b),\ \lambda\in\dR_-. \end{equation*} Moreover, it follows from~\eqref{eq:phi_1} that \begin{equation}\label{eq:phi_ineq}
c_1(x,\lambda)\ge 1+|\lambda|\int_0^xR_2(t)\,dR_1(t), \quad
c_2(x,\lambda)\ge |\lambda|R_2(x). \end{equation} Therefore, the relations \eqref{eq:c_12} hold since \begin{equation*}
\int_0^x R_2(t)\,dR_1(t) > 0 \ \text{for}\ x\in(a_1,b)
\quad \text{and} \quad
R_2(x)>0 \ \text{for}\ x\in(a,b). \end{equation*}
The proof of \eqref{eq:s_12} is similar.
\textbf{4.} The identity \eqref{eq:Liouv} yields \begin{equation}\label{eq:cs12Id}
\frac{s_2(x,\lambda)}{c_2(x,\lambda)}-\frac{s_1(x,\lambda)}{c_1(x,\lambda)}=\frac{1}{c_1(x,\lambda)c_2(x,\lambda)} \end{equation} This proves the inequality~\eqref{eq:sc1}.
It follows from \eqref{eq:IntSys}, \eqref{eq:in-by-parts}, \eqref{eq:quotient-rule}, and \eqref{eq:Liouv+} that \begin{equation*}
d\left(\frac{s_1(x,\lambda)}{c_1(x,\lambda)}\right) =
\frac{c_{1+}(x,\lambda)s_2(x,\lambda)-c_2(x,\lambda)s_{1+}(x,\lambda)}{c_1(x,\lambda) c_{1+}(x,\lambda)} dR_1(x) =
\frac{1}{c_1(x,\lambda) c_{1+}(x,\lambda)}\,dR_1(x) \end{equation*} and hence \begin{equation}\label{eq:c1s1}
\frac{s_1(x,\lambda)}{c_1(x,\lambda)}=\int_0^x\frac{1}{c_1(t,\lambda) c_{1+}(t,\lambda)}\,dR_1(t). \end{equation} Since $c_1(x,\lambda)$, $ c_{1+}(x,\lambda)>0$ for $\lambda\in\dR_-$ and $x\in[0,b)$, the function $\frac{s_1(x,\lambda)}{c_1(x,\lambda)}$ is increasing on $[0,b)$.
Similarly, by~\eqref{eq:IntSys}, \eqref{eq:in-by-parts}, \eqref{eq:quotient-rule}, and \eqref{eq:Liouv++} \begin{equation}\label{eq:cs2}
d\left(\frac{c_2(x,\lambda)}{s_2(x,\lambda)}\right) = \frac{-\lambda}{s_2(x,\lambda) s_{2+}(x,\lambda)}\,dR_2(x), \quad x\in[0,b) \end{equation} and hence the function $\frac{c_2(x,\lambda)}{s_2(x,\lambda)}$ is increasing on $[0,b)$. This proves (iv). Notice, that the function $\frac{s_2(x,\lambda)}{c_2(x,\lambda)}$ is not defined on $[0,a]$.
\end{proof}
\subsection{The Evans-Everitt condition in the limit circle case} \begin{proposition} \label{prop:1}
The system $S[R_1,R_2]$ is limit circle at $b$ if and only if $1, R_1\in {\mathcal L}^2(R_2)$. \end{proposition} \begin{proof}
Using the well-known procedure from~\cite[Theorem 5.6.1]{Atk64} (see also \cite[Theorem 4.5]{Str19}) one can show that $S[R_1,R_2]$ is limit circle at $b$ if and only if $c_1(x,0)$ and $s_1(x,0) $ belong to ${\mathcal L}^2(R_2)$.
Substitution of $\lambda=0$ to~\eqref{eq:IntSys} immediately provides $c_2(x,0) = 0$, $s_2(x,0) = 1 $ and hence $c_1(x,0) = 1$, $s_1(x,0) = R_1(x)$. \end{proof}
If the system $S[R_1,R_2]$ is regular at $b$, then the following limits exist: \begin{equation}\label{eq:lim_1}
c_1(b,\lambda)=\lim_{t\to b}c_1(t,\lambda),\quad s_1(b,\lambda)=\lim_{t\to b}s_1(t,\lambda), \end{equation} \begin{equation}\label{eq:lim_2}
c_2(b,\lambda)=\lim_{t\to b}c_2(t,\lambda),\quad s_2(b,\lambda)=\lim_{t\to b}s_2(t,\lambda). \end{equation}
Assume now that the system $S[R_1,R_2]$ is in the limit circle at $b$. One can check (see \cite[Section 10.7]{KacK68}, \cite[Theorem 3.8]{Str18}) that for every element $\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \in T_{\max}$ the limit \begin{equation} \label{eq:f2d}
u_2(b) = u_2(0) - \int_0^b f\,dR_2 \end{equation} exists and is well defined. Therefore, the limits~\eqref{eq:lim_2} exist.
\begin{lemma}\label{lem:psi}
Let the system $S[R_1,R_2]$ be limit circle at $b$.
Then for every $\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \in T_N^*$ one has $u_2\in {\mathcal L}^2(R_1)$ and the following two equalities hold:
\begin{equation} \label{eq:2.26}
\lim_{x\to b} u_1(x) = u_1(0) + (f,R_1),
\end{equation}
\begin{equation} \label{eq:LC-SLP}
\lim_{x\to b} u_1(x) u_2(x) = 0.
\end{equation}
If $\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \in T_{\max}$ and the endpoint $b$ is singular then $\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \in T_N^*$ provided \eqref{eq:LC-SLP} holds. \end{lemma} \begin{proof}
Let $\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \in T_N^*$.
Applying the integration-by-parts formula~\eqref{eq:in-by-parts} to the first line of~\eqref{eq:sys_lin_rel} one gets
\begin{equation} \label{eq:u1_ibp}
u_1(x) = u_1(0) + u_2(x) R_1(x) + \int_0^x R_1(t) f(t)\, dR_2(t).
\end{equation}
We recall that in the limit circle case $1, R_1 \in {\mathcal L}^2(R_2)$ and $f \in {\mathcal L}^2(R_2)$ by the assumption of the lemma.
The condition $u_2(b)=0$ provides $u_2(x) = \int_x^b f\, dR_2$ and hence~\eqref{eq:u1_ibp} can be rewritten as
\begin{equation} \label{eq:l1}
u_1(x) = u_1(0) + (f,R_1) - \int_x^b (R_1(t)-R_1(x)) f(t)\, dR_2(t).
\end{equation}
Note the following estimation:
\begin{equation} \label{eq:l2}
\begin{split}
\left| \int_x^b (R_1(t)-R_1(x)) f(t)\, dR_2(t) \right| &\le
\int_x^b (R_1(t)-R_1(x)) |f(t)|\, dR_2(t) \\
&\le \int_x^b R_1 |f|\, dR_2 \to 0\quad \text{as}\quad x \to b.
\end{split}
\end{equation}
Now~\eqref{eq:2.26} follows from \eqref{eq:l1} and \eqref{eq:l2}, and~\eqref{eq:LC-SLP} finally follows from~\eqref{eq:2.26}.
The claim $u_2\in {\mathcal L}^2(R_1)$ for $\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \in T_N^*$ follows from~\eqref{eq:2.26} and the first Green's identity~\eqref{eq:G1}
\begin{equation}\label{eq:Green1LC}
\begin{split}
\int_0^b f(t) \conj{u_1(t)}\,dR_2(t)
&= \int_0^b |u_2|^2 dR_1(t) - \lim_{x\to b} u_2(x) \conj{u_1(x)} + u_2(0) \conj{u_1(0)}\\
&= \int_0^b |u_2|^2 dR_1(t) + u_2(0) \conj{u_1(0)}.
\end{split}
\end{equation}
Now assume that the endpoint $b$ is singular and $\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \in T_{\max}$.
From~\eqref{eq:f2d} we have $u_2(b)=a$ where $a \in {\mathbb C}$.
In the limit circle case the singular endpoint $b$ implies $R_1(b)=\infty$.
If $a\neq0$ then from~\eqref{eq:sys_lin_rel} we get $u_1(b)=\pm\infty$ and hence~\eqref{eq:LC-SLP} does not hold. \end{proof} \begin{remark} The condition \eqref{eq:LC-SLP} for Sturm-Liouville operators in the limit circle case was introduced and studied by Evans and Everitt in~\cite{EvEver}. We will call it the Evans--Everitt condition. \end{remark}
\subsection{Boundary triples for integral systems in the limit circle case} \begin{definition}[see~\cite{Ben89,Kost13}] \label{def:5.1}
The function $m(\lambda,b,\infty)$ from~\eqref{eTWeyl} for which the solution
\begin{equation} \label{eTWeyl2}
\psi^N(t,\lambda)=s(t,\lambda)-m(\lambda,b,\infty){c}(t,\lambda), \qquad t \in I,
\end{equation}
satisfies the condition
\begin{equation} \label{eq:m_functLC}
\psi^N_2(b,\lambda) = 0,
\end{equation}
is called the \emph{Neumann $m$-function} of the system $S[R_1,R_2]$ on $I$ subject to the boundary condition~\eqref{eq:m_functLC}. \end{definition}
It follows from~\eqref{eTWeyl} and the condition $\psi_2^N(b,\lambda)=0$ that $s_2(b,\lambda)-m(\lambda,b,\infty)c_2(b,\lambda)=0$ which proves the formula \begin{equation}\label{eq:WFA_N2}
m(\lambda,b,\infty)=
\frac{s_2(b,\lambda)}{c_2(b,\lambda)}. \end{equation}
We will show below that the function $ m(\lambda,b,\infty)$ is a Weyl function of a one-dimensional symmetric extension $T_N$ of the linear relation $T_{\min}$ defined by \begin{equation}\label{eq:A_N}
T_{N} = \left\{
\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \colon
(u_1,u_2,f) \in \cT,\ u_1(0)=u_2(0)=u_2(b)=0 \right\}. \end{equation} As follows from~\eqref{eq:G2} the adjoint linear relation $T_N^*$ is of the form \begin{equation}
T_N^* = \left\{
\bm{u}=\begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \colon
(u_1,u_2,f) \in \cT:\, u_2(b)=0 \right\}. \end{equation}
\begin{proposition}\label{prop:BT_IS_N}
Let the system $S[R_1,R_2]$ be singular and limit circle at $b$, let $T_N$ be defined by \eqref{eq:A_N}, and let $m(\lambda,b,\infty)$ be the Neumann $m$-function of the system $S[R_1,R_2]$ given by~\eqref{eq:WFA_N2}.
Then: \begin{enumerate} \item [(i)] $T_N$ is a symmetric nonnegative linear relation in $L^2(R_2)$ with deficiency indices $(1,1)$. \item [(ii)] The triple $\Pi^N = (\dC,\Gamma_0^N,\Gamma_1^N)$, where \begin{equation}\label{eq:GammaA_N}
\Gamma_0^N \bm{u} = u_2(0), \quad \Gamma_1^N \bm{u} = -u_1(0), \quad \bm{u} \in T_N^*, \end{equation} is a boundary triple for $T_N^*$. \item[(iii)] The Weyl function $m_N(\lambda)$ of $T_N$ corresponding to the boundary triple $\Pi^N$ coincides with the Neumann $m$-function $m(\lambda,b,\infty)$. \item[(iv)] The Weyl function $m_N(\lambda)$ of $T_N$ coincides with the principal Titchmarsh-Weyl coefficient $q(\lambda)$ of the system $S[R_1,R_2]$, belongs to the Stieltjes class ${\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}$, and \begin{equation}\label{eq:lim}
\lim_{\lambda\to-\infty}m_N(\lambda)=R_{1+}(a), \end{equation} where $a=\inf\supp dR_2$. \item[(v)] The Weyl function $m_N(\lambda)$ of $T_N$ admits the representation
\begin{equation}\label{eq:Pol_m}
m_N(\lambda)=-\frac{1}{R_2(b)\cdot\lambda}+\widetilde m(\lambda);
\end{equation} where $\widetilde m$ is a function from ${\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}$ such that $\lim_{y\to 0}y\widetilde m(iy)=0$. \end{enumerate} \end{proposition} \begin{proof} \textbf{1.} To show ${\textrm (i)}$ and ${\textrm (ii)}$, let the tuples $(u_1,u_2,f)$ and $(v_1,v_2,g)$ satisfy the system~\eqref{eq:sys_lin_rel} and assume that $u_2(b)=v_2(b)=0$, i.e. $\bm{u}, \bm{v} \in T_N^*$. Let $\mu\in\dR$. By formula~\eqref{eq:Liouv} at least one of the values $c_2(b,\mu)$ and $s_2(b,\mu)$ is not equal to 0. Assume that $c_1(b,\mu)\ne 0$ and let us set $c(x) \coloneqq c(x,\mu)$. Due to the identity \begin{equation}\label{eq:Wron_Id}
[u,v]_b = c_2(b,\mu)^{-1}
\left\{ [u(b),c(b,\mu)] \conj{v_2(b)} - u_2(b) [\conj{v(b)},c(b,\mu)] \right\} \end{equation} the second Green's identity \eqref{eq:G2} is of the form \begin{equation}\label{eq:GreenN}
\int_0^b (f\conj{v_1} - u_1\conj{g})\, dR_2(t) =
[u,\conj{v}]_b - [u,\conj{v}]_0
=u_2(0)\conj{v_1(0)}-u_1(0)\conj{v_2(0)}. \end{equation} By Definition~\ref{def:btriple}
the boundary triple for $T_N^*$ can be taken as $\Pi^N = (\dC,\Gamma_0^N,\Gamma_1^N)$, with $\Gamma_0^N, \Gamma_1^N$ given in \eqref{eq:GammaA_N}.
It follows from the first Green's identity \eqref{eq:Green1LC} and Lemma~\ref{lem:psi} that for every $(\pi u_1,\pi f)^T\in T_N$ \begin{equation} \label{eq:Green1_}
\int_0^b f(t)\conj{u_1(t)}\,dR_2(t)=\int_0^b |u_2|^2dR_1(t)\ge 0. \end{equation}
\textbf{2.} Now ${\textrm (iii)}$ is shown. The defect subspace $\sN_\lambda(T_N^*)$ is spanned by the function $\psi_1^N(\cdot,\lambda)$, where $\psi^N(t,\lambda)$ is the Weyl solution from \eqref{eTWeyl2} corresponding to the Neumann $m$-function $m(\lambda,b,\infty)$. Denote $\bm{u}^N(t,\lambda)=(\psi_1^N(\cdot,\lambda),\lambda\psi_1^N(\cdot,\lambda))^T\in\widehat \sN_\lambda(T_N^*)$. Using the formulas~\eqref{eTWeyl2} and~\eqref{eq:GammaA_N} one obtains \begin{equation*} \Gamma_1^N\bm{u}^N(\cdot,\lambda)=-\psi_1^N(0,\lambda)=m(\lambda,b,\infty),\quad \Gamma_0^N\bm{u}^N(\cdot,\lambda)=\psi^N_2(0,\lambda)=1 \end{equation*} and hence by~\eqref{def:Weyl} the Weyl function $m_N(\lambda)$ is of the form \begin{equation}
m_N(\lambda)=\frac{\Gamma_1^N\bm{u}^N(\cdot,\lambda)}{\Gamma_0^N\bm{u}^N(\cdot,\lambda)}
=m(\lambda,b,\infty). \end{equation} Therefore, the Weyl function $m_N(\lambda)$ coincides with the Neumann $m$-function $m(\lambda,b,\infty)$.
\textbf{3.} The inclusion $m_N\in{\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}$ follows from Lemma~\ref{lem:2.4}, since the functions $s_2(x,\lambda)$ and $c_2(x,\lambda)$ are positive for $\lambda<0$ and the function $ m_N(\lambda)$ admits a holomorphic nonnegative continuation on $\dR_-$.
Let $a=\inf\supp R_2$ and $a_1=\inf(\supp R_1\cap(a,b))$. Then by Assumption~\ref{assum:surj} $a_1<b$ and due to~\eqref{eq:IntSys} and Lemma~\ref{lem:2.4} (iii) \begin{equation*}
c_1(x,\lambda)\equiv 1\ \text{for}\ x\le a_1
\quad \text{and} \quad
\lim_{\lambda\to-\infty} c_1(x,\lambda)=\infty\ \text{for}\ x\ge a_1. \end{equation*} Now we must consider two cases. In case if $c_1(\cdot,\lambda)$ has a jump at point $a_1$, which is only possible if $a_1>a$, we get \begin{equation}
\frac{1}{c_1(x,\lambda) c_{1+}(x,\lambda)} \to \chi_{[0,a_1)}(x)
\quad \text{as} \quad \lambda \to -\infty \end{equation} and hence by the Lebesgue bounded convergence theorem one obtains from~\eqref{eq:c1s1} \begin{equation}\label{eq:c1s1_Lim}
\lim_{\lambda\to-\infty} \frac{s_1(x,\lambda)}{c_1(x,\lambda)} =
\int_0^x \frac{1}{c_1(x,\lambda) c_{1+}(x,\lambda)}\, dR_1 =
\int_{[0,a_1)}\,dR_1 = R_1(a_1) = R_{1+}(a). \end{equation} The last equality in~\eqref{eq:c1s1_Lim} follows from $a_1>a$ and the definition of the points $a$, $a_1$.
In case if $c_1(\cdot,\lambda)$ has no jump at point the $a_1$, which is possible either if $a_1=a$ or $a_1>a$ and $R_1$ has no jump at $a_1$, we get \begin{equation}
\frac{1}{c_1(x,\lambda) c_{1+}(x,\lambda)} \to \chi_{[0,a_1]}(x)
\quad \text{as} \quad \lambda \to -\infty \end{equation} and similarly to~\eqref{eq:c1s1_Lim} \begin{equation}
\lim_{\lambda\to-\infty} \frac{s_1(x,\lambda)}{c_1(x,\lambda)} =
R_{1+}(a_1) = R_{1+}(a). \end{equation}
Since $R_1(b)+R_2(b)=+\infty$ it follows from \eqref{eq:phi_ineq} that $\lim_{x\to b} c_1(x,\lambda) c_2(x,\lambda) = +\infty$ for all $\lambda\in\dR_-$ and hence it follows from~\eqref{eq:cs12Id} that \begin{equation*}
q(\lambda)=\lim_{x\to b}\frac{s_1(x,\lambda)}{c_1(x,\lambda)} =
\lim_{x\to b} \frac{s_2(x,\lambda)}{c_2(x,\lambda)} = m_N(\lambda),
\quad \lambda\in\dR_-. \end{equation*} Since $q$ and $m_N$ are holomorphic on $\dC\setminus\dR_+$ this proves that $q(\lambda)\equiv m_N(\lambda)$, and (iv) is shown.
\textbf{4.} Now we prove ${\textrm (v)}$. It follows from~\eqref{eq:IntSys} that \begin{equation}\label{eq:s2}
s_2(x,\lambda)=1-\lambda\int_0^x s_1(x,\lambda)\,dR_2(t), \quad
c_2(x,\lambda)=-\lambda\int_0^x c_1(x,\lambda)\,dR_2(t) \end{equation} and by~\eqref{eq:WFA_N2} that \begin{equation}
m_N(\lambda)=\frac{1-\lambda\int_0^b s_1(x,\lambda)\,dR_2(t)}{-\lambda\int_0^b c_1(x,\lambda)\,dR_2(t)},\quad \lambda\in\dC\setminus\dR. \end{equation} Moreover, for $\lambda<0$ the functions $s_1(x,\lambda)$ and $c_1(x,\lambda)$ are positive and increasing on $(0,b)$ and $c_2(0,\lambda)=1$, hence \begin{equation}
\int_0^b c_1(x,\lambda)\,dR_2(t)>R_2(b), \quad\int_0^b s_1(x,\lambda)\,dR_2(t)>0. \end{equation} Since $c_1(x,\lambda)\to c_1(x,0)\equiv 1$ and $s_1(x,\lambda)\to s_1(x,0)=R_1(x)$ as $\lambda\to 0-$ and these convergences are monotone and uniform on $[0,b]$ one finds that \begin{equation}
\int_0^b c_1(t,\lambda)\,dR_2(t)\to R_2(b),\quad \int_0^b s_1(x,\lambda)\,dR_2(t)\to \int_0^b R_1(t)\,dR_2(t), \quad\text{as}\quad \lambda\to 0-. \end{equation} Therefore, \begin{equation}
\lambda m_N(\lambda)\to -\frac{1}{R_2(b)},\quad \text{as}\quad\lambda\to 0- \end{equation} and thus $m_N(\lambda)$ admits the representation~\eqref{eq:Pol_m}. \end{proof}
\subsection{Integral systems in the regular case} \label{subsec:q-r}
Assume that the system $S[R_1,R_2]$ is regular at~$b$, i.e. $R_1(b)+R_2(b)<\infty$. Then for every tuple $(u_1,u_2,f) \in \cT$ it follows from~\eqref{eq:f2d} that the function $u_2$ is bounded and hence the limit \begin{equation} \label{eq:f1d}
u_1(b) = u_1(0) + \int_0^b u_2\,dR_1 \end{equation} exists and well defined. Therefore, the limits~\eqref{eq:lim_1} exist.
\begin{definition}\label{def:5.1D}{\textrm (see~\cite{Ben89,Kost13})} The function $m(\lambda,b,0)$ for which the solution \begin{equation} \label{eTWeylND} \psi^{ND}(t,\lambda)=s(t,\lambda)-m(\lambda,b,0){c}(t,\lambda), \qquad t \in I, \end{equation}
satisfies the condition \begin{equation}\label{eq:m_functLCD} \psi^{ND}_1(b,\lambda)=0 \end{equation} is called the \emph{Neumann $m$-function} of the system $S[R_1,R_2]$ on $I$ subject to the boundary condition~\eqref{eq:m_functLCD}. \end{definition} It follows from~\eqref{eTWeyl} and the condition $\psi^{ND}_1(b,\lambda)=0$ that $s_1(b,\lambda)-m(\lambda,b,0)c_1(b,\lambda)=0$ which yields the formula \begin{equation}\label{eq:WFA_N1}
m(\lambda,b,0)=
\frac{s_1(b,\lambda)}{c_1(b,\lambda)} \end{equation} and hence the Neumann $m$-function $ m(\lambda,b,0)$ coincides with the principal Titchmarsh-Weyl coefficient $q(\lambda)$ of the system $S[R_1,R_2]$.
Let $T_D$ be a symmetric extension of the linear relation $T_{\min}$ defined by \begin{equation}\label{eq:A_D}
T_{D} = \left\{ \bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \colon (u_1,u_2,f) \in \cT,\ u_1(0) = u_2(0) = u_1(b) = 0 \right\}. \end{equation} As follows from~\eqref{eq:G2} the adjoint linear relation $T_D^*$ is of the form
\begin{equation}
T_D^* = \left\{\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix} \colon (u_1,u_2,f) \in \cT:\, u_1(b)=0 \right\}.
\end{equation}
\begin{proposition}[cf. \cite{Str18}] \label{prop:BT_IS_D}
Let the system $S[R_1,R_2]$ be regular at $b$, and let $T_D$ be defined by~\eqref{eq:A_D}.
Then: \begin{enumerate} \item[(i)] $T_D$ is a symmetric nonnegative linear relation in $L^2(R_2)$ with deficiency indices $(1,1)$ and $u_2\in\ L^2(R_1)$ for all $\bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix}\in T_D^*$;
\item[(ii)] the triple $\Pi^{ND} = (\dC,\Gamma_0^{ND},\Gamma_1^{ND})$, where \begin{equation}\label{eq:GammaA_ND}
\Gamma_0^{ND} \bm{u} = u_2(0), \quad \Gamma_1^{ND} \bm{u} = -u_1(0), \quad \bm{u} \in T_D^*, \end{equation} is a boundary triple for $T_D^*$. \item[(iii)] The Weyl function $m_{ND}(\lambda)$ of $T_D$ corresponding to the boundary triple $\Pi^{ND}$ coincides with $m(\lambda,b,0)$. \item[(iv)] The Weyl function $m_{ND}$ of $T_D$ belongs to the Stieltjes class ${\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}$ and coincides with the principal Titchmarsh-Weyl coefficient $q(\lambda)$ of the system $S[R_1,R_2]$. \end{enumerate} \end{proposition} \begin{proof} \textbf{1.} To show ${\textrm (i)}$ and ${\textrm (ii)}$, let the tuples $(u_1,u_2,f)$ and $(v_1,v_2,g)$ satisfy the system \eqref{eq:sys_lin_rel} and assume that $u_1(b)=v_1(b)=0$, i.e. $\bm{u}, \bm{v} \in T_D^*$. Let $\mu\in\dR$. By the Liouville-Ostrogradskii formula~\eqref{eq:Liouv} at least one of the values $c_1(b,\mu)$ and $s_1(b,\mu)$ is not equal to 0. Assume that $c_1(b,\mu)\ne 0$ and let us set $c(x) \coloneqq c(x,\mu)$.
Due to the identity \begin{equation}\label{eq:Wron_IdD}
[u,v]_b = c_1(b,\mu)^{-1}\left\{[u(b),c(b,\mu)]\conj{v_1(b)} - u_1(b)[\conj{v(b)},c_(b,\mu)]\right\} \end{equation} the Green's identity \eqref{eq:G2} is of the form~\eqref{eq:GreenN}. By Definition~\ref{def:btriple} the boundary triple for $T_D^*$ can be taken as $\Pi^{ND} = (\dC,\Gamma_0^{ND},\Gamma_1^{ND})$, with $\Gamma_0^{ND},\Gamma_1^{ND}$ given in \eqref{eq:GammaA_ND}.
It follows from the the first Green's identity \eqref{eq:G1} and Lemma~\ref{lem:psi} that for every $\bm{u} \in T_D$ the identity~\eqref{eq:Green1_} holds and thus the linear relation $T_D$ is nonnegative.
\textbf{2.} Now ${\textrm (iii)}$ is shown. The defect subspace $\sN_\lambda(T_D)$ is spanned by the function $\psi_1^{ND}(\cdot,\lambda)$ determined by~\eqref{eTWeylND}. Denote $\bm{u}^{ND}(t,\lambda)=(\psi_1^{ND}(\cdot,\lambda),\lambda\psi_1^{ND}(\cdot,\lambda))^T\in\widehat \sN_\lambda(T_{D}^*)$. Using the formulae~\eqref{eTWeyl} and~\eqref{eq:csK} one obtains \begin{equation*}
\Gamma_1^{ND}\bm{u}^{ND}(\cdot,\lambda) = -\psi_1^{ND}(0,\lambda) = m(\lambda,b,0), \quad
\Gamma_0^{ND}\bm{u}^{ND}(\cdot,\lambda) = \psi^{ND}_2(0,\lambda) = 1 \end{equation*}
and hence the Weyl function $m_{ND}(\lambda)$ is of the form \begin{equation}
m_{ND}(\lambda) =
\frac{\Gamma_1^{ND} \bm{u}^{ND}(\cdot,\lambda)}{\Gamma_0^{ND} \bm{u}^{ND}(\cdot,\lambda)} =
m(\lambda,b,0). \end{equation} Therefore, the Weyl function $m_{ND}(\lambda)$ coincides with the Neumann $m$-function $m(\lambda,b,0)$.
\textbf{3.} Finally we prove ${\textrm (iv)}$. The inclusion $m_{ND}\in{\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}$ follows from Lemma~\ref{lem:2.4}. The equality $m_{ND}(\lambda)\equiv q(\lambda)$, $\lambda\in\dC\setminus\dR$, is implied by~\eqref{eq:WFA_N1}. \end{proof} \begin{remark} \label{rem:Param} The functions $R_1$ and $R_2$ are not uniquely defined by the principal Titchmarsh-Weyl coefficient of the system $S[R_1,R_2]$. As was shown in \cite[Lemma 2.12]{Kost13} if functions $\widetilde R_1(\xi)$ and $\widetilde R_2(\xi)$ are connected by \begin{equation*}
\widetilde R_1(\xi) = R_1(x(\xi)), \quad \widetilde R_2(\xi) = R_2(x(\xi)), \quad \xi\in[0,\beta]. \end{equation*}
where $x(\xi)$ is an increasing function on the interval $[0,\beta]$, such that $x(0)=0$ and $x(\beta)=b$, then the principal Titchmarsh-Weyl coefficient $\widetilde q$ of the system \begin{equation}\label{eq:IntSysW}
\widetilde u(\xi,\lambda) = \widetilde u(0,\lambda) -J \int_0^\xi
\begin{bmatrix} \lambda d\widetilde R_2(\tau) & 0 \\ 0 & d\widetilde R_1(\tau)\end{bmatrix} \widetilde u(\tau,\lambda),\quad \xi\in[0,\beta]. \end{equation}
coincides with the principal Titchmarsh-Weyl coefficient $q$ of the system $S[R_1,R_2]$.
Therefore we can always assume that for regular systems $S[R_1,R_2]$ the parameter $x$ ranges over a finite interval $[0,b]$, $b<\infty$. \end{remark}
\begin{remark}\label{rem:MP} As is known, see~\cite[Section A13]{KacK68}, a truncated moment problem can be reduced to a regular integral system $S[R_1,R_2]$
with \begin{gather*}
R_1(x) = x, \quad R_2(x) = \sum_{j=0}^{n-1} m_j H (x-x_j), \quad x\in[0,x_n],\\
x_j=\sum_{j=1}^j l_i, \quad m_{j-1}, l_j>0, \quad 1\le j\le n. \end{gather*} where $H(x)$ is the Heaviside function.
The corresponding monodromy matrix $U(x_n,\lambda)$ is of the form \begin{equation}
U(x_n,\lambda) =\prod_{j=1}^n U_{x_{j-1}}(x_j,\lambda),\quad \text{where}\quad
U_{x_{j-1}}(x_j,\lambda) =
\begin{bmatrix}
1 - \lambda l_j m_{j-1} & l_j \\
-\lambda m_{j-1} & 1
\end{bmatrix}. \end{equation} The system $S[R_1,R_2]$ satisfies Assumption~\ref{assum:surj} if $n>1$. If $n=1$ then $R_2(x)=H(x)$, $x\in[0,l_1]$, $L^2(R_2)=\dC$, the system $S[R_1,R_2]$ is of the form \begin{equation}\label{eq:1_dim}
u_1(x) = u_1(0) + x u_2(x), \quad
u_2(x) = u_2(0) - \lambda u_1(0) m_0, \quad
x\in(0,l_1] \end{equation} and does not satisfy the Assumption~\ref{assum:surj}. However, in this case one can still introduce a boundary triple $(\dC,\Gamma_0,\Gamma_1)$ for $T_{\max}=\dC\times\dC$ by \begin{equation}\label{eq:BT_1dim}
\Gamma_0\bm{u}=u_1(0),\quad \Gamma_1\bm{u}=f(0), \quad
\bm{u} = \begin{bmatrix} u_1 \\ f \end{bmatrix} \in T_{\max} \end{equation} and the corresponding Weyl function is $m(\lambda)=m_0\lambda$.
The system $S[\widetilde R_1,\widetilde R_2]$ with $\widetilde R_1(x)=l_1H(x-1)$, $\widetilde R_2(x)=m_0 H(x)$, $x\in[0,2]$ is equivalent to the system $S[R_1,R_2]$ in the sense that its Weyl function corresponding to the boundary triple~\eqref{eq:BT_1dim} coincides with $m(\lambda)=m_0\lambda$ and the monodromy matrix $\widetilde U(2,\lambda)$ of this system coincides with $U(l_1,\lambda)$. The advantage of system $S[\widetilde R_1,\widetilde R_2]$ is that the elementary factors of $\widetilde U(2,\lambda)$ from its factorization \begin{equation*}
\widetilde U(2,\lambda) = U^{(1)}(\lambda)U^{(0)}(\lambda), \quad
U^{(1)}(\lambda) =
\begin{pmatrix}
1 & l_1 \\
0 & 1
\end{pmatrix}, \quad
U^{(0)}(\lambda) =
\begin{pmatrix}
1 & 0 \\
-\lambda m_0 & 1
\end{pmatrix} \end{equation*} can be also treated as monodromy matrices of systems $S[0,\widetilde R_2]$ on the interval $[0,1]$ and $S[\widetilde R_1,0]$ on $[1,2]$, respectively. \end{remark}
\section{Integral systems in the limit point case} \subsection{The strong limit point condition} The next lemma is an analog of one result in~\cite[Lemma]{Ever76} in the case of integral systems.
\begin{lemma} \label{lem:log}
Let $f$ be a (not necessarily strictly) monotone function on $[b_0,b)$ such that either $f(x) \to \pm\infty$ or $f(x) \to 0$ as $x\to b$ and let $f(x)\neq0$ on $[b_0,b)$. Then
\begin{equation} \label{eq:intdf_f}
\lim_{x\to b} \int_{b_0}^x df/f = \pm \infty.
\end{equation} \end{lemma} \begin{proof}
We will prove the lemma in the case $f>0$, $f\to 0$. The proof in the other cases is similar.
Let $D_f $ be the set of the points of discontinuity of $f$ on $[b_0,b)$.
One can write
\begin{equation} \label{eq:log3.1}
\int\limits_{[b_0,x)} \frac{df}{f} = \int\limits_{[b_0,x)\setminus D_f} \frac{df}{f} + \int\limits_{[b_0,x)\cap D_f} \frac{df}{f}.
\end{equation}
Notice that both the integrals on the right hand side of~\eqref{eq:log3.1} are negative, therefore if one of them diverges (as $x\to b$) then the assertion of the lemma holds.
Let $D_f = \{x_n\}_{n=0}^{\infty}$. Consider the following inequality
\begin{equation} \label{eq:log3.3}
\frac{f_+(x_n) - f_-(x_n)}{f(x_n)} \le \frac{f_+(x_n) - f_-(x_n)}{f_-(x_n)} = \frac{f_+(x_n)}{f_-(x_n)} - 1 < 0
\end{equation}
and the associated series
\begin{equation} \label{eq:log3.6}
\sum_{n=0}^{\infty} \left( \frac{f_+(x_n)}{f_-(x_n)} - 1 \right).
\end{equation}
If series~\eqref{eq:log3.6} diverges then the following integral
\begin{equation} \label{eq:log3.4}
\int\limits_{[b_0,b)\cap D_f} \frac{df}{f} = \sum_{x_n\in D_f} \frac{f_+(x_n) - f_-(x_n)}{f(x_n)}
\end{equation}
diverges as well, so the assertion of the lemma holds immediately.
Assume now that series~\eqref{eq:log3.6} converges and denote $a_n \coloneqq 1 - f_+(x_n)/f_-(x_n)$. Notice that the measure $d\log(f)$ is absolutely continuous with respect to $df$ and therefore there exists the Radon-Nikodym derivative $d\log(f)/df \in L^1(df)$ which has a representative (see \cite[5.3, formula (3.5)]{BerUsShe})
\begin{equation} \label{eq:log3.radon-nikodym}
\frac{d\log(f)}{df} = \left\{
\begin{array}{ll}
1/f(x), & x \in [b_0,b)\setminus D_f,\\
(\log f_+(x) - \log f_-(x))/(f_+(x)-f_-(x)), & x \in D_f.
\end{array} \right.
\end{equation}
Now we get by the Radon-Nikodym theorem
\begin{equation}
\log \frac{f_-(x)}{f_+(b_0)} = \int\limits_{[b_0,x)} \frac{d\log(f)}{df}\,df = \int\limits_{[b_0,x)\setminus D_f} \frac{df}{f} + \int\limits_{D_f} \frac{\log f_+(x) - \log f_-(x)}{f_+(x)-f_-(x)}\,df
\end{equation}
and hence
\begin{equation} \label{eq:log3.5}
\int\limits_{[b_0,x)\setminus D_f} \frac{df}{f} = \log \frac{f_-(x)}{f_+(b_0)} + \sum_{x_n\in D_f} \log \frac{f_-(x_n)}{f_+(x_n)}.
\end{equation}
One can see from the following inequality
\begin{equation}
0 < \log \frac{f_-(x_n)}{f_+(x_n)} \le \frac{f_-(x_n) - f_+(x_n)}{f_+(x_n)} = \frac{a_n}{1-a_n}
\end{equation}
that the series
\begin{equation}
\sum_{n=1}^{\infty} \log \frac{f_-(x_n)}{f_+(x_n)}
\end{equation}
converges provided the series $\sum_{1}^{\infty} a_n$ converges. Hence we obtain that the integral on the left hand side of~\eqref{eq:log3.5} diverges which completes the proof. \end{proof}
\begin{definition}[\cite{Ever66,Ever73,Ever76}] \label{def:SLP_D}
Let the system $S[R_1,R_2]$ be singular at $b$. It is said to be in the \emph{strong limit point} case if
\begin{equation} \label{eq:def_SLP}
\lim_{x\to b} u_1(x)v_2(x) = 0\quad\text{for any }\quad
(u_1,u_2, f),\ (v_1,v_2, g) \in \cT;
\end{equation}
and it is said to have \emph{the Dirichlet property} if
\begin{equation} \label{eq:def_D}
\int_0^b |u_2(t)|^2dR_1(t) <\infty\quad\text{ for any } \quad (u_1,u_2, f)\in\cT.
\end{equation} \end{definition}
\begin{theorem} \label{thm:SLP}
Let the system $S[R_1,R_2]$ be singular at $b$. Then the following statements are equivalent:
\begin{enumerate}
\item[(LP)] The system $S[R_1,R_2]$ is in the limit point case.
\item[(D)] The system $S[R_1,R_2]$ has the Dirichlet property.
\item[(SLP$^*$)] For any $(u_1,u_2, f) \in \cT$ the following equality holds
\begin{equation} \label{eq:def_SLP_var}
\lim_{x\to b} u_1(x)u_2(x) = 0.
\end{equation}
\item[(SLP)] The system $S[R_1,R_2]$ is in the strong limit point case.
\end{enumerate} \end{theorem} \begin{proof}
Without loss of generality we assume here that the functions $u_1$, $u_2$, and $f$ are real-valued. By the first Green's identity~\eqref{eq:G1} one obtains \begin{equation*}
\int_0^x u_2^2\, dR_1 = \int_0^x f u_1\, dR_2 + u_1 u_2|_0^x, \end{equation*} and hence \begin{equation}
\lim_{x\to b} u_1(x) u_2(x) = d \end{equation} where $d \in \mathbb{R}$ if the Dirichlet property holds and $d=+\infty$ otherwise.
Let us start with the implication (LP) $\Rightarrow$ (D). For this purpose we assume the contrary i.e. the system $S[R_1,R_2]$ is in the limit point case but $d=+\infty$. Notice, that the functions $R_1$ and $R_2$ do not have common points of discontinuity, therefore neither do the functions $u_1$ and $u_2$. It implies that both $u_1$ and $u_2$ preserve their signs on some interval $[b_0,b)$ (otherwise they would have to share a jump from a positive to a negative value or vice versa), so the function $u_1$ is either positive and increasing or negative and decreasing. If $1 \notin {\mathcal L}^2(R_2)$ then it immediately results as $u_1 \notin {\mathcal L}^2(R_2)$. In the case if $1 \in {\mathcal L}^2(R_2)$ (and hence $R_1 \notin {\mathcal L}^2(R_2)$) the implication $f \in {\mathcal L}^2(R_2) \Rightarrow f \in {\mathcal L}^1(R_2)$ is valid and hence (see~\eqref{eq:f2d}) there exists a finite limit $u_2(b) \coloneqq \lim_{x\to b} u_2(x)$. The limit $u_2(b)$ must be zero, otherwise from \begin{equation}
|u_1(x) - u_1(b_0)| = \left| \int_{b_0}^x u_2\,dR_1 \right| \ge \frac{|u_2(b)|}{2} (R_1(x)-R_1(b_0)) \end{equation} one gets $u_1 \notin {\mathcal L}^2(R_2)$. One can see that $1/u_2 \notin {\mathcal L}^2(R_2)$. Indeed, if $1/u_2 \in {\mathcal L}^2(R_2)$ then the integral \begin{equation}
\int_0^x \frac{f}{u_2} dR_2 = - \int_0^x \frac{du_2}{u_2} \end{equation}
converges as $x\to b$, which contradicts to Lemma~\ref{lem:log}. Since $d=+\infty$, the estimate $|u_1| > 1/|u_2|$ hold on some interval $[b_0,b)$ and provides again $u_1 \notin {\mathcal L}^2(R_2)$. This completes the proof of the implication (LP) $\Rightarrow$ (D).
Now let us prove the implication (D) $\Rightarrow$ (SLP*). We first will show that $d=0$. In the case $1 \in {\mathcal L}^2(R_2)$ the reasoning of the previous paragraph can be used to show that $u_1 \notin {\mathcal L}^2(R_2)$ for every non-zero $d$. In the case $1 \notin {\mathcal L}^2(R_2)$ the reasoning above shows again that $u_1 \notin {\mathcal L}^2(R_2)$ for every $d>0$. Therefore we assume $d<0$ and get that $u_1$ is either positive and decreasing or negative and increasing on some interval $[b_0,b)$, namely $u_1 \to 0$ as $x\to b$. From $|u_1 u_2| > |d|/2$ on $[b_0,b)$ (with a possible change of point $b_0$) we obtain the following inequality \begin{equation}
\int_{b_0}^b u_2^2\,dR_1 = \int_{b_0}^b u_2\,df_1 >
\frac{d}{2} \int_{b_0}^b \frac{du_1}{u_1} = +\infty. \end{equation} The left hand side converges by our assumption but the right hand side diverges due to Lemma~\ref{lem:log}. This contradiction proves that $d=0$. Thus, implication (D) $\Rightarrow$ (SLP) is valid.
As is known (see \cite[Theorem 4.3]{Str19}), the system $S[R_1,R_2]$ is in the limit point case if and only if for every $(u_1,u_2, f)$ and $(v_1,v_2, g)$ from $\cT$ \begin{equation} \label{eq:LP-W_crit}
\lim_{x\to b} [u,v]_x = \lim_{x\to b} (u_1(x)v_2(x) - u_2(x)v_1(x)) = 0. \end{equation}
In order to prove the implication (SLP*) $\Rightarrow$ (SLP) we notice first that by Lemma~\ref{lem:psi} the system $S[R_1,R_2]$ cannot be in the limit circle case since~\eqref{eq:def_SLP_var} holds for every $(u_1,u_2, f) \in \cT$. The condition~\eqref{eq:def_SLP} follows from~\eqref{eq:def_SLP_var}, \eqref{eq:LP-W_crit} and the following equality (cf.~\cite{Ever76}) \begin{equation}
2 u_1(x)v_2(x) = (u_1 + v_1)(u_2 + v_2) + [u,v]_x = 0. \end{equation}
Assume that the statement (SLP) holds, i.e. condition~\eqref{eq:def_SLP} is satisfied for every $(u_1,u_2, f)$ and $(v_1,v_2, g)$ from $\cT$. Then, clearly, \eqref{eq:LP-W_crit} holds for every $(u_1,u_2, f)$ and $(v_1,v_2, g)$ from $\cT$ and hence the system $S[R_1,R_2]$ is in the limit point case. This proves the implication (SLP) $\Rightarrow$ (LP). \end{proof}
\begin{remark} In the case of absolutely continuous $R_1$ and $R_2$ the implication $(LP)\Rightarrow(SLP)$ for the system $S[R_1,R_2]$ was proved in~\cite{Kalf74}, see also~\cite{Ever76}. \end{remark} \subsection{Boundary triples for integral systems in the limit point case} \begin{definition}\label{def:Neumann_m} Let the system $S[R_1,R_2]$ be in the limit point case at $b$. Then for each $\lambda\in\nC\setminus\nR$ there is a unique coefficient $m_N(\lambda)$, such that \begin{equation} \label{eTWeyl_1}
\psi_1(\cdot,\lambda) = s_1(\cdot,\lambda) - m_N(\lambda) c_1 (\cdot,\lambda)\in {\mathcal L}^2(R_2). \end{equation} The function $m_N$ is called the \emph{Neumann $m$-function} of the system \eqref{eq:IntSys} on $I$ and the function $\psi(t,\lambda)$ is called the Weyl solution of the system $S[R_1,R_2]$ on $I$. \end{definition}
Let us collect some statements concerning boundary triples for $S^*$, which were partially formulated in~\cite{Str18,Str19}. \begin{proposition}\label{prop:BT_IS_LP}
Let the system $S[R_1,R_2]$ be in the limit point case at $b$, and let $T=T_{min}$. Then:
\begin{enumerate}
\item [(i)] $T$ is a symmetric nonnegative operator in $L^2(R_2)$ with deficiency indices $(1,1)$.
\item [(ii)] The triple $\Pi = (\dC,\Gamma_0,\Gamma_1)$, where
\begin{equation}\label{eq:BTLP}
\Gamma_0 \bm{u} = u_2(0), \quad \Gamma_1 \bm{u} = -u_1(0), \quad \bm{u} \in T^*,
\end{equation}
is a boundary triple for $T^*$.
\item[(iii)] The defect subspace $\sN_\lambda(T)$ is spanned by the Weyl solution $\psi_1(t,\lambda)$, and the Weyl function $m(\lambda)$ of $T$ corresponding to the boundary triple $\Pi$ coincides with the Neumann $m$-function of the system $S[R_1,R_2]$ on $I$:
\begin{equation}\label{eq:WFLP}
m(\lambda)=- \frac{\psi_1(0,\lambda)}{\psi_2(0,\lambda)} = m_N(\lambda).
\end{equation}
\item[(iv)] The Weyl function $m(\lambda)$ of $T$ corresponding to the boundary triple $\Pi$ coincides with the principal Titchmarsh-Weyl coefficient $q(\lambda)$ of the system $S[R_1,R_2]$ on $I$ and belongs to the Stieltjes class ${\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}$.
\item[(v)] If $R_2(b)<\infty$ then the Weyl function $m_N$ of $T_N$ admits the representation
\begin{equation}\label{eq:Pol_mLP}
m_N(\lambda) = -\frac{1}{R_2(b)\cdot\lambda} + \widetilde m(\lambda);
\end{equation}
where $\widetilde m$ is a function from ${\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}$ such that $\lim_{y\downarrow 0} y \widetilde m(iy) = 0$.
\end{enumerate} \end{proposition} \begin{proof} \textbf{1.} At first we show ${\textrm (i)-(ii)}$. Since \eqref{eq:IntSys} is in the limit point case at $b$, \begin{equation*}
\lim_{x\to b} [u,\conj{v}]_x = 0 \quad \text{for} \quad \bm{u} = \begin{bmatrix} \pi u_1 \\ \pi f \end{bmatrix},\ \bm{v} = \begin{bmatrix} \pi v_1 \\ \pi g \end{bmatrix}\in T_{\max} \end{equation*} and hence the generalized Green's identity \eqref{eq:G2} is of the form \begin{equation}\label{eq:GreenLP}
\int_0^b (f\conj{v_1} - u_1\conj{g})\, dR_2(t) = -[u,\conj{v}]_0 = u_2(0)\conj{v_1(0)} - u_1(0)\conj{v_2(0)}. \end{equation} Therefore, the triple $\Pi$ in~\eqref{eq:BTLP} is a boundary triple for $T^*$.
It follows from the first Green's identity \eqref{eq:G1} and Lemma~\ref{lem:psi} that for every $\bm{u} \in T$ the identity~\eqref{eq:Green1_} holds and thus the linear relation $T$ is nonnegative.
\textbf{2.} Now ${\textrm (iii)}$ is shown. In the limit point case there is only one linearly independent solution $\psi(\cdot,\lambda)$ of the system $S[R_1,R_2]$ such that $\psi_1(\cdot,\lambda)\in L^2(R_2)$, see~\eqref{eTWeyl_1}, and hence the defect subspace $\sN_\lambda(T^*)$ is spanned by the function $\psi_1(\cdot,\lambda)$. Denote $\bm{u}(t,\lambda)=(\psi_1(\cdot,\lambda),\lambda\psi_1(\cdot,\lambda))^T\in\widehat \sN_\lambda(T^*)$. It follows from~\eqref{eq:BTLP} that \begin{equation*}
\Gamma_0\bm{u}(\cdot,\lambda) = \psi_2(0,\lambda) = 1,\quad
\Gamma_1\bm{u}(\cdot,\lambda) = -\psi_1(0,\lambda) = m_N(\lambda), \end{equation*} This yields formula~\eqref{eq:WFLP}.
\textbf{3.} Now we show ${\textrm (iv)}$. If $\lambda\in\dR_-$ then it follows from Lemma~\ref{lem:2.4} that the function $\frac{s_1(x,\lambda)}{c_1(x,\lambda)}$ is increasing and bounded from above. Therefore, the following limit \begin{equation}\label{eq:mLP}
q(\lambda) \coloneqq \lim_{x\to b}\frac{s_1(x,\lambda)}{c_1(x,\lambda)} \end{equation} exists and is nonnegative for every $\lambda\in\dR_-$. By Stieltjes-Vitaly theorem the function $q$ is holomorphic on $\dC\setminus[0,\infty)$. The function $q$ belongs to the Stieltjes class ${\mathcal S}} \def\cT{{\mathcal T}} \def\cU{{\mathcal U}$, since it is nonnegative for every $\lambda\in\dR_-$. Since $\frac{s_1(x,\lambda)}{c_1(x,\lambda)}$ belongs to the Weyl disc $D_x(\lambda)$ and the system $S[R_1,R_2]$ is limit point at $b$, for every $\lambda\in\dC_+\cup\dC_-$ the following equality holds \begin{equation}
q(\lambda)=\lim_{x\to b}\frac{s_1(x,\lambda)}{c_1(x,\lambda)}=m_N(\lambda). \end{equation}
\textbf{4.} Assume that $R_2(b)<+\infty$. Let us consider the family of von Neumann $m$-functions $m_N^x(\lambda)=\frac{s_2(x,\lambda)}{c_2(x,\lambda)}$ converging to $m_N(\lambda)$ as $x\to b-$. Due to equality~\eqref{eq:cs2} \begin{equation}\label{eq:c2s2}
\frac{1}{ m_N^x(\lambda)}= \frac{c_2(x,\lambda)}{s_2(x,\lambda)}=\int_0^x\frac{-\lambda}{s_2(x,\lambda) s_{2+}(x,\lambda)}\,dR_2(x). \end{equation} Since $s_2(x,\lambda)\ge 1$ for $x\in[0,b)$ and $\lambda\in\dR_-$ there exists the limit \begin{equation*} \frac{-1}{\lambda m_N(\lambda)}=\lim_{x\to b}\frac{-c_2(x,\lambda)}{\lambda s_2(x,\lambda)}=\int_0^b\frac{1}{s_2(x,\lambda) s_{2+}(x,\lambda)}\,dR_2(x) \end{equation*} Due to Lemma~\ref{lem:2.4} \begin{equation*}
\lim_{\lambda\downarrow 0}\frac{1}{s_2(x,\lambda) s_{2+}(x,\lambda)} = 1,
\quad \text{and} \quad
\left|\frac{1}{s_2(x,\lambda) s_{2+}(x,\lambda)}\right|\le 1
\quad \text{for} \quad x\in[a,b). \end{equation*} Hence one obtains by the Lebesgue bounded convergence Theorem \begin{equation}
\lim_{\lambda\to 0} \frac{1}{-\lambda m_N(\lambda)} = \int_{[0,b)}dR_2 = R_2(b). \end{equation} This implies (v).
\end{proof}
\subsection{The canonical singular continuation of a regular integral system} If the integral system $S[R_1,R_2]$ is regular at $b$ then due to Remark~\ref{rem:Param} we can assume without loss of generality that $b<\infty$. \begin{definition}
For a regular system $S[R_1,R_2]$ with $b<\infty$ we define the extended functions
\begin{equation}\label{eq:SingCont}
\widetilde R_1(x) \coloneqq \left\{
\begin{array}{cc}
R_1(x) :& x\in [0,b], \\
R_1(b) :& x\in [b,\infty),
\end{array}\right. \quad
\widetilde R_2(x) \coloneqq \left\{
\begin{array}{cc}
R_2(x) :& x\in [0,b], \\
R_2(b)+x-b :& x\in [b,\infty).
\end{array}\right.
\end{equation}
The integral system $S[\widetilde R_1,\widetilde R_2]$ corresponding to \begin{equation} \label{eq:ContIntSys}
\widetilde u(x,\lambda) = \widetilde u(0,\lambda)+\int_0^x
\begin{bmatrix}
0 & d\widetilde{R}_1(t) \\
-\lambda d\widetilde{R}_2(t) & 0
\end{bmatrix} \widetilde u(t,\lambda),\quad x\in [0, \infty)
\end{equation}
will be called \emph{the canonical singular continuation} of a regular integral system $S[R_1,R_2]$. \end{definition}
\begin{proposition}\label{prop:SingCont} Let the integral system $S[R_1,R_2]$, see~\eqref{eq:IntSys}, be regular at $b<\infty$. Then the principal Titchmarsh-Weyl coefficient $\widetilde q$ of its canonical singular continuation $S[\widetilde R_1,\widetilde R_2]$ coincides with the principal Titchmarsh-Weyl coefficient $q$ of the system $S[R_1,R_2]$: \begin{equation}\label{eq:wt_q}
\widetilde q(\lambda) = q(\lambda), \quad \lambda\in\dC\setminus\dR. \end{equation} \end{proposition} \begin{proof} Let the pair $u_1,u_2$ satisfy the integral system $S[R_1,R_2]$ for some $\lambda\in\dC\setminus\dR$ and let $\widetilde u_1,\widetilde u_2$ be the continuations of $u_1,u_2$ to the interval $[0,+\infty)$ given by \begin{equation}\label{eq:wt_uu}
\left\{
\begin{aligned}
\widetilde u_1(x,\lambda)&=u_1(b,\lambda),\quad x\in(b,\infty), \\
\widetilde u_2(x,\lambda)&=u_2(b,\lambda)-\lambda u_1(b,\lambda)(x-b),\quad x\in(b,\infty).
\end{aligned}
\right. \end{equation} Then the pair $\widetilde u_1,\widetilde u_2$ satisfies the integral system~\eqref{eq:ContIntSys}. If $c_1,c_2$ and $s_1,s_2$ are solutions of ~\eqref{eq:IntSys} according to the initial conditions~\eqref{eq:csK} then the continuations $\widetilde c_1,\widetilde c_2$ and $\widetilde s_1,\widetilde s_2$ are solutions of the integral system~\eqref{eq:ContIntSys} with the same initial conditions~\eqref{eq:csK}.
In view of~\eqref{eq:wt_uu} the principal Titchmarsh-Weyl coefficient $\widetilde q$ of the canonical singular continuation $S[\widetilde R_1,\widetilde R_2]$ is of the form \begin{equation*}
\widetilde q(\lambda) =
\lim_{x\to\infty}\frac{\widetilde s_1(x,\lambda)}{\widetilde c_1(x,\lambda)} =
\lim_{x\to\infty}\frac{s_1(x,\lambda)}{c_1(x,\lambda)} =
q(\lambda). \end{equation*} \end{proof}
\section{Dual integral systems} \label{sec:dual_strings} \begin{definition}\label{def:dual_systems}
\emph{The dual system} $\widehat S[R_1,R_2]$ to a singular system $S[R_1,R_2]$ is defined by changing the roles of $R_1$ and $R_2$ in~\eqref{eq:IntSys}, that is $\hat S[R_1,R_2] = S[R_2,R_1]$ and \begin{equation} \label{eq:DualIntSys}
\widehat u(x,\lambda) = \widehat u(0,\lambda) + \int_0^x
\begin{bmatrix}
0 & dR_1(t) \\
-\lambda dR_2(t) & 0
\end{bmatrix} \widehat u(t,\lambda), \quad x \in [0,b). \end{equation} In case the system $S[R_1,R_2]$ is regular, we will denote by $\widehat S[R_1,R_2]$ the dual to its canonical singular continuation:
$\hat S[R_1,R_2]=S[\widetilde R_2, \widetilde R_1]$. \end{definition}
Let $\widehat{s}(\cdot,\lambda)$ and $\widehat{c}(\cdot,\lambda)$ be the unique solutions of \eqref{eq:DualIntSys} satisfying the initial conditions \begin{equation}
\widehat{c}_1(0,\lambda) = 1, \ \widehat c_2(0,\lambda) = 0,
\quad \text{and} \quad
\widehat{s}_1(0,\lambda) = 0, \ \widehat s_2(0,\lambda) = 1. \end{equation}
\begin{theorem} \label{thm:2} Let $U(x,\lambda)$ and $\widehat U(x,\lambda)$ be the fundamental matrices of the system $S[R_1,R_2]$ and its dual system $\widehat S[R_1,R_2]$ respectively. Let $m_N$ and $\widehat m_N$ be the Neumann $m$-functions of the systems $S[R_1,R_2]$ and $\widehat S[R_1,R_2]$ in the sense of Definitions~\ref{def:5.1}, \ref{def:Neumann_m}. Then:
\begin{enumerate}
\item [(i)] The matrices $U(x,\lambda)$ and $\widehat U(x,\lambda)$ are related by \begin{equation}\label{eq:Fund_dir_dual}
\widehat U(x,\lambda) = D(\lambda)^{-1} U(x,\lambda) D(\lambda),
\quad \text{where} \quad
D(\lambda) =
\begin{pmatrix}
0 & -\lambda^{-1} \\
1 & 0
\end{pmatrix}. \end{equation} \item [(ii)] If the system $S[R_1,R_2]$ is singular at $b$,
then \begin{equation}\label{eq:whm_mLP}
\widehat m_N(\lambda)=-\frac{1}{\lambda m_{N}(\lambda )}. \end{equation}
\item [(iii)] If $S[R_1,R_2]$ is regular at $b$, then \begin{equation}\label{eq:whm_m}
\widehat m_N(\lambda )=\frac{\widehat s_2(b,\lambda )}{\widehat c_2(b,\lambda )}=-\frac{c_1(b,\lambda )}{\lambda s_1(b,\lambda )}
= -\frac{1}{\lambda m_{ND}(\lambda )}, \end{equation} where $m_{ND}(\lambda )$ is the Neumann $m$-function of system $S[R_1,R_2]$, subject to the boundary condition $u_1(b)=0$, see Definition~\ref{def:5.1D}.
\item [(iv)] The principal Titchmarsh-Weyl coefficients $q$ and $\widehat q$ of $S[R_1,R_2]$ and $\widehat S[R_1,R_2]$ are connected by the equality \begin{equation}\label{eq:mS_whmS2}
\widehat q(\lambda) = -\frac{1}{\lambda q(\lambda)},
\quad \lambda\in\dC\setminus\dR_+. \end{equation} \end{enumerate} \end{theorem} \begin{proof} \textbf{1.} At first (i) is shown. A straightforward calculation shows that the solutions $\widehat{s}(\cdot,\lambda)$ and $\widehat{c}(\cdot,\lambda)$ of \eqref{eq:DualIntSys} are related to the solutions ${s}(\cdot,\lambda)$ and ${c}(\cdot,\lambda)$ of \eqref{eq:IntSys} by the equalities \begin{equation}\label{eq:wh_cs}
\begin{bmatrix}
\widehat{c}_1(\cdot,\lambda) \\
\widehat{c}_2(\cdot,\lambda)
\end{bmatrix} = \begin{bmatrix}
{s}_2(\cdot,\lambda) \\
-\lambda{s}_1(\cdot,\lambda)
\end{bmatrix},
\quad
\begin{bmatrix}
\widehat{s}_1(\cdot,\lambda) \\
\widehat{s}_2(\cdot,\lambda)
\end{bmatrix}
= \begin{bmatrix}
-\lambda^{-1} {c}_2(\cdot,\lambda) \\
{c}_1(\cdot,\lambda)
\end{bmatrix}. \end{equation} The equality~\eqref{eq:Fund_dir_dual} follows from \eqref{eq:wh_cs}.
System $S[R_1,R_2]$ is regular at $b$ if and only if both $S[R_1,R_2]$ are in the limit circle case at $b$. Therefore the proof of (ii) can be splitted into the following three cases \textbf{2--4}.
\textbf{2.} \emph{Both $S[R_1,R_2]$ and $\widehat S[R_1,R_2]$ are in the limit point case at $b$$:$}\\ Let $m_N$ be the Neumann $m$-function of the systems $S[R_1,R_2]$, see Definition~\ref{def:Neumann_m}, and let $\psi_1(\cdot,\lambda)$ be the corresponding Weyl solution of the system $S[R_1,R_2]$. Then the vector function \begin{equation}\label{eq:wh_psi}
\widehat\psi(\cdot,\lambda) \coloneqq
\begin{bmatrix}
\widehat{s}_1(\cdot,\lambda ) \\
\widehat{s}_2(\cdot,\lambda )
\end{bmatrix}
+\frac{1}{\lambda m_N(\lambda )}
\begin{bmatrix}
\widehat{c}_1(\cdot,\lambda ) \\
\widehat{c}_2(\cdot,\lambda )
\end{bmatrix}
= \begin{bmatrix}
-\lambda^{-1} {c}_2(\cdot,\lambda )+ \lambda^{-1} m_N(\lambda)^{-1} s_2(\cdot,\lambda ) \\
{c}_1(\cdot,\lambda ) - m_N(\lambda)^{-1} s_1(\cdot,\lambda )
\end{bmatrix} \end{equation} is a solution of the system \eqref{eq:DualIntSys}. Moreover, due to Lemma~\ref{thm:SLP} $\widehat \psi_{1}(\cdot,\lambda)= \frac{1}{\lambda m_N(\lambda)} \psi_{2}(\cdot,\lambda)$ belongs to $L^2(R_1)$. Therefore, $\widehat\psi_1(\cdot,\lambda)$ is the Weyl solution of the system $\widehat S[R_1,R_2]$ and the function $-\frac{1}{\lambda m_N(\lambda)}$ is the Neumann $m$-function of the system~$\widehat S[R_1,R_2]$.
\textbf{3.} \emph{$S[R_1,R_2]$ is in the limit circle case and $\widehat S[R_1,R_2]$ is in the limit point case at $b$$:$}\\ Let the function $\psi^N$ be defined by~\eqref{eTWeyl2}. Since \eqref{eq:IntSys} is in the limit circle case it follows from Lemma~\ref{lem:psi} that $\psi^N_2\in L^2(R_1)$. Hence, $\widehat\psi(\cdot,\lambda)$ is a solution of the system $\widehat S[R_1,R_2]$, such that $\widehat \psi_{1}(\cdot,\lambda)=\frac{1}{\lambda m_N(\lambda )}\psi^N_{2}(\cdot,\lambda)\in L^2(R_1)$. Therefore, $\widehat \psi_1$ is the {\em Weyl solution} of the system $\widehat S[R_1,R_2]$ and the function $-\frac{1}{\lambda m_N(\lambda )}$ is the Neumann $m$-function of the systems $\widehat S[R_1,R_2]$.
\textbf{4.} \emph{$S[R_1,R_2]$ is in the limit point case and $\widehat S[R_1,R_2]$ is in the limit circle case at $b$$:$}\\ As was shown on Step \textbf{3} the Neumann $m$-function $\widehat m_{N}(\lambda )$ of the systems $\widehat S[R_1,R_2]$ subject to the boundary condition $\widehat\psi_{2}(b,\lambda)=0$ is connected with the Neumann $m$-function $m_{N}(\lambda )$ of the system $S[R_1,R_2]$ by the equality \begin{equation*}
m_N(\lambda) = -\frac{1}{\lambda \widehat m_{N}(\lambda)} \end{equation*} which is equivalent to~\eqref{eq:whm_mLP}.
\textbf{5.} Now (iii) is shown. Let $m_{ND}(\lambda )$ be the Neumann $m$-function of the system $S[R_1,R_2]$, subject to the boundary condition~\eqref{eq:m_functLCD} and let $\psi_1^{ND}(\cdot,\lambda )$ be the corresponding {\em Weyl solution} of the system $S[R_1,R_2]$ defined by~\eqref{eTWeylND}. By definition $\psi^{ND}_{1}(b,\lambda )=0$. Then the vector function \begin{equation*}
\widehat\psi(\cdot,\lambda) \coloneqq
\begin{bmatrix}
\widehat{s}_1(\cdot,\lambda) \\
\widehat{s}_2(\cdot,\lambda)
\end{bmatrix}
+\frac{1}{\lambda m_{ND}(\lambda)}
\begin{bmatrix}
\widehat{c}_1(\cdot,\lambda) \\
\widehat{c}_2(\cdot,\lambda)
\end{bmatrix}
= -\frac{1}{m_{ND}(\lambda)}
\begin{bmatrix}
-\frac{1} {\lambda }\left(s_2(\cdot,\lambda)- m_{ND}(\lambda) c_2(\cdot,\lambda)\right) \\
s_1(\cdot,\lambda)- m_{ND}(\lambda) c_1(\cdot,\lambda)
\end{bmatrix} \end{equation*} is a solution of the system \eqref{eq:DualIntSys} such that $\widehat \psi_{2}(b,\lambda)=\psi^{ND}_{1}(b,\lambda)=0$.
Therefore, the function $\frac{-1}{\lambda m_{ND}(\lambda )}$ is the Neumann $m$-function of the systems $\widehat S[R_1,R_2]$, subject to the boundary condition $\widehat \psi_{2}(b,\lambda)=0$.
\textbf{6.} Finally (iv) is shown. If the integral system $S[R_1,R_2]$ is singular at $b$ then the Neumann $m$-function $m_N$ (resp. $\widehat m_N$) coincides with the principal Titchmarsh-Weyl coefficient $q$ of the system~$S[R_1,R_2]$ (resp. $\widehat q$ of the system~ $\widehat S[R_1,R_2]$), see Propositions~\ref{prop:BT_IS_N}, \ref{prop:BT_IS_LP}. Therefore, \eqref{eq:mS_whmS2} is implied by \eqref{eq:whm_mLP}.
If the system $S[R_1,R_2]$ is regular at $b$ then by Propositions~\ref{prop:SingCont} $q$ coincides with the principal Titchmarsh-Weyl coefficient $\widetilde q$ of the canonical singular continuation $S[\widetilde R_1,\widetilde R_2]$ of the system $S[R_1,R_2]$ to $[0,+\infty)$, see~\eqref{eq:SingCont}. By the statement of the above paragraph the principal Titchmarsh-Weyl coefficient $\widehat q$ of the dual system $S[\widetilde R_2,\widetilde R_1]$ is of the form \begin{equation*}
\widehat q(\lambda) = -\frac{1}{\lambda \widetilde q(\lambda)} =
-\frac{1}{\lambda q(\lambda )}, \end{equation*} and \eqref{eq:mS_whmS2} is shown.
\end{proof}
Since the relation of duality for integral systems is reflexive one derives from the proof of Theorem~\ref{thm:2} the following statement. \begin{corollary}\label{cor:psi} Let the system $S[R_1,R_2]$ be in the limit point case and let $\widehat S[R_1,R_2]$ be in the limit circle case at $b$. Let $\psi_1(\cdot,\lambda)$ be the corresponding Weyl solution of the system $S[R_1,R_2]$. Then \begin{equation}\label{eq:psi_0}
\lim_{x\to b} \psi_1(x,\lambda) = 0. \end{equation} \end{corollary}
\begin{proof}
As it was mentioned in the proof of Theorem~\ref{thm:2} (Step \textbf{3}), the Weyl solution $\psi(\cdot,\lambda)$ of the system $S[R_1,R_2]$ is connected with the Weyl solution $\widehat\psi^N(\cdot,\lambda)$ of the dual system \eqref{eq:IntSys} by the equality $\psi_{1}(\cdot,\lambda)=\frac{1}{\lambda\widehat m_N(\lambda )}\widehat\psi^N_{2}(\cdot,\lambda)$.
Since $\widehat\psi^N_{2}(b,\lambda)=0$ one obtains~\eqref{eq:psi_0}. \end{proof} \begin{remark}
Formula~\eqref{eq:whm_mLP} was proven in \cite{KWW06} for Krein strings and in \cite{Kost13} for integral systems.
However, in \cite{Kost13} it was overlooked that in the regular case the formula \eqref{eq:whm_mLP} fails to hold and should be replaced by~\eqref{eq:whm_m}. \end{remark}
\end{document} | arXiv |
home chevron_right Learning centerchevron_rightEnergy storage and conversionchevron_rightBatterychevron_rightQCM: Why measure at overtones matters
Topic 10 min read
QCM: Why measure at overtones matters
Latest updated: November 23, 2021
As previously mentioned [1], common QCM systems measure the fundamental resonant frequency changes and sometimes the resistance, which can somehow be related to the dissipative properties of the film or medium.
Advanced systems based on impedance or ring down measurements give access to the true dissipation or half bandwidth or quality factor and allow measurements at several overtones[1]. The number of allowed overtones depends on the bandwidth of the instrument and on the fundamental resonant frequency of the quartz.
The overtone order must be odd (1, 3, 5…) to ensure an antisymmetric pattern of motion of the quartz. If the overtone order is even (2, 4, 6…) the deformation is symmetric and there is no current between the electrodes [4].
The BluQCM QSD can measure up to the $13^{\text{th}}$ overtone with 5 MHz quartz crystals. With 50 MHz HFF sensors, only the $3^{\text{rd}}$ overtone can be measured.
Figure 1 shows the vibrational movement of quartz at the fundamental frequency $f_0$ (Top) and the third harmonic $3f_0$ (Bottom).
Figure 1: Quartz at the fundamental frequency $f_0$ (Top) and the third harmonic $3f_0$ (Bottom).
Why measure overtones or harmonics? Ensuring the validity of the Sauerbrey equation.
Measuring at harmonics gives another way of ensuring that the film coating the bare electrode is rigid. If the value $\Delta f_n/n$ is constant for each harmonic, the film can be considered rigid.
Measurements at overtones are more reliable
The value $\Delta f_n/n$, where $f_n$ is the resonant frequency at the $n^{th}$ harmonic, should theoretically be constant with the harmonic order. However, it has been noted [3] that it was constant only after the $3^{\text{rd}}$ harmonic, which then can be considered to be a more reliable measurement of the resonant frequency change. This can be seen in Fig.2 taken from [5].
Figure 2: The absolute frequency of several 5 MHz crystals–#60, 66, and 67 (two measurements): (solid shapes) measured on various overtones with a QCM-D instrument in air, compared with the values of (open shapes) $f_n/n$ obtained theoretically. All frequencies are scaled by the overtone order.
Measurements at harmonics are needed for advanced data interpretation
Dissipation and resonant frequency shift are needed at several harmonics when one needs to study and characterize visco-elastic homogeneous thin films, particles, molecules, or conformation.
As long as the studied object is the growth or deposition of a rigid thin film or Sauerbrey thin film or elastic film, the mechanical response does not depend on the frequency. This is not the case when the object of study is a visco-elastic film, that is to say, a film where the energy of the wave is both elastically transferred and viscously dissipated. In this case, the response of the system composed of the quartz, the electrodes, the film, and the medium (air or liquid) depends on the input frequency.
A parallel can be made between the electrical transfer function of a system, for example, the impedance, and the mechanical transfer function of a system, for example, the shear modulus. In the same manner that the impedance $Z(\omega)$ (with $\omega=2\pi f$) allows the calculation of the current response from the potential input, the shear modulus $G(\omega)$ allows the calculation of the stress response from the strain input. In a visco-elastic film, the shear modulus is a frequency-dependent complex function.
$G(\omega)=\mathrm{Re}(G(\mathrm{\omega}))+j\;\mathrm{Im}(G(\mathrm {\omega}))$
More details can be found in [4].
[1] Quartz Crystal Microbalance: Measurement principles
[2] https://en.wikipedia.org/wiki/Overtone
[3] https://en.wikipedia.org/wiki/Harmonic
[4] D. Johannsmann, in "The quartz crystal microbalance in soft matter research", Springer, 2015
[5] I. Reviakine, A. N. Morozov, F. F. Rossetti, J. Appl. Phys. 95 (2004) 7712.
1. Overtones and harmonics are similar terms although it seems that overtones are mainly used in the field of acoustics. Furthermore, whereas "harmonics" mean multiple integers of the fundamental frequency, including the fundamental frequency, "overtones" seem to mean any multiple of and over the fundamental frequency, excluding the fundamental frequency [2,3].
QCM-D overtones dissipation Quartz Crystal Microbalance Sauerbrey equation viscoelastic films particles BluQCM
BluQCM QSD
The BluQCM QSD is a single channel, compact and modular instrument. Its low footprint and lightweight makes it particularly suitable for crowded labs. It is available as standalone, with temperature control or/and flow control.
Sensors and cells
Unique design for quick/easy resonator positioning. Optimized contact helps generate stable, reliable measurements. | CommonCrawl |
How many different positive integers divisible by 4 can be formed using each of the digits 1, 2, 3, and 4 at most once, and no other digits? For example, 12 counts, but 512 does not.
The only one-digit integer divisible by $4$ that we can construct is $4$.
We can construct $3$ two-digit integers divisible by $4$: $12$, $32$, and $24$.
An integer is divisible by $4$ if its rightmost two digits are divisible by $4$. Thus we can append either or both of the remaining two digits to any of these two-digit integers and preserve divisibility by $4$. For each, there are $2$ ways to choose one digit to append, and $2$ ways to order the digits if we append both of them. Thus we get $4$ more integers for each, or $12$ total.
The full number is $12+3+1=\boxed{16}$ integers. | Math Dataset |
Research | Open | Published: 07 January 2016
A new iterative algorithm for the sum of infinite m-accretive mappings and infinite $\mu_{i}$-inversely strongly accretive mappings and its applications to integro-differential systems
Li Wei1 &
Ravi P Agarwal2,3
Fixed Point Theory and Applicationsvolume 2016, Article number: 7 (2016) | Download Citation
A new three-step iterative algorithm for approximating the zero point of the sum of an infinite family of m-accretive mappings and an infinite family of $\mu_{i}$-inversely strongly accretive mappings in a real q-uniformly smooth and uniformly convex Banach space is presented. The computational error in each step is being considered. A strong convergence theorem is proved by means of some new techniques, which extend the corresponding work by some authors. The relationship between the zero point of the sum of an infinite family of m-accretive mappings and an infinite family of $\mu_{i}$-inversely strongly accretive mappings and the solution of one kind variational inequalities is investigated. As an application, an integro-differential system is exemplified, from which we construct an infinite family of m-accretive mappings and an infinite family of $\mu_{i}$-inversely strongly accretive mappings. Moreover, the iterative sequence of the solution of the integro-differential systems is obtained.
Throughout this paper, we assume that E is a real Banach space with norm $\Vert \cdot \Vert $ and $E^{*}$ is the dual space of E. We use '→' and '⇀' (or '$w-\lim$') to denote strong and weak convergence either in E or in $E^{*}$, respectively. We denote the value of $f \in E^{*}$ at $x \in E$ by $\langle x,f \rangle$.
A Banach space E is said to be uniformly convex if, for each $\varepsilon\in(0,2]$, there exists $\delta> 0$ such that
$$\Vert x\Vert = \Vert y\Vert = 1,\qquad \Vert x - y\Vert \geq \varepsilon\quad \Rightarrow\quad \biggl\Vert \frac{x+y}{2}\biggr\Vert \leq1-\delta. $$
A Banach space E is said to be smooth if
$$\lim_{t \rightarrow0}\frac{\Vert x+ty\Vert -\Vert x\Vert }{t} $$
exists for each $x , y \in\{z \in E : \Vert z\Vert = 1\}$.
In addition, we define a function $\rho_{E}: [0,+\infty) \rightarrow [0,+\infty)$ called the modulus of smoothness of E as follows:
$$\rho_{E}(t) = \sup \biggl\{ \frac{1}{2}\bigl(\Vert x+y\Vert + \Vert x-y\Vert \bigr)-1 : x,y \in E, \Vert x\Vert =1, \Vert y\Vert \leq t\biggr\} . $$
It is well known that E is uniformly smooth if and only if $\frac{\rho_{E}(t)}{t}\rightarrow0$, as $t \rightarrow0$. Let $q > 1$ be a real number. A Banach space E is said to be q-uniformly smooth if there exists a positive constant C such that $\rho_{E}(t)\leq Ct^{q}$. It is obvious that q-uniformly smooth Banach space must be uniformly smooth.
An operator $B: E\rightarrow E^{*}$ is said to be monotone [1] if $\langle u - v, Bu - Bv\rangle\geq0$, for all $u,v \in D(B)$. The monotone operator B is said to be maximal monotone if the graph of B, $G(B)$, is not contained properly in any other monotone subset of $E \times E^{*}$. An operator $B: E \rightarrow E^{*} $ is said to be coercive if $\lim_{n \rightarrow \infty}\frac{\langle x_{n}, Bx_{n}\rangle}{ \Vert x_{n}\Vert }= +\infty$ for $\{x_{n}\} \subset D(B)$ such that $\lim_{n \rightarrow \infty} \Vert x_{n}\Vert = +\infty$.
A single-valued mapping $F:D(F) = E \rightarrow E^{*}$ is said to be hemi-continuous [1] if $w-\lim_{t\rightarrow0}F(x+ty) = Fx$, for any $x,y\in E$. A single-valued mapping $F:D(F) = E \rightarrow E^{*}$ is said to be demi-continuous [1] if $w-\lim_{n\rightarrow\infty}Fx_{n} = Fx$, for any sequence $\{x_{n}\}$ strongly convergent to x in E.
Following from [1] or [2], the function h is said to be a proper convex function on E if h is defined from E onto $(-\infty, +\infty]$, h is not identically +∞ such that $h((1-\lambda)x+\lambda y)\leq(1-\lambda)h(x)+\lambda h(y)$, whenever $x,y \in E$ and $0 \leq\lambda\leq1$. h is said to be strictly convex if $h((1-\lambda)x+\lambda y)< (1-\lambda)h(x)+\lambda h(y)$, for all $0 < \lambda< 1$ and $x,y \in E$ with $x \neq y$, $h(x) <+\infty$ and $h(y) < +\infty$. The function $h: E \rightarrow(-\infty, +\infty]$ is said to be lower-semi-continuous on E if $\liminf_{y \rightarrow x}h(y) \geq h(x)$, for any $x \in E$. Given a proper convex function h on E and a point $x \in E$, we denote by $\partial h(x)$ the set of $x^{*} \in E^{*}$ such that $h(x) \leq h(y)+\langle x - y, x^{*}\rangle$ for any $y \in E$. Such elements $x^{*}$ are called subgradients of h at x, and $\partial h(x)$ is called the subdifferential of h at x.
For $q > 1$, the generalized duality mapping $J_{q}: E \rightarrow 2^{E^{*}}$ is defined by
$$J_{q}x : = \bigl\{ f \in E^{*}: \langle x, f\rangle= \Vert x\Vert ^{q}, \Vert f\Vert = \Vert x\Vert ^{q-1}\bigr\} ,\quad x \in E. $$
In particular, $J: = J_{2}$ is called the normalized duality mapping and $J_{q}(x) = \Vert x\Vert ^{q-2}J(x)$ for $x \neq0$. It is well known that if E is smooth, then $J_{q}$ is single-valued. If E is reduced to the Hilbert space H, then $J_{q} \equiv I$ is the identity mapping. It can be seen from [2] that the normalized duality mapping J has the following properties:
if E is uniformly smooth, then J is norm-to-norm uniformly continuous on each bounded subset in E;
the reflexivity of E and strict convexity of $E^{*}$ imply that J is single-valued, monotone, and demi-continuous.
In the following, we still denote by J and $J_{q}$ the single-valued normalized duality mapping and the single-valued generalized duality mapping.
For a mapping $T: D(T) \sqsubseteq E \rightarrow E$, we use $\operatorname {Fix}(T)$ to denote the fixed point set of it; that is, $\operatorname {Fix}(T) : = \{x\in D(T): Tx = x\}$.
Let $T : D(T) \sqsubseteq E \rightarrow E$ be a mapping. Then T is said to be
non-expansive if
$$\Vert Tx - Ty\Vert \leq \Vert x-y\Vert \quad \mbox{for } \forall x,y \in D(T); $$
k-Lipschitz if there exists $k > 0$ such that
$$\Vert Tx - Ty\Vert \leq k \Vert x - y\Vert \quad \mbox{for } \forall x,y \in D(T); $$
in particular, if $0 < k < 1$, then T is called a contraction and if $k = 1$, then T reduces to a non-expansive mapping;
accretive if, for all $x, y \in D(T)$, there exists $j_{q}(x-y) \in J_{q}(x-y)$ such that
$$\bigl\langle Tx - Ty, j_{q}(x-y)\bigr\rangle \geq0; $$
μ-inversely strongly accretive if, for all $x, y \in D(T)$, there exists $j_{q}(x-y) \in J_{q}(x-y)$ such that
$$\bigl\langle Tx - Ty, j_{q}(x-y)\bigr\rangle \geq\mu \Vert Tx - Ty \Vert ^{q} $$
for some $\mu> 0$;
m-accretive if T is accretive and $R(I+\lambda T) = E$ for $\forall\lambda> 0$;
strongly positive(see [3]) if $D(T) = E$ where E is a real smooth Banach space and there exists $\overline{\gamma} > 0$ such that
$$\langle Tx,Jx\rangle\geq\overline{\gamma} \Vert x\Vert ^{2}\quad \mbox{for } \forall x \in E; $$
$$\Vert aI-bT\Vert = \sup_{\Vert x\Vert \leq1}\bigl\vert \bigl\langle (aI - bT)x, J(x)\bigr\rangle \bigr\vert , $$
where I is the identity mapping and $a \in [0,1]$, $b \in[-1,1]$;
demiclosed at p if whenever $\{x_{n}\}$ is a sequence in $D(T)$ such that $x_{n} \rightharpoonup x \in D(T)$ and $Tx_{n} \rightarrow p$ then $Tx =p$;
strongly accretive if, for all $x, y \in D(T)$, there exists $j(x-y) \in J(x-y)$ such that
$$\bigl\langle Tx - Ty, j(x-y)\bigr\rangle \geq\epsilon \Vert x - y\Vert ^{2} $$
for some $\epsilon> 0$.
For the accretive mapping A, we use $N(A)$ to denote the set of zero points of it; that is, $N(A): = \{x \in D(A) : Ax = 0\}$. If A is accretive, then we can define, for each $r>0$, a single-valued mapping $J_{r}^{A} : R(I+rA)\rightarrow D(A)$ by $J_{r}^{A} : = (I+rA)^{-1}$, which is called the resolvent of A [1]. It is well known that $J^{A}_{r}$ is non-expansive and $N(A) = \operatorname {Fix}(J_{r}^{A})$.
Let C be a nonempty, closed and convex subset of E and Q be a mapping of E onto C. Then Q is said to be sunny [4] if $Q(Q(x)+t(x-Q(x))) = Q(x)$, for all $x \in E$ and $t \geq 0$.
A mapping Q of E into E is said to be a retraction [4] if $Q^{2} = Q$. If a mapping Q is a retraction, then $Q(z) = z$ for every $z \in R(Q)$, where $R(Q)$ is the range of Q.
A subset C of E is said to be a sunny non-expansive retract of E [4] if there exists a sunny non-expansive retraction of E onto C and it is called a non-expansive retract of E if there exists a non-expansive retraction of E onto C.
Many practical problems can be reduced to finding zeros of the sum of two accretive operators; that is, $0 \in(A+B)x$. Forward-backward splitting algorithms, which have recently received much attention to many mathematicians, were proposed by Lions and Mercier [5], by Passty [6], and, in a dual form for convex programming, by Han and Lou [7].
The classical forward-backward splitting algorithm is given in the following way:
$$ x_{n+1} = (I+r_{n} B)^{-1}(I-r_{n} A)x_{n},\quad n \geq0. $$
Based on iterative algorithm (1), much work has been done for finding $x \in H$ such that $x \in N(A+B)$, where A and B are μ-inversely strongly accretive mapping and m-accretive mapping defined in the Hilbert space H, respectively. In 2015, Wei et al. extended the related work from the Hilbert space to the real smooth and uniformly convex Banach space and presented the following iterative algorithm with errors [8]:
$$ \textstyle\begin{cases} x_{0}\in C \quad \text{chosen arbitrarily},\\ y_{n} = Q_{C}[(1-\alpha_{n})(x_{n}+e_{n})],\\ z_{n}= (1- \beta_{n})x_{n} + \beta_{n} [a_{0}y_{n} + \sum_{i = 1}^{N} a_{i} J_{r_{n,i}}^{A_{i}}(y_{n}-r_{n,i}B_{i}y_{n})],\\ x_{n+1}=\gamma_{n} \eta f(x_{n})+(I-\gamma_{n}T)z_{n},\quad n \geq0, \end{cases} $$
where C is a nonempty, closed, and convex sunny non-expansive retract of E, $Q_{C}$ is the sunny non-expansive retraction of E onto C, $\{e_{n}\}\subset E $ is the error sequence, $\{A_{i}\}_{i = 1}^{N}$ is a finite family of m-accretive mappings and $\{B_{i}\}_{i = 1}^{N}$ is a finite family of μ-inversely strongly accretive mappings. $T : E \rightarrow E$ is a strongly positive linear bounded operator with coefficient γ̅ and $f : E \rightarrow E $ is a contraction with coefficient $k \in(0,1)$. $J^{A_{i}}_{r_{n,i}}= (I+r_{n,i}A_{i})^{-1}$, for $i = 1,2, \ldots, N$, $\sum_{m = 0}^{N}a_{m} = 1$, $0 < a_{m} < 1$, for $m = 0 , 1, 2, \ldots, N$. Then $\{x_{n}\}$ is proved to converge strongly to $p_{0} \in\bigcap_{i = 1}^{N} N(A_{i}+B_{i})$, which solves the variational inequality
$$\bigl\langle (T-\eta f)p_{0}, J(p_{0}-z)\bigr\rangle \leq0, $$
for $\forall z\in\bigcap_{i = 1}^{N} N(A_{i}+B_{i})$ under some conditions.
The implicit midpoint rule (IMR) is one of the powerful numerical methods for solving ordinary differential equations, which is extensively studied recently by Alghamdi et al. They presented the following implicit midpoint rule for approximating the fixed point of a non-expansive mapping in a Hilbert space in [9]:
$$ x_{0} \in H,\quad x_{n+1}= (1-\alpha_{n})x_{n} + \alpha_{n} T\biggl(\frac {x_{n}+x_{n+1}}{2}\biggr),\quad n \geq0, $$
where T is non-expansive from H to H. If $\operatorname {Fix}(T) \neq \emptyset$, then they proved that $\{x_{n}\}$ converges weakly to $p_{0} \in \operatorname {Fix}(T)$, under some conditions.
Inspired by the work in [8] and [9], we shall present the following iterative algorithm with errors in a real q-uniformly smooth and uniformly convex Banach space:
$$ \textstyle\begin{cases} x_{0}\in C \quad \text{chosen arbitrarily},\\ y_{n} = Q_{C}[(1-\alpha_{n})(x_{n}+e'_{n})],\\ z_{n}= \delta_{n}y_{n} + \beta_{n}\sum_{i = 1}^{\infty}a_{i} J_{r_{n,i}}^{A_{i}}[\frac{y_{n}+z_{n}}{2}-r_{n,i}B_{i}(\frac{y_{n}+z_{n}}{2})] +\zeta_{n} e''_{n},\\ x_{n+1}=\gamma_{n} \eta f(x_{n})+(I-\gamma_{n}T)z_{n}+e'''_{n}, \quad n \geq0, \end{cases} $$
where C is a nonempty, closed, and convex sunny non-expansive retract of E, $Q_{C}$ is the sunny non-expansive retraction of E onto C, $\{e'_{n}\}$, $\{e''_{n}\}$, and $\{e'''_{n}\}$ are three error sequences, $A_{i}: C \rightarrow E$ is m-accretive and $B_{i}: C \rightarrow E$ is $\mu_{i}$-inversely strongly accretive, where $i \in \mathbb{N^{+}}$. $T : E \rightarrow E$ is a strongly positive linear bounded operator with coefficient γ̅ and $f : E \rightarrow E $ is a contraction with coefficient $k \in(0,1)$. $J_{r_{n,i}}^{A_{i}} = (I + r_{n,i} A_{i} )^{-1}$, for $i \in\mathbb {N^{+}}$. $\sum_{m = 1}^{\infty}a_{m} = 1$, $0 < a_{m} < 1$, for $m \in \mathbb{N^{+}}$. $\delta_{n} + \beta_{n} +\zeta_{n} \equiv1$, for $n \geq 0$. More details of iterative algorithm (A) will be presented in Section 2. Then $\{x_{n}\}$ is proved to converge strongly to $p_{0} \in\bigcap_{i = 1}^{\infty} N(A_{i}+B_{i})$, which is also a solution of one kind of variational inequality.
Our main contributions are:
A new three-step iterative algorithm is designed by combining the ideas of famous iterative algorithms such as proximal methods, Halpern methods, convex combination methods, viscosity methods, and implicit midpoint methods.
The assumption that 'the duality mapping J is weakly sequentially continuous' or 'J is weakly sequentially continuous at zero' in most of the existing related work is deleted.
'$B_{i}$ is μ-inversely strongly accretive' in most of the related work is replaced by '$B_{i}$ is $\mu _{i}$-inversely strongly accretive', which makes the discussion more general. Moreover, the design of the iterative algorithm is extended from finite case of the sum of m-accretive mappings and μ-inversely strongly accretive mappings to the infinite case.
The discussion is undertaken in the frame of a real q-uniformly smooth and uniformly convex Banach space, which is more general than that in a Hilbert space.
In each step of the three-step iterative algorithm, computational error is being considered - that is, we consider three error sequences $\{e'_{n}\}$, $\{e''_{n}\}$, and $\{e'''_{n}\}$.
All of the three sequences $\{y_{n}\}$, $\{z_{n}\}$, and $\{x_{n}\}$ constructed in the new iterative algorithm are proved to be strongly convergent to the zero point of the sum of an infinite family of m-accretive mappings and an infinite family of $\mu_{i}$-inversely strongly accretive mappings.
(vii)
The connection between the zero point of the sum of m-accretive mappings and $\mu_{i}$-inversely strongly accretive mappings and the solution of one kind variational inequalities is being set up.
(viii)
In Section 3, the application of the main result in Section 2 to one kind integro-differential systems is demonstrated, from which we can see the connections between variational inequalities, integro-differential equations, and iterative algorithms.
Next, we list some results we need in the sequel.
Lemma 1
(see [2])
Let E be a Banach space and C be a nonempty closed and convex subset of E. Let $f: C \rightarrow C$ be a contraction. Then f has a unique fixed point $u \in C$.
(see [10])
Let E be a real uniformly convex Banach space, C be a nonempty, closed, and convex subset of E and $T: C \rightarrow E$ be a non-expansive mapping such that $\operatorname {Fix}(T) \neq \emptyset$, then $I-T$ is demiclosed at zero.
In a real Banach space E, the following inequality holds:
$$\Vert x+y\Vert ^{2} \leq \Vert x\Vert ^{2} + 2\bigl\langle y, j(x+y)\bigr\rangle , \quad \forall x,y \in E, $$
where $j(x+y) \in J(x+y)$.
Let $\{a_{n}\}$ and $\{c_{n}\}$ be two sequences of nonnegative real numbers satisfying
$$a_{n+1}\leq(1-t_{n})a_{n} + b_{n}+c_{n}, \quad \forall n \geq0, $$
where $\{t_{n}\}\subset(0,1)$ and $\{b_{n}\}$ is a number sequence. Assume that $\sum_{n=0}^{\infty}t_{n} = +\infty$, $\limsup_{n \rightarrow\infty} \frac{b_{n}}{t_{n}} \leq0$, and $\sum_{n=0}^{\infty}c_{n} < +\infty$. Then $\lim_{n \rightarrow\infty }a_{n} = 0$.
Let E be a Banach space and let A be an m-accretive mapping. For $\lambda>0$, $\mu>0$, and $x \in E$, one has
$$J_{\lambda}^{A}x = J_{\mu}^{A}\biggl( \frac{\mu}{\lambda}x+\biggl(1-\frac{\mu}{\lambda}\biggr)J_{\lambda}^{A} x\biggr), $$
where $J_{\lambda}^{A} = (I+\lambda A)^{-1}$ and $J_{\mu}^{A} = (I+\mu A)^{-1}$.
Let E be a real Banach space and let C be a nonempty, closed, and convex subset of E. Suppose $A: C \rightarrow E$ is a single-valued mapping and $B: E \rightarrow 2^{E}$ is m-accretive. Then
$$\operatorname {Fix}\bigl((I+rB)^{-1}(I-rA)\bigr) = N(A+B) \quad \textit{for } \forall r > 0. $$
Assume T is a strongly positive bounded operator with coefficient $\overline{\gamma} > 0$ on a real smooth Banach space E and $0 < \rho\leq \Vert T\Vert ^{-1}$. Then $\Vert I-\rho T \Vert \leq1- \rho\overline{\gamma}$.
Let E be a real strictly convex Banach space and let C be a nonempty closed and convex subset of E. Let $T_{m}: C \rightarrow C$ be a non-expansive mapping for each $m \geq1$. Let $\{a_{m}\}$ be a real number sequence in (0,1) such that $\sum_{m = 1}^{\infty}a_{m} = 1$. Suppose that $\bigcap_{m=1}^{\infty} \operatorname {Fix}(T_{m}) \neq\emptyset$. Then the mapping $\sum_{m = 1}^{\infty}a_{m} T_{m}$ is non-expansive with $\operatorname {Fix}(\sum_{m = 1}^{\infty}a_{m}T_{m}) = \bigcap_{m = 1}^{\infty} \operatorname {Fix}(T_{m})$.
Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space E with constant $K_{q}$. Let the mapping $A: C\rightarrow E$ be a μ-inversely strongly accretive mapping. Then the following inequality holds:
$$\bigl\Vert (I-rA)x-(I-rA)y\bigr\Vert ^{q} \leq \Vert x-y\Vert ^{q} - r\bigl(q \mu- K_{q} r^{q-1}\bigr)\Vert Ax-Ay\Vert ^{q}. $$
In particular, if $0 < r \leq(\frac{q \mu}{K_{q}})^{\frac{1}{q-1}}$, then $(I-rA)$ is non-expansive.
Strong convergence theorems
Lemma 10
Let E be a real uniformly smooth and uniformly convex Banach space and C be a nonempty, closed, and convex sunny non-expansive retract of E, and let $Q_{C}$ be the sunny non-expansive retraction of E onto C. Let $f : E \rightarrow E$ be a fixed contractive mapping with coefficient $k \in(0,1)$, $T: E \rightarrow E$ be a strongly positive linear bounded operator with coefficient γ̅ and $U : C \rightarrow C$ be a non-expansive mapping. Suppose that $0 < \eta< \frac{\overline{\gamma}}{2k}$ and $\operatorname {Fix}(U) \neq\emptyset$. If for each $t \in(0,1)$, define $T_{t} : E \rightarrow E$ by
$$ T_{t} x : = t \eta f(x) + (I-t T)UQ_{C}x, $$
then $T_{t}$ has a fixed point $x_{t}$, for each $0 < t \leq \Vert T\Vert ^{-1}$, which is convergent strongly to the fixed point of U, as $t \rightarrow0$. That is, $\lim_{t\rightarrow0}x_{t} = p_{0} \in \operatorname {Fix}(U)$. Moreover, $p_{0}$ satisfies the following variational inequality: for $\forall z \in \operatorname {Fix}(U)$,
$$ \bigl\langle (T-\eta f)p_{0}, J(p_{0}-z)\bigr\rangle \leq0. $$
Copying Steps 1 to 5 of Lemma 8 in [8], we have the following results:
$T_{t}$ is a contraction, for $0 < t < \Vert T\Vert ^{-1}$.
$T_{t}$ has a unique fixed point $x_{t}$.
$\{x_{t}\}$ is bounded, for $0 < t < \Vert T\Vert ^{-1}$.
$x_{t} - UQ_{C}x_{t} \rightarrow0$, as $t \rightarrow0$.
If the inequality (5) has a solution, then the solution must be unique.
Finally, we are to show that $x_{t} \rightarrow p_{0} \in \operatorname {Fix}(U)$, as $t \rightarrow0$, which satisfies the variational inequality (5).
Assume $t_{n} \rightarrow0$. Set $x_{n} : = x_{t_{n}}$ and define $\mu: E \rightarrow\mathbb{R}$ by
$$\mu(x) = \operatorname {LIM}\Vert x_{n} - x\Vert ^{2},\quad x \in E, $$
where LIM is the Banach limit on $l^{\infty}$. Let
$$K = \Bigl\{ x \in E : \mu(x) = \min_{x \in E}\operatorname {LIM}\Vert x_{n} - x\Vert ^{2}\Bigr\} . $$
It is easily seen that K is a nonempty, closed, convex, and bounded subset of E. Since $x_{n} - UQ_{C}x_{n} \rightarrow0$, for $x \in K$,
$$\mu(UQ_{C}x) = \operatorname {LIM}\Vert x_{n} - UQ_{C}x \Vert ^{2}\leq \operatorname {LIM}\Vert x_{n} - x\Vert ^{2} = \mu(x), $$
it follows that $UQ_{C}(K) \subset K$; that is, K is invariant under $UQ_{C}$. Since a uniformly smooth Banach space has the fixed point property for non-expansive mappings, $UQ_{C}$ has a fixed point, say $p_{0}$, in K. That is, $UQ_{C} p_{0} = p_{0} \in C$, which ensures that $p_{0} = Up_{0}$ from the definition of U and then $p_{0} \in \operatorname {Fix}(U)$. Since $p_{0}$ is also a minimizer of μ over E, it follows that, for $t \in(0,1)$
$$\begin{aligned} 0 \leq{}&\frac{\mu(p_{0}+\eta t f(p_{0})-tTp_{0})-\mu(p_{0})}{t} \\ ={}& \operatorname {LIM}\frac{\Vert x_{n}-p_{0}-\eta t f(p_{0})+tTp_{0}\Vert ^{2}-\Vert x_{n} - p_{0}\Vert ^{2}}{t} \\ ={}& \operatorname {LIM}\frac{ \langle x_{n}-p_{0}-\eta t f(p_{0}) +t Tp_{0}, J(x_{n}-p_{0}-\eta t f(p_{0}) +t Tp_{0})\rangle -\Vert x_{n} - p_{0}\Vert ^{2}}{t} \\ ={}& \operatorname {LIM}\bigl( \bigl\langle x_{n}-p_{0}, J \bigl(x_{n}-p_{0}-\eta t f(p_{0}) +t Tp_{0}\bigr)\bigr\rangle \\ &{}+ t\bigl\langle Tp_{0}-\eta f(p_{0}), J \bigl(x_{n}-p_{0}-\eta t f(p_{0}) +t Tp_{0}\bigr)\bigr\rangle -\Vert x_{n} - p_{0} \Vert ^{2}\bigr)/{t}. \end{aligned}$$
Since E is uniformly smooth, then by letting $t \rightarrow0$, we find the two limits above can be interchanged and obtain
$$ \operatorname {LIM}\bigl\langle \eta f(p_{0})-Tp_{0}, J(x_{n} - p_{0})\bigr\rangle \leq0. $$
Since $x_{t} - p_{0} = t(\eta f(x_{t})-Tp_{0})+(I-tT)(UQ_{C}x_{t} - p_{0})$, then
$$\begin{aligned} \Vert x_{t} - p_{0}\Vert ^{2} ={}& t \bigl\langle \eta f(x_{t}) - Tp_{0}, J(x_{t}-p_{0}) \bigr\rangle + \bigl\langle (I-tT) (UQ_{C}x_{t}-p_{0}), J(x_{t}-p_{0})\bigr\rangle \\ \leq{}& t \eta\bigl\langle f(x_{t}) - f(p_{0}), J(x_{t}-p_{0})\bigr\rangle +t \bigl\langle \eta f(p_{0}) - Tp_{0}, J(x_{t}-p_{0}) \bigr\rangle \\ &{}+\Vert I -tT\Vert \Vert x_{t} -p_{0}\Vert ^{2} \\ \leq{}&\bigl[1-t(\overline{\gamma}-\eta k)\bigr]\Vert x_{t} -p_{0}\Vert ^{2}+t \bigl\langle \eta f(p_{0}) - Tp_{0}, J(x_{t}-p_{0})\bigr\rangle . \end{aligned}$$
$$\Vert x_{t} - p_{0}\Vert ^{2} \leq \frac{1}{\overline{\gamma}-\eta k}\bigl\langle \eta f(p_{0}) - Tp_{0}, J(x_{t}-p_{0})\bigr\rangle . $$
Hence by (6)
$$\operatorname {LIM}\Vert x_{n} - p_{0}\Vert ^{2} \leq \frac{1}{\overline{\gamma}-\eta k}\operatorname {LIM}\bigl\langle \eta f(p_{0}) - Tp_{0}, J(x_{n}-p_{0})\bigr\rangle \leq0, $$
which implies that $\operatorname {LIM}\Vert x_{n} - p_{0}\Vert ^{2}=0$, and then there exists a subsequence which is still denoted by $\{x_{n}\}$ such that $x_{n} \rightarrow p_{0}$.
Next, we shall show that $p_{0}$ solves the variational inequality (5).
Since $x_{t} = t\eta f(x_{t})+(I-tT)UQ_{C}x_{t}$, $(T- \eta f)x_{t} = -\frac {1}{t}(I-tT)(I-UQ_{C})x_{t}$. For $\forall z \in \operatorname {Fix}(U)$,
$$\begin{aligned} & \bigl\langle (T-\eta f)x_{t},J(x_{t} - z)\bigr\rangle \\ &\quad = -\frac {1}{t}\bigl\langle (I-tT) (I-UQ_{C})x_{t}, J(x_{t}-z)\bigr\rangle \\ &\quad = -\frac{1}{t} \bigl\langle (I-UQ_{C})x_{t}-(I-UQ_{C})z, J(x_{t}-z)\bigr\rangle + \bigl\langle T(I-UQ_{C})x_{t}, J(x_{t}-z)\bigr\rangle \\ &\quad = -\frac{1}{t}[\Vert x_{t}-z\Vert ^{2} - \bigl\langle UQ_{C}x_{t}-UQ_{C}z, J(x_{t}-z)\bigr\rangle +\bigl\langle T(I-UQ_{C})x_{t}, J(x_{t}-z)\bigr\rangle \\ &\quad \leq\bigl\langle T(I-UQ_{C})x_{t}, J(x_{t}-z)\bigr\rangle . \end{aligned}$$
Taking the limits on both sides of the above inequality, $\langle (T-\eta f)p_{0}, J(p_{0}-z)\rangle\leq0$ since $x_{n} \rightarrow p_{0}$ and J is uniformly continuous on each bounded subsets of E.
Thus $p_{0}$ satisfies the variational inequality (5).
Now assume there exists another subsequence $\{x_{m}\}$ of $\{x_{t}\}$ satisfying $x_{m} \rightarrow q_{0}$. Then result (d) implies that $UQ_{C}x_{m} \rightarrow q_{0}$. From Lemma 2, we know that $I-UQ_{C}$ is demiclosed at zero, then $q_{0} = UQ_{C}q_{0}$, which ensures that $q_{0}\in \operatorname {Fix}(U)$. Repeating the above proof, we can also know that $q_{0}$ solves the variational inequality (5). Thus $p_{0} = q_{0}$ by using the result of (e).
Hence $x_{t} \rightarrow p_{0}$, as $t \rightarrow0$, which is the unique solution of the variational inequality (5).
This completes the proof. □
Theorem 11
Let E be a real q-uniformly smooth Banach space with constant $K_{q}$ and also be a uniformly convex Banach space. Let C be a nonempty, closed, and convex sunny non-expansive retract of E, and $Q_{C}$ be the sunny non-expansive retraction of E onto C. Let $f : E \rightarrow E$ be a contraction with coefficient $k \in(0,1)$, $T: E \rightarrow E$ be a strongly positive linear bounded operator with coefficient γ̅. Let $A_{i}: C \rightarrow E$ be m-accretive mappings, $B_{i}: C \rightarrow E$ be $\mu_{i}$-inversely strongly accretive mappings, for $i \in\mathbb{N^{+}}$. Let $D : = \bigcap_{i = 1}^{\infty}N(A_{i}+B_{i})\neq\emptyset$. Suppose $0 < \eta< \frac{\overline{\gamma}}{2k}$. Suppose $\{\alpha_{n}\}$, $\{\delta_{n}\}$, $\{\beta_{n}\}$, $\{\zeta_{n}\}$, $\{\gamma_{n}\} \subset(0,1)$, and $\{r_{n,i}\}\subset(0,+\infty)$ for $i \in\mathbb{N^{+}}$. Suppose $\{a_{i}\}_{i = 1}^{\infty}\subset(0,1)$ with $\sum_{i = 1}^{\infty}a_{i} = 1$, $\{e''_{n}\}\subset C$, and $\{e'_{n}\}, \{e'''_{n}\} \subset E$ are three error sequences. Let $\{x_{n}\}$ be generated by the iterative algorithm (A). Further suppose that the following conditions are satisfied:
$\sum_{n=0}^{\infty} \alpha_{n} <+\infty$;
$\sum_{n=0}^{\infty} \gamma_{n} = \infty$, $\gamma_{n} \rightarrow0$, as $n \rightarrow\infty$ and $\sum_{n=1}^{\infty} \vert \gamma_{n} -\gamma_{n-1}\vert <+\infty$;
$\sum_{n=0}^{\infty} \vert r_{n+1,i} - r_{n,i}\vert < +\infty$, $0 < \varepsilon\leq r_{n,i}\leq (\frac{q\mu_{i}}{K_{q}})^{\frac{1}{q-1}}$, for $n \geq0$ and $i \in\mathbb{N^{+}}$;
$\delta_{n} + \beta_{n} + \zeta_{n} \equiv1$, for $n \geq0$, $\sum_{n=1}^{\infty} \vert \delta_{n} -\delta_{n-1}\vert <+\infty$, $\sum_{n=1}^{\infty} \vert \beta_{n}-\beta_{n-1}\vert <+\infty$, $\sum_{n=0}^{\infty}\frac{\zeta_{n}}{\beta_{n}}<+\infty$, and $\beta_{n} \rightarrow1$, as $n \rightarrow\infty$;
$\sum_{n=0}^{\infty} \Vert e'_{n}\Vert < +\infty$, $\sum_{n=0}^{\infty} \Vert e''_{n}\Vert < +\infty$, $\sum_{n=0}^{\infty} \Vert e'''_{n}\Vert < +\infty$.
Then three sequences $\{x_{n}\}$, $\{y_{n}\}$, and $\{z_{n}\}$ converge strongly to the unique element $p_{0} \in D$, which satisfies the following variational inequality: for $\forall y \in D$,
$$ \bigl\langle (T - \eta f)p_{0}, J(p_{0} - y)\bigr\rangle \leq0. $$
We shall split the proof into seven steps.
Step 1. $\{x_{n}\}$ is well defined.
In fact, it suffices to show that $\{z_{n}\}$ is well defined.
For $t,s,r \in(0,1)$ and $t+s+r \equiv1$, define $U_{t,s,r}: C \rightarrow C$ by $U_{t,s,r} x: = tu + sU(\frac{u+x}{2})+rv$, where $U: C \rightarrow C$ is non-expansive for $x,u, v \in C$. Then
$$\Vert U_{t,s,r} x - U_{t,s,r}y\Vert \leq s\biggl\Vert \frac{u+x}{2}-\frac{u+y}{2}\biggr\Vert \leq\frac{s}{2} \Vert x-y\Vert . $$
Thus $U_{t,s,r}$ is a contraction, which ensures from Lemma 1 that there exists $x_{t,s,r}\in C$ such that $U_{t,s,r} x_{t,s,r} = x_{t,s,r}$. That is, $x_{t,s,r} = tu + sU(\frac{u+x_{t,s,r}}{2})+rv$.
Since $J_{r_{n,i}}^{A_{i}}(I-r_{n,i} B_{i})$ is non-expansive in view of Lemma 9 and $\sum_{i = 1}^{\infty}a_{i} = 1$, $\sum_{i = 1}^{\infty}a_{i}J_{r_{n,i}}^{A_{i}}(I-r_{n,i} B_{i})$ is non-expansive, which implies that $\{z_{n}\}$ is well defined, and then $\{x_{n}\}$ is well defined.
Step 2. $D:= \bigcap_{i = 1}^{\infty}N(A_{i}+B_{i}) = \operatorname {Fix}(\sum_{i = 1}^{\infty}a_{i}J_{r_{n,i}}^{A_{i}}(I-r_{n,i} B_{i}))$.
Lemma 6 implies that $N(A_{i}+B_{i}) = \operatorname {Fix}(J_{r_{n,i}}^{A_{i}}(I-r_{n,i} B_{i}))$, where $i\in\mathbb{N^{+}}$. Then Lemma 8 ensures that $\bigcap_{i = 1}^{\infty}N(A_{i}+B_{i}) = \bigcap_{i = 1}^{\infty} \operatorname {Fix}(J_{r_{n,i}}^{A_{i}}(I-r_{n,i} B_{i})) = \operatorname {Fix}(\sum_{i = 1}^{\infty}a_{i}J_{r_{n,i}}^{A_{i}}(I-r_{n,i} B_{i}))$.
Step 3. $\{x_{n}\}$, $\{y_{n}\}$, and $\{z_{n}\}$ are all bounded.
$\forall p \in D$, we see that, for $n \geq0$,
$$ \Vert y_{n} - p\Vert \leq(1-\alpha_{n})\Vert x_{n}-p\Vert +(1-\alpha_{n})\bigl\Vert e'_{n}\bigr\Vert +\alpha_{n} \Vert p \Vert . $$
Therefore, for $p \in D$ and $n \geq0$, we have
$$\begin{aligned} \Vert z_{n} - p\Vert &\leq\delta_{n}\Vert y_{n}-p\Vert +\beta_{n}\Biggl\Vert \sum _{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}} \biggl[\frac{y_{n}+z_{n}}{2}-r_{n,i}B_{i}\biggl( \frac {y_{n}+z_{n}}{2}\biggr)\biggr]-p\Biggr\Vert +\zeta_{n}\bigl\Vert e''_{n}-p\bigr\Vert \\ &\leq \biggl(\delta_{n}+\frac{\beta_{n}}{2}\biggr)\Vert y_{n}-p\Vert +\frac{\beta_{n}}{2}\Vert z_{n}-p\Vert + \zeta _{n}\bigl\Vert e''_{n}-p \bigr\Vert \\ &\leq \biggl(1-\frac{\beta_{n}}{2}\biggr)\Vert y_{n}-p\Vert + \frac{\beta_{n}}{2}\Vert z_{n}-p\Vert +\zeta_{n}\bigl\Vert e''_{n}-p\bigr\Vert , \end{aligned}$$
$$ \Vert z_{n} - p\Vert \leq \Vert y_{n}-p\Vert + \frac{2\zeta_{n}}{2-\beta_{n}}\bigl\Vert e''_{n}-p\bigr\Vert \leq \Vert y_{n} - p\Vert +2\bigl\Vert e''_{n}\bigr\Vert +\frac{2\zeta_{n}}{2-\beta_{n}} \Vert p\Vert . $$
Noticing (8) and (9), using Lemma 7, we have, for $n \geq0$,
$$ \begin{aligned}[b] \Vert x_{n+1} - p\Vert \leq{}& \gamma_{n}\eta k \Vert x_{n}- p\Vert + \gamma _{n}\bigl\Vert \eta f(p) - Tp\bigr\Vert + (1-\gamma_{n} \overline{\gamma})\Vert z_{n} - p\Vert + \bigl\Vert e'''_{n}\bigr\Vert \\ \leq{}&\bigl[1-\gamma_{n} (\overline{\gamma}- k \eta)\bigr]\Vert x_{n} -p\Vert + \gamma_{n} \bigl\Vert \eta f(p) - Tp \bigr\Vert \\ &{}+ \bigl\Vert e'_{n}\bigr\Vert + 2\bigl\Vert e''_{n}\bigr\Vert +\bigl\Vert e'''_{n}\bigr\Vert + \alpha_{n} \Vert p\Vert +\frac{2\zeta_{n}}{2-\beta_{n}}\Vert p\Vert . \end{aligned} $$
By using the inductive method, we can easily get the following result from (10):
$$\begin{aligned} \Vert x_{n+1}-p\Vert \leq{}& \max\biggl\{ \Vert x_{0} - p\Vert , \frac{\Vert \eta f(p) -Tp\Vert }{\overline{\gamma}-k \eta} \biggr\} + \sum_{k=0}^{n}\bigl\Vert e'_{k}\bigr\Vert +2 \sum_{k=0}^{n} \bigl\Vert e''_{k}\bigr\Vert \\ & {}+ \sum_{k=0}^{n}\bigl\Vert e'''_{k}\bigr\Vert +\Vert p \Vert \Biggl(\sum_{k=0}^{n} \alpha_{k}+\sum_{k=0}^{n} \frac{2\zeta_{k}}{2-\beta_{k}}\Biggr). \end{aligned}$$
Therefore, from assumptions (i), (iv), and (v), we know that $\{x_{n}\}$ is bounded. Then $\{y_{n}\}$ and $\{z_{n}\}$ are bounded in view of (8) and (9), respectively.
Let $u_{n,i} = (I - r_{n,i}B_{i})(\frac{y_{n}+z_{n}}{2})$, then $\{u_{n,i}\}$ is bounded in view of Lemma 9, for $n \geq0$ and $i \in\mathbb{N^{+}}$.
Since $\Vert \sum_{i = 1}^{\infty}a_{i}J_{r_{n,i}}^{A_{i}}u_{n,i}\Vert \leq \Vert \sum_{i = 1}^{\infty}a_{i}J_{r_{n,i}}^{A_{i}}u_{n,i} - \sum_{i = 1}^{\infty}a_{i}J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})p\Vert +\Vert p\Vert \leq \Vert \frac{y_{n}+z_{n}}{2}-p\Vert +\Vert p\Vert $ in view of Step 2, then $\{\sum_{i = 1}^{\infty}a_{i}J_{r_{n,i}}^{A_{i}}u_{n,i}\}$ is bounded. Moreover, we can easily know that $\{f(x_{n})\}$, $\{Tz_{n}\}$, $\{B_{i}(\frac{y_{n}+z_{n}}{2})\}$, and $\{J_{r_{n,i}}^{A_{i}}u_{n,i}\}$ are all bounded, for $n \geq0$ and $i\in\mathbb{N^{+}}$.
Set $M' = \sup\{ \Vert u_{n,i}\Vert , \Vert x_{n}\Vert , \Vert Tz_{n}\Vert , \Vert y_{n}\Vert , \Vert f(x_{n})\Vert , \Vert J_{r_{n,i}}^{A_{i}}u_{n,i}\Vert , \Vert \sum_{i = 1}^{\infty}a_{i}J_{r_{n,i}}^{A_{i}}u_{n,i}\Vert , \Vert B_{i}(\frac{y_{n}+z_{n}}{2})\Vert : n \geq0, i \in\mathbb{N^{+}}\}$. Then $M'$ is a positive constant.
Step 4. $\lim_{n \rightarrow\infty} \Vert x_{n+1} - x_{n} \Vert = 0$.
In fact, if $r_{n,i}\leq r_{n+1,i}$, then, using Lemma 5,
$$ \begin{aligned}[b] &\bigl\Vert J_{r_{n+1,i}}^{A_{i}}u_{n+1,i}-J_{r_{n,i}}^{A_{i}}u_{n,i} \bigr\Vert \\ &\quad \leq \Vert u_{n+1,i}-u_{n,i}\Vert +\frac{r_{n+1,i}-r_{n,i}}{\varepsilon} \bigl\Vert J_{r_{n+1,i}}^{A_{i}}u_{n+1,i}-u_{n,i}\bigr\Vert \\ & \quad \leq \Vert u_{n+1,i}-u_{n,i}\Vert +2 \frac{r_{n+1,i}-r_{n,i}}{\varepsilon}M'. \end{aligned} $$
If $r_{n+1,i}\leq r_{n,i}$, then imitating the proof of (11), we have
$$ \bigl\Vert J_{r_{n+1,i}}^{A_{i}}u_{n+1,i}-J_{r_{n,i}}^{A_{i}}u_{n,i} \bigr\Vert \leq \Vert u_{n+1,i}-u_{n,i}\Vert +2 \frac{r_{n,i}-r_{n+1,i}}{\varepsilon}M'. $$
Combining (11) and (12), we have, for $n \geq0$ and $i \in\mathbb{N^{+}}$,
$$ \bigl\Vert J_{r_{n+1,i}}^{A_{i}}u_{n+1,i}-J_{r_{n,i}}^{A_{i}}u_{n,i} \bigr\Vert \leq \Vert u_{n+1,i}-u_{n,i}\Vert +2 \frac{\vert r_{n,i}-r_{n+1,i}\vert }{\varepsilon}M'. $$
Then in view of Lemma 9
$$ \begin{aligned}[b] \Vert u_{n+1,i}-u_{n,i}\Vert ={}& \biggl\Vert (I-r_{n+1,i}B_{i}) \biggl(\frac {y_{n+1}+z_{n+1}}{2}- \frac{y_{n}+z_{n}}{2}\biggr)\biggr\Vert \\ &{} + \vert r_{n,i}-r_{n+1,i}\vert \biggl\Vert B_{i}\biggl(\frac{y_{n}+z_{n}}{2}\biggr)\biggr\Vert \\ \leq{}& \biggl\Vert \frac{y_{n+1}-y_{n}}{2}\biggr\Vert +\biggl\Vert \frac{z_{n+1}-z_{n}}{2}\biggr\Vert +\vert r_{n,i}-r_{n+1,i}\vert M'. \end{aligned} $$
In view of (13) and (14), we have
$$\begin{aligned} &\Vert z_{n+1}-z_{n}\Vert \\ &\quad \leq \delta_{n+1}\Vert y_{n+1}-y_{n} \Vert +\vert \delta_{n+1}-\delta_{n}\vert \Vert y_{n}\Vert +\vert \beta _{n+1}-\beta_{n} \vert \Biggl\Vert \sum_{i = 1}^{\infty}a_{i}J_{r_{n,i}}^{A_{i}}u_{n,i} \Biggr\Vert \\ &\qquad {}+\beta_{n+1} \Biggl\Vert \sum_{i = 1}^{\infty}a_{i}J_{r_{n+1,i}}^{A_{i}}u_{n+1,i} - \sum_{i = 1}^{\infty}a_{i}J_{r_{n,i}}^{A_{i}}u_{n,i} \Biggr\Vert + \bigl\Vert \zeta _{n+1}e''_{n+1}- \zeta_{n} e''_{n}\bigr\Vert \\ &\quad \leq \biggl(\delta_{n+1}+\frac{\beta_{n+1}}{2}\biggr)\Vert y_{n+1}-y_{n}\Vert +\vert \beta_{n+1}-\beta _{n}\vert M'+\vert \delta_{n+1}- \delta_{n}\vert M'+\frac{\beta_{n+1}}{2} \Vert z_{n+1} - z_{n}\Vert \\ &\qquad {}+\biggl(1+\frac{2}{\varepsilon}\biggr)\beta_{n+1} \vert r_{n,i}-r_{n+1,i}\vert M' +\bigl\Vert \zeta_{n+1}e''_{n+1}- \zeta_{n} e''_{n}\bigr\Vert , \end{aligned} $$
$$\begin{aligned} \Vert z_{n+1}-z_{n}\Vert \leq{}& \frac{\delta_{n+1}+\frac{\beta_{n+1}}{2}}{1-\frac{\beta_{n+1}}{2}} \Vert y_{n+1}-y_{n}\Vert + \frac{2\vert \beta_{n+1}-\beta_{n}\vert M'}{2-\beta_{n+1}} \\ &{}+\frac{2\vert \delta _{n+1}-\delta_{n}\vert M'}{2-\beta_{n+1}} +\frac{2(1+\frac{2}{\varepsilon})\beta_{n+1} \vert r_{n,i}-r_{n+1,i}\vert M'}{2-\beta_{n+1}} +\frac{2\Vert \zeta_{n+1}e''_{n+1}-\zeta_{n} e''_{n}\Vert }{2-\beta_{n+1}} \\ \leq{}&\Vert y_{n+1}-y_{n}\Vert + 2\vert \beta_{n+1}-\beta_{n}\vert M'+2\vert \delta_{n+1}-\delta_{n}\vert M' \\ &{}+2\biggl(1+\frac{2}{\varepsilon}\biggr)\beta_{n+1} \vert r_{n,i}-r_{n+1,i}\vert M' +2\bigl\Vert \zeta_{n+1}e''_{n+1}- \zeta_{n} e''_{n}\bigr\Vert . \end{aligned}$$
On the other hand,
$$ \begin{aligned}[b]\Vert y_{n+1}-y_{n}\Vert \leq{}& (1- \alpha_{n+1})\Vert x_{n+1}-x_{n}\Vert +\vert \alpha_{n+1}-\alpha_{n}\vert \Vert x_{n}\Vert \\ &{}+(1-\alpha_{n+1})\bigl\Vert e'_{n+1}-e'_{n} \bigr\Vert +\vert \alpha_{n+1}-\alpha_{n}\vert \bigl\Vert e'_{n}\bigr\Vert . \end{aligned} $$
Thus in view of (15) and (16), we have, for $n \geq1$,
$$ \begin{aligned}[b] &\Vert x_{n+1}-x_{n}\Vert \\ &\quad \leq\gamma_{n} \eta\bigl\Vert f(x_{n})-f(x_{n-1}) \bigr\Vert + \eta \vert \gamma_{n}-\gamma_{n-1}\vert \bigl\Vert f(x_{n-1})\bigr\Vert \\ &\qquad {} + \Vert I-\gamma_{n}T\Vert \Vert z_{n} - z_{n-1}\Vert +\vert \gamma_{n}-\gamma_{n-1} \vert \Vert Tz_{n-1}\Vert +\bigl\Vert e'''_{n}-e'''_{n-1} \bigr\Vert \\ & \quad \leq\gamma_{n} \eta k\Vert x_{n}-x_{n-1} \Vert + \eta \vert \gamma_{n}-\gamma_{n-1}\vert \bigl\Vert f(x_{n-1})\bigr\Vert + (1-\gamma_{n}\overline{ \gamma})\Vert z_{n} - z_{n-1}\Vert \\ &\qquad {} +\vert \gamma_{n}-\gamma_{n-1}\vert \Vert Tz_{n-1}\Vert +\bigl\Vert e'''_{n}-e'''_{n-1} \bigr\Vert \\ & \quad \leq\bigl[1-\gamma_{n}(\overline{\gamma}- \eta k)\bigr] \Vert x_{n}-x_{n-1}\Vert +(1+\eta) M'\vert \gamma_{n}-\gamma_{n-1}\vert +\bigl\Vert e'''_{n}-e'''_{n-1} \bigr\Vert \\ &\qquad {}+(1-\gamma_{n}\overline {\gamma})\biggl[M' \vert \alpha_{n}-\alpha_{n-1}\vert + 2M' \vert \beta_{n}-\beta_{n-1}\vert +2M'\vert \delta_{n}-\delta_{n-1}\vert \\ &\qquad {}+2M'\biggl(1+\frac{2}{\varepsilon }\biggr)\vert r_{n,i}-r_{n-1,i}\vert +\bigl\Vert e'_{n} \bigr\Vert + 2\bigl\Vert e'_{n-1}\bigr\Vert +2\bigl\Vert e''_{n}\bigr\Vert +2\bigl\Vert e''_{n-1}\bigr\Vert \biggr]. \end{aligned} $$
Using Lemma 4, we have from (17) $\lim_{n \rightarrow\infty} \Vert x_{n+1} - x_{n} \Vert = 0$.
Step 5. $\lim_{n \rightarrow\infty} \Vert y_{n}-z_{n}\Vert = 0$, $\lim_{n \rightarrow\infty} \Vert \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})(\frac{y_{n}+z_{n}}{2})-z_{n}\Vert = 0$ and $\lim_{n \rightarrow\infty} \Vert \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n}-y_{n}\Vert = 0$.
Since both $\{x_{n}\}$ and $\{Tz_{n}\}$ are bounded and $\gamma_{n} \rightarrow0$, as $n \rightarrow+\infty$,
$$x_{n+1}- z_{n} = \gamma_{n} \bigl(\eta f(x_{n})-Tz_{n}\bigr)+e'''_{n} \rightarrow0, \quad \mbox{as } n \rightarrow +\infty. $$
In view of Step 4, $x_{n}- z_{n} \rightarrow0$, as $n \rightarrow+\infty$. Since $\alpha_{n} \rightarrow0$, $\Vert y_{n}- Q_{C}x_{n}\Vert \leq\alpha_{n}\Vert x_{n}\Vert +(1-\alpha_{n})\Vert e'_{n}\Vert \rightarrow0$, as $n \rightarrow+\infty$. Therefore
$$y_{n}- z_{n} = y_{n} - Q_{C} z_{n} = y_{n} -Q_{C}x_{n} + Q_{C}x_{n} - Q_{C} z_{n} \rightarrow0,\quad \mbox{as } n \rightarrow+\infty. $$
Since $\delta_{n} + \beta_{n} + \zeta_{n} \equiv1$, $\beta_{n} \rightarrow1$, as $n \rightarrow\infty$, and $\{\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})(\frac{y_{n}+z_{n}}{2})\}$ is bounded,
$$\begin{aligned} & \Biggl\Vert z_{n} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i}) \biggl(\frac{y_{n}+z_{n}}{2}\biggr)\Biggr\Vert \\ & \quad \leq\delta_{n}\Biggl\Vert y_{n} - \sum _{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i}) \biggl(\frac{y_{n}+z_{n}}{2}\biggr)\Biggr\Vert \\ &\qquad {}+\zeta_{n}\Biggl\Vert e''_{n}- \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i}) \biggl( \frac{y_{n}+z_{n}}{2}\biggr)\Biggr\Vert \rightarrow 0, \end{aligned}$$
as $n \rightarrow+\infty$. Using the above facts, we have
$$\begin{aligned} & \Biggl\Vert \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n}-y_{n} \Biggr\Vert \\ &\quad \leq\Biggl\Vert \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n}- \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i}) \biggl( \frac{y_{n}+z_{n}}{2}\biggr)\Biggr\Vert \\ &\qquad {} +\Biggl\Vert \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i}) \biggl(\frac{y_{n}+z_{n}}{2}\biggr)-z_{n}\Biggr\Vert +\Vert z_{n}-y_{n}\Vert \rightarrow 0,\quad \mbox{as } n \rightarrow\infty. \end{aligned} $$
Step 6. $\limsup_{n\rightarrow+\infty}\langle\eta f(p_{0})-Tp_{0}, J(x_{n+1}-p_{0})\rangle\leq0$, where $p_{0} \in D$, which is the unique solution of the variational inequality (7).
Since $\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i}): C \rightarrow C$ is non-expansive, using Lemma 10, we know that there exists $z_{t}$ such that $z_{t} = t\eta f(z_{t})+(I-tT)\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})Q_{C}z_{t}$ for $t \in (0,\Vert T\Vert ^{-1})$. Moreover, $z_{t} \rightarrow p_{0} \in D$, as $t \rightarrow 0$, which is the unique solution of the variational inequality (7).
Since $\Vert z_{t}\Vert \leq \Vert z_{t} - p_{0} \Vert +\Vert p_{0}\Vert $, $\{z_{t}\}$ is bounded, as $t \rightarrow0$. Using Lemma 3, we have
$$\begin{aligned} &\Vert z_{t} - y_{n}\Vert ^{2} \\ &\quad = \Biggl\Vert z_{t} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n}+ \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} - y_{n}\Biggr\Vert ^{2} \\ &\quad \leq\Biggl\Vert z_{t} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr\Vert ^{2} + 2\Biggl\langle \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} - y_{n}, J(z_{t} - y_{n}) \Biggr\rangle \\ & \quad = \Biggl\Vert t\eta f(z_{t}) + (I-tT)\sum _{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})Q_{C}z_{t} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr\Vert ^{2} \\ &\qquad{} + 2\Biggl\langle \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} - y_{n}, J(z_{t} - y_{n}) \Biggr\rangle \\ &\quad \leq\Biggl\Vert \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})Q_{C}z_{t} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr\Vert ^{2} \\ &\qquad {}+ 2t\Biggl\langle \eta f(z_{t}) - T\sum _{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})Q_{C}z_{t}, J\Biggl(z_{t} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr) \Biggr\rangle \\ &\qquad {}+ 2\Biggl\langle \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} - y_{n}, J(z_{t} - y_{n}) \Biggr\rangle \\ &\quad \leq \Vert z_{t} - y_{n}\Vert ^{2} \\ &\qquad {}+ 2t\Biggl\langle \eta f(z_{t}) - T\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})Q_{C}z_{t}, J\Biggl(z_{t} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr) \Biggr\rangle \\ &\qquad {} + 2 \Biggl\Vert \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} - y_{n}\Biggr\Vert \Vert z_{t} - y_{n} \Vert , \end{aligned}$$
$$\begin{aligned} & t\Biggl\langle T\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})Q_{C}z_{t}- \eta f(z_{t}), J\Biggl(z_{t} - \sum _{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr) \Biggr\rangle \\ &\quad \leq\Biggl\Vert \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} - y_{n}\Biggr\Vert \Vert z_{t}-y_{n}\Vert . \end{aligned} $$
So, $\lim_{t \rightarrow0}\limsup_{n\rightarrow+\infty}\langle T\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})Q_{C}z_{t}-\eta f(z_{t}), J(z_{t} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I- r_{n,i}B_{i})y_{n}) \rangle\leq 0$ in view of Step 5.
Since $z_{t} \rightarrow p_{0}$, $\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})Q_{C}z_{t} \rightarrow\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i}) Q_{C}p_{0} = p_{0}$, as $t \rightarrow0$. Noticing the following fact:
$$\begin{aligned} & \Biggl\langle Tp_{0}-\eta f(p_{0}), J \Biggl(p_{0} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr)\Biggr\rangle \\ &\quad = \Biggl\langle Tp_{0}-\eta f(p_{0}), J \Biggl(p_{0} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr) - J\Biggl(z_{t} -\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr)\Biggr\rangle \\ &\qquad {}+ \Biggl\langle Tp_{0}-\eta f(p_{0}), J \Biggl(z_{t} -\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr)\Biggr\rangle \\ &\quad = \Biggl\langle Tp_{0}-\eta f(p_{0}), J \Biggl(p_{0} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr) - J\Biggl(z_{t} -\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr)\Biggr\rangle \\ &\qquad {} + \Biggl\langle Tp_{0}-\eta f(p_{0})-T\sum _{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})Q_{C}z_{t}+ \eta f(z_{t}), \\ &\qquad {} J\Biggl(z_{t} -\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr)\Biggr\rangle \\ & \qquad {}+ \Biggl\langle T\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})Q_{C}z_{t}- \eta f(z_{t}), J\Biggl(z_{t} -\sum _{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n} \Biggr)\Biggr\rangle , \end{aligned}$$
we have $\limsup_{n\rightarrow+\infty}\langle Tp_{0}-\eta f(p_{0}), J(p_{0} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n})\rangle\leq0$.
Since $\langle Tp_{0}-\eta f(p_{0}), J(p_{0} - x_{n+1})\rangle= \langle Tp_{0}-\eta f(p_{0}), J(p_{0} - x_{n+1})-J(p_{0} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n})\rangle+ \langle Tp_{0}-\eta f(p_{0}), J(p_{0} - \sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I-r_{n,i}B_{i})y_{n})\rangle$ and $x_{n+1}-\sum_{i = 1}^{\infty} a_{i} J_{r_{n,i}}^{A_{i}}(I- r_{n,i}B_{i})y_{n} \rightarrow 0$ in view of Step 5, then $\limsup_{n \rightarrow\infty} \langle\eta f(p_{0})-Tp_{0}, J(x_{n+1}-p_{0}) \rangle\leq0$.
Step 7. $x_{n} \rightarrow p_{0}$, as $n \rightarrow+\infty$, where $p_{0} \in D$ is the same as that in Step 6.
Let $M'' = \sup \{\Vert (1-\alpha_{n})(x_{n}+e_{n})-p_{0}\Vert , \Vert x_{n}-p_{0}\Vert , M'\Vert p_{0}\Vert , \Vert e''_{n}-p_{0}\Vert ^{2}: n \geq0\}$. By using Lemma 3 again, we have
$$ \begin{aligned}[b] &\Vert y_{n} - p_{0}\Vert ^{2} \\ &\quad \leq(1-\alpha_{n})^{2} \Vert x_{n} - p_{0}\Vert ^{2} +2 \bigl\langle (1-\alpha_{n})e'_{n}- \alpha_{n}p_{0}, J\bigl[(1-\alpha_{n}) \bigl(x_{n}+e'_{n}\bigr)-p_{0}\bigr] \bigr\rangle . \end{aligned} $$
$$\begin{aligned} \Vert z_{n} - p_{0}\Vert ^{2} &\leq\delta_{n} \Vert y_{n}-p_{0} \Vert ^{2}+\beta_{n} \biggl\Vert \frac{y_{n}+z_{n}}{2}-p_{0} \biggr\Vert ^{2}+\zeta _{n}\bigl\Vert e''_{n}-p_{0}\bigr\Vert ^{2} \\ & \leq \biggl(\delta_{n}+\frac{\beta_{n}}{2}\biggr)\Vert y_{n}-p_{0}\Vert ^{2}+\frac{\beta_{n}}{2} \Vert z_{n}-p_{0}\Vert ^{2}+ \zeta_{n}\bigl\Vert e''_{n}-p_{0} \bigr\Vert ^{2}, \end{aligned} $$
combining (18), we have
$$ \begin{aligned}[b] &\Vert z_{n} - p_{0}\Vert ^{2} \\ &\quad \leq \Vert y_{n}-p_{0}\Vert ^{2}+2 \zeta_{n}\bigl\Vert e''_{n}-p_{0} \bigr\Vert ^{2} \\ &\quad \leq(1-\alpha_{n})^{2} \Vert x_{n} - p_{0}\Vert ^{2} +2 \bigl\langle (1-\alpha_{n})e'_{n}- \alpha_{n}p_{0}, J\bigl[(1-\alpha_{n}) \bigl(x_{n}+e'_{n}\bigr)-p_{0}\bigr] \bigr\rangle \\ &\qquad {}+2\zeta_{n}\bigl\Vert e''_{n}-p_{0} \bigr\Vert ^{2}. \end{aligned} $$
Using (19) and Lemma 3, we have, for $n \geq0$,
$$\begin{aligned} &\Vert x_{n+1} - p_{0}\Vert ^{2} \\ &\quad = \bigl\Vert \gamma_{n}\bigl(\eta f(x_{n})-Tp_{0} \bigr)+(I- \gamma_{n} T) (z_{n} - p_{0})+e'''_{n} \bigr\Vert ^{2} \\ &\quad \leq(1-\gamma_{n} \overline{\gamma})^{2}\Vert z_{n}-p_{0}\Vert ^{2}+2\gamma_{n} \bigl\langle \eta f(x_{n})-Tp_{0}, J(x_{n+1}-p_{0}) \bigr\rangle \\ &\qquad {}+ 2\bigl\langle e'''_{n}, J(x_{n+1}-p_{0})\bigr\rangle \\ &\quad \leq(1-\gamma_{n} \overline{\gamma})^{2}(1- \alpha_{n})^{2}\Vert x_{n}-p_{0} \Vert ^{2}+ 2\bigl\langle e'''_{n}, J(x_{n+1}-p_{0})\bigr\rangle \\ &\qquad {}+2\gamma_{n}\eta\bigl\langle f(x_{n})-f(p_{0}), J(x_{n+1}-p_{0})-J(x_{n}-p_{0})\bigr\rangle \\ &\qquad {} +2\gamma_{n} \eta\bigl\langle f(x_{n})- f(p_{0}), J(x_{n}-p_{0})\bigr\rangle +2 \gamma_{n}\bigl\langle \eta f(p_{0})-Tp_{0}, J(x_{n+1}-p_{0})\bigr\rangle \\ & \qquad {}+ 2(1-\gamma_{n} \overline{\gamma})^{2}(1- \alpha_{n})\bigl\langle e'_{n}, J\bigl[(1- \alpha_{n}) \bigl(x_{n}+e'_{n} \bigr)-p_{0}\bigr]\bigr\rangle \\ &\qquad {} - 2\alpha_{n} (1-\gamma_{n} \overline{ \gamma})^{2} \bigl\langle p_{0}, J\bigl[(1-\alpha _{n}) \bigl(x_{n}+e'_{n} \bigr)-p_{0}\bigr]\bigr\rangle + 2(1-\gamma_{n} \overline{ \gamma})^{2} \zeta_{n} \bigl\Vert e''_{n}-p_{0} \bigr\Vert ^{2} \\ & \quad \leq\bigl[1-\gamma_{n} (\overline{\gamma}-2\eta k )\bigr] \Vert x_{n}-p_{0}\Vert ^{2}+2M'' \bigl[\bigl\Vert e'_{n}\bigr\Vert + \bigl\Vert e'''_{n}\bigr\Vert +(1- \gamma_{n} \overline{\gamma})^{2}(\alpha_{n} + \zeta_{n})\bigr] \\ & \qquad {}+2\gamma_{n} \bigl[\bigl\langle \eta f(p_{0})-Tp_{0}, J(x_{n+1}-p_{0})\bigr\rangle +\eta \Vert x_{n}-p_{0}\Vert \Vert x_{n+1}-x_{n} \Vert \bigr]. \end{aligned}$$
Let $\delta_{n}^{(1)} = \gamma_{n}(\overline{\gamma}-2\eta k)$, $\delta_{n}^{(2)} = 2\gamma_{n}[\langle\eta f(p_{0})-Tp_{0}, J(x_{n+1}-p_{0})\rangle+\eta \Vert x_{n}-p_{0}\Vert \Vert x_{n+1}-x_{n}\Vert ]$, $\delta_{n}^{(3)} = 2M''[\Vert e'_{n}\Vert + \Vert e'''_{n}\Vert +(1-\gamma_{n} \overline{\gamma})^{2}(\alpha_{n} +\zeta_{n})]$. Then (20) can be simplified as $\Vert x_{n+1}-p_{0}\Vert ^{2} \leq (1-\delta_{n}^{(1)})\Vert x_{n}-p_{0}\Vert ^{2} + \delta_{n}^{(2)}+\delta_{n}^{(3)}$.
From the assumptions (i), (ii), (iv), and (v), the results of Steps 1, 4, and 6 and Lemma 4, we know that $x_{n} \rightarrow p_{0}$, as $n \rightarrow+\infty$.
Combine the result of Step 5, $y_{n} \rightarrow p_{0}$ and $z_{n} \rightarrow p_{0}$, as $n \rightarrow\infty$.
Remark 12
The assumptions imposed on the real number sequences in Theorem 11 are reasonable if we take $\alpha_{n} = \frac{1}{n^{2}}$, $\gamma_{n} = \frac{1}{n}$, $\delta_{n} = 1-\frac{1}{n^{2}}-\frac{n}{n+1}$, $\beta_{n} = \frac{n}{n+1}$, and $\zeta_{n} = \frac{1}{n^{2}}$ for $n \geq0$.
Three sequences $\{x_{n}\}$, $\{y_{n}\}$, and $\{z_{n}\}$ are proved to be strongly convergent to the zero point of the sum of an infinite family of m-accretive mappings and an infinite family of $\mu_{i}$-inversely strongly accretive mappings. The strongly convergent point $p_{0}$ is the unique solution of a variational inequality.
Compared to the previous work, the computational error is considered in each step and the work on finding zero point of the sum of a finite family of m-accretive mappings and an finite family of μ-inversely strongly accretive mapping is extended to the infinite case. Compared to the work in [8], the construction of $z_{n}$ in the iterative algorithm (A) is implicit and a different $B_{i}$ corresponds to a different $\mu_{i}$, which makes the iterative algorithm (A) more general. Moreover, the assumption that 'the normalized duality mapping J is weakly sequentially continuous at zero' is deleted.
If $e'_{n} = e''_{n} = e'''_{n} \equiv0$, then iterative algorithm (A) becomes an accurate one.
If $C \equiv E$, then the iterative algorithm (A) becomes the following one:
$$\textstyle\begin{cases} x_{0}\in E,\\ y_{n} = (1-\alpha_{n})(x_{n}+e'_{n}),\\ z_{n}= \delta_{n}y_{n} + \beta_{n} \sum_{i = 1}^{\infty}a_{i} J_{r_{n,i}}^{A_{i}}[\frac {y_{n}+z_{n}}{2}-r_{n,i}B_{i}(\frac{y_{n}+z_{n}}{2})] +\zeta_{n} e''_{n},\\ x_{n+1}=\gamma_{n} \eta f(x_{n})+(I-\gamma_{n}T)z_{n}+e'''_{n}, \quad n \geq0. \end{cases} $$
Integro-differential systems and iterative algorithms
In this section, we have five purposes: (1) based on one kind nonlinear integro-differential system, construct an infinite family of m-accretive mappings and an infinite family of $\mu_{i}$-inversely strongly accretive mappings; (2) prove that under some conditions, the nonlinear integro-differential systems discussed exist solutions; (3) show the connections between the solution of the integro-differential systems and the zero point of the sum of an infinite family of m-accretive mappings and an infinite family of $\mu_{i}$-inversely strongly accretive mappings; (4) construct the iterative approximate sequence of the solution of the integro-differential systems; (5) demonstrate the relationship between the solution of the nonlinear integro-differential systems and the solution of one kind variational inequalities.
Discussion of integro-differential systems
We shall study the following nonlinear integro-differential systems involving the generalized $p_{i}$-Laplacian:
$$ \textstyle\begin{cases} \frac{\partial u^{(i)}(x,t)}{\partial t} -\operatorname {div}[(C(x,t)+\vert Du^{(i)}\vert ^{2})^{\frac{p_{i}-2}{2}}Du^{(i)}] + \varepsilon \vert u^{(i)}\vert ^{r_{i}-2}u^{(i)} \\ \quad {}+ g(x,u^{(i)}, Du^{(i)})+ a\frac{\partial}{\partial t} \int_{\Omega}u^{(i)}\,dx = f(x,t),\quad (x,t) \in\Omega\times(0,T), \\ -\langle\vartheta,(C(x,t)+\vert Du^{(i)}\vert ^{2})^{\frac{p_{i}-2}{2}}Du^{(i)} \rangle\in\beta_{x}(u^{(i)}),\quad (x,t) \in\Gamma\times(0,T),\\ u^{(i)}(x,0) = u^{(i)}(x, T),\quad x \in\Omega, i \in\mathbb{N^{+}}, \end{cases} $$
where Ω is a bounded conical domain of a Euclidean space $R^{N}$ ($N\geq1$), Γ is the boundary of Ω with $\Gamma\in C^{1}$ [17] and ϑ denotes the exterior normal derivative to Γ. $\langle\cdot,\cdot\rangle$ and $\vert \cdot \vert $ denote the Euclidean inner-product and Euclidean norm in $R^{N}$, respectively. T is a positive constant. $Du^{(i)} = (\frac{\partial u^{(i)}}{\partial x_{1}}, \frac{\partial u^{(i)}}{\partial x_{2}}, \ldots, \frac{\partial u^{(i)}}{\partial x_{N}})$ and $x = (x_{1}, x_{2}, \ldots, x_{N}) \in\Omega$. $\beta_{x}$ is the subdifferential of $\varphi_{x}$, where $\varphi_{x}= \varphi(x,\cdot):R\rightarrow R$ for $x\in\Gamma$. a and ε are non-expansive constants, $0 \leq C(x,t) \in \bigcap_{i = 1}^{\infty}V_{i} : = \bigcap_{i = 1}^{\infty} L^{p_{i}}(0 , T; W^{1,p_{i}}(\Omega))$, $f(x,t)\in\bigcap_{i = 1}^{\infty} W_{i} : = \bigcap_{i = 1}^{\infty} L^{\max\{p_{i},p_{i}'\}}(0,T; L^{\max\{p_{i},p_{i}'\}}(\Omega))$ and $g:\Omega\times R^{N+1} \rightarrow R$ are given functions.
Our discussion of (21) is based on the following assumptions, some of which can be found in [18–20].
$\{p_{i}\}_{i=1}^{\infty}$ is a real number sequence with $\frac{2N}{N+1} < p_{i} < +\infty$, $\{\mu_{i}\}_{i=1}^{\infty}$ is any real number sequence in $(0,1]$ and $\{r_{i}\}_{i=1}^{\infty}$ is a real number sequence satisfying $\frac{2N}{N+1} < r_{i} \leq \min\{p_{i},p_{i}'\} < +\infty$. $\frac{1}{p_{i}}+\frac{1}{p'_{i}} = 1$ and $\frac{1}{r_{i}}+\frac{1}{r'_{i}} = 1$ for $i \in\mathbb{N^{+}}$.
Green's formula is available.
For each $x\in\Gamma, \varphi_{x}= \varphi(x,\cdot):R\rightarrow R$ is a proper, convex and lower- semi-continuous function and $\varphi_{x}(0)=0$.
$0 \in\beta_{x}(0)$ and for each $t \in R$, the function $x \in\Gamma\rightarrow(I+\lambda\beta_{x})^{-1}(t)\in R$ is measurable for $\lambda > 0$.
Suppose that $g:\Omega\times R^{N+1} \rightarrow R$ satisfies the following conditions:
Carathéodory's conditions;
growth condition:
$$\bigl\vert g(x,r_{1},\ldots,r_{N+1})\bigr\vert ^{\max\{p_{i},p_{i}'\}}\leq\bigl\vert h_{i}(x,t)\bigr\vert ^{p_{i}}+ b_{i} \vert r_{1}\vert ^{p_{i}}, $$
where $(r_{1}, r_{2}, \ldots, r_{N+1})\in R^{N+1} $, $h_{i}(x,t)\in W_{i}$, and $b_{i}$ is a positive constant, for $i \in\mathbb{N^{+}}$;
monotone condition: g is monotone in the following sense:
$$\bigl(g(x,r_{1},\ldots,r_{N+1})-g(x,t_{1}, \ldots,t_{N+1})\bigr)\geq(r_{1} - t_{1}) , $$
for all $x \in\Omega$ and $(r_{1},\ldots,r_{N+1}),(t_{1},\ldots,t_{N+1})\in R^{N+1}$.
For $i \in\mathbb{N^{+}}$, let $V^{*}_{i}$ denote the dual space of $V_{i}$. The norm in $V_{i}$, $\Vert \cdot \Vert _{V_{i}}$, is defined by
$$\bigl\Vert u(x,t)\bigr\Vert _{V_{i}} = \biggl( \int_{0}^{T} \bigl\Vert u(x,t)\bigr\Vert _{W^{1,p_{i}}(\Omega)} ^{p_{i}}\,dt\biggr)^{\frac{1}{p_{i}}},\quad u(x,t) \in V_{i}. $$
For $i \in\mathbb{N^{+}}$, define the operator $B_{i}: V_{i} \rightarrow V^{*}_{i}$ by
$$\langle w,B_{i}u \rangle= \int_{0}^{T} \int_{\Omega}\bigl\langle \bigl(C(x,t)+\vert Du\vert ^{2}\bigr)^{\frac {p_{i}-2}{2}}Du, Dw\bigr\rangle \,dx\,dt +\varepsilon \int_{0}^{T} \int_{\Omega} \vert u\vert ^{r_{i}-2}uw \,dx\,dt, $$
for $u,w \in V_{i}$. Then $B_{i}$ is maximal monotone and coercive, where $i \in\mathbb{N^{+}}$.
For $i \in\mathbb{N^{+}}$, define the function $\Phi_{i}: V_{i} \rightarrow R $ by
$$\Phi_{i}(u)= \int_{0}^{T} \int_{\Gamma}\varphi_{x}\bigl(u|_{\Gamma}(x,t) \bigr)\,d\Gamma(x)\,dt, $$
for $u(x,t) \in V_{i}$. Then $\Phi_{i}$ is a proper, convex and lower-semi-continuous mapping on $V_{i}$. Therefore, the subdifferential $\partial\Phi_{i}: V_{i}\rightarrow V^{*}_{i}$ is maximal monotone.
For $i \in\mathbb{N^{+}}$, define $S_{i}: D(S_{i}) = \{u(x,t) \in V_{i}: \frac{\partial u }{\partial t} \in V^{*}_{i}, u(x,0) = u(x,T)\} \rightarrow V^{*}_{i}$ by
$$S_{i}u = \frac{\partial u }{\partial t}+ a\frac{\partial}{\partial t} \int_{\Omega}u \,dx. $$
Then $S_{i}$ is linear maximal monotone operator possessing a dense domain in $V_{i}$, where $i \in\mathbb{N^{+}}$.
Definition 20
For $i \in\mathbb{N^{+}}$, define a mapping $A_{i}: W_{i} \rightarrow2^{W_{i}}$ as follows:
$$D(A_{i})=\bigl\{ u \in W_{i} |\mbox{ there exists an }f \in W_{i}\mbox{ such that }f \in B_{i}u + \partial \Phi_{i}(u) + S_{i}u\bigr\} . $$
For $u \in D(A_{i})$, we set $A_{i}u =\{f\in W_{i} | f \in B_{i}u + \partial \Phi_{i}(u) + S_{i}u\}$.
The mapping $A_{i}: W_{i} \rightarrow2^{W_{i}}$ is m-accretive, where $i \in\mathbb{N^{+}}$.
Similar to the proof of Lemmas 3.5 and 3.7 in [18] or the proof of Proposition 2.5 in [19], we have $R(I+\lambda A_{i})= W_{i}$, for $\forall\lambda>0$.
Let $J_{i}: W_{i} \rightarrow W_{i}^{*}$ denote the generalized duality mapping. Then, for $u(x,t) \in W_{i}$,
$$J_{i} u = \textstyle\begin{cases} \vert u\vert ^{p_{i}-1}\operatorname {sgn}u, & p_{i} \geq2,\\ \vert u\vert ^{p'_{i}-1}\operatorname {sgn}u,& 1< p_{i} < 2. \end{cases} $$
In fact, if $p_{i} \geq2$, then $\langle u, J_{i} u\rangle= \int^{T}_{0}\int_{\Omega} \vert u\vert ^{p_{i}}\,dx\,dt = \Vert u\Vert _{W_{i}}^{p_{i}}$ and $\Vert J_{i}u\Vert _{W_{i}^{*}}= (\int^{T}_{0}\int_{\Omega} \vert u\vert ^{(p_{i}-1)p'_{i}}\,dx\,dt)^{\frac{1}{p'_{i}}} = \Vert u\Vert _{W_{i}}^{\frac{p_{i}}{p'_{i}}}= \Vert u\Vert _{W_{i}}^{p_{i}-1}$. Thus $J_{i} u = \vert u\vert ^{p_{i}-1}\operatorname {sgn}u$, if $p_{i} \geq2$. Similarly, $J_{i} u = \vert u\vert ^{p'_{i}-1}\operatorname {sgn}u$, if $1< p_{i}<2$.
By using a similar method as that of Proposition 2.4 in [19], we can prove that for any $u,v\in D(A_{i})$, $\langle A_{i}u - A_{i}v, J_{i}(u-v)\rangle\geq0$. Thus $A_{i}$ is accretive. The result follows. This completes the proof. □
Noticing Proposition 21, an infinite family of m-accretive mappings $\{A_{i}\}_{i = 1}^{\infty}$ is constructed.
Define $C_{i} : D(C_{i}) = L^{\max\{p_{i},p'_{i}\}}(0,T;W^{1,\max\{p_{i},p'_{i}\}}(\Omega))\subset W_{i} \rightarrow W_{i}$ by
$$(C_{i}u) (x,t) = g(x,u,Du)-f(x,t) $$
for $\forall u(x,t) \in D(C_{i})$ and $f(x,t)$ is the same as that in (21), where $i \in\mathbb{N^{+}}$.
The mapping $C_{i}: D(C_{i})\subset W_{i} \rightarrow W_{i}$ is continuous and strongly accretive. If, further assume that $g(x,r_{1},\ldots,r_{N+1}) \equiv r_{1}$, then $C_{i}$ is $\mu_{i}$-inversely strongly accretive, where $i \in\mathbb{N^{+}}$.
Similar to Proposition 2.6 in [19], we know that for $u \in D(C_{i})$, $x \rightarrow g(x, u, Du)$ is measurable on Ω, and then $C_{i}$ is everywhere defined and continuous.
Our next discussion is divided into two cases.
Case 1. $p_{i} \geq2$. From assumption 5, we know that
$$\begin{aligned}[b] & \bigl\langle C_{i}u - C_{i}v, \widetilde{J_{i}}(u-v)\bigr\rangle \\ &\quad = \int_{0}^{T} \int_{\Omega} \bigl(g(x,u,Du) - g(x, v, Dv)\bigr)\Vert u-v \Vert ^{2-p_{i}}_{W_{i}}\vert u-v\vert ^{p_{i}-1} \operatorname {sgn}(u-v) \,dx\,dt \\ &\quad \geq \Vert u-v\Vert ^{2-p_{i}}_{W_{i}} \int_{0}^{T} \int_{\Omega} \vert u-v\vert ^{p_{i}}\,dx\,dt = \Vert u-v\Vert ^{2}_{W_{i}}, \end{aligned} $$
where $\widetilde{J_{i}}: W_{i} \rightarrow W_{i}^{*}$ is the normalized duality mapping, which implies that $C_{i}$ is strongly accretive.
If, furthermore, $g(x,r_{1},\ldots,r_{N+1}) \equiv r_{1}$, since $\{\mu_{i}\} \subset(0,1]$, then we have
$$\bigl\langle C_{i}u - C_{i}v, J_{i}(u-v)\bigr\rangle = \int_{0}^{T} \int_{\Omega} \vert u-v\vert ^{p_{i}}\,dx\,dt = \Vert C_{i}u-C_{i}v\Vert ^{p_{i}}_{W_{i}}\geq \mu_{i}\Vert C_{i}u-C_{i}v\Vert _{W_{i}}^{p_{i}}, $$
where $J_{i}: W_{i} \rightarrow W_{i}^{*}$ is the generalized duality mapping in Proposition 21, which implies that $C_{i}$ is $\mu_{i}$-inversely strongly accretive.
Case 2. $1< p_{i} < 2$. Similar to Case 1, the result follows.
Noticing Lemma 24, we have constructed an infinite family of $\mu_{i}$-inversely strongly accretive mappings.
(1) If $w(x,t) \in\partial\Phi_{i}(u)$, then $w(x,t) \in\partial\beta_{x}(u)$, a.e. on $\Gamma\times (0,T)$. (2) $\langle\varphi, \partial\Phi_{i}(u)\rangle\equiv0$, $\forall\varphi\in C_{0}^{\infty}(0, T; \Omega)$.
Let E be a smooth Banach space, let $A: D(A) \subset E \rightarrow2^{E}$ be an m-accretive mapping, and $S: D(S) \subset E \rightarrow E $ be a continuous and strongly accretive mapping with $\overline{D(A)}\subset D(S)$. Then, for any $z \in E$, the equation $z \in Sx+\lambda Ax$ has a unique solution $x_{\lambda}$, $\lambda> 0$.
For $f(x,t)\in\bigcap_{i = 1}^{\infty}W_{i}$, there exists unique $u^{(i)} \in W_{i}$ satisfying the following:
$\frac{\partial u^{(i)}(x,t)}{\partial t} -\operatorname {div}[(C(x,t)+\vert D u^{(i)}\vert ^{2})^{\frac{p_{i}-2}{2}}Du^{(i)}] + \varepsilon \vert u^{(i)}\vert ^{r_{i}-2}u^{(i)} + g(x,u^{(i)},Du^{(i)})+ a \frac{\partial}{\partial t} \int_{\Omega}u^{(i)}\,dx = f(x,t)$, $(x,t) \in\Omega\times(0,T)$;
$-\langle\vartheta,(C(x,t)+\vert Du^{(i)}\vert ^{2})^{\frac{p_{i}-2}{2}}Du^{(i)} \rangle\in\beta_{x}(u^{(i)}(x,t))$, $(x,t) \in\Gamma\times (0,T)$;
$u^{(i)}(x,0) = u^{(i)}(x,T)$, $x \in\Omega$, where $i \in\mathbb{N^{+}}$.
Using Proposition 21, Lemmas 24 and 27, we know that for $\theta\in W_{i}$, there exists unique $u^{(i)}(x,t)\in D(A_{i})\subset W_{i}$ such that
$$ \theta= C_{i}u^{(i)} + A_{i}u^{(i)}. $$
Then, for $\varphi\in C^{\infty}_{0}(0,T;\Omega)$, we have
$$\langle\varphi, \theta\rangle= \bigl\langle \varphi, C_{i}u^{(i)} \bigr\rangle + \bigl\langle \varphi, B_{i}u^{(i)}\bigr\rangle + \bigl\langle \varphi, \partial\Phi_{i}\bigl(u^{(i)}\bigr)\bigr\rangle + \bigl\langle \varphi, S_{i}u^{(i)}\bigr\rangle , $$
$$\begin{aligned} & \int_{0}^{T} \int_{\Omega} f \varphi \,dx\,dt \\ &\quad = \int_{0}^{T} \int_{\Omega}\frac{\partial u^{(i)}}{\partial t }\varphi \,dx\,dt + a \int_{0}^{T} \int_{\Omega} \biggl(\frac{\partial}{\partial t} \int_{\Omega}u^{(i)}\,dx\biggr) \varphi \,dx\,dt \\ &\qquad {}+ \int_{0}^{T} \int_{\Omega}\bigl\langle \bigl(C(x,t)+\bigl\vert Du^{(i)}\bigr\vert ^{2}\bigr)^{\frac{p_{i}-2}{2}}Du^{(i)}, D\varphi\bigr\rangle \,dx\,dt + \varepsilon \int_{0}^{T} \int_{\Omega}\bigl\vert u^{(i)}\bigr\vert ^{r_{i}-2} u^{(i)}\varphi \,dx\,dt \\ &\qquad {}+ \int_{0}^{T} \int_{\Omega} g\bigl(x,u^{(i)},Du^{(i)}\bigr) \varphi \,dx\,dt \\ &\quad = \int_{0}^{T} \int_{\Omega}\frac{\partial u^{(i)}}{\partial t }\varphi \,dx\,dt + a \int_{0}^{T} \int_{\Omega} \biggl(\frac{\partial}{\partial t} \int_{\Omega}u^{(i)}\,dx\biggr) \varphi \,dx\,dt \\ &\qquad {}+ \int_{0}^{T} \int_{\Omega}-\operatorname {div}\bigl[\bigl(C(x,t)+\bigl\vert Du^{(i)}\bigr\vert ^{2}\bigr)^{\frac {p_{i}-2}{2}}Du^{(i)} \bigr]\varphi \,dx\,dt + \varepsilon \int_{0}^{T} \int_{\Omega}\bigl\vert u^{(i)}\bigr\vert ^{r_{i}-2} u^{(i)} \varphi \,dx\,dt \\ &\qquad {}+ \int_{0}^{T} \int_{\Omega} g\bigl(x,u^{(i)}, Du^{(i)}\bigr) \varphi \,dx\,dt. \end{aligned}$$
Therefore, from the property of the generalized function, we know that (a) is true.
From the definition of $S_{i}$, we know that (c) is trivial.
By using the results of (a), the Green's formula and (22), we have, for $w \in W_{i}$,
$$\begin{aligned} & \int^{T}_{0} \int_{\Gamma}\bigl\langle \vartheta, \bigl(C(x,t)+\bigl\vert Du^{(i)}\bigr\vert ^{2}\bigr)^{\frac{p_{i}-2}{2}}Du^{(i)} \bigr\rangle w d\Gamma(x)\,dt \\ &\quad = \int_{0}^{T} \int_{\Omega} \operatorname {div}\bigl[\bigl(C(x,t)+\bigl\vert Du^{(i)}\bigr\vert ^{2}\bigr)^{\frac{p_{i}-2}{2}}Du^{(i)} \bigr] w \,dx\,dt \\ &\qquad {}+ \int_{0}^{T} \int_{\Omega}\bigl\langle \bigl(C(x,t)+\bigl\vert Du^{(i)}\bigr\vert ^{2}\bigr)^{\frac{p_{i}-2}{2}}Du^{(i)}, Dw\bigr\rangle \,dx\,dt \\ &\quad = \int_{0}^{T} \int_{\Omega}g\bigl(x,u^{(i)},Du^{(i)} \bigr)w\,dx\,dt+ \int_{0}^{T} \int_{\Omega}\frac{\partial u^{(i)}}{\partial t} w \,dx\,dt\\ &\qquad {}+ \int_{0}^{T} \int_{\Omega} \biggl(a \frac{\partial}{\partial t} \int_{\Omega}u^{(i)}\,dx\biggr) w \,dx\,dt + \varepsilon \int_{0}^{T} \int_{\Omega}\bigl\vert u^{(i)}\bigr\vert ^{r_{i}-2} u^{(i)}w \,dx\,dt\\ &\qquad {}+ \int_{0}^{T} \int_{\Omega}\bigl\langle \bigl(C(x,t)+\bigl\vert Du^{(i)}\bigr\vert ^{2}\bigr)^{\frac{p_{i}-2}{2}}Du^{(i)}, Dw\bigr\rangle \,dx\,dt - \int_{0}^{T} \int_{\Omega}f(x,t) w \,dx\,dt \\ &\quad = \int_{0}^{T} \int _{\Omega}-\partial\Phi_{i}\bigl(u^{(i)} \bigr) w \,dx\,dt. \end{aligned}$$
Thus $- \langle\vartheta, (C(x,t)+\vert Du^{(i)}\vert ^{2})^{\frac{p_{i}-2}{2}}Du^{(i)}\rangle \in \partial\Phi_{i}(u^{(i)})$. In view of Lemma 26, (b) follows.
Applications of iterative algorithms to integro-differential systems
If $\varepsilon\equiv0$, $g(x,r_{1},\ldots,r_{N+1}) \equiv r_{1}$ and $f(x,t) \equiv k $, were k is a constant, then $u(x,t)\equiv k$ is the unique solution of the integro-differential system (21). Moreover, $\{u(x,t)\in\bigcap_{i = 1}^{\infty}W_{i}| u(x,t)\equiv k\textit{ satisfying }\mbox{(21)}\} = \bigcap_{i = 1}^{\infty}N(A_{i}+C_{i})$.
From Theorem 28, we know that (21) has a unique solution for this special case. It is easy to check that $u(x,t)\equiv k $ satisfies (21), which implies that $u(x,t)\equiv k $ is the unique solution of (21) for this special case.
Next, we show that $\bigcap_{i = 1}^{\infty}N(A_{i}+C_{i})$ is a singleton in this special case.
In fact, if $A_{i} u +C_{i} u \equiv0$ and $A_{i} v+C_{i} v \equiv0$, then $A_{i} u + u \equiv A_{i}v + v$, which implies that $0 \leq \langle A_{i} u -A_{i} v, J_{i}(u-v)\rangle= \langle v - u, J_{i}(u-v)\rangle\leq0$, and then $u(x,t) \equiv v(x,t)$. That is, $\bigcap_{i = 1}^{\infty}N(A_{i}+C_{i})$ is a singleton.
The result $u(x,t)\equiv k \in\bigcap_{i = 1}^{\infty}N(A_{i}+C_{i})$ follows from the definitions of $A_{i}$ and $C_{i}$, which implies that $\{u(x,t)\in\bigcap_{i = 1}^{\infty}W_{i}| u(x,t)\equiv k\mbox{ satisfying (21)}\} = \bigcap_{i = 1}^{\infty}N(A_{i}+C_{i})$.
Combining the results of Proposition 21, Lemma 24, and Theorem 29, we set up the relationship between the solution of one kind integro-differential systems and the zero point of the sum of infinite m-accretive mappings and infinite $\mu_{i}$-inversely strongly accretive mappings.
Set $p:= \inf_{i \in N^{+}}(\min\{p_{i},p_{i}'\})$ and $q:= \sup_{i \in N^{+}}(\max\{p_{i},p_{i}'\})$.
Let $E:= L^{\min\{p,p'\}}(0,T; L^{\min\{p,p'\}}(\Omega))$, where $\frac{1}{p}+\frac{1}{p'} = 1$.
Let $X:= L^{\max\{q,q'\}}(0,T; W^{1,\max\{q,q'\}}(\Omega))$, where $\frac{1}{q}+\frac{1}{q'} = 1$.
Then $E = L^{p}(0,T;L^{p}(\Omega))$, $X = L^{q}(0,T;W^{1,q}(\Omega))$ and $X \subset W_{i} \subset E$, $\forall i \in\mathbb{N^{+}}$. Our next discussion of Theorem 32 will be based on X and E.
Suppose $A_{i}$ and $C_{i}$ are the same as those in Proposition 21 and Lemma 24, respectively. Let $f : E\rightarrow E$ be a fixed contractive mapping with coefficient $k \in(0,1)$ and $T: E \rightarrow E$ be any strongly positive linear bounded operator with coefficient γ̅. Suppose that $0 < \eta< \frac{\overline{\gamma}}{2k}$, $\{\alpha_{n}\}$, $\{\delta_{n}\}$, $\{\beta_{n}\}$, $\{\zeta_{n}\}$, $\{\gamma_{n}\} \subset(0,1)$ and $\{r_{n,i}\}\subset(0,+\infty)$ for $i\in\mathbb {N^{+}}$. Suppose $\{a_{i}\}_{i = 1}^{\infty}\subset(0,1)$ with $\sum_{i = 1}^{\infty}a_{i} = 1$, $\{e''_{n}\}\subset X $, and $\{e'_{n}\}, \{e'''_{n}\} \subset E$. Furthermore, suppose that the following conditions are satisfied:
$\sum_{n=0}^{\infty} \gamma_{n} = \infty$, $\gamma_{n} \rightarrow0$, as $n \rightarrow\infty$, and $\sum_{n=1}^{\infty} \vert \gamma_{n} -\gamma_{n-1}\vert <+\infty$;
$\sum_{n=0}^{\infty} \vert r_{n+1,i} - r_{n,i}\vert < +\infty$ and $0 < \varepsilon\leq r_{n,i}\leq (\frac{p\mu_{i}}{K_{p}})^{\frac{1}{p-1}}$, for $n \geq0$ and $i\in\mathbb{N^{+}}$;
Let $\{u_{n}\}$ be generated by the iterative algorithm (C)
$$ \textstyle\begin{cases} u_{0}(x,t) \in X,\quad \text{chosen arbitrarily},\\ v_{n}(x,t)= Q_{X}[(1-\alpha_{n})(u_{n}(x,t)+e_{n}')],\\ w_{n}(x,t) = \delta_{n} v_{n}(x,t) + \beta_{n} \sum_{i=1}^{\infty}a_{i}J_{r_{n,i}}^{A_{i}} [\frac{w_{n}+v_{n}}{2}-r_{n,i}C_{i}(\frac{w_{n}+v_{n}}{2})]+\zeta_{n}e''_{n},\\ u_{n+1}(x,t)=\gamma_{n} \eta f(u_{n})+(I-\gamma_{n}T)w_{n}(x,t)+ e_{n}''', \quad n \geq 0. \end{cases} $$
If, in the integro-differential systems (21), $\varepsilon\equiv0$, $g(x,r_{1},\ldots,r_{N+1})\equiv r_{1}$, and $f(x,t)\equiv k$, then three sequences $\{u_{n}(x,t)\}$, $\{v_{n}(x,t)\}$, and $\{w_{n}(x,t)\}$ converge strongly to the unique solution $u(x,t)$ of (21), which is also the unique element in $\bigcap_{i=1}^{\infty }N(A_{i}+C_{i})$ and satisfies the following variational inequality: for $\forall y \in \bigcap_{i=1}^{\infty}N(A_{i}+C_{i})$,
$$\bigl\langle (T - \eta f)u(x,t), J\bigl(u(x,t) - y\bigr)\bigr\rangle \leq0. $$
From the work done in this section, we can find the connection between integro-differential systems, variational inequalities, and iterative algorithms. This may emphasize the significance of the work in this paper.
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This research was supported by the National Natural Science Foundation of China (11071053), Natural Science Foundation of Hebei Province (No. A2014207010), Key Project of Science and Research of Hebei Educational Department (ZH2012080), and Key Project of Science and Research of Hebei University of Economics and Business (2015KYZ03).
The authors wish to thank the referees for their helpful comments, which notably improved the presentation of this manuscript.
School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, 050061, China
Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX, 78363, USA
Ravi P Agarwal
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
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Correspondence to Li Wei.
All the authors contributed equally. All authors read and approve the final manuscript.
m-accretive mapping
$\mu_{i}$-inversely strongly accretive mapping
integro-differential systems | CommonCrawl |
How many different prime factors are in the prime factorization of $117\cdot119$?
Both these numbers look like they could be primes, but are actually not. $117=3^2\cdot13$, and $119=7\cdot17$. That gives $\boxed{4}$ distinct primes in the prime factorization. | Math Dataset |
Sample size determination
Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power. In complicated studies there may be several different sample sizes: for example, in a stratified survey there would be different sizes for each stratum. In a census, data is sought for an entire population, hence the intended sample size is equal to the population. In experimental design, where a study may be divided into different treatment groups, there may be different sample sizes for each group.
Sample sizes may be chosen in several ways:
• using experience – small samples, though sometimes unavoidable, can result in wide confidence intervals and risk of errors in statistical hypothesis testing.
• using a target variance for an estimate to be derived from the sample eventually obtained, i.e., if a high precision is required (narrow confidence interval) this translates to a low target variance of the estimator.
• using a target for the power of a statistical test to be applied once the sample is collected.
• using a confidence level, i.e. the larger the required confidence level, the larger the sample size (given a constant precision requirement).
Introduction
Larger sample sizes generally lead to increased precision when estimating unknown parameters. For example, if we wish to know the proportion of a certain species of fish that is infected with a pathogen, we would generally have a more precise estimate of this proportion if we sampled and examined 200 rather than 100 fish. Several fundamental facts of mathematical statistics describe this phenomenon, including the law of large numbers and the central limit theorem.
In some situations, the increase in precision for larger sample sizes is minimal, or even non-existent. This can result from the presence of systematic errors or strong dependence in the data, or if the data follows a heavy-tailed distribution.
Sample sizes may be evaluated by the quality of the resulting estimates. For example, if a proportion is being estimated, one may wish to have the 95% confidence interval be less than 0.06 units wide. Alternatively, sample size may be assessed based on the power of a hypothesis test. For example, if we are comparing the support for a certain political candidate among women with the support for that candidate among men, we may wish to have 80% power to detect a difference in the support levels of 0.04 units.
Estimation
Estimation of a proportion
A relatively simple situation is estimation of a proportion. For example, we may wish to estimate the proportion of residents in a community who are at least 65 years old.
The estimator of a proportion is ${\hat {p}}=X/n$, where X is the number of 'positive' e.g., the number of people out of the n sampled people who are at least 65 years old). When the observations are independent, this estimator has a (scaled) binomial distribution (and is also the sample mean of data from a Bernoulli distribution). The maximum variance of this distribution is 0.25, which occurs when the true parameter is p = 0.5. In practice, since p is unknown, the maximum variance is often used for sample size assessments. If a reasonable estimate for p is known the quantity $p(1-p)$ may be used in place of 0.25.
For sufficiently large n, the distribution of ${\hat {p}}$ will be closely approximated by a normal distribution.[1] Using this and the Wald method for the binomial distribution, yields a confidence interval of the form
$\left({\widehat {p}}-Z{\sqrt {\frac {0.25}{n}}},\quad {\widehat {p}}+Z{\sqrt {\frac {0.25}{n}}}\right)$ ,
where Z is a standard Z-score for the desired level of confidence (1.96 for a 95% confidence interval).
If we wish to have a confidence interval that is W units total in width (W/2 on each side of the sample mean), we will solve
$Z{\sqrt {\frac {0.25}{n}}}=W/2$
for n, yielding the sample size
$n={\frac {Z^{2}}{W^{2}}}$ , in the case of using .5 as the most conservative estimate of the proportion. (Note: W/2 = margin of error.)
In the figure below one can observe how sample sizes for binomial proportions change given different confidence levels and margins of error.
Otherwise, the formula would be $Z{\sqrt {\frac {p(1-p)}{n}}}=W/2$ , which yields $n={\frac {4Z^{2}p(1-p)}{W^{2}}}$.
For example, if we are interested in estimating the proportion of the US population who supports a particular presidential candidate, and we want the width of 95% confidence interval to be at most 2 percentage points (0.02), then we would need a sample size of (1.96)2/ (0.022) = 9604. It is reasonable to use the 0.5 estimate for p in this case because the presidential races are often close to 50/50, and it is also prudent to use a conservative estimate. The margin of error in this case is 1 percentage point (half of 0.02).
The foregoing is commonly simplified
$\left({\widehat {p}}-1.96{\sqrt {\frac {0.25}{n}}},{\widehat {p}}+1.96{\sqrt {\frac {0.25}{n}}}\right)$
will form a 95% confidence interval for the true proportion. If this interval needs to be no more than W units wide, the equation
$4{\sqrt {\frac {0.25}{n}}}=W$
can be solved for n, yielding[2][3] n = 4/W2 = 1/B2 where B is the error bound on the estimate, i.e., the estimate is usually given as within ± B. For B = 10% one requires n = 100, for B = 5% one needs n = 400, for B = 3% the requirement approximates to n = 1000, while for B = 1% a sample size of n = 10000 is required. These numbers are quoted often in news reports of opinion polls and other sample surveys. However, the results reported may not be the exact value as numbers are preferably rounded up. Knowing that the value of the n is the minimum number of sample points needed to acquire the desired result, the number of respondents then must lie on or above the minimum.
Estimation of a mean
When estimating the population mean using an independent and identically distributed (iid) sample of size n, where each data value has variance σ2, the standard error of the sample mean is:
${\frac {\sigma }{\sqrt {n}}}.$
This expression describes quantitatively how the estimate becomes more precise as the sample size increases. Using the central limit theorem to justify approximating the sample mean with a normal distribution yields a confidence interval of the form
$\left({\bar {x}}-{\frac {Z\sigma }{\sqrt {n}}},\quad {\bar {x}}+{\frac {Z\sigma }{\sqrt {n}}}\right)$ ,
where Z is a standard Z-score for the desired level of confidence (1.96 for a 95% confidence interval).
If we wish to have a confidence interval that is W units total in width (W/2 being the margin of error on each side of the sample mean), we would solve
${\frac {Z\sigma }{\sqrt {n}}}=W/2$
for n, yielding the sample size
$n={\frac {4Z^{2}\sigma ^{2}}{W^{2}}}$.
For example, if we are interested in estimating the amount by which a drug lowers a subject's blood pressure with a 95% confidence interval that is six units wide, and we know that the standard deviation of blood pressure in the population is 15, then the required sample size is ${\frac {4\times 1.96^{2}\times 15^{2}}{6^{2}}}=96.04$, which would be rounded up to 97, because the obtained value is the minimum sample size, and sample sizes must be integers and must lie on or above the calculated minimum.
Required sample sizes for hypothesis tests
A common problem faced by statisticians is calculating the sample size required to yield a certain power for a test, given a predetermined Type I error rate α. As follows, this can be estimated by pre-determined tables for certain values, by Mead's resource equation, or, more generally, by the cumulative distribution function:
Tables
[4]
Power
Cohen's d
0.20.50.8
0.25 84146
0.50 1933213
0.60 2464016
0.70 3105020
0.80 3936426
0.90 5268534
0.95 65110542
0.99 92014858
The table shown on the right can be used in a two-sample t-test to estimate the sample sizes of an experimental group and a control group that are of equal size, that is, the total number of individuals in the trial is twice that of the number given, and the desired significance level is 0.05.[4] The parameters used are:
• The desired statistical power of the trial, shown in column to the left.
• Cohen's d (= effect size), which is the expected difference between the means of the target values between the experimental group and the control group, divided by the expected standard deviation.
Mead's resource equation
Mead's resource equation is often used for estimating sample sizes of laboratory animals, as well as in many other laboratory experiments. It may not be as accurate as using other methods in estimating sample size, but gives a hint of what is the appropriate sample size where parameters such as expected standard deviations or expected differences in values between groups are unknown or very hard to estimate.[5]
All the parameters in the equation are in fact the degrees of freedom of the number of their concepts, and hence, their numbers are subtracted by 1 before insertion into the equation.
The equation is:[5]
$E=N-B-T,$
where:
• N is the total number of individuals or units in the study (minus 1)
• B is the blocking component, representing environmental effects allowed for in the design (minus 1)
• T is the treatment component, corresponding to the number of treatment groups (including control group) being used, or the number of questions being asked (minus 1)
• E is the degrees of freedom of the error component and should be somewhere between 10 and 20.
For example, if a study using laboratory animals is planned with four treatment groups (T=3), with eight animals per group, making 32 animals total (N=31), without any further stratification (B=0), then E would equal 28, which is above the cutoff of 20, indicating that sample size may be a bit too large, and six animals per group might be more appropriate.[6]
Cumulative distribution function
Let Xi, i = 1, 2, ..., n be independent observations taken from a normal distribution with unknown mean μ and known variance σ2. Consider two hypotheses, a null hypothesis:
$H_{0}:\mu =0$
and an alternative hypothesis:
$H_{a}:\mu =\mu ^{*}$
for some 'smallest significant difference' μ* > 0. This is the smallest value for which we care about observing a difference. Now, if we wish to (1) reject H0 with a probability of at least 1 − β when Ha is true (i.e. a power of 1 − β), and (2) reject H0 with probability α when H0 is true, then we need the following:
If zα is the upper α percentage point of the standard normal distribution, then
$\Pr({\bar {x}}>z_{\alpha }\sigma /{\sqrt {n}}\mid H_{0})=\alpha $
and so
'Reject H0 if our sample average (${\bar {x}}$) is more than $z_{\alpha }\sigma /{\sqrt {n}}$'
is a decision rule which satisfies (2). (This is a 1-tailed test.)
Now we wish for this to happen with a probability at least 1 − β when Ha is true. In this case, our sample average will come from Normal distribution with mean μ*. Therefore, we require
$\Pr({\bar {x}}>z_{\alpha }\sigma /{\sqrt {n}}\mid H_{a})\geq 1-\beta $
Through careful manipulation, this can be shown (see Statistical power Example) to happen when
$n\geq \left({\frac {z_{\alpha }+\Phi ^{-1}(1-\beta )}{\mu ^{*}/\sigma }}\right)^{2}$
where $\Phi $ is the normal cumulative distribution function.
Stratified sample size
With more complicated sampling techniques, such as stratified sampling, the sample can often be split up into sub-samples. Typically, if there are H such sub-samples (from H different strata) then each of them will have a sample size nh, h = 1, 2, ..., H. These nh must conform to the rule that n1 + n2 + ... + nH = n (i.e., that the total sample size is given by the sum of the sub-sample sizes). Selecting these nh optimally can be done in various ways, using (for example) Neyman's optimal allocation.
There are many reasons to use stratified sampling:[7] to decrease variances of sample estimates, to use partly non-random methods, or to study strata individually. A useful, partly non-random method would be to sample individuals where easily accessible, but, where not, sample clusters to save travel costs.[8]
In general, for H strata, a weighted sample mean is
${\bar {x}}_{w}=\sum _{h=1}^{H}W_{h}{\bar {x}}_{h},$
with
$\operatorname {Var} ({\bar {x}}_{w})=\sum _{h=1}^{H}W_{h}^{2}\operatorname {Var} ({\bar {x}}_{h}).$[9]
The weights, $W_{h}$, frequently, but not always, represent the proportions of the population elements in the strata, and $W_{h}=N_{h}/N$. For a fixed sample size, that is $n=\sum n_{h}$,
$\operatorname {Var} ({\bar {x}}_{w})=\sum _{h=1}^{H}W_{h}^{2}\operatorname {Var} ({\bar {x}}_{h})\left({\frac {1}{n_{h}}}-{\frac {1}{N_{h}}}\right),$[10]
which can be made a minimum if the sampling rate within each stratum is made proportional to the standard deviation within each stratum: $n_{h}/N_{h}=kS_{h}$, where $S_{h}={\sqrt {\operatorname {Var} ({\bar {x}}_{h})}}$ and $k$ is a constant such that $\sum {n_{h}}=n$.
An "optimum allocation" is reached when the sampling rates within the strata are made directly proportional to the standard deviations within the strata and inversely proportional to the square root of the sampling cost per element within the strata, $C_{h}$:
${\frac {n_{h}}{N_{h}}}={\frac {KS_{h}}{\sqrt {C_{h}}}},$[11]
where $K$ is a constant such that $\sum {n_{h}}=n$, or, more generally, when
$n_{h}={\frac {K'W_{h}S_{h}}{\sqrt {C_{h}}}}.$[12]
Qualitative research
Sample size determination in qualitative studies takes a different approach. It is generally a subjective judgment, taken as the research proceeds.[13] One approach is to continue to include further participants or material until saturation is reached.[14] The number needed to reach saturation has been investigated empirically.[15][16][17][18]
There is a paucity of reliable guidance on estimating sample sizes before starting the research, with a range of suggestions given.[16][19][20][21] A tool akin to a quantitative power calculation, based on the negative binomial distribution, has been suggested for thematic analysis.[22][21]
See also
• Design of experiments
• Engineering response surface example under Stepwise regression
• Cohen's h
References
1. NIST/SEMATECH, "7.2.4.2. Sample sizes required", e-Handbook of Statistical Methods.
2. "Inference for Regression". utdallas.edu.
3. "Confidence Interval for a Proportion" Archived 2011-08-23 at the Wayback Machine
4. Chapter 13, page 215, in: Kenny, David A. (1987). Statistics for the social and behavioral sciences. Boston: Little, Brown. ISBN 978-0-316-48915-7.
5. Kirkwood, James; Robert Hubrecht (2010). The UFAW Handbook on the Care and Management of Laboratory and Other Research Animals. Wiley-Blackwell. p. 29. ISBN 978-1-4051-7523-4. online Page 29
6. Isogenic.info > Resource equation by Michael FW Festing. Updated Sept. 2006
7. Kish (1965, Section 3.1)
8. Kish (1965), p. 148.
9. Kish (1965), p. 78.
10. Kish (1965), p. 81.
11. Kish (1965), p. 93.
12. Kish (1965), p. 94.
13. Sandelowski, M. (1995). Sample size in qualitative research. Research in Nursing & Health, 18, 179–183
14. Glaser, B. (1965). The constant comparative method of qualitative analysis. Social Problems, 12, 436–445
15. Francis, Jill J.; Johnston, Marie; Robertson, Clare; Glidewell, Liz; Entwistle, Vikki; Eccles, Martin P.; Grimshaw, Jeremy M. (2010). "What is an adequate sample size? Operationalising data saturation for theory-based interview studies" (PDF). Psychology & Health. 25 (10): 1229–1245. doi:10.1080/08870440903194015. PMID 20204937. S2CID 28152749.
16. Guest, Greg; Bunce, Arwen; Johnson, Laura (2006). "How Many Interviews Are Enough?". Field Methods. 18: 59–82. doi:10.1177/1525822X05279903. S2CID 62237589.
17. Wright, Adam; Maloney, Francine L.; Feblowitz, Joshua C. (2011). "Clinician attitudes toward and use of electronic problem lists: A thematic analysis". BMC Medical Informatics and Decision Making. 11: 36. doi:10.1186/1472-6947-11-36. PMC 3120635. PMID 21612639.
18. Mason, Mark (2010). "Sample Size and Saturation in PhD Studies Using Qualitative Interviews". Forum Qualitative Sozialforschung. 11 (3): 8.
19. Emmel, N. (2013). Sampling and choosing cases in qualitative research: A realist approach. London: Sage.
20. Onwuegbuzie, Anthony J.; Leech, Nancy L. (2007). "A Call for Qualitative Power Analyses". Quality & Quantity. 41: 105–121. doi:10.1007/s11135-005-1098-1. S2CID 62179911.
21. Fugard AJB; Potts HWW (10 February 2015). "Supporting thinking on sample sizes for thematic analyses: A quantitative tool" (PDF). International Journal of Social Research Methodology. 18 (6): 669–684. doi:10.1080/13645579.2015.1005453. S2CID 59047474.
22. Galvin R (2015). How many interviews are enough? Do qualitative interviews in building energy consumption research produce reliable knowledge? Journal of Building Engineering, 1:2–12.
General references
• Bartlett, J. E., II; Kotrlik, J. W.; Higgins, C. (2001). "Organizational research: Determining appropriate sample size for survey research" (PDF). Information Technology, Learning, and Performance Journal. 19 (1): 43–50.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Kish, L. (1965). Survey Sampling. Wiley. ISBN 978-0-471-48900-9.
• Smith, Scott (8 April 2013). "Determining Sample Size: How to Ensure You Get the Correct Sample Size". Qualtrics. Retrieved 19 September 2018.
• Israel, Glenn D. (1992). "Determining Sample Size". University of Florida, PEOD-6. Retrieved 29 June 2019.
• Rens van de Schoot, Milica Miočević (eds.). 2020. Small Sample Size Solutions (Open Access): A Guide for Applied Researchers and Practitioners. Routledge.
Further reading
• NIST: Selecting Sample Sizes
• ASTM E122-07: Standard Practice for Calculating Sample Size to Estimate, With Specified Precision, the Average for a Characteristic of a Lot or Process
External links
• A MATLAB script implementing Cochran's sample size formula
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| Wikipedia |
Viterbi algorithm
The Viterbi algorithm is a dynamic programming algorithm for obtaining the maximum a posteriori probability estimate of the most likely sequence of hidden states—called the Viterbi path—that results in a sequence of observed events, especially in the context of Markov information sources and hidden Markov models (HMM).
The algorithm has found universal application in decoding the convolutional codes used in both CDMA and GSM digital cellular, dial-up modems, satellite, deep-space communications, and 802.11 wireless LANs. It is now also commonly used in speech recognition, speech synthesis, diarization,[1] keyword spotting, computational linguistics, and bioinformatics. For example, in speech-to-text (speech recognition), the acoustic signal is treated as the observed sequence of events, and a string of text is considered to be the "hidden cause" of the acoustic signal. The Viterbi algorithm finds the most likely string of text given the acoustic signal.
History
The Viterbi algorithm is named after Andrew Viterbi, who proposed it in 1967 as a decoding algorithm for convolutional codes over noisy digital communication links.[2] It has, however, a history of multiple invention, with at least seven independent discoveries, including those by Viterbi, Needleman and Wunsch, and Wagner and Fischer.[3] It was introduced to Natural Language Processing as a method of part-of-speech tagging as early as 1987.
Viterbi path and Viterbi algorithm have become standard terms for the application of dynamic programming algorithms to maximization problems involving probabilities.[3] For example, in statistical parsing a dynamic programming algorithm can be used to discover the single most likely context-free derivation (parse) of a string, which is commonly called the "Viterbi parse".[4][5][6] Another application is in target tracking, where the track is computed that assigns a maximum likelihood to a sequence of observations.[7]
Extensions
A generalization of the Viterbi algorithm, termed the max-sum algorithm (or max-product algorithm) can be used to find the most likely assignment of all or some subset of latent variables in a large number of graphical models, e.g. Bayesian networks, Markov random fields and conditional random fields. The latent variables need, in general, to be connected in a way somewhat similar to a hidden Markov model (HMM), with a limited number of connections between variables and some type of linear structure among the variables. The general algorithm involves message passing and is substantially similar to the belief propagation algorithm (which is the generalization of the forward-backward algorithm).
With the algorithm called iterative Viterbi decoding one can find the subsequence of an observation that matches best (on average) to a given hidden Markov model. This algorithm is proposed by Qi Wang et al. to deal with turbo code.[8] Iterative Viterbi decoding works by iteratively invoking a modified Viterbi algorithm, reestimating the score for a filler until convergence.
An alternative algorithm, the Lazy Viterbi algorithm, has been proposed.[9] For many applications of practical interest, under reasonable noise conditions, the lazy decoder (using Lazy Viterbi algorithm) is much faster than the original Viterbi decoder (using Viterbi algorithm). While the original Viterbi algorithm calculates every node in the trellis of possible outcomes, the Lazy Viterbi algorithm maintains a prioritized list of nodes to evaluate in order, and the number of calculations required is typically fewer (and never more) than the ordinary Viterbi algorithm for the same result. However, it is not so easy to parallelize in hardware.
Pseudocode
This algorithm generates a path $X=(x_{1},x_{2},\ldots ,x_{T})$, which is a sequence of states $x_{n}\in S=\{s_{1},s_{2},\dots ,s_{K}\}$ that generate the observations $Y=(y_{1},y_{2},\ldots ,y_{T})$ with $y_{n}\in O=\{o_{1},o_{2},\dots ,o_{N}\}$, where $N$ is the number of possible observations in the observation space $O$.
Two 2-dimensional tables of size $K\times T$ are constructed:
• Each element $T_{1}[i,j]$ of $T_{1}$ stores the probability of the most likely path so far ${\hat {X}}=({\hat {x}}_{1},{\hat {x}}_{2},\ldots ,{\hat {x}}_{j})$ with ${\hat {x}}_{j}=s_{i}$ that generates $Y=(y_{1},y_{2},\ldots ,y_{j})$.
• Each element $T_{2}[i,j]$ of $T_{2}$ stores ${\hat {x}}_{j-1}$ of the most likely path so far ${\hat {X}}=({\hat {x}}_{1},{\hat {x}}_{2},\ldots ,{\hat {x}}_{j-1},{\hat {x}}_{j}=s_{i})$ $\forall j,2\leq j\leq T$
The table entries $T_{1}[i,j],T_{2}[i,j]$ are filled by increasing order of $K\cdot j+i$:
$T_{1}[i,j]=\max _{k}{(T_{1}[k,j-1]\cdot A_{ki}\cdot B_{iy_{j}})}$,
$T_{2}[i,j]=\operatorname {argmax} _{k}{(T_{1}[k,j-1]\cdot A_{ki}\cdot B_{iy_{j}})}$,
with $A_{ki}$ and $B_{iy_{j}}$ as defined below. Note that $B_{iy_{j}}$ does not need to appear in the latter expression, as it's non-negative and independent of $k$ and thus does not affect the argmax.
Input
• The observation space $O=\{o_{1},o_{2},\dots ,o_{N}\}$,
• the state space $S=\{s_{1},s_{2},\dots ,s_{K}\}$,
• an array of initial probabilities $\Pi =(\pi _{1},\pi _{2},\dots ,\pi _{K})$ such that $\pi _{i}$ stores the probability that $x_{1}=s_{i}$,
• a sequence of observations $Y=(y_{1},y_{2},\ldots ,y_{T})$ such that $y_{t}=o_{i}$ if the observation at time $t$ is $o_{i}$,
• transition matrix $A$ of size $K\times K$ such that $A_{ij}$ stores the transition probability of transiting from state $s_{i}$ to state $s_{j}$,
• emission matrix $B$ of size $K\times N$ such that $B_{ij}$ stores the probability of observing $o_{j}$ from state $s_{i}$.
Output
• The most likely hidden state sequence $X=(x_{1},x_{2},\ldots ,x_{T})$
function VITERBI$(O,S,\Pi ,Y,A,B):X$
for each state $i=1,2,\ldots ,K$ do
$T_{1}[i,1]\leftarrow \pi _{i}\cdot B_{iy_{1}}$
$T_{2}[i,1]\leftarrow 0$
end for
for each observation $j=2,3,\ldots ,T$ do
for each state $i=1,2,\ldots ,K$ do
$T_{1}[i,j]\gets \max _{k}{(T_{1}[k,j-1]\cdot A_{ki}\cdot B_{iy_{j}})}$
$T_{2}[i,j]\gets \arg \max _{k}{(T_{1}[k,j-1]\cdot A_{ki}\cdot B_{iy_{j}})}$
end for
end for
$z_{T}\gets \arg \max _{k}{(T_{1}[k,T])}$
$x_{T}\leftarrow s_{z_{T}}$
for $j=T,T-1,\ldots ,2$ do
$z_{j-1}\leftarrow T_{2}[z_{j},j]$
$x_{j-1}\leftarrow s_{z_{j-1}}$
end for
return $X$
end function
Restated in a succinct near-Python:
function viterbi$(O,S,\Pi ,Tm,Em):best\_path$ Tm: transition matrix Em: emission matrix
$trellis\leftarrow matrix(length(S),length(O))$ To hold probability of each state given each observation
$pointers\leftarrow matrix(length(S),length(O))$ To hold backpointer to best prior state
for s in $range(length(S))$: Determine each hidden state's probability at time 0…
$trellis[s,0]\leftarrow \Pi [s]\cdot Em[s,O[0]]$
for o in $range(1,length(O))$: …and after, tracking each state's most likely prior state, k
for s in $range(length(S))$:
$k\leftarrow \arg \max(trellis[k,o-1]\cdot Tm[k,s]\cdot Em[s,o]\ {\mathsf {for}}\ k\ {\mathsf {in}}\ range(length(S)))$
$trellis[s,o]\leftarrow trellis[k,o-1]\cdot Tm[k,s]\cdot Em[s,o]$
$pointers[s,o]\leftarrow k$
$best\_path\leftarrow list()$
$k\leftarrow \arg \max(trellis[k,length(O)-1]\ {\mathsf {for}}\ k\ {\mathsf {in}}\ range(length(S)))$ Find k of best final state
for o in $range(length(O)-1,-1,-1)$: Backtrack from last observation
$best\_path.insert(0,S[k])$ Insert previous state on most likely path
$k\leftarrow pointers[k,o]$ Use backpointer to find best previous state
return $best\_path$
Explanation
Suppose we are given a hidden Markov model (HMM) with state space $S$, initial probabilities $\pi _{i}$ of being in the hidden state $i$ and transition probabilities $a_{i,j}$ of transitioning from state $i$ to state $j$. Say, we observe outputs $y_{1},\dots ,y_{T}$. The most likely state sequence $x_{1},\dots ,x_{T}$ that produces the observations is given by the recurrence relations[10]
${\begin{aligned}V_{1,k}&=\mathrm {P} {\big (}y_{1}\ |\pi _{k}{\big )}\cdot \pi _{k},\\V_{t,k}&=\max _{x\in S}\left(\mathrm {P} {\big (}y_{t}\ |\pi _{k}{\big )}\cdot a_{x,k}\cdot V_{t-1,x}\right).\end{aligned}}$
Here $V_{t,k}$ is the probability of the most probable state sequence $\mathrm {P} {\big (}x_{1},\dots ,x_{t},y_{1},\dots ,y_{t}{\big )}$ responsible for the first $t$ observations that have $k$ as its final state. The Viterbi path can be retrieved by saving back pointers that remember which state $x$ was used in the second equation. Let $\mathrm {Ptr} (k,t)$ be the function that returns the value of $x$ used to compute $V_{t,k}$ if $t>1$, or $k$ if $t=1$. Then
${\begin{aligned}x_{T}&=\arg \max _{x\in S}(V_{T,x}),\\x_{t-1}&=\mathrm {Ptr} (x_{t},t).\end{aligned}}$
Here we're using the standard definition of arg max.
The complexity of this implementation is $O(T\times \left|{S}\right|^{2})$. A better estimation exists if the maximum in the internal loop is instead found by iterating only over states that directly link to the current state (i.e. there is an edge from $k$ to $j$). Then using amortized analysis one can show that the complexity is $O(T\times (\left|{S}\right|+\left|{E}\right|))$, where $E$ is the number of edges in the graph.
Example
Consider a village where all villagers are either healthy or have a fever, and only the village doctor can determine whether each has a fever. The doctor diagnoses fever by asking patients how they feel. The villagers may only answer that they feel normal, dizzy, or cold.
The doctor believes that the health condition of the patients operates as a discrete Markov chain. There are two states, "Healthy" and "Fever", but the doctor cannot observe them directly; they are hidden from the doctor. On each day, there is a certain chance that a patient will tell the doctor "I feel normal", "I feel cold", or "I feel dizzy", depending on the patient's health condition.
The observations (normal, cold, dizzy) along with a hidden state (healthy, fever) form a hidden Markov model (HMM), and can be represented as follows in the Python programming language:
obs = ("normal", "cold", "dizzy")
states = ("Healthy", "Fever")
start_p = {"Healthy": 0.6, "Fever": 0.4}
trans_p = {
"Healthy": {"Healthy": 0.7, "Fever": 0.3},
"Fever": {"Healthy": 0.4, "Fever": 0.6},
}
emit_p = {
"Healthy": {"normal": 0.5, "cold": 0.4, "dizzy": 0.1},
"Fever": {"normal": 0.1, "cold": 0.3, "dizzy": 0.6},
}
In this piece of code, start_p represents the doctor's belief about which state the HMM is in when the patient first visits (all the doctor knows is that the patient tends to be healthy). The particular probability distribution used here is not the equilibrium one, which is (given the transition probabilities) approximately {'Healthy': 0.57, 'Fever': 0.43}. The transition_p represents the change of the health condition in the underlying Markov chain. In this example, a patient who is healthy today has only a 30% chance of having a fever tomorrow. The emit_p represents how likely each possible observation (normal, cold, or dizzy) is, given the underlying condition (healthy or fever). A patient who is healthy has a 50% chance of feeling normal; one who has a fever has a 60% chance of feeling dizzy.
A patient visits three days in a row, and the doctor discovers that the patient feels normal on the first day, cold on the second day, and dizzy on the third day. The doctor has a question: what is the most likely sequence of health conditions of the patient that would explain these observations? This is answered by the Viterbi algorithm.
def viterbi(obs, states, start_p, trans_p, emit_p):
V = [{}]
for st in states:
V[0] [st] = {"prob": start_p[st] * emit_p[st] [obs[0]], "prev": None}
# Run Viterbi when t > 0
for t in range(1, len(obs)):
V.append({})
for st in states:
max_tr_prob = V[t - 1] [states[0]] ["prob"] * trans_p[states[0]] [st] * emit_p[st] [obs[t]]
prev_st_selected = states[0]
for prev_st in states[1:]:
tr_prob = V[t - 1] [prev_st] ["prob"] * trans_p[prev_st] [st] * emit_p[st] [obs[t]]
if tr_prob > max_tr_prob:
max_tr_prob = tr_prob
prev_st_selected = prev_st
max_prob = max_tr_prob
V[t] [st] = {"prob": max_prob, "prev": prev_st_selected}
for line in dptable(V):
print(line)
opt = []
max_prob = 0.0
best_st = None
# Get most probable state and its backtrack
for st, data in V[-1].items():
if data["prob"] > max_prob:
max_prob = data["prob"]
best_st = st
opt.append(best_st)
previous = best_st
# Follow the backtrack till the first observation
for t in range(len(V) - 2, -1, -1):
opt.insert(0, V[t + 1] [previous] ["prev"])
previous = V[t + 1] [previous] ["prev"]
print ("The steps of states are " + " ".join(opt) + " with highest probability of %s" % max_prob)
def dptable(V):
# Print a table of steps from dictionary
yield " " * 5 + " ".join(("%3d" % i) for i in range(len(V)))
for state in V[0]:
yield "%.7s: " % state + " ".join("%.7s" % ("%lf" % v[state] ["prob"]) for v in V)
The function viterbi takes the following arguments: obs is the sequence of observations, e.g. ['normal', 'cold', 'dizzy']; states is the set of hidden states; start_p is the start probability; trans_p are the transition probabilities; and emit_p are the emission probabilities. For simplicity of code, we assume that the observation sequence obs is non-empty and that trans_p[i] [j] and emit_p[i] [j] is defined for all states i,j.
In the running example, the forward/Viterbi algorithm is used as follows:
viterbi(obs,
states,
start_p,
trans_p,
emit_p)
The output of the script is
$ python viterbi_example.py
0 1 2
Healthy: 0.30000 0.08400 0.00588
Fever: 0.04000 0.02700 0.01512
The steps of states are Healthy Healthy Fever with highest probability of 0.01512
This reveals that the observations ['normal', 'cold', 'dizzy'] were most likely generated by states ['Healthy', 'Healthy', 'Fever']. In other words, given the observed activities, the patient was most likely to have been healthy on the first day and also on the second day (despite feeling cold that day), and only to have contracted a fever on the third day.
The operation of Viterbi's algorithm can be visualized by means of a trellis diagram. The Viterbi path is essentially the shortest path through this trellis.
Soft output Viterbi algorithm
The soft output Viterbi algorithm (SOVA) is a variant of the classical Viterbi algorithm.
SOVA differs from the classical Viterbi algorithm in that it uses a modified path metric which takes into account the a priori probabilities of the input symbols, and produces a soft output indicating the reliability of the decision.
The first step in the SOVA is the selection of the survivor path, passing through one unique node at each time instant, t. Since each node has 2 branches converging at it (with one branch being chosen to form the Survivor Path, and the other being discarded), the difference in the branch metrics (or cost) between the chosen and discarded branches indicate the amount of error in the choice.
This cost is accumulated over the entire sliding window (usually equals at least five constraint lengths), to indicate the soft output measure of reliability of the hard bit decision of the Viterbi algorithm.
See also
• Expectation–maximization algorithm
• Baum–Welch algorithm
• Forward-backward algorithm
• Forward algorithm
• Error-correcting code
• Viterbi decoder
• Hidden Markov model
• Part-of-speech tagging
• A* search algorithm
References
1. Xavier Anguera et al., "Speaker Diarization: A Review of Recent Research", retrieved 19. August 2010, IEEE TASLP
2. 29 Apr 2005, G. David Forney Jr: The Viterbi Algorithm: A Personal History
3. Daniel Jurafsky; James H. Martin. Speech and Language Processing. Pearson Education International. p. 246.
4. Schmid, Helmut (2004). Efficient parsing of highly ambiguous context-free grammars with bit vectors (PDF). Proc. 20th Int'l Conf. on Computational Linguistics (COLING). doi:10.3115/1220355.1220379.
5. Klein, Dan; Manning, Christopher D. (2003). A* parsing: fast exact Viterbi parse selection (PDF). Proc. 2003 Conf. of the North American Chapter of the Association for Computational Linguistics on Human Language Technology (NAACL). pp. 40–47. doi:10.3115/1073445.1073461.
6. Stanke, M.; Keller, O.; Gunduz, I.; Hayes, A.; Waack, S.; Morgenstern, B. (2006). "AUGUSTUS: Ab initio prediction of alternative transcripts". Nucleic Acids Research. 34 (Web Server issue): W435–W439. doi:10.1093/nar/gkl200. PMC 1538822. PMID 16845043.
7. Quach, T.; Farooq, M. (1994). "Maximum Likelihood Track Formation with the Viterbi Algorithm". Proceedings of 33rd IEEE Conference on Decision and Control. Vol. 1. pp. 271–276. doi:10.1109/CDC.1994.410918.{{cite conference}}: CS1 maint: multiple names: authors list (link)
8. Qi Wang; Lei Wei; Rodney A. Kennedy (2002). "Iterative Viterbi Decoding, Trellis Shaping, and Multilevel Structure for High-Rate Parity-Concatenated TCM". IEEE Transactions on Communications. 50: 48–55. doi:10.1109/26.975743.
9. A fast maximum-likelihood decoder for convolutional codes (PDF). Vehicular Technology Conference. December 2002. pp. 371–375. doi:10.1109/VETECF.2002.1040367.
10. Xing E, slide 11.
General references
• Viterbi AJ (April 1967). "Error bounds for convolutional codes and an asymptotically optimum decoding algorithm". IEEE Transactions on Information Theory. 13 (2): 260–269. doi:10.1109/TIT.1967.1054010. (note: the Viterbi decoding algorithm is described in section IV.) Subscription required.
• Feldman J, Abou-Faycal I, Frigo M (2002). "A fast maximum-likelihood decoder for convolutional codes". Proceedings IEEE 56th Vehicular Technology Conference. Vol. 1. pp. 371–375. CiteSeerX 10.1.1.114.1314. doi:10.1109/VETECF.2002.1040367. ISBN 978-0-7803-7467-6. S2CID 9783963.
• Forney GD (March 1973). "The Viterbi algorithm". Proceedings of the IEEE. 61 (3): 268–278. doi:10.1109/PROC.1973.9030. Subscription required.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 16.2. Viterbi Decoding". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
• Rabiner LR (February 1989). "A tutorial on hidden Markov models and selected applications in speech recognition". Proceedings of the IEEE. 77 (2): 257–286. CiteSeerX 10.1.1.381.3454. doi:10.1109/5.18626. S2CID 13618539. (Describes the forward algorithm and Viterbi algorithm for HMMs).
• Shinghal, R. and Godfried T. Toussaint, "Experiments in text recognition with the modified Viterbi algorithm," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-l, April 1979, pp. 184–193.
• Shinghal, R. and Godfried T. Toussaint, "The sensitivity of the modified Viterbi algorithm to the source statistics," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-2, March 1980, pp. 181–185.
External links
• Implementations in Java, F#, Clojure, C# on Wikibooks
• Tutorial on convolutional coding with viterbi decoding, by Chip Fleming
• A tutorial for a Hidden Markov Model toolkit (implemented in C) that contains a description of the Viterbi algorithm
• Viterbi algorithm by Dr. Andrew J. Viterbi (scholarpedia.org).
Implementations
• Mathematica has an implementation as part of its support for stochastic processes
• Susa signal processing framework provides the C++ implementation for Forward error correction codes and channel equalization here.
• C++
• C#
• Java
• Java 8
• Julia (HMMBase.jl)
• Perl
• Prolog
• Haskell
• Go
• SFIHMM includes code for Viterbi decoding.
| Wikipedia |
Charlotte Barnum
Charlotte Cynthia Barnum (May 17, 1860 – March 27, 1934), mathematician and social activist, was the first woman to receive a Ph.D. in mathematics from Yale University.[1]
Charlotte Barnum
Born(1860-05-17)May 17, 1860
Phillipston, Massachusetts
DiedMarch 27, 1934(1934-03-27) (aged 73)
Middletown, Connecticut
NationalityAmerican
Alma materVassar College
Yale University
Scientific career
Institutions
• Carleton College
• U.S. Naval Observatory
• U.S. Coast and Geodetic Survey
• U.S. Department of Agriculture as an editor for the Biological Survey
ThesisFunctions Having Lines or Surfaces of Discontinuity (1895)
Doctoral advisorUnknown
Early life and education
Charlotte Barnum was born in Phillipston, Massachusetts, the third of four children of the Reverend Samuel Weed Barnum (1820–1891) and Charlotte Betts (1823–1899). Education was important in her family: two uncles had received medical degrees from Yale and her father had graduated from there with a Bachelor of Arts and a Bachelor of Divinity. Her brothers Samuel and Thomas would both graduate from Yale, and her sister Clara would attend Yale graduate school after graduating from Vassar.[2]
After graduating from Hillhouse High School in New Haven, Connecticut Charlotte attended Vassar College, where she graduated in 1881. From 1881 to 1886 she taught at a boys’ preparatory school, Betts Academy, in Stamford, Connecticut and at Hillhouse High School. She also did computing work for the Yale Observatory 1883–1885 and worked on a revision of James Dwight Dana’s System of Mineralogy. Charlotte was an editorial writer for Webster's International Dictionary from 1886 to 1890, and then taught astronomy at Smith College for the academic year 1889–90.
In 1890 Charlotte applied for graduate studies at Johns Hopkins University, but was turned down because they did not accept women. She persisted and with the support of Simon Newcomb, professor of mathematics and astronomy at the university, she won approval to attend lectures without enrollment and without charge. Two years later, she moved to New Haven to pursue her graduate studies at Yale. In 1895 she was the first woman to receive a Ph.D. in mathematics from that institution. Her thesis was titled "Functions Having Lines or Surfaces of Discontinuity". The identity of her adviser is unclear from the record.[2][3]
Later career
After receiving her Ph.D., Charlotte Barnum taught at Carleton College in Northfield, Minnesota for one year. She then left academia, and did civilian and governmental applied mathematics and editorial work the remainder of her career.
In 1898 she joined the American Academy of Actuaries and until 1901 worked as an actuarial computer for the Massachusetts Mutual Life Insurance Company, Springfield, Massachusetts and the Fidelity Mutual Life Insurance Company in Philadelphia, Pennsylvania.
In 1901 she moved to Washington D.C. to work as a computer for US Naval Observatory. She subsequently did the same work for the tidal division of the US Coast and Geodetic Survey until 1908 and then was editorial assistant in the biological survey section of the US Department of Agriculture through 1913.
She left government employment and returned to New Haven in 1914 where she did editorial work for Yale Peruvian Expeditions, the Yale University secretary's office, and the Yale University Press.
Starting in 1917 she worked in various organizations and academic institutions in Connecticut, New York and Massachusetts as an editor, actuary and teacher. All her life she was involved in social and charitable organizations and activities. In 1934 she died in Middletown, Connecticut of meningitis at the age of seventy-three.[2][3][4][5]
Memberships
One of the first women members of the American Mathematical Society[6]
Fellow, American Academy of Actuaries (AAAS)[2]
Fellow, American Association for the Advancement of Science[4]
Alumnae Member, Vassar College chapter of Phi Beta Kappa[4]
Women's Joint Legislative Commission (for equal rights)[2]
National Conference of Charities (now the National Conference on Social Welfare)[2]
Publications
1911: “The Girl Who Lives at Home: Two Suggestions to Trade Union Women,” (Life and Labor, Volume 1, 1911) p. 346.[2]
References
1. Green, Judy; LaDuke, Jeanne (2009). Pioneering Women in American Mathematics — The Pre-1940 PhD's. History of Mathematics. Vol. 34. American Mathematical Society, The London Mathematical Society. pp. 136–137. ISBN 978-0-8218-4376-5.
2. Judy Green and Jeanne LaDuke, “Supplementary Material for Pioneering Women in American Mathematics: The Pre-1940 PhD’s,” 473–477 http://www.ams.org/publications/authors/books/postpub/hmath-34-PioneeringWomen.pdf.
3. Riddle, Larry (2010), "Charlotte Barnum", Biographies of Women Mathematicians, Agnes Scott College, retrieved 2012-09-30
4. “Charlotte Cynthia Barnum, Ph.D., 1895.” (Yale Obituary Record 1933 – 1934) pp. 204–205. http://digital.library.yale.edu/cdm/compoundobject/collection/1004_8/id/12679/rec/4
5. Leonard, John William, ed. (1914), Woman's Who's Who of America: A Biographical Dictionary of Contemporary Women of the United States and Canada, 1914–1915, New York: American Commonwealth Company, p. 77.
6. “Members of the Society,” (American Mathematical Society Annual Register, Published by the Society, New York, January 1900) p. 10: https://play.google.com/books/reader?id=olI_AQAAMAAJ&printsec=frontcover&output=reader&hl=en
External links
• Charlotte and Clara Barnum Papers, Vassar College Archives and Special Collections Library
• Biography Center
• Vassar College Mathematics Department History
• Charlotte Barnum at Find a Grave
• Shelby L. Eaton: “Women in Mathematics in the United States: 1866–1900,” August 21, 1997
• Charlotte Barnum Purported image of Charlotte Barnum
• Mathematics Genealogy Project
Authority control: Academics
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Lara Alcock
Lara Alcock is a British mathematics educator. She is a reader in mathematics education at Loughborough University, head of the Mathematics Education Centre at Loughborough, and the author of several books on mathematics.[1][2] Alcock won the Selden Prize for her research in mathematics education[3] and the inaugural John Blake University Teaching Medal in 2021.[4] Alcock is a National Teaching Fellow.[5]
Lara Alcock
Alma materUniversity of Warwick (BSc, MSc, PhD)
AwardsNational Teaching Fellowship (2015)
Scientific career
FieldsMathematical thinking
Proof comprehension
Mathematical reading[1]
InstitutionsLoughborough University
Rutgers University
Essex University
ThesisCategories, definitions and mathematics : student reasoning about objects in analysis (2001)
Doctoral advisorAdrian Simpson
Websitelaraalcock.com
Education
Alcock earned bachelor's and master's degrees in mathematics at the University of Warwick, and in 2001 completed a PhD in mathematics education at Warwick.[2][3] Her PhD research on Categories, definitions and mathematics: Student reasoning about objects in analysis, was supervised by Adrian Simpson.[6]
Career and research
After working as an assistant professor at Rutgers University in New Jersey, she returned to the UK as a teaching fellow at Essex University. She moved to Loughborough in 2007.[2][7][8][9][3]
Publications
Alcock's publications[1] include:
• Ideas from Mathematics Education: An Introduction for Mathematicians (with Adrian Simpson, Higher Education Academy, 2009)
• How to Study for a Mathematics Degree / How to Study as a Mathematics Major (UK/US; Oxford University Press, 2013); Wie man erfolgreich Mathematik studiert (translated into German by Bernhard Gerl, Springer Spektrum, 2017)[10][11][12][13]
• How to Think about Analysis (Oxford University Press, 2014)[14]
• Mathematics Rebooted: A Fresh Approach to Understanding (Oxford University Press, 2017)[7]
Awards and honours
Alcock is the 2012 winner of the Annie and John Selden Prize for research in undergraduate mathematics education, given by the Mathematical Association of America.[3] She was named a National Teaching Fellow by the Higher Education Academy in 2015.[5]
Alcock was the inaugural winner of the Institute of Mathematics and its Applications John Blake University Teaching Medal in 2021 [4]
References
1. Lara Alcock publications indexed by Google Scholar
2. Centre Staff: Dr Lara Alcock, Loughborough University Mathematics Education Centre, retrieved 2018-09-12
3. 2012 Selden Prize Winner, Mathematical Association of America, retrieved 2018-09-12
4. "Dr Lara Alcock wins the inaugural IMA John Blake University Teaching Medal". Institute of Mathematics & its Applications. Retrieved 8 August 2021.
5. Grove, Jack (11 June 2015), "National Teaching Fellows of 2015 are named: Fifty-five people working in universities have been named as the latest winners of the sector's top honour for teaching and learning", Times Higher Education
6. Alcock, Lara (2001). Categories, definitions and mathematics : student reasoning about objects in analysis (PhD thesis). University of Warwick. OCLC 921054748. EThOS uk.bl.ethos.246820.
7. Reviews of Mathematics Rebooted: Bultheel, Adhemar (February 2018), Review, European Mathematical Society
8. Stenger, Allen (April 2018), "Review", MAA Reviews
9. Grove, Michael (January 2019), "Review" (PDF), Newsletter of the London Mathematical Society, 480: 39–40
10. Foster, Colin (July 2014), The Mathematical Gazette, 98 (542): 377–378, doi:10.1017/s0025557200001625{{citation}}: CS1 maint: untitled periodical (link)
11. Storr, Graham (2014), The Mathematical Gazette, 98 (543): 547, doi:10.1017/s0025557200008457, S2CID 184177892{{citation}}: CS1 maint: untitled periodical (link)
12. Selden, Annie (September 2013), "Review", MAA Reviews
13. Panse, Anja (2018), Mathematische Semesterberichte, 65 (1): 133–135, doi:10.1007/s00591-018-0219-6, S2CID 125487245{{citation}}: CS1 maint: untitled periodical (link)
14. Huddy, Stanley R. (January 2015), "Review of How to Think about Analysis", MAA Reviews
Authority control
International
• ISNI
• VIAF
National
• Germany
• Israel
• United States
• Czech Republic
Academics
• Google Scholar
• zbMATH
Other
• IdRef
| Wikipedia |
\begin{document}
\date{\today}
\title{Baire theorem and hypercyclic algebras}
\author[F. Bayart]{Fr\'ed\'eric Bayart} \email{[email protected]} \author[F. Costa J\'unior]{Fernando Costa J\'unior} \email{Fernando.Vieira\underline{ }Costa\underline{ }[email protected],} \author[D. Papathanasiou]{Dimitris Papathanasiou} \email{[email protected]} \thanks{The first and the second author were partially supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front)} \address{Laboratoire de Mathématiques Blaise Pascal UMR 6620 CNRS, Université Clermont Auvergne, Campus universitaire des Cézeaux, 3 place Vasarely, 63178 Aubière Cedex, France.}
\subjclass{47A16}
\keywords{hypercyclic operators, weighted shifts, convolution operators, hypercyclic algebras}
\begin{abstract} The question of whether a hypercyclic operator $T$ acting on a Fréchet algebra $X$ admits or not an algebra of hypercyclic vectors (but 0) has been addressed in the recent literature. In this paper we give new criteria and characterizations in the context of convolution operators acting on $H(\mathbb C)$ and backward shifts acting on a general Fréchet sequence algebra. Analogous questions arise for stronger properties like frequent hypercyclicity. In this trend we give a sufficient condition for a weighted backward shift to admit an upper frequently hypercyclic algebra and we find a weighted backward shift acting on $c_0$ admitting a frequently hypercyclic algebra for the coordinatewise product. The closed hypercyclic algebra problem is also covered. \end{abstract}
\maketitle
\section{Introduction}
Among the many problems in linear dynamics, understanding the structure of the set of hypercyclic vectors is a major one. Let us introduce the relevant definitions. Let $(X,T)$ be a linear dynamical system, namely $X$ is a topological vector space and $T$ is a bounded linear operator on $X$. A vector $x\in X$ with dense orbit under $T$ is called a hypercyclic vector, and we denote by $HC(T)$ the set of hypercyclic vectors for $T$: $$ HC(T)=\{x\in X: \{x, Tx, T^2x,\dots \} \,\, \mbox{is dense in} \,\, X\}. $$ This set $HC(T)$ possesses interesting properties. When $X$ is a Baire space, its nonemptyness implies its residuality, preventing it from being a non-trivial proper linear subspace of $X$. However, it is well known that, whenever $HC(T)$ is nonempty, then $HC(T)\cup \{0\}$ contains a dense linear manifold (see \cite{bourdon}). In many cases (not always) $HC(T)\cup \{0\}$ contains a closed, infinite dimensional linear subspace (see \cite{GoRo, peterson,shkarin, menet}). These properties reflect that we are working in a linear space.
Suppose now that $X$ has a richer structure: it is an $F$-algebra, namely a metrizable and complete topological algebra. It is natural to ask whether $HC(T)\cup\{0\}$ also contains a non-trivial subalgebra of $X$. Such an algebra will be called a \emph{hypercyclic algebra}. The pioneering work in that direction has been done independently by Shkarin in \cite{shkarin} and by Bayart and Matheron in \cite{BM09}: they showed that the derivation operator $D:f\mapsto f'$, acting on the Fr\'echet algebra $H(\mathbb C)$ of entire functions endowed with the pointwise multiplication, supports a hypercyclic algebra. However, this is not the case for all hypercyclic operators acting on an $F$-algebra: for instance, as pointed out in \cite{ACPS07}, the translation operators, acting on $H(\mathbb C)$, do not support a hypercyclic algebra. Recent papers (see e.g. \cite{Bayhcalg,bes1,BCP18,BP18,FalGre18,FalGre19}) give other examples of operators admitting a hypercyclic algebra.
Our aim, in this paper, is to shed new light on this problem and to study how it interacts with popular problems arising in linear dynamics. We are particularly interested in two questions.
\subsection{Existence of hypercyclic algebras}
All examples in the literature of operators supporting a hypercyclic algebra are generalizations of $D$. There are several ways to extend it. You may see $D$ as a special convolution operator acting on $H(\mathbb C)$. By \cite{GoSh91}, such an operator may be written $\phi(D)$, where $\phi$ is an entire function with exponential type; if $\phi$ is not constant, then $\phi(D)$ is hypercyclic. When $|\phi(0)|<1$, the existence of hypercyclic algebras is well-understood since \cite{Bayhcalg}: such an algebra does exist if and only if $\phi$ is not a multiple of an exponential function. When $|\phi(0)|=1$, sufficient conditions are given in \cite{Bayhcalg} or in \cite{BesErnstPrieto}
but almost nothing, except a very specific example, is known when $|\phi(0)|>1$. We partly fill this gap by proving the existence of a hypercyclic algebra when $\phi$ goes to zero along some half-line.
\begin{theorem}\label{thm:convintro} Let $\phi$ be a nonconstant entire function with exponential type, not a multiple of an exponential function.
Assume that $|\phi(0)|>1$ and that there exists some $w\in\mathbb C$ such that $|\phi(tw)|\to 0$ as $t\to+\infty$. Then $\phi(D)$ supports a hypercyclic algebra. \end{theorem} In particular, we shall see that if $\phi(z)=P(z)e^z$ for some non-constant polynomial $P$, then $\phi(D)$ supports a hypercyclic algebra.
Another way to generalize $D$ is to see it as a weighted backward shift acting on $H(\mathbb C)$ considered as a sequence space. This was explored in \cite{FalGre18}. The general context is that of a Fr\'echet sequence algebra $X$. Precisely we assume that $X$ is a subspace of the space $\omega=\mathbb C^{\mathbb N_0}$ of all complex sequences, whose topology is induced by a
non-decreasing sequence of seminorms $(\|\cdot\|_q)_{q\geq 1}$ and that $X$ is endowed with a product $\cdot$ such that, for all $x,y\in X$, all $q\geq 1$,
$$\|x\cdot y\|_q\leq \|x\|_q\times \|y\|_q.$$ There are two natural products on a Fr\'echet sequence space: the coordinatewise product and the convolution or Cauchy product. It is clear that $\ell_p$ and $c_0$ are Fréchet sequence algebras for the coordinatewise product, and that $\ell_1$ is also a Fréchet sequence algebra for the convolution product. Endowing $H(\mathbb C)$ with
\[ \left\| \sum_{n\geq 0}a_n z^n \right\|_q =\sum_{n\geq 0}|a_n| q^n \] and $\omega$ with
\[ \left\| (x_n) \right\|_q=\sum_{n=0}^q |x_n|,\]
we also obtain that $H(\mathbb C)$ and $\omega$ are Fréchet sequence algebras for both products (on $H(\mathbb C)$, the Cauchy product of $f$ and $g$ is nothing else but the product of the two functions $f$ and $g$). Another interesting source of examples for us will be the sequence spaces $X=\{(x_n)\in\omega: \gamma_n x_n\to 0\}$ endowed with $\|x\|=\sup_n \gamma_n |x_n|$, where $(\gamma_n)\in\mathbb R_+^{\mathbb N_0}$. Provided $\gamma_n\geq 1$ for all $n$, $X$ is a Fr\'echet sequence algebra for the coordinatewise product.
Given a sequence of nonzero complex numbers $w=(w_n)_{n\in\mathbb N}$, the (unilateral) weighted backward shift $B_w$ with weight $w$ is defined by $$B_w(x_0,x_1,\dots)=(w_1x_1,w_2x_2,\dots).$$ The weight $w$ will be called \emph{admissible} (for $X$) if $B_w$ is a bounded operator on $X$. It is known that, provided the canonical basis $(e_n)$ is a Schauder basis of $X$, $B_w$ is hypercyclic if and only if there exists a sequence $(n_k)$ such that for all $l\in\mathbb N$, $\big((w_{l+1}\cdots w_{n_k+l})^{-1}e_{n_k+l}\big)$ goes to zero.
Let us first assume that $X$ is a Fr\'echet algebra under the coordinatewise product. Under a supplementary technical condition on $X$, a sufficient condition on $w$ is given in \cite{FalGre18}
so that $B_w$ supports a hypercyclic algebra. It turns out that we shall give a very natural characterization of this property when $X$ admits a continous norm. We recall that a Fr\'echet space $(X,(\|\cdot\|_q))$ admits a continuous norm if there exists a norm $\|\cdot\|:X\to\mathbb R$ that is continuous for the topology of $X$, namely there exists $q\in\mathbb N$ and $C>0$ with $\|x\|\leq C\|x\|_q$ for all $x\in X$. In particular, for any $q$ large enough, $\|\cdot\|_q$ itself is a norm.
\begin{theorem}\label{thm:mainws} Let $X$ be a Fr\'echet sequence algebra for the coordinatewise product and with a continuous norm. Assume that $(e_n)$ is a Schauder basis for $X$. Let also $B_w$ be a bounded weighted shift on $X$. The following assumptions are equivalent. \begin{enumerate}[(i)] \item $B_w$ supports a dense and not finitely generated hypercyclic algebra. \item There exists a sequence of integers $(n_k)$ such that for all $\gamma>0$, for all $l\in\mathbb N$, $\big((w_{l+1}\cdots w_{n_k+l})^{-\gamma}e_{n_k+l}\big)$ tends to zero. \end{enumerate} \end{theorem}
In particular, this theorem implies that on $\ell_p$ or $c_0$, any hypercyclic weighted shift supports a hypercyclic algebra.
When $X$ is a Fr\'echet algebra for the Cauchy product, it is shown in \cite{FalGre18} that, under additional technical assumptions on $X$, $B_w$ supports a hypercyclic algebra as soon as it is mixing, namely as soon as $(w_1\cdots w_n)^{-1}e_n$ tends to zero. We shall improve that theorem by showing that any hypercyclic backward shift on a Fréchet sequence algebra for the Cauchy product supports a hypercyclic algebra. We will only require a supplementary assumption on $X$ (to be regular) which is less strong than the assumption required in \cite{FalGre18}.
\begin{theorem}\label{thm:wscauchy}
Let $X$ be a regular Fréchet sequence algebra for the Cauchy product and let $B_w$ be a bounded weighted shift on $X$. The following assertions are equivalent.
\begin{enumerate}[(i)]
\item $B_w$ is hypercyclic.
\item $B_w$ supports a dense and not finitely generated hypercyclic algebra.
\end{enumerate} \end{theorem}
In particular, if we compare this statement with Theorem \ref{thm:mainws}, we see that, for the convolution product, we do not need extra assumptions on the weight, which was not the case for the coordinatewise product. Even if the proofs of Theorems \ref{thm:mainws}, \ref{thm:wscauchy} share some similarities (the latter one being much more difficult), they also have strong differences, the main one being that, under the coordinatewise product, any power of $x\in\omega$ keeps the same support, which is far from being the case if we work with the Cauchy product.
We also point out that this detailed study of the existence of hypercyclic algebras for weighted shifts has interesting applications. For instance, working with a bilateral shift, it will allow us to exhibit an invertible operator on a Banach algebra supporting a hypercyclic algebra and such that its inverse does not (see Example \ref{ex:inverse}).
\subsection{Frequently and upper frequently hypercyclic algebras}
Another fruitful subject in linear dynamics is frequent and upper frequent hypercyclicity. We say that $T$ is frequently hypercyclic (resp. upper frequently hypercyclic) if there exists a vector $x\in X$ such that, for all $U\subset X$ open and non-empty, the set $\{n\in\mathbb N: T^n x\in U\}$ has positive lower density (resp. positive upper density). Again, linearity allows to give a nice criterion to prove that an operator is (upper) frequently hypercyclic and gives rise to nice examples. For instance, if $X$ is a Fr\'echet sequence space and $B_w$ is a bounded weighted shift acting on $X$, it is known that the unconditional convergence of $\sum_{n\geq 1}(w_1\cdots w_n)^{-1}e_n$ implies that $B_w$ is frequently hypercyclic. Moreover, in some spaces (for instance, on $\ell_p$-spaces), this condition is even necessary for the upper frequent hypercyclicity of $B_w$.
Of course, it is natural to ask if a (upper) frequently hypercyclic operator defined on an $F$-algebra $X$ admits a (upper) frequently hypercyclic algebra, namely an algebra consisting only, except 0, of (upper) frequently hypercyclic vectors. Falc\'o and Grosse-Erdmann have shown recently (\cite{FalGre19}) that this is not always the case: for instance, $\lambda B$, $\lambda>1$, acting on any $\ell_p$ space ($1\leq p<+\infty$) or on $c_0$, endowed with the coordinatewise product, does not admit a frequently hypercyclic algebra. Nevertheless, this leaves open the possibility for $\lambda B$, $\lambda>1$, to admit an upper frequently hypercyclic algebra.
We shall give two general results implying that a weighted shift on a Fréchet sequence algebra admits an upper frequently hypercyclic algebra. The first one deals with Fréchet sequence algebras endowed with the coordinatewise product. In view of Theorem \ref{thm:mainws}, the natural extension of the above result for the existence of an upper frequently hypercyclic vector is to ask now for the unconditional convergence of the series $\sum_{n\geq 1}(w_1\cdots w_n)^{-1/m}e_n$ for all $m\geq 1$. This is sufficient!
\begin{theorem}\label{thm:ufhcws} Let $X$ be a Fr\'echet sequence algebra for the coordinatewise product and with a continuous norm. Assume that $(e_n)$ spans a dense subspace of $X$. Let also $B_w$ be a bounded weighted shift on $X$ such that, for all $m\geq 1$, $\sum_{n\geq 1}(w_1\cdots w_n)^{-1/m}e_n$ converges unconditionally. Then $B_w$ admits an upper frequently hypercyclic algebra. \end{theorem}
In particular, for $\lambda>1$, $\lambda B$ admits on any $\ell_p$-space ($1\leq p<+\infty$) and on $c_0$ an upper frequently hypercyclic algebra. This last result was independently obtained by Falc\'o and Grosse-Erdmann in \cite{FalGre19} in a different context (they concentrate themselves on $\lambda B$ but allow different notions of hypercyclicity) and with a completely different proof.
Regarding Fréchet sequence algebras endowed with the convolution product, we also have been able to get a general statement (see the forthcoming Theorem \ref{thm:ufhcconvolution}). Its main feature is that we will only need the unconditional convergence of $\sum_{n\geq 1}(w_1\cdots w_n)^{-1}e_n$ and a technical condition to ensure the existence of an upper frequently hypercyclic algebra. As a corollary, we can state the following. \begin{corollary} \begin{enumerate}[(i)] \item Let $X=\ell_1$ endowed with the convolution product and let $\lambda>1$. Then $\lambda B$ admits an upper frequently hypercyclic algebra. \item Let $X=H(\mathbb C)$ endowed with the convolution product. Then $D$ admits an upper frequently hypercyclic algebra. \end{enumerate} \end{corollary}
Coming back to our initial problem, we show that it is possible to exhibit a weighted shift supporting a frequently hypercyclic algebra. The place to do this will be $c_0$ endowed with the coordinatewise product; of course, the weight sequence will be much more complicated than that of the Rolewicz operator.
\begin{theorem}\label{thm:afhcc0}
There exists a weight $(w_n)$ such that $B_w$, acting on $c_0$ endowed with the coordinatewise product, supports a frequently hypercyclic algebra. \end{theorem}
The proof of this theorem will need the construction of disjoint subsets of $\mathbb N$ with positive lower density and with some other extra properties, which seems interesting by itself.
\subsection{Organization of the paper} Up to now, there were two ways to produce hypercyclic algebras: by a direct construction (this is the method devised in \cite{shkarin} and in \cite{FalGre18}) or by using a Baire argument (this method was initiated in \cite{BM09}). In this paper, we improve the latter. We first give in Section \ref{sec:criterion} a general result for the existence of a hypercyclic algebra, enhancing the main lemma proved in \cite{BM09}. This general theorem will be suitable to our new examples of operators supporting a hypercyclic algebra. Next, we adapt the Baire argument to produce upper frequently hypercyclic algebras as well. Since the set of frequently hypercyclic vectors is always meagre, Theorem \ref{thm:afhcc0} cannot be proved using such an argument; it follows from a careful construction both of the weight and of the algebra.
We finish in the last section by making some remarks and asking some questions. In particular, we give a negative answer to a question raised by Shkarin about the existence of a closed hypercyclic algebra for the derivation operator.
\subsection{Notations} The symbol $\mathbb N$ will stand for the set of positive integers, whereas $\mathbb N_0=\mathbb N\cup\{0\}$. We shall denote by $\mathcal P_f(A)$ the set of finite subsets of a given set $A$.
For $x=\sum_{n=0}^{+\infty}x_n e_n\in \omega$, the support of $x$ is equal to $\text{supp}(x)=\{n\in\mathbb N_0: x_n\neq 0\}$. The notation $c_{00}$ will denote the set of sequences in $\omega$ with finite support.
For $u\in X^d$ and $\alpha\in\mathbb N_0^d$, $u^\alpha$ will mean $u_1^{\alpha_1}\cdots u_d^{\alpha_d}$. If $z$ is any complex number and $m\in\mathbb N$, $z^{1/m}$ will denote any $m$th root of $z$.
When working on a Fréchet space $(X,\|\cdot\|_p)$, it is often convenient to endow $X$ with an $F$-norm $\|\cdot\|$ defining the topology of $X$ (see \cite[Section 2.1]{GePeBook}). Such an $F$-norm can be defined by the formula
\[ \|x\|=\sum_{p=1}^{+\infty}\frac1{2^p}\min(1,\|x\|_p). \] In particular, an $F$-norm satisfies the triangle inequality and the inequality \begin{equation}\label{eq:fnorm}
\forall \lambda\in\mathbb C,\ \forall x\in X,\ \|\lambda x\|\leq (|\lambda|+1)\|x\|, \end{equation} a property which replaces the positive homogeneity of the norm.
We finally recall some results on unconditional convergence in Fr\'echet spaces (see for instance \cite[Appendix A]{GePeBook}). A series $\sum_{n=0}^{+\infty}x_n$
in a Fréchet space $X$ is called unconditionally convergent if for any bijection $\pi:\mathbb N_0\to\mathbb N_0$, the series $\sum_{n=0}^{+\infty}x_{\pi(n)}$ is convergent. This amounts to saying that, for any $\varepsilon>0$, there is some $N\in\mathbb N$ such that, whenever $\sup_n |\alpha_n|\leq 1$, the series $\sum_{n=0}^{+\infty}\alpha_n x_n$ converges and
\[\left \|\sum_{n=N}^{+\infty}\alpha_n x_n\right\|<\varepsilon. \]
\section{A general criterion} \subsection{A transitivity criterion to get hypercyclic algebras} \label{sec:criterion}
We first give a general statement which may be thought of as a Birkhoff transitivity theorem for hypercyclic algebras. This criterion will be the main ingredient for the results from Section \ref{sec:convolution} and some from Section \ref{sec:ws}. A different version of this approach will give rise to a new criterion for the existence of upper frequently hypercyclic algebras in Section \ref{sec:fhc}.
\begin{theorem}\label{thm:generalcriterion}
Let $T$ be a continuous operator on a separable commutative $F$-algebra $X$ and let $d\geq 1$. Assume that for any $A\subset \mathbb N_0^d \backslash\{(0,\dots,0)\}$ finite and non-empty,
for any non-empty open subsets $U_1,\dots,U_d,V$ of $X$, for any neighbourhood $W$ of $0$,
there exist $u=(u_1,\dots,u_d)\in U_1\times\cdots\times U_d$, $\beta\in A$ and $N\geq 1$ such that $T^N(u^\beta)\in V$ and $T^N(u^\alpha)\in W $ for all $\alpha\in A$, $\alpha\neq\beta$.
Then the set of $d$-tuples that generate a hypercyclic algebra for $T$ is residual in $X^d$. Moreover, if the assumptions are satisfied for all $d\geq 1$, then $T$ admits a dense and not finitely generated hypercyclic algebra. \end{theorem}
\begin{proof}
Let $(V_k)$ be a basis of open neighbourhoods of $X$. For $A\in \mathcal P_f(\mathbb N_0^d)$, $A\neq\varnothing$, $(0,\dots,0)\notin A$, for $s,k\geq 1$, for $\beta\in A$, define \begin{align*}
E(A, \beta,s)&=\left\{\sum_{\alpha\in A}\hat P(\alpha)z^\alpha\in\mathbb C[z_1,\dots,z_d]:\ \hat P(\beta)=1\textrm{ and }\sup_{\alpha\in A} |\hat P(\alpha)|\leq s\right\}\\
\mathcal A(A,\beta,s,k)&=\left\{u\in X^d:\ \forall P\in E(A,\beta,s),\ \exists N\geq 1,\ T^N(P(u))\in V_k\right\}. \end{align*} The sets $E(A,\beta,s)$ are compact subsets of $\mathbb C[z_1,\dots,z_d]$. By continuity of the maps $(u,P)\mapsto T^N(P(u))$, this implies that each set $\mathcal A(A,\beta,s,k)$ is open. Let us show that, for all $A$, $k$ and $s$, $\bigcup_{\beta\in A}\mathcal A(A,\beta,s,k)$ is dense in $X^d$. Indeed, pick $U_1,\dots,U_d$ non-empty open subsets of $X$. Let $V\subset V_k$ and $W=B(0,\varepsilon)$ be a neighbourhood of $0$ such that $V+B\big(0,(s+1)\textrm{card}(A)\varepsilon\big)\subset V_k$. The assumptions of the proposition give the existence of $u=(u_1,\dots,u_d)\in U_1\times\cdots\times U_d$, $\beta\in A$ and $N\geq 1$. We claim that $u$ belongs to $\mathcal A(A,\beta,s,k)$. Indeed, $$T^N P(u)=\sum_{\alpha\neq\beta} \hat P(\alpha)T^N(u^\alpha)+T^N(u^\beta)\in V+B\big(0,(s+1)\textrm{card}(A)\varepsilon\big)\subset V_k$$ (observe that we have used \eqref{eq:fnorm}). Hence, $\bigcap_{A,s,k}\bigcup_\beta \mathcal A(A,\beta,s,k)$ is a residual subset of $X^d$. Pick $u\in \bigcap_{A,s,k}\bigcup_\beta \mathcal A(A,\beta,s,k)$.
We show that for all non-zero polynomials $P\in \mathbb C[z_1,\dots,z_d]$ with $P(0)=0$, $P(u)$ belongs to $HC(T)$. We set $A=\left\{\alpha:\ \hat P(\alpha)\neq 0\right\}$ and we first prove that $\bigcup_{\beta\in A}\textrm{Orb}\left(T,\frac1{\hat P(\beta)}P(u)\right)$ is dense. Let us fix some $k$
and let us set $s=\sup_{\alpha,\beta\in A} |\hat P(\alpha)|/|\hat P(\beta)|$. Let $\beta\in A$ be such that $u\in\mathcal A(A,\beta,s,k)$. Define $Q=P/\hat P(\beta)$. Then $Q$ belongs to $E(A,\beta,s)$ so that there exists $N\geq 1$ satisfying \[ T^N\left(\frac1{\hat P(\beta)}P(u)\right)=T^N \big(Q(u)\big)\in V_k. \] By the Bourdon-Feldman theorem, we deduce that there is some $\beta_0\in A$ such that $\textrm{Orb}\left(T,\frac 1{\hat P(\beta_0)}P(u)\right)$ is dense in $X$. Since any non-zero multiple of a hypercyclic vector remains hypercyclic, we finally deduce that $P(u)$ is a hypercyclic vector for $T$.
The modification to obtain dense and infinitely generated algebras is easy. For $A\in\mathcal P_f(\mathbb N_0^d)$, $A\neq\varnothing$, $(0,\dots,0)\notin A$, we now let $$ \mathcal A(A,\beta,s,k)=\left\{u\in X^\mathbb N:\ \forall P\in E(A,\beta,s),\ \exists N\geq 1,\ T^N(P(u))\in V_k\right\}$$ and we still consider the set $\bigcap_{A,s,k}\bigcup_{\beta\in A}\mathcal A(A,\beta,s,k)$ where now the intersection runs over all non-empty and finite sets $A\subset \mathbb N^d\backslash\{(0,\dots,0)\}$ with $d\geq 1$ arbitrary. This intersection is still residual in $X^\mathbb N$. We also know from \cite{BP18} that the set of $u$ in $X^\mathbb N$ that induce a dense algebra in $X$ is residual in $X^\mathbb N$. Hence we may pick $u\in X^\mathbb N$ belonging to $\bigcap_{A,s,k}\bigcup_\beta\mathcal A(A,\beta,s,k)$ and inducing a dense algebra in $X$. It is plain that for any non-zero polynomial $P$ with $P(0)=0$, $P(u)$ is hypercyclic for $T$.
It remains to show that the algebra generated by $u$ is not finitely generated. Assume on the contrary that it is generated by a finite number of $P_1(u),\dots,P_p(u)$. In particular, it is generated by a finite number of $u_1,\dots,u_q$. Then there exists a polynomial $Q\in \mathbb C[z_1,\dots,z_q]$ such that $Q(0)=0$ and $u_{q+1}=Q(u_1,\dots,u_q)$. Define $P(z)=z_{q+1}-Q(z)$. Then $P$ is a non-zero polynomial with $P(0)=0$. Nevertheless, $P(u)=0$, which contradicts the fact that $P(u)$ is a hypercyclic vector for $T$. \end{proof}
\begin{remark}
Theorem \ref{thm:generalcriterion} remains true if the algebra is not commutative. This is clear if $d=1$. For the remaining cases,
we have to replace in the proof polynomials in $d$ commutative variables by polynomials in $d$ non-commutative variables.
Details are left to the reader. \end{remark}
We point out that, unlike \cite[Lemma 3.1]{BP18}, in the previous theorem, the index $\beta$ may depend on $A$, $U_1,\dots,U_d,V$ and $W$. We will never use this possibility: we will only need that $\beta$ may depend on $A$ and we will denote $\beta=\beta_A$. For this particular case, we could give an easier proof avoiding the use of the Bourdon-Feldman theorem (see the proof of \cite[Lemma 3.1]{BP18}).
Let us give a couple of corollaries. The first one comes from \cite[Remark 5.28]{BM09} and was the key lemma in \cite{BM09}, \cite{BCP18} or \cite{Bayhcalg} to get hypercyclic algebras.
\begin{corollary}\label{cor:thmgenmax} Let $T$ be a continuous operator on a separable $F$-algebra $X$. Assume that, for any pair $(U,V)$ of non-empty open sets in $X$, for any open neighbourhood $W$ of zero, and for any positive integer $m$, one can find $u\in U$ and an integer $N$ such that $T^N(u^n)\in W$ for all $n<m$ and $T^N(u^m)\in V$. Then $T$ admits a hypercyclic algebra. \end{corollary} \begin{proof} It is straightforward to show that the assumptions of Theorem \ref{thm:generalcriterion} with $d=1$ and $\beta_A=\max A$ are satisfied. \end{proof}
In this work, we will often use the inverse choice for $\beta_A$. \begin{corollary}\label{cor:thmgenmin} Let $T$ be a continuous operator on a separable $F$-algebra $X$. Assume that, for any pair $(U,V)$ of non-empty open sets in $X$, for any open neighbourhood $W$ of zero, and for any positive integers $m_0<m_1$, one can find $u\in U$ and an integer $N$ such that $T^N(u^m)\in W$ for all $m\in\{m_0+1,\dots,m_1\}$ and $T^N(u^{m_0})\in V$. Then $T$ admits a hypercyclic algebra. \end{corollary} \begin{proof} This is now Theorem \ref{thm:generalcriterion} with $d=1$ and $\beta_A=\min A$. \end{proof}
\subsection[Algebras not contained in a finitely generated algebra]{Countably generated, free hypercyclic algebras.}
The conclusion of Theorem \ref{thm:generalcriterion} about infinitely generated hypercyclic algebras does not prevent the possibility for such an algebra to be contained in a finitely generated algebra. Furthermore, it is well known that every at least two generated algebra contains an infinitely generated subalgebra. For all of the examples of this paper we may avoid this scenario thanks to the following result. We notice that due to \cite[Corollary 2.7]{BP18}, a countably generated free algebra is not contained in a finitely generated one.
\begin{corollary}\label{generators}
Let $X$ be a separable commutative $F$-algebra that contains a dense freely generated subalgebra.
Let $T$ be a continuous operator on $X$ and let $d\geq 1$. Assume that for any $A\subset \mathbb N_0^d \backslash\{(0,\dots,0)\}$ finite and non-empty,
for any non-empty open subsets $U_1,\dots,U_d,V$ of $X$, for any neighbourhood $W$ of $0$,
there exist $u=(u_1,\dots,u_d)\in U_1\times\cdots\times U_d$, $\beta\in A$ and $N\geq 1$ such that $T^N(u^\beta)\in V$ and $T^N(u^\alpha)\in W$ for all $\alpha\in A$, $\alpha\neq\beta$.
Then $T$ admits a $d$-generated, free hypercyclic algebra. Moreover, if the assumptions are satisfied for all $d\geq 1$, then $T$ admits a dense, countably generated, free hypercyclic algebra. \end{corollary}
\begin{proof} By Theorem~\ref{thm:generalcriterion} the set of $d$-tuples generating a hypercyclic algebra for $T$ is residual in $X^d$. By \cite[Proposition 2.4]{BP18}, the set of $u$ in $X^d$ that induce a $d$-generated, free algebra is residual in $X^d$. For the conclusion we just need to pick an element in the intersection of those two sets.
For the second claim, we consider $X^{\mathbb N}$ endowed with the product topology. By the assumption, for each $N\in \mathbb N$, the set $$ H_N=\{ (u_n)\in X^{\mathbb N}: (u_1,\dots ,u_N) \,\, \mbox{generates a hypercyclic algebra for} \,\, T\} $$ is residual in $X^{\mathbb N}$ and hence, by the Baire category theorem, the set $H=\bigcap_{N=1}^{\infty}H_N$ is residual as well. The algebra generated by any $u\in H$ is hypercyclic for $T$. Furthermore, by \cite[Proposition 2.4]{BP18}, the set of sequences of $X^{\mathbb N}$ which generate a dense and free algebra is also residual. The conclusion follows by one more application of the Baire category theorem. \end{proof}
Hence, we need to provide for our examples a dense and freely generated subalgebra. This is quite easy for a Fréchet sequence algebra endowed with the Cauchy product: provided $\textrm{span}(e_n)$ is dense, it always contain a dense and freely generated subalgebra, namely the unital algebra generated by the sequence $e_1$. This covers the case of $H(\mathbb C)$ and of the disc algebra. This is slightly more difficult for a Fréchet sequence algebra endowed with the coordinatewise product.
\begin{lemma}\label{free} Let $X$ be a Fr\'echet sequence algebra, endowed with the pointwise product, and for which the sequence $(e_n)$ is a Schauder basis. Then $X$ has a dense freely generated subalgebra. \end{lemma}
\begin{proof}
Let $(b_n)\subset (0,1)$ be such that the series $\sum_{n=0}^{\infty}b_n \|e_n\|_n$ converges. Consider the sequence of natural numbers $(a_n)$ such that $a_0=0$ and $a_n=a_{n-1}+n, n\in \mathbb N$ and define $c_n:=\min_{l\in [a_{m-1},a_m)}b_l$, if $n\in [a_{m-1},a_m)$. The series $\sum_{n=0}^{\infty}c_ne_n$ converges absolutely since $c_n\leq b_n,\ n\in \mathbb N$.
Define a sequence $(\lambda_n)\subset (0,1)$ inductively as follows: choose $\lambda_0 \in (0,1)$ and, for $n\in \mathbb N,$ take $\lambda_n \in (0,1)\setminus \{ \lambda_0^{p_0}\dots \lambda_{n-1}^{p_{n-1}}: p_0,\dots ,p_{n-1}\in \mathbb Q \}$. Observe that if $\lambda=(\lambda_0,\dots ,\lambda_p),$ $p\in \mathbb N _0$ and $\alpha \neq \beta \in \mathbb N _0^p$, then $\lambda ^{\alpha}\neq \lambda ^{\beta}$. Let now, for each $n\in \mathbb N _0$, $$ g_n=e_n+\sum_{k=1}^{\infty}\lambda_n^{n+k}c_{n+k}e_{n+k}, $$
where the convergence of the series is ensured by the convergence of $\sum_{n=0}^{+\infty}b_n\|e_n\|_n$ and the inequality $\lambda_n^{n+k}c_{n+k}<b_{n+k}$. We claim that the algebra generated by the $g_n,\ n\in \mathbb N $ is dense and free.
First, we show that the sequence $\{g_n: n\in \mathbb N \}$ is algebraically independent. For that reason, let \begin{equation} \label{1} \sum_{\alpha \in A}a_{\alpha}g^{\alpha}=0, \end{equation} where $A=\{ \alpha(1),\dots ,\alpha(q) \}\subset \mathbb N_0^p$, $g=(g_1,\dots ,g_p)\in X^p$, and $p\in \mathbb N$. If we consider the coordinate $N=n+k$ in equation \eqref{1}, we get the following equation which holds for all $N$ sufficiently large. \begin{equation} \label{2} \sum_{i=1}^qa_{\alpha(i)}(\lambda_1^Nc_N)^{\alpha_1(i)}\dots (\lambda_p^Nc_N)^{\alpha_p(i)}=0. \end{equation} Now we may choose $N$ sufficiently big such that $N, \dots ,N+q-1\in [a_{m-1},a_m)$ for some $m$, which means that $c_N=\dots =c_{N+q-1}=b_m$. Equation \eqref{2} then becomes $$ \sum_{i=1}^qa_{\alpha(i)}(\lambda_1^Mb_m)^{\alpha_1(i)}\dots (\lambda_p^Mb_m)^{\alpha_p(i)}=0, $$ where $M$ ranges over $N,\dots ,N+q-1$. Setting $A$ the matrix $$ \begin{bmatrix} (\lambda_1^Nb_m)^{\alpha_1(1)}\cdots (\lambda_p^Nb_m)^{\alpha_p(1)} &\dots &(\lambda_1^Nb_m)^{\alpha_1(q)}\cdots (\lambda_p^Nb_m)^{\alpha_p(q)} \\ \vdots &\ddots &\vdots \\ (\lambda_1^{N+q-1}b_m)^{\alpha_1(1)}\cdots (\lambda_p^{N+q-1}b_m)^{\alpha_p(1)} &\dots &(\lambda_1^{N+q-1}b_m)^{\alpha_1(q)}\cdots (\lambda_p^{N+q-1}b_m)^{\alpha_p(q)} \end{bmatrix}, $$ we find the matrix equality $$A\begin{bmatrix} a_{\alpha(1)}\\ \vdots \\ a_{\alpha(q)} \end{bmatrix} =0. $$ The determinant of the square matrix $A$, after making use of the Vandermonde identity, is $$
\prod_{i=1}^q \prod_{j=1}^p (b_m\lambda_j^N)^{\alpha_j(i)}\prod_{i>j}[(\lambda_1^{\alpha_1(i)}\dots \lambda_p^{\alpha_p(i)})-(\lambda_1^{\alpha_1(j)}\dots \lambda_p^{\alpha_p(j)})]\neq 0. $$ Hence, we get that $a_{\alpha(i)}=0$ for all $i=1,\dots ,q$.
Next, we show that the algebra generated by $\{g_n: n\in \mathbb N \}$ is dense in $X$. We will show that the elements $e_n, n\in \mathbb N$, are in the closure of this algebra. Let us fix $n\in\mathbb N$ and observe that, for all $p\in\mathbb N$, $$ g_n^p-e_n=\sum_{k=1}^{\infty}\lambda_n^{(n+k)p}c_{n+k}^p e_{n+k}. $$ Fix $q\in\mathbb N$ and let $\varepsilon>0$. There exists $N\geq q$ such that, for all $p\in\mathbb N$, $$
\left \| \sum_{k>N}\lambda_n^{(n+k)p}c_{n+k}^pe_{n+k}\right \|_q\leq \sum_{k>N}b_{n+k}\|e_{n+k}\|_q <\varepsilon. $$ Since furthermore $$ \sum_{k=1}^N\lambda_n^{(n+k)p}c_{n+k}^p e_{n+k}\xrightarrow[p\rightarrow \infty]{} 0, $$ we conclude that $g_n^p\xrightarrow[p\rightarrow \infty]{}e_n$. \end{proof}
We conclude this subsection by comparing Corollary \ref{generators} with \cite[Remark 3.4]{BP18}. Corollary \ref{generators} allows the index $\beta$ to depend on $A, U_1,\dots ,U_d,V$ and $W$ providing, at least theoretically, an extra flexibility and range of application for the result. Practically, throughout the paper, $\beta$ will depend only on $A$ in which case Corollary \ref{generators} and \cite[Remark 3.4]{BP18} coincide. We were unable to find an example where Corollary \ref{generators} applies while \cite[Remark 3.4]{BP18} does not.
\section{Convolution operators with $|\phi(0)|>1$}
\subsection{Operators with many eigenvectors}
\label{sec:convolution}
In this section we shall deduce Theorem \ref{thm:convintro} from a more general assertion on operators having many eigenvalues. As Theorem \ref{thm:generalcriterion} does for Corollary \ref{cor:thmgenmax}, this generalized statement also includes \cite[Theorem 2.1]{Bayhcalg} as a particular case. Before stating and proving it, let us add some notation.
Given $p,d\in\mathbb N$, we denote each set $\{1,...,p\}$ by $I_p$, each $d$-tuple $(j_1,...,j_d)\in I_p^d$ by the multi-index $\textbf{j}\in I_p^d$ and each product $a_{j_1}\cdots a_{j_m}$ by the symbol $a_{\textbf{j}}$. We allow $d=0$ with the convention that, in this case, $a_{\textbf{j}}=1$.
\begin{theorem}\label{versionconvol} Let $X$ be an $F$-algebra and let $T\in\mathcal{L}(X)$. Assume that there exist a function $E:\mathbb C\to X$ and an entire function $\phi:\mathbb C\to\mathbb C$ satisfying the following assumptions: \begin{enumerate}[(a)]
\item for all $\lambda\in\mathbb C$, $TE(\lambda)=\phi(\lambda)E(\lambda)$;
\item for all $\lambda, \mu\in\mathbb C$, $E(\lambda)E(\mu)=E(\lambda+\mu)$;
\item for all $\Lambda \subset\mathbb C$ with an accumulation point, the linear span of $\{E(\lambda):\lambda\in\Lambda\}$ is dense in $X$;
\item $\phi$ is not a multiple of an exponential function;
\item for all $I\in\mathcal{P}_f(\mathbb N)\backslash\{\varnothing\}$, there exist $m\in I$ and $a,b\in\mathbb C$ such that $|\phi(mb)|>1$ and, for all $n\in I$ and $d\in\{0,...,n\}$, with $(n,d)\neq(m,m)$, $|\phi(db+(n-d)a)|<|\phi(mb)|^{d/m}.$ \end{enumerate} Then $T$ supports a hypercyclic algebra. \end{theorem}
The proof of this result follows the lines of that of Theorem 2.1 in \cite{Bayhcalg}, replacing Corollary \ref{cor:thmgenmax} by the more general Theorem \ref{thm:generalcriterion}. For the sake of completeness, we include the details.
\begin{proof} Let $(U,V, W)$ be a triple of non-empty open sets in $X$, with $0\in W$, and let $I\in\mathcal{P}_f(\mathbb N)\backslash\{\varnothing\}$. By the hypothesis there are $m\in I$ and $a,b\in\mathbb C$ satisfying (e).
Define $w_0:=mb$ and let $\delta>0$ be small enough and $w_1,w_2\in B(w_0,\delta)$ so that \begin{enumerate}[(i)]
\item $|\phi|>1$ on $B(w_0,\delta)$;
\item $t\mapsto \log|\phi(tw_1+(1-t)w_2)|$ is strictly convex (the existence of $w_1,w_2\in B(w_0,\delta)$ comes from \cite[Lemma 2.2]{Bayhcalg} and is a consequence of (d)); \item for all $n\in I$ and $d\in\{0,...,n\}$, with $(n,d)\neq(m,m)$, and for all $\lambda_1,...,\lambda_d\in B(w_0,\delta)$ and $\gamma_1,...,\gamma_{n-d}\in B(a,\delta)$,
\begin{equation}\label{starcond0} \biggl|\phi\biggl(\frac{\lambda_1+\cdots+\lambda_d}{m}+\gamma_1+\cdots+\gamma_{n-d}\biggr)\biggr|<\left(|\phi(\lambda_1)|\times\cdots\times |\phi(\lambda_d)|\right)^{1/m}. \end{equation} \end{enumerate} The last condition can be satisfied because \[ \frac{\lambda_1+\cdots+\lambda_d}{m}+\gamma_1+\cdots+\gamma_{n-d}=db+(n-d)a+z,\] where the size of $z$ can be controlled through $\delta$. Now, since $B(a,\delta)$ and $[w_1,w_2]$ have accumulation points, we can find $p,q\in\mathbb N$, $a_1,...,a_p,b_1,...,b_q\in\mathbb C$, $\gamma_1,...,\gamma_p\in B(a,\delta)$ and $\lambda_1,...,\lambda_q\in[w_1,w_2]$ with \[\sum_{l=1}^pa_lE(\gamma_l)\in U~~\text{and}~~\sum_{j=1}^qb_jE(\lambda_j)\in V.\] For some big $N\in\mathbb N$ (which will be determined later in the proof) and each $j\in\{1,...,q\}$, let $c_j:=c_j(N)$ be any complex number satisfying \[c_j^{m}(N)=\frac{b_j}{\phi(\lambda_j)^N}\] and define \[u:=u(N)=\sum_{l=1}^pa_lE(\gamma_l)+\sum_{j=1}^qc_jE\biggl(\frac{\lambda_j}{m}\biggr).\] For the powers of $u$ we have the formula \begin{equation}\label{formulaun} u^n=\sum_{d=0}^n\sum_{\substack{\textbf{l}\in I_p^{n-d}\\ \textbf{j}\in I_q^d}}\alpha(\textbf{l},\textbf{j},d,n)a_{\textbf{l}}c_{\textbf{j}}E\biggl(\gamma_{l_1}+\cdots+\gamma_{l_{n-d}}+ \frac{\lambda_{j_1}+\cdots+\lambda_{j_d}}{m}\biggl). \end{equation} We claim that, if $N$ is taken large enough, $u=u(N)$ satisfies the conditions of the general criterion with $d=1$, what will complete the proof.
That $u\in U$ for large $N$ is clear since, from (i), $c_j(N)\to 0$ as $N\to\infty$. Applying $T^N$ to $u^n$ we see that we need to study the behaviour (as $N$ grows) of \begin{equation}\label{study0} c_{\textbf{j}}(N)\biggl[\phi\biggl(\gamma_{l_1}+\cdots+\gamma_{l_{n-d}}+\frac{\lambda_{j_1}+\cdots+\lambda_{j_d}}{m}\biggl)\biggl]^N. \end{equation} For $n\in I\backslash\{m\}$ we have that (\ref{study0}) goes to 0 as $N$ grows by the inequality (\ref{starcond0}) and the definition of $c_j$, $j=1,...,q$. This way we get $T^N(u^n)\in W$ for all $n\in I\backslash\{m\}$ if $N$ is large enough. Now let us consider the case $n=m$. We have \begin{align*}
u^{m} &= \sum_{d=0}^{m-1}\sum_{\substack{\textbf{l}\in I_p^{m-d}\\ \textbf{j}\in I_q^d}} \alpha(\textbf{l},\textbf{j},d,m)a_{\textbf{l}}c_{\textbf{j}} E\biggl(\gamma_{l_1}+\cdots+\gamma_{l_{m-d}}+\frac{\lambda_{j_1}+\cdots+\lambda_{j_d}}{m}\biggl) \\
&+ \sum_{\textbf{j}\in I_q^m\backslash D_q} \alpha(\textbf{j},m)c_{\textbf{j}}E\biggl(\frac{\lambda_{j_1}+\cdots+\lambda_{j_m}}{m}\biggl)\\
&+\sum_{j=1}^qc_j^{m}E(\lambda_j)\\
&=:v_1+v_2+v_3, \end{align*} where \[ v_1:=\sum_{d=0}^{m-1}\sum_{\substack{\textbf{l}\in I_p^{m-d}\\ \textbf{j}\in I_q^d}} \alpha(\textbf{l},\textbf{j},d,m)a_{\textbf{l}}c_{\textbf{j}} E\biggl(\gamma_{l_1}+\cdots+\gamma_{l_{m-d}}+\frac{\lambda_{j_1}+\cdots+\lambda_{j_d}}{m}\biggl), \] \[v_2:=\sum_{\substack{\textbf{j}\in I_q^{m}}\backslash D_q}\alpha(\textbf{j},m)c_{\textbf{j}}E\biggl(\frac{\lambda_{j_1}+\cdots+\lambda_{j_{m}}}{m}\biggl),~~~~v_3:=\sum_{j=1}^qc_j^mE(\lambda_j) \]
and $D_q$ is the diagonal of $I_q^{m}$, that is, the set of all $m$-tuples $(j,...,j)$ with $1\leq j\leq q$. Again we have $T^N(v_1)\to 0$ as $N\to\infty$ from (\ref{starcond0}). Furthermore, since $t\in[0,1]\mapsto\log|\phi(tw_1+(1-t)w_2)|$ is strictly convex, we have \[
\biggl|\phi\biggl(\frac{\lambda_{j_1}+\cdots+\lambda_{j_{m}}}{m}\biggl)\biggl|<|\phi(\lambda_{j_1})|^{1/m}\cdots|\phi(\lambda_{j_{m}})|^{1/m}. \] From this we conclude that $$\begin{array}{l}
\displaystyle |c_{\textbf{j}}(N)|\cdot\Biggl|\phi\Biggl(\frac{\lambda_{j_1}+\cdots+\lambda_{j_{m}}}{m}\Biggl)\Biggr|^N \\
\displaystyle \quad\quad = \Biggl|\frac{b_{\textbf{j}}^{1/m}}{|\phi(\lambda_{j_1})|^{N/m}\cdots|\phi(\lambda_{j_n})|^{N/m}}\Biggr|\cdot \Biggl|\phi\Biggl(\frac{\lambda_{j_1}+\cdots+\lambda_{j_{m}}}{m}\Biggr)\Biggr|^N\\
\displaystyle \quad\quad =\bigl|b_{\textbf{j}}^{1/m}\bigl|\cdot\left|\frac{\phi\Bigl(\frac{\lambda_{j_1}+\cdots+\lambda_{j_{m}}}{m}\Bigr)}{|\phi(\lambda_{j_1})|^{1/m}\cdots|\phi(\lambda_{j_{m}})|^{1/m}}\right|^N\to0~~\text{as}~~N\to\infty, \end{array} $$ what shows that $T^N(v_2)$ also tends to 0 as $N\to\infty$. Finally, by the definition of $c_j$, $j=1,...,q$, we get \[ T^Nv_3=\sum_{j=1}^qb_jE(\lambda_j)\in V \] for all $N\in\mathbb N$. This completes the proof. \end{proof}
We now deduce a more readable corollary, when the entire function $\phi$ is ``well behaved'' in some half-line of the complex plane (like in the Figure \ref{figure} for example).
\begin{corollary}\label{wellbehaved} Let $X$ be an $F$-algebra and let $T\in\mathcal{L}(X)$. Assume that there exist a function $E:\mathbb C\to X$ and an entire function $\phi:\mathbb C\to\mathbb C$ satisfying the following assumptions: \begin{enumerate}[(a)]
\item for all $\lambda\in\mathbb C$, $TE(\lambda)=\phi(\lambda)E(\lambda)$;
\item for all $\lambda, \mu\in\mathbb C$, $E(\lambda)E(\mu)=E(\lambda+\mu)$;
\item for all $\Lambda \subset\mathbb C$ with an accumulation point, the linear span of $\{E(\lambda):\lambda\in\Lambda\}$ is dense in $X$;
\item $\phi$ is not a multiple of an exponential function;
\item there exist $v\in\mathbb C$ and a real number $p>0$ such that $|\phi(v)|>1$ and $|\phi(tv)|\leq1$ for all $t>p$. \end{enumerate} Then $T$ supports a hypercyclic algebra. \end{corollary}
\begin{figure}
\caption{Graph of $t\mapsto |\phi(tv)|$}
\label{figure}
\end{figure}
For the proof of Corollary \ref{wellbehaved}, we are going to need the following simple lemma.
\begin{lemma}\label{lemmawellbehaved}
Let $\phi$ be entire and $\Lambda\subset\mathbb R$ have an accumulation point in it. If $|\phi(t)|=1$ for all $t\in\Lambda$ then $|\phi(t)|=1$ for all $t\in\mathbb R$. \end{lemma} \begin{proof}
Since $|\phi(t)|=1$ for all $t\in\Lambda\subset\mathbb R$, we have $\overline{\phi(\overline{t})}\cdot\phi(t)=\overline{\phi(t)}\cdot\phi(t)=1$, hence \[ \overline{\phi(\overline{t})}=\phi(t)^{-1}, \forall t\in\Lambda\subset\mathbb R. \] Since this is a holomorphic equality ($t\mapsto\overline{\phi(\overline{t})}$ is entire), it extends to the whole complex plane. In particular it holds for all $t\in\mathbb R$, that is, \[ \overline{\phi(t)}=\overline{\phi(\overline{t})}=\phi(t)^{-1}, \forall t\in\mathbb R, \] thus \[
|\phi(t)|=1, \forall t\in\mathbb R, \] as we wanted. \end{proof}
\begin{proof}[Proof of Corollary \ref{wellbehaved}] We may assume without loss of generality that $v=1$. Let $t_0>0$ be the smallest positive real number such that $|\phi(t)|\leq1$ for all $t\geq t_0$. We just need to prove that condition (e) of Theorem \ref{versionconvol} is satisfied.
So let $I\in\mathcal{P}_f(\mathbb N)\backslash\{\varnothing\}$ be arbitrary and set $m=\min I$. We can find $t_1<t_0$ near enough to $t_0$ so that $|\phi(t_1)|>1$ and $t_1+\frac{t_1}{m}>t_0$.
Letting $b:=\frac{t_1}{m}$ we have $|\phi(mb)|>1$. Now fix $a_0>t_0$ and take $\epsilon\in(0,1/2)$ such that $a_0-\epsilon>t_0$.
There exists $a\in[a_0-\epsilon,a_0+\epsilon]$ such that $|\phi(db+(n-d)a)|<1$ for all $n\in I$ and $d\in\{0,...,n\}$ with $(n,d)\neq(m,m)$. In fact, if this is not the case then,
for each $a\in[a_0-\epsilon,a_0+\epsilon]$, we can find a point $t_a=d_ab+(n_a-d_a)a$, with $n_a\in I$, $d_a\in\{0,...,n_a\}$ and $(n_a,d_a)\neq(m,m)$, such that $|\phi(t_a)|\geq 1$.
Since $m=\min(I)$, $b=t_1/m$ and $a>t_0$, we get $t_a>t_0$ so that $|\phi(t_a)|\leq 1$.
This way, varying $a\in[a_0-\epsilon,a_0+\epsilon]$ we find infinitely many points $t_a$ within $[t_0,\max I(a_0+\varepsilon)]$
in which $|\phi|$ assumes the value 1. The set $\Lambda$ composed by these points is infinite, closed and subset of the compact $[t_0,\max (I)(\alpha_0+\epsilon)]$, hence $\Lambda$ has an accumulation point.
By Lemma \ref{lemmawellbehaved} we conclude that $|\phi(t)|=1$ for all $t\in\mathbb R$, which contradicts the fact that $|\phi(t_1)|>1$. This completes the proof. \end{proof}
\subsection{Applications to convolution operators}
We now observe that we may apply Corollary \ref{wellbehaved} to convolution operators $\phi(D)$ with $|\phi(0)|>1$, where $E(\lambda)(z)=e^{\lambda z}$. This yields immediately Theorem \ref{thm:convintro}.
We may also apply Corollary \ref{wellbehaved} to handle the case $|\phi(0)|=1$.
\begin{corollary} Let $P\in\mathbb C[z]$ be a non-constant polynomial and let $\phi(z)=P(z)e^z$. Then $\phi(D)$ supports a hypercyclic algebra. \end{corollary} \begin{proof}
The case $|P(0)|<1$ is done in \cite{Bayhcalg}, the case $|P(0)|>1$ is settled by Theorem \ref{thm:convintro}. It remains to consider the case $|P(0)|=1$. Since $|P(it)|$ tends to $+\infty$ as $t$ tends to $+\infty$, there exists $t_0\in\mathbb R$ such that $|\phi(it_0)|>1$. By continuity of $|\phi|$, there exists $v=|v|e^{i\theta}$ with $\theta\in (\pi/2,3\pi/2)$ such that $|\phi(v)|>1$. Now, because $v$ lies in the left half-plane, $|\phi(tv)|$ tends to $0$ as $t$ tends to $+\infty$. We may conclude with Corollary \ref{wellbehaved}. \end{proof}
We finish this section by pointing out that Theorem \ref{versionconvol} can also handle functions which do not satisfy the properties described above.
\begin{example}
The convolution operator induced by $\phi(z)=\frac 12e^z+e^{iz}-\frac 14$ supports a hypercylic algebra (let us observe that $|\phi(0)|>1$ and that $\phi$ does not tend to $0$ along any ray). Indeed, for any $I\in\mathcal P_f(\mathbb N)\backslash\{\varnothing\}$ we choose $m=\max(I)$ and take $a=k(2\pi i)$ and $b=k2\pi$ for some large integer $k$. Let $n\in I$ and $d\in\{0,\dots,n\}$ with $(n,d)\neq (m,m)$. Then $$\phi(db+(n-d)a)=\frac 12 e^{2dk\pi}+e^{-2(n-d)k\pi}-\frac 14.$$ In particular,
$$|\phi(mb)|=\frac 12e^{2mk\pi}+\frac 34>1.$$ When $d=0$,
$$|\phi(na)|=\left|e^{-2nk\pi}+\frac 14\right|<1.$$ Finally,
$$\left|\phi(db+(n-d)a)\right|\leq \frac12 e^{2dk\pi}+\frac 34$$ and we have, for all $d=1,\dots,m-1$, $$\left(\frac 12e^{2dk\pi}+\frac 34\right)^m\leq \left(\frac 12e^{2mk\pi}+\frac 34\right)^d$$ if $k$ is large enough. \end{example}
\begin{remark} Combining the previous arguments with that of \cite[Section 6]{Bayhcalg}, under the assumptions of Theorem \ref{thm:convintro}, $\phi(D)$ admits a dense, countably generated, free hypercyclic algebra. \end{remark}
\section{Weighted shifts on Fréchet sequence algebras}
\label{sec:ws}
\subsection{Fr\'echet sequence algebras with the coordinatewise product}
We begin with the proof of Theorem \ref{thm:mainws}. We first explain where the property of admitting a continuous norm comes into play.
\begin{lemma} Let $X$ be a Fr\'echet sequence algebra for the coordinatewise product and with a continuous norm. Then the sequence $(e_n)$ is bounded below. \end{lemma} \begin{proof}
Let $q\geq 1$ be such that $\|\cdot\|_q$ is a norm on $X$. Then for all $n\in\mathbb N_0$,
$$0<\|e_n\|_q=\|e_n\cdot e_n\|_q\leq \|e_n\|_q^2$$
which shows that $\|e_n\|_q\geq 1$. \end{proof}
We shall prove the following precised version of Theorem \ref{thm:mainws}. \begin{theorem}\label{thm:wsprecised} Let $X$ be a Fr\'echet sequence algebra for the coordinatewise product and with a continuous norm. Assume that $(e_n)$ is a Schauder basis for $X$. Let also $B_w$ be a bounded weighted shift on $X$. The following assumptions are equivalent. \begin{enumerate}[(i)] \item $B_w$ supports a dense, countably generated, free hypercyclic algebra. \item $B_w$ supports a hypercyclic algebra. \item For all $m\geq 1$, there exists $x\in X$ such that $x^m$ is a hypercyclic vector for $B_w$. \item For all $m\geq 1$, for all $L\in\mathbb N$, there exists a sequence of integers $(n_k)$ such that, for all $l=0,\dots,L$, $((w_{l+1}\cdots w_{n_k+l})^{-1/m}e_{n_k+l})$ tends to zero. \item There exists a sequence of integers $(n_k)$ such that, for all $\gamma>0$ and for all $l\in\mathbb N$, $\big((w_{l+1}\cdots w_{n_k+l})^{-\gamma}e_{n_k+l}\big)$ tends to zero. \end{enumerate} \end{theorem}
\begin{proof}
The implications $(i)\implies (ii)$ and $(ii)\implies (iii)$ are trivial. The proof of $(iii)\implies (iv)$ mimics that of the necessary condition for hypercyclicity. Let $m\geq 1$ and $x\in X$ be such that $x^m\in HC(B_w)$. Write $x=\sum_{n=0}^{+\infty}x_n e_n$. Since $(e_n)$ is a Schauder basis, the sequence $(x_n e_n)$ goes to zero. Moreover, there exists a sequence of integers $(n_k)$ such that $(B_w^{n_k}(x^m))_k$ goes to $e_0+\cdots+e_L$. Since convergence in $X$ implies coordinatewise convergence, for all $l=0,\dots,L$, $(w_{l+1}\cdots w_{n_k+l}x_{n_k+l}^m)$ converges to $1$. Hence the sequences $\big((w_{l+1}\cdots w_{n_k+l})^{1/m}x_{n_k+l}\big)$ are bounded below. Writing $$(w_{l+1}\cdots w_{n_k+l})^{-1/m}e_{n_k+l}=\frac{1}{(w_{l+1}\cdots w_{n_k+l})^{1/m}x_{n_k+l}}\cdot x_{n_k+l}e_{n_k+l}$$ we get the result.
To prove that $(iv)\implies (v)$, observe that a diagonal argument ensure the existence of a sequence $(n_k)$ such that, for all $m\geq 1$ and all $l\in\mathbb N$, the sequence $((w_{l+1}\cdots w_{n_k+l})^{-1/m}e_{n_k+l})$ tends to zero. Now we can conclude by observing that, since the sequence $(e_n)$ is bounded below, if $\big((w_{l+1}\cdots w_{n_k+l})^{-1/m}e_{n_k+l}\big)$ tends to zero for some $m$, then $(w_{l+1}\cdots w_{n_k+l})$ tends to $+\infty$ and, in particular, $\big((w_{l+1}\cdots w_{n_k+l})^{-\gamma}e_{n_k+l}\big)$ tends to zero for all $\gamma\geq 1/m$.
It remains to prove the most difficult implication, $(v)\implies (i)$. We start by fixing a sequence of integers $(n_k)$ such that for all $\gamma>0$ and for all $l\in\mathbb N$, $\big((w_{l+1}\cdots w_{n_k+l})^{-\gamma}e_{n_k+l}\big)$ goes to zero. We intend to apply Theorem \ref{thm:generalcriterion}. Thus, let $d\geq 1$ and $A\subset\mathbb N_0^d \backslash\{(0,\dots,0)\}$ be finite and non-empty. For $\alpha\in A$ we define the linear form $L_\alpha$ on $\mathbb R^d$ by $L_\alpha(\kappa)=\sum_{j=1}^d \alpha_j\kappa_j$. Since $L_\alpha$ and $L_{\alpha'}$ coincide only on a hyperplane for $\alpha\neq \alpha'$, there exist $\kappa\in(0,+\infty)^d$ and $\beta=\beta_A\in A$ such that $0<L_{\beta}(\kappa)<L_\alpha(\kappa)$ for all $\alpha\neq\beta$, $\alpha\in A$. Without loss of generality, we may assume that $L_{\beta}(\kappa)=1$.
Let now $U_1,\dots,U_d,V$ be non-empty open subsets of $X$ and let $W$ be a neighbourhood of zero. Let $x_1,\dots,x_d$ belonging respectively to $U_1,\dots,U_d$ with finite support and let $y=\sum_{l=0}^p y_l e_l$ belonging to $V$. We set, for $j=1,\dots,d$, $$u_j:=u_j(n_k)=x_j+\sum_{l=0}^p \frac{y_l^{\kappa_j}}{(w_{l+1}\cdots w_{n_k+l})^{\kappa_j}}e_{n_k+l}.$$ Our assumption implies that, provided $n_k$ is large enough, $u_j$ belongs to $U_j$ for all $j=1,...,d$. Moreover, again if $n_k$ is large enough (larger than the size of the support of each $x_j$), for all $\alpha\in A$, \[ B_w^{n_k}(u^\alpha)=\sum_{l=0}^p \frac{y_l^{L_\alpha(\kappa)}}{(w_{l+1}\cdots w_{n_k+l})^{L_\alpha(\kappa)-1}}e_{l}. \] In particular, for $\alpha=\beta$, $B_w^{n_k}(u^{\beta})=y\in V$ whereas, for $\alpha\neq\beta$, since $L_\alpha(\kappa)-1>0$ and since the sequences $(w_{l+1}\cdots w_{n_k+l})$ tend to $+\infty$, we get $B_w^{n_k}(u^\alpha)\in W$ provided $n_k$ is large enough. Hence, $B_w$ admits a dense and not finitely generated hypercyclic algebra. \end{proof}
As recalled in the introduction, the hypercyclicity of $B_w$ on $X$ is equivalent to the existence of a sequence $(n_k)$ such that, for all $l\in\mathbb N$, $((w_{l+1}\cdots w_{n_k+l})^{-1}e_{n_k+l})$ tends to zero. It is well known that this last condition is equivalent to the following one, which seems much weaker: there exists a sequence of integers $(n_k)$ such that $((w_1\cdots w_{n_k})^{-1}e_{n_k})$ tends to zero. In view of this and of Theorem \ref{thm:wsprecised}, it is tempting to conjecture that $B_w$ supports a hypercyclic algebra if and only if there exists a sequence of integers $(n_k)$ such that, for all $\gamma>0$, $( (w_1\cdots w_{n_k})^{-\gamma}e_{n_k})$ tends to zero. Unfortunately, this is not the case, as the following example points out.
\begin{example}
Let $X=\{(x_n)\in\omega: |x_n|a_n\to 0\}$ where $a_{2n}=1$ and $a_{2n+1}=2^n$ endowed with $\|x\|=\sup_n |x_n|a_n$ and let $w$ be the weight such that $w_1\cdots w_{2n}=2^{n-1}$
and $w_1\cdots w_{2n+1}=2^{2n}$. Then $w$ is an admissible weight on $X$, $( (w_1\cdots w_{2n})^{-\gamma}e_{2n})$ tends to zero for all $\gamma>0$ but $B_w$ does not admit a hypercyclic
algebra. \end{example}
\begin{proof}
We first observe that, endowed with the coordinatewise product, $X$ is a Fréchet sequence algebra (since $a_n\geq 1$ for all $n$). To prove that $w$ is admissible, it suffices to observe
that $w_k \|e_{k-1}\|\leq 2 \|e_k\|$ for all $k$. The construction of $w$ ensures that $w_{2n}=2^{-(n-1)}$ and $w_{2n+1}=2^{n+1}$. Hence the previous inequality is clearly satisfied
if we separate the case $k$ even and $k$ odd. Moreover for all $\gamma>0$,
\[ (w_1\cdots w_{2n})^{-\gamma} \|e_{2n}\|=2^{-\gamma(n-1)}\xrightarrow{n\to+\infty}0. \]
To prove that $B_w$ does not support a hypercyclic algebra, it suffices to observe that, for all $n\geq 1$, $(w_1\cdots w_{2n+1})^{-1/2}\|e_{2n+1}\|=1$, which implies that condition (v) of Theorem
\ref{thm:wsprecised} cannot be satisfied. \end{proof}
Nevertheless, if we add an extra assumption on $X$, then we get the expected result.
\begin{corollary}
Let $X$ be a Fr\'echet sequence algebra for the coordinatewise product and with a continuous norm.
Assume that $(e_n)$ is a Schauder basis for $X$. Assume also that, for all admissible weights $w$, for all $\gamma>0$, $w^\gamma$ is admissible.
Let $B_w$ be a bounded weighted shift on $X$. The following assumptions are equivalent. \begin{enumerate}[(i)] \item $B_w$ supports a hypercyclic algebra. \item For all $\gamma>0$, there exists a sequence $(n_k)$ such that $((w_1\cdots w_{n_k})^{-\gamma} e_{n_k})$ tends to zero. \end{enumerate} \end{corollary}
\begin{proof}
We assume that (ii) is satisfied and we show that, for all $\gamma>0$ and for all $L\in\mathbb N$, there exists a sequence $(m_k)$ such that
$( (w_1\cdots w_{m_k+l})^{-\gamma}e_{m_k+l})$ tends to zero. An application of Theorem \ref{thm:wsprecised} will then allow to conclude.
It is easy to get this sequence $m_k$. Indeed, it is sufficient to set $m_k=n_k-L$, since in that case
\[ (w_1\cdots w_{m_k+l})^{-\gamma}e_{m_k+l}=(B_{w^\gamma})^{L-l}\left((w_1\cdots w_{n_k})^{-\gamma}e_{n_k}\right)\]
which goes to zero by continuity of $B_{w^\gamma}$. \end{proof}
We may observe that our favorite sequence spaces (namely unweighted $\ell_p$-spaces or $H(\mathbb C)$) satisfy the assumptions of the last corollary. We may also observe that on unweighted $\ell_p$-spaces as well as on any Fréchet sequence algebra with a continuous norm such that $(e_n)$ is bounded, the convergence of $\big((w_1\cdots w_{n_k})^{-\gamma}e_{n_k}\big)$ to zero is equivalent to the convergence of $(w_1\cdots w_{n_k})$ to $+\infty$. Hence, we may formulate the following corollary.
\begin{corollary}\label{cor:wslp}
Let $X$ be a Fr\'echet sequence algebra for the coordinatewise product and with a continuous norm.
Assume that $(e_n)$ is a Schauder basis for $X$ and that $(e_n)$ is bounded. Assume also that, for all admissible weights $w$, for all $\gamma>0$, $w^\gamma$ is admissible.
Let $B_w$ be a bounded weighted shift on $X$. The following assumptions are equivalent. \begin{enumerate}[(i)] \item $B_w$ supports a hypercyclic algebra. \item $B_w$ is hypercyclic. \item There exists a sequence $(n_k)$ such that $(w_1\cdots w_{n_k})$ tends to $+\infty$. \end{enumerate} \end{corollary}
\begin{remark} On $H(\mathbb C)$, the sequence $(z^n)$ is unbounded. Nevertheless, any hypercyclic weighted shift $B_w$ on $H(\mathbb C)$ supports a hypercyclic algebra. Indeed, for a sequence of integers $(n_k)$, \begin{align*} &(w_1\cdots w_{n_k})^{-1}z^{n_k}\textrm{ tends to $0$ in }H(\mathbb C)\\ \iff&\forall q\geq 1,\ (w_1\cdots w_{n_k})^{-1}q^{n_k}\textrm{ tends to $0$}\\ \iff&\forall q\geq 1,\ \forall \gamma>0,\ (w_1\cdots w_{n_k})^{-\gamma}q^{n_k}\textrm{ tends to $0$}\\ \iff&\forall \gamma>0,\ (w_1\cdots w_{n_k})^{-\gamma}z^{n_k}\textrm{ tends to $0$ in }H(\mathbb C). \end{align*} \end{remark}
\begin{remark} Theorem \ref{thm:wsprecised} points out one difficulty when dealing with hypercyclic algebras: to admit a hypercyclic algebra is not a property preserved by similarity.
Indeed, let $X=\{(x_n)\in\omega : |x_n|2^n\to 0\}$ endowed with $\|x\|=\sup_n |x_n|2^n$ and let $w$ be the weight such that $w_1\cdots w_n=n\cdot 2^n$ for all $n\geq 1$. Then $(w_1\cdots w_n)^{-1}2^n$ goes to zero whereas $(w_1\cdots w_n)^{-1/2}2^n$ tends to $+\infty$, showing that $B_w$ is hypercyclic but that no square vector $x^2$ belongs to $HC(B_w)$.
Let now $(\rho_n)$ be defined by $\rho_1=1$ and $\rho_n=n/(n-1)$ for $n\geq 2$. Then $B_w$ acting on $X$ is similar to $B_\rho$ acting on $c_0$, the similarity being given by $S:X\to c_0,\ (x_n)\mapsto (2^n x_n)$. But $B_\rho$ admits a hypercyclic algebra, which is not the case of $B_w$. Of course, the problem is that $S$ is not a morphism of algebra. \end{remark}
When $X$ does not admit a continuous norm, one cannot apply Theorem \ref{thm:wsprecised}. The space $\omega$ is the prototypal example of a Fr\'echet space without a continuous norm (in fact, by a result of Bessaga and Pelczinski \cite{BesPel57}, a Fréchet space fails to admit a continous norm if and only if it has a subspace isomorphic to $\omega$) and we shall now concentrate on this space. On $\omega$, for all weight sequences $w=(w_n)$, the weighted shift $B_w$ is bounded, hypercyclic and satisfies $(iv)$. If the sequences $(w_l\cdots w_{n+l})$ converge to $+\infty$ for all $\ell\geq 0$, then an easy modification of the proof of the previous theorem shows that $B_w$ admits a hypercyclic algebra. On the other hand, if the sequences $(w_l\cdots w_{n+l})$ converge to $0$ for all $\ell\geq 0$, then we may modify the previous proof using Corollary \ref{cor:thmgenmax} instead of Corollary \ref{cor:thmgenmin} to prove that $B_w$ still admits a hypercyclic algebra. A completely different case is that of the unweighted shift $B$. It is a hypercyclic multiplicative operator on $\omega$. By \cite[Theorem 16]{BCP18}, $B$ supports a hypercyclic algebra if and only if for each nonconstant polynomial $P\in\mathbb C[X]$ with $P(0)=0$, the map $\tilde P:\omega\to\omega,\ x\mapsto P(x)$ has dense range. This is clearly true.
We now show that every weighted shift on $\omega$ admits a hypercyclic algebra showing that, coordinate by coordinate, $B_w$ behaves like one of the three previous models. \begin{theorem}\label{thm:algebraomega} Every weighted shift $B_w$ on $\omega$ endowed with the coordinatewise product supports a hypercyclic algebra. \end{theorem} \begin{proof} For $V$ a non-empty open subset of $\omega$, $I\subset\mathbb N$ finite and non-empty and $s>0$, let us define \begin{align*}
E(I,s)=\Big\{P\in\mathbb C[z]:\ & |\hat P(\min I)|\geq 1/s,\ |\hat P(\max I)|\geq 1/s,\\
&|\hat P(n)|\leq s\textrm{ for all }n\in\mathbb N,\\ &\hat P(n)=0\textrm{ when }n\notin I\Big\} \end{align*} \begin{align*} \mathcal A(I,s,V)=\Big\{u\in \omega:\ \forall P\in E(I,s),\ \exists N\geq 1,\ T^N(P(u))\in V\Big\}. \end{align*} As in the proof of Theorem \ref{thm:generalcriterion}, it is enough to prove that each set $\mathcal A(I,s,V)$ is dense and open. The last property follows easily from the compactness of $E(I,s)$. Thus, let us fix $I,s$ and $V$ and let us prove that $\mathcal A(I,s,V)$ is dense. We set $m_0=\min(I)$ and $m_1=\max(I)$. Let $U$ be a non-empty open subset of $\omega$. Let $p\in\mathbb N_0$, $u_0,\dots,u_p,v_0,\dots,v_p\in \mathbb C$ and $\varepsilon>0$ be such that, for all $x,y\in\omega$,
\[ |x_l-u_l|<\varepsilon\textrm{ for all }l=0,\dots,p\textrm{ implies }x\in U, \]
\[ |y_l-v_l|<\varepsilon\textrm{ for all }l=0,\dots,p\textrm{ implies }y\in V.\] Let us first look at the sequence $(w_1\cdots w_n)$. Three possibilities (which are not mutually exclusive) can occur: \begin{itemize} \item either $(w_1\cdots w_n)$ is bounded and bounded below; \item or it admits a subsequence going to zero; \item or it admits a subsequence going to $+\infty$. \end{itemize} Thus, we get the existence of a subsequence $(w_1\cdots w_{n_k})$ going to $a_0\in [0,+\infty]$. We then do the same with $(w_2\cdots w_{n_k+1})$ and so on. By successive extractions, we get the existence of a sequence of integers $(n_k)$ (we can assume that $n_{k+1}-n_k>p$ for all $k$ and that $n_0>p$) and of $a_0,\dots,a_p\in [0,+\infty]$ such that, for all $l=0,\dots,p$, $(w_{l+1}\cdots w_{n_k+l})$ tends to $a_l$. We set $A_1=\{l\in\{0,\dots,p\}:\ a_l=+\infty\}$, $A_2=\{l\in\{0,\dots,p\}:\ a_l=0\}$ and $A_3=\{l\in\{0,\dots,p\}:\ a_l\in(0,+\infty)\}$.
We fix now $(\alpha(k))$, $(\beta(k))$ two sequences of non-zero complex numbers and $(z(k))$ a sequence in $\mathbb C^{p+1}$ such that $(\alpha(k),\beta(k),z(k))$ is dense in $\mathbb C^{p+3}$. We set $$x=u+\sum_{k=0}^{+\infty}y(k)$$ where, for $l=0,\dots,p$, $$y_{n_k+l}(k)= \left\{ \begin{array}{cl} \displaystyle \frac{v_l^{1/m_0}}{\alpha(k)^{1/m_0}(w_{l+1}\cdots w_{n_k+l})^{1/m_0}}&\textrm{provided }l\in A_1,\\[0.5cm] \displaystyle \frac{v_l^{1/m_1}}{\beta(k)^{1/m_1}(w_{l+1}\cdots w_{n_k+l})^{1/m_1}}&\textrm{provided }l\in A_2,\\[0.5cm] z_l(k)&\textrm{ provided }l\in A_3 \end{array} \right.$$ and $y_i(k)=0$ if $i\neq n_k,\dots,n_k+p$.
We claim that $x\in U\cap \mathcal A(I,s,V)$. The definition of $\varepsilon$ and $p$ ensure that $x\in U$. Let $P\in E(I,s)$. There exists an increasing function $\phi:\mathbb N\to\mathbb N$ such that $\alpha(\phi(k))\to \hat P(m_0)$, $\beta(\phi(k))\to \hat P(m_1)$ and $a_l P(z_l(\phi(k)))\to v_l$ for all $l\in A_3$. We claim that $(B_w^{n_{\phi(k)}}(P(x)))$ belongs to $V$ provided $k$ is large enough. It suffices to prove that for $l=0,\dots,p$, the $l$-th coordinate of $B_w^{n_{\phi(k)}}(P(x))$ tends to $v_l$. Assume first that $l\in A_1$. This $l$-th coordinate is equal to $$w_{l+1}\cdots w_{n_{\phi(k)}+l}P\left(\frac{v_l^{1/m_0}}{\alpha(\phi(k))^{1/m_0}(w_{l+1}\cdots w_{n_{\phi(k)}+l})^{1/m_0}}\right).$$ Now, since $w_{l+1}\cdots w_{n_{\phi(k)}+l}$ tends to $+\infty$, and $m_0=\min(I)$, $$w_{l+1}\cdots w_{n_{\phi(k)}+l}P\left(\frac{v_l^{1/m_0}}{\alpha(\phi(k))^{1/m_0}(w_{l+1}\cdots w_{n_{\phi(k)}+l})^{1/m_0}}\right)=\hat P(m_0)\frac{v_l}{\alpha(\phi(k))}+o(1)$$ and this tends to $v_l$. When $l\in A_2$, the proof is similar since now, because $w_{l+1}\cdots w_{n_{\phi(k)}+l}$ tends to 0, and $m_1=\max(I)$, $$w_{l+1}\cdots w_{n_{\phi(k)}+l}P\left(\frac{v_l^{1/m_1}}{\beta(\phi(k))^{1/m_1}(w_{l+1}\cdots w_{n_{\phi(k)}+l})^{1/m_1}}\right)=\hat P(m_1)\frac{v_l}{\beta(\phi(k))}+o(1)$$ and this also goes to $v_l$. Finally, when $l\in A_3$, the $l$-th coordinate of $B_w^{n_{\phi(k)}}(P(x))$ is equal to $w_{l+1}\cdots w_{n_{\phi(k)}+l}P(z_l(\phi(k)))$ which tends again to $v_l$. \end{proof}
Theorem \ref{thm:algebraomega} has an analogue (with a completely different proof!) if we endow $\omega$ with the Cauchy product: see \cite[Corollary 3.9]{FalGre18}. We also point out that the existence of a continuous norm is an important assumption in several problems in linear dynamics, for instance for the existence of a closed infinite dimensional subspace of hypercyclic vectors (see \cite{menet}).
\subsection[Bilateral shifts]{Bilateral shifts on Fréchet sequence algebras with the coordinatewise product}
In this section, we investigate the case of bilateral shifts on a Fréchet sequence algebra $X$ on $\mathbb Z$; namely, $X$ is a subset of $\mathbb C^\mathbb Z$ endowed with the coordinatewise product under which it is an $F$-algebra. We intend to give an analogue of Theorem \ref{thm:wsprecised} for bilateral shifts on $X$. The statement and the methods are close to what happens for unilateral shifts. Since we do not want to give an exhaustive list of examples in this work, there is an extra interest for looking at bilateral shifts: a small subtility appears in this case, since the condition that appears is not symmetric for the positive part of the weight and for the negative one. This will lead us to an interesting example of a hypercyclic operator $T$ supporting a hypercyclic algebra such that $T^{-1}$ does not.
\begin{theorem}\label{thm:bilateralshifts} Let $X$ be a Fréchet sequence algebra on $\mathbb Z$ for the coordinatewise product, with a continuous norm. Assume that $(e_n)_{n\in\mathbb Z}$ is a Schauder basis for $X$. Let also $B_w$ be a bounded bilateral shift on $X$ such that, for all $\gamma\in(0,1)$, $B_{w^\gamma}$ is bounded. The following assertions are equivalent. \begin{enumerate}[(i)] \item $B_w$ supports a hypercyclic algebra. \item For all $m\geq 1$, for all $L\in\mathbb N$, there exists a sequence of integers $(n_k)$ such that, for all $l=-L,\dots,L$, $\big( (w_{l+1}\cdots w_{n_k+l})^{-1/m}e_{n_k+l}\big)$ and $\big( w_{l}\cdots w_{-n_k+l+1}e_{-n_k+l}\big)$ tend to zero. \end{enumerate} \end{theorem} \begin{proof} $(ii)\implies (i)$. We intend to apply Corollary \ref{cor:thmgenmin}. Let $1\leq m_0<m_1$, let $U,V$ be nonempty open subsets of $X$ and let $W$ be a neighbourhood of zero. Let $x,y$ belonging to $U$ and $V$ respectively, with finite support contained in $[-p,p]$. Write $y=\sum_{l=-p}^p y_le_l$ and let $(n_k)$ be the sequence given in (ii) for $m=m_0$ and $L=p$. Define $$u:=u(n_k)=x+\sum_{l=-p}^p \frac{y_l^{1/m_0}}{(w_{l+1}\cdots w_{n_k+l})^{1/m_0}}e_{n_k+l}.$$ Provided $k$ is large enough, $u$ belongs to $U$. Moreover, for $m\in\{m_0,\dots,m_1\}$, $$B_w^{n_k}(u^m)=\sum_{l=-p}^p w_l\cdots w_{-n_k+l+1}x_l^m e_{-n_k+l}+ \sum_{l=-p}^p \frac{y_l^{m/m_0}}{(w_{l+1}\cdots w_{n_k+l})^{\frac m{m_0}-1}}e_l.$$ For all values of $m$, it is clear that $$\sum_{l=-p}^p w_l\cdots w_{-n_k+l+1}x_l^me_{-n_k+l}\xrightarrow{k\to+\infty}0.$$ Hence, for $m=m_0$ and provided $k$ is large enough, $B_w^{n_k}(u^{m_0})$ belongs to $V$. Furthermore, if $m>m_0$, since each sequence $(w_{l+1}\cdots w_{n_k+l})^{-1}$ goes to zero (recall that $(e_n)$ is bounded below), then $B_w^{n_k}(u^m)$ belongs to $W$ for large values of $k$, showing that $B_w$ admits a hypercyclic algebra.
$(i)\implies (ii)$. The proof is slightly more difficult than for unilateral shifts. Fix $m$ and $L$ and let $x\in X$ be such that $x^m\in HC(B_w)$. Let $(s_k)$ be an increasing sequence of integers such that $B_w^{s_k}(x^m)$ tends to $e_{-L}+\cdots+e_L$. We fix some $s\in\mathbb N$ (which can be taken equal to some $s_{k_0}$) such that, for all $l=-L,\dots,L$, the $l$-th coordinate of $B_w^s(x)$ is not equal to zero. We then consider $y\in X$ defined by $y_l=(w_{l+1}\cdots w_{l+s})^{1/m}x_{l+s}$ (namely, $y=B_{w^{1/m}}^s(x)$) and we set $n_k=s_k-s$. It is easy to check that $B_w^{n_k}(y^m)=B_w^{s_k}(x^m)$. Hence, it goes to $e_{-L}+\cdots +e_L$. This implies that \begin{itemize} \item for all $l=-L,\cdots, L$, $$w_l\cdots w_{-n_k+l+1}y_l^m e_{-n_k+l}\textrm{ tends to }0.$$ \item for all $l=-L,\cdots,L$, $$w_{l+1}\cdots w_{n_k+l}y_{n_k+l}^m\textrm{ tends to }1.$$ \end{itemize} We conclude as in the unilateral case, using that $y_l$ is never equal to zero for $l=-L,\cdots,L$. \end{proof} We can then state corollaries similar to what happens in the unilateral case.
\begin{corollary}
Let $X$ be a Fr\'echet sequence algebra on $\mathbb Z$ for the coordinatewise product and with a continuous norm.
Assume that $(e_n)_{n\in\mathbb Z}$ is a Schauder basis for $X$. Assume also that, for all admissible weights $w$, for all $\gamma\in (0,1)$, $w^\gamma$ is admissible.
Let $B_w$ be a bounded bilateral weighted shift on $X$. The following assumptions are equivalent. \begin{enumerate}[(i)] \item $B_w$ supports a hypercyclic algebra. \item For all $\gamma>0$, there exists a sequence $(n_k)$ such that $((w_1\cdots w_{n_k})^{-\gamma} e_{n_k})$ tends to zero and $(w_{-1}\cdots w_{-n_k}e_{-n_k})$ tends to $0$. \end{enumerate} \end{corollary}
\begin{corollary}
Let $X$ be a Fr\'echet sequence algebra on $\mathbb Z$ for the coordinatewise product and with a continuous norm.
Assume that $(e_n)$ is a Schauder basis for $X$ and that $(e_n)$ is bounded. Assume also that, for all admissible weights $w$, for all $\gamma\in(0,1)$, $w^\gamma$ is admissible.
Let $B_w$ be a bounded bilateral weighted shift on $X$. The following assumptions are equivalent. \begin{enumerate}[(i)] \item $B_w$ supports a hypercyclic algebra. \item $B_w$ is hypercyclic. \item There exists a sequence $(n_k)$ such that $(w_1\cdots w_{n_k})$ and $(w_{-1}\cdots w_{-n_k})$ tend to $+\infty$. \end{enumerate} \end{corollary}
On the contrary, the nonsymmetry of the conditions in (ii) of Theorem \ref{thm:bilateralshifts} proves to be useful to get the following example. \begin{example}\label{ex:inverse} There exists an invertible operator $T$ on a Banach algebra such that $T$ supports a hypercyclic algebra and $T^{-1}$ does not. \end{example}
\begin{proof} Let
$$X=\left\{x\in\mathbb C^\mathbb Z:\ x_n (|n|+1)\xrightarrow{n\to\pm \infty}0\right\},$$ endowed with
$$\|x\|=\sup_n |x_n| (|n|+1).$$
Equipped with the coordinatewise product, $X$ is a Fréchet sequence algebra. Let $w$ be the weight defined by $w_0=1$, $w_n=2$ and $w_{-n}=n^2/(n+1)^2$ for $n>0$. For all $\gamma>0$, the weighted shift $B_{w^\gamma}$ is bounded on $X$. Moreover, it satisfies the assumptions of Theorem \ref{thm:bilateralshifts} with $(n_k)$ equal to the whole sequence of integers. In particular, $w_{-1}\cdots w_{-n} \|e_{-n}\|=(n+1)^{-1}$ tends to zero.
It is plain that $B_w$ is invertible and that its inverse is the forward shift $F_\rho$, defined by $F_\rho(e_n)=\rho_{n+1}e_{n+1}$ with $\rho_n=1/w_n$. Assume that $F_\rho$ supports a hypercyclic algebra. Then we apply the symmetrized version of Theorem \ref{thm:bilateralshifts} adapted to forward shifts with $m=2$ to get the existence of a sequence $(n_k)$ such that $(\rho_{-1}\cdots \rho_{-n_k})^{-1/2}e_{-n_k}$ tends to zero. This is impossible since
$$\left\|(\rho_{-1}\cdots \rho_{-n_k})^{-1/2}e_{-n_k}\right\|\sim_{k\to+\infty}\frac{n_k}{n_k^{2\times 1/2}}=1.$$ \end{proof}
\color{black}
\subsection{Fréchet sequence algebras for the convolution product}
This subsection is devoted to the proof of Theorem \ref{thm:wscauchy}. We first have to give the meaning of a regular Fréchet sequence algebra.
Let $(X,(\|\cdot\|_q))$ be such a Fréchet sequence algebra for the Cauchy product. We will say that $X$ is \emph{regular} provided that it satisfies the following three properties: \begin{enumerate}[(a)]
\item $X$ admits a continuous norm;
\item $(e_n)$ is a Schauder basis of $X$;
\item for any $r\geq 1$, there exists $q\geq 1$ and $C>0$ such that, for all $n,k\geq 0$,
\[ \|e_n\|_r\cdot \|e_k\|_r\leq C\|e_{n+k}\|_q.\] \end{enumerate}
Let us make some comments on these assumptions. Conditions (a) and (b) are standard in this work. We shall use (a) by assuming that $\|e_n\|_q>0$ for all $n\in\mathbb N_0$ and all $q>0$. Regarding (c), it should be thought as a reverse inequality for the continuity of the product in $X$. Observe also that $H(\mathbb C)$ and $\ell_1$ are clearly regular. However, this is not the case of all Fréchet sequence algebras for the Cauchy product. Pick for instance any sequence $(a_n)$ of positive real numbers such that, for all $n,p,q\in\mathbb N_0$, with $n=p+q$, $a_n\leq a_p\cdot a_q$
and $a_n^2/a_{2n}\to +\infty$ (the sequence $a_n=1/n!$ does the job). Then the Banach space $X=\{x\in\omega:\|x\|_X=\sum_{n\geq 0} a_n |x_n|<+\infty\}$ is a Fréchet sequence algebra for the Cauchy product which does satisfy (c).
A consequence of (c) is the following technical lemma which will be crucial later. \begin{lemma}\label{lem:wscauchy}
Let $X$ be a regular Fréchet sequence algebra for the Cauchy product and let $(w_n)$ be an admissible weight sequence on $X$. Then, for all $M\geq 1$,
for all $r\geq 1$, and for all $\rho \geq 0$, there exist $C>0$ and $q\geq r$ such that, for all $n\geq M$, for all $u<v$ in $\{n-M,\dots,n\}$, for all $k\in\{n-M+\rho,\dots,n+\rho \}^{v-u}$,
\begin{equation}\label{eq:wscauchy} \prod_{j=1}^{v-u} w_{k_j} \|e_u\|_r\leq C \|e_v\|_q.
\end{equation}
\end{lemma}
Before to proceed with the proof, let us comment the statement of Lemma \ref{lem:wscauchy}. The inequality \eqref{eq:wscauchy} is nothing else than the continuity of $B_w$ if we assume that $k_j=u+j$ for $j=1,\dots,v-u$. The regularity of $X$ (and more precisely the third condition) will imply that we may slightly move the indices $k_j$.
\begin{proof}
Let us fix $M\geq 1$ and observe that $v-u$ may only take the values $1,\dots,M$. Then, upon doing a finite induction and taking suprema, we need only to prove that,
for all $r\geq 1$, and for all $\rho \geq 0$, there exist $C>0$ and $q\geq r$ such that, for all $n\geq M$, for all $u\in\{n-M,\dots,n-1\}$, for all $k\in \{n-M+\rho,\dots,n+\rho \}$,
\begin{equation}\label{eq:lemwscauchy}
w_k \|e_u\|_r\leq C \|e_{u+1}\|_q,
\end{equation}
a property which should be thought as a strong version of the continuity of $B_w$. Assume first that $u\geq k-1$. Then, writing $e_u=e_{k-1}\cdot e_{u-(k-1)}$
and using the continuity of the product and of $B_w$, we get the existence of $C>0$ and $q_1\geq r$ such that
\begin{align*}
w_k \|e_u\|_r&\leq w_k \|e_{k-1}\|_r \cdot \|e_{u-(k-1)}\|_r \\
&\leq C_1 \|e_k\|_{q_1} \cdot \|e_{u-(k-1)}\|_{q_1}.
\end{align*}
We now use property (c) for $r=q_1$ to deduce the existence of $C_2>0$ and $q_2\geq q_1$ such that
\[ w_k \|e_u\|_r \leq C_1C_2 \|e_{u+1}\|_{q_2}. \]
Hence, \eqref{eq:lemwscauchy} is proved for $q=q_2$ and $C=C_1C_2$.
If we assume that $u<k-1$, then the argument is completely similar by exchanging the place where we use the continuity of the product and property (c). Precisely,
\begin{align*}
w_k \|e_u\|_r&=w_k\frac{\|e_u\|_r\cdot \|e_{(k-1)-u}\|_r}{\|e_{(k-1)-u}\|_r}\\
&\leq C_1 w_k \frac{\| e_{k-1} \|_{q_1}}{\|e_{(k-1)-u}\|_r}\\
&\leq C_1 C_2 \frac{\|e_k\|_{q_2}}{\|e_{k-(u+1)}\|_r}\\
&\leq C_1 C_2 \frac{\|e_{k-(u+1)}\|_{q_2}}{\|e_{k-(u+1)}\|_r}\|e_{u+1}\|_{q_2}.
\end{align*}
Hence, \eqref{eq:lemwscauchy} is proved for $q=q_2$ and
\[ C=\max \left\{C_1C_2 \frac{\|e_l\|_{q_2}}{\|e_l\|_{r}}:1\leq l\leq M+\rho -1 \right\}. \] \end{proof}
Lemma \ref{lem:wscauchy} will be used through the following more particular form. \begin{corollary}\label{cor:wscauchy}
Let $X$ be a regular Fréchet sequence algebra for the Cauchy product and let $(w_n)$ be an admissible weight sequence on $X$. Then, for all $m\geq 1$, for all $N\geq 1$, for
all $r\geq 1$, and for all $\rho \geq 0$, there exist $C>0$ and $q\geq 1$ such that, for all $n\geq mN$, for all $s\in\{1,\dots,N\}$,
\[ (w_{n-s+1+\rho })^{m-1}\cdots (w_{n-1+\rho })^{m-1}(w_{n+\rho} )^{m-1} \|e_{n-ms+m\rho}\|_r \leq C\|e_{n-s+\rho}\|_q. \] \end{corollary} \begin{proof}
We apply the previous lemma with $M=mN$ to get $q'\geq r$ and $C'>0$ such that
\[(w_{n-s+1+\rho })^{m-1}\cdots (w_{n-1+\rho })^{m-1}(w_{n+\rho} )^{m-1} \|e_{n-ms}\|_r \leq {C'}\|e_{n-s}\|_{q'}.\] Now, using property (c) and the continuity of the product on a Fréchet algebra, we get $q\geq q'\geq r$ and $C''>0$ with \begin{align*}
&(w_{n-s+1+\rho })^{m-1}\cdots (w_{n-1+\rho })^{m-1}(w_{n+\rho} )^{m-1} \|e_{n-ms+m\rho}\|_r \\ &\quad\quad\quad\quad\leq (w_{n-s+1+\rho })^{m-1}\cdots (w_{n-1+\rho })^{m-1}(w_{n+\rho} )^{m-1} \|e_{n-ms}\|_r\|e_\rho\|_r\|e_{(m-1)\rho}\|_r \\ &\quad\quad\quad\quad\leq C'\|e_{n-s}\|_{q'}\|e_\rho\|_{q'}\|e_{(m-1)\rho}\|_r \\ &\quad\quad\quad\quad\leq C'C''\|e_{n-s+\rho}\|_q\|e_{(m-1)\rho}\|_r\\&\quad\quad\quad\quad = C\|e_{n-s+\rho}\|_q, \end{align*}
where $C=C'C''\|e_{(m-1)\rho}\|_r$. \end{proof}
Before proceeding with the proof of Theorem \ref{thm:wscauchy}, let us make some comments on the difference between the coordinatewise product and the convolution product. Let $P(z)=\sum_{m\in I}\hat P(m)z^m$ be a nonzero polynomial, $x,y$ with finite support. The work done in Section \ref{sec:criterion} shows that it is important for us to find $u$ close to $x$ and $N$ such that $B_w^N(P(u))$ is close to $y$. In both cases, $u$ will be of the form $u=x+z$, where $z$ has finite support and $\min(\text{supp}(z))\gg \max(\text{supp}(x))$. For the coordinatewise product, each $u^m$ has the same support as $u$. Moreover, since $z$ has to be small, $z^m$ becomes smaller as $m$ increases. Hence, in $P(u)$, the most important term was $u^{m_0}$, where $m_0=\min(I)$ (we assume $\hat P(m)\neq 0\iff m\in I$) and it was natural to apply Corollary \ref{cor:thmgenmin}.
Regarding the convolution product, the support of $u^m$ is now moving to the right: $\max(\text{supp}(u^{m+1}))\geq \max(\text{supp}(u^m))$. By using a specific translation term (an idea coming from \cite{shkarin} and \cite{FalGre19}), we will arrange the choice of $N$ such that $B_w^N(u^m)=0$ when $m<m_1:=\max(I)$ and $B_w^N(u^{m_1})$ is close to $y$. This explains why we will rather use Corollary \ref{cor:thmgenmax}.
\begin{proof}[Proof of Theorem \ref{thm:wscauchy}]
We start with a hypercyclic weighted shift $B_w$ and prove that $B_w$ supports a dense hypercyclic algebra which is not contained in a finitely generated algebra. Let $d\geq1$, $A\in\mathcal P_f(\mathbb N_0^d)\backslash\{\varnothing\}$ and $U_1,...,U_d,$ $V,W\subset X$ be open and non-empty, with $0\in W$. We choose $\beta=\max A$ under the lexicographical order, say $\beta=(m,\beta_2,...,\beta_d)$. Upon interchanging the coordinates in $\mathbb N^d$, we may and will assume that $m>0$. Let $x_1,...,x_d$ belonging respectively to $U_1,...,U_d$ and let $y=\sum_{l=0}^py_le_l$ belonging to $V$. We can find $r\geq1$, $\delta>0$ and a ball $B\subset W$ for the seminorm $\|\cdot\|_r$ and with radius $\delta$ such that $y+B\subset V$ and $x_i+B\subset U_i$, for all $i=1,...,d$.
Since $\beta>\alpha$ for all $\alpha\in A\backslash\{\beta\}$ under the lexicographical order,
we may find integers $s_1,\dots,s_d$ with $s_i>4p$ such that \begin{equation}\label{cond:s} (m-\alpha_1)s_1+(\beta_2-\alpha_2)s_2+\cdots+(\beta_d-\alpha_d)s_d>3p,\text{ for all }\alpha\in A\setminus \{ \beta \}. \end{equation} The procedure to do this is the following. First find $s_d$ such that $(\beta _d-\alpha _d)s_d>3p$, for all $\alpha _d<\beta _d$, $\alpha_d \in \pi _d(A)$. Next, find $s_{d-1}$ such that $(\beta _{d-1}-\alpha_{d-1})s_{d-1}+(\beta_d-\alpha_d)s_d>3p$, for all $\alpha_{d-1}<\beta_{d-1}\in \pi_{d-1}(A)$ and $\alpha_d \in \pi_d(A)$. Continuing inductively, after finitely many steps we define $s_2$ such that $(\beta_2-\alpha_2)s_2+\dots +(\beta_d-\alpha_d)s_d>3p$, for all $\alpha_2<\beta_2$, $\alpha_2\in \pi_2(A)$, and $\alpha_i\in \pi_i(A)$, for $i=3,\dots ,d$. Finally we chose $s_1$ large enough so that $(m-\alpha_1)s_1+(\beta_2-\alpha_2)s_2+\cdots+(\beta_d-\alpha_d)s_d>3p$, for all $\alpha_1<m$, $\alpha_1 \in \pi_1(A)$, $\alpha_i \in \pi_i(A)$, for $i=2,\dots ,p$. This way we get (\ref{cond:s}).
These $s_i$ being fixed, we now choose positive real numbers $\eta_2,\dots,\eta_d$ such that \begin{equation}\label{cond:eta}
\|\eta_ie_{s_i}\|_r<\delta,\text{ for all }i=2,...,d. \end{equation}
We will distinguish two cases in order to apply Corollary \ref{generators}. The most difficult one is $m>1$, an assumption that we now make. We set $\rho=\beta_2s_2+\cdots+\beta_ds_d$ and we consider a sequence $(J_k)$ going to $+\infty$ such that, for all $l=0,\dots,p$, \[ (w_1\cdots w_{mJ_k-3p+l+\rho})^{-1}e_{mJ_k-3p+l+\rho}\xrightarrow{k\to+\infty}0. \] Indeed, let $(m_k)$ be a sequence of integers such that $(w_1\cdots w_{m_k})^{-1}e_{m_k}$ goes to zero and $m_k\geq m+\rho$ for all $k$. Define $J_k$ as the single integer such that $m_k -m < m J_k-2p+\rho\leq m_k$. Then, for all $l=0,\dots,p$, \[ (w_1\cdots w_{mJ_k-3p+\rho+l})^{-1} e_{mJ_k-3p+\rho+l}=B_w^{ m_k-mJ_k+3p-\rho-l}\big( (w_1\cdots w_{m_k})^{-1} e_{m_k}\big) \] which tends to zero by the continuity of $B_w$ and because $$0\leq m_k-mJ_k+3p-\rho-l\leq m+p.$$
We now proceed with the construction of the vectors $u_1,\dots,u_d$ required to apply Corollary \ref{generators}. We set, for $k$ large enough, $N=mJ_k-3p+\rho$ and \begin{gather*}
\varepsilon=\max_{0\leq l\leq p}\left(\frac{\|e_{J_k-3p+l}\|_r}{w_1\cdots w_{mJ_k-3p+l+\rho}}\right)^{\frac{1}{2(m-1)}}\times \min\left(\frac{1}{\|e_{J_k}\|_r},\frac{1}{(w_1\cdots w_{mJ_k+\rho})^{1/m}}\right)^{\frac{1}{2}},\\ d_j=\frac{w_1\cdots w_jy_j}{\eta_2^{\beta_2}\cdots \eta_d^{\beta_d}m\varepsilon^{m-1}w_1\cdots w_{mJ_k-3p+j+\rho}}. \end{gather*} We also define \begin{gather*} u_1=x_1+\sum_{j=0}^pd_je_{J_k-3p+j}+\varepsilon e_{J_k},\\ u_i=x_i+\eta_ie_{s_i},\text{ for }i=2,...,d. \end{gather*} Let us postpone the proof of the following facts. \begin{gather}
\label{condconv:1} \varepsilon \|e_{J_k}\|_r\to0\text{ as }k\to+\infty, \\
\label{condconv:2} |d_j|\cdot \|e_{J_k-3p+j+\rho}\|_r\to0\text{ as }k\to+\infty,\text{ for all }j=0,...,p, \\ \label{condconv:3} \varepsilon^mw_1\cdots w_{mJ_k+\rho}\to0\text{ as }k\to+\infty. \end{gather} From (\ref{condconv:1}) and (\ref{condconv:2}) we get $u_1\in U_1$ if $k$ is large enough and from (\ref{cond:eta}) we get $u_i\in U$ for $i=2,...,d$. We claim that $u^\alpha\in\ker B_w^N$ for all $\alpha\in A$ with $\alpha\neq\beta$. In fact, for a given $\alpha\in A\backslash\{\beta\}$, say $\alpha=(\alpha_1, ..., \alpha_d)$, we have \begin{gather*} \max \left(\text{supp}(u^\alpha)\right) \leq \alpha_1J_k+\alpha_2s_2+\cdots+\alpha_ds_d, \end{gather*} so the claim follows by (\ref{cond:s}) since $\alpha_1\leq m$ and for $k$ large enough, $J_k\geq s_1$.
Finally, for the main power $\beta$ we write $$u^\beta=z+\sum_{j=0}^p \eta_2^{\beta_2}\cdots \eta_d^{\beta_d}m\varepsilon^{m-1}d_j e_{mJ_k-3p+j+\rho}+\varepsilon^m \eta_2^{\beta_2}\cdots \eta_d^{\beta_d}e_{mJ_k+\rho},$$ where the maximum of the support of $z$ is less than $N=mJ_k-3p+\rho$. Indeed, a term in $z$ can come \begin{itemize} \item either from a term in $u_1^{\beta_1}$ with support in $[0,mJ_k-4p]$ so that the maximum of the support of this term is at most $mJ_k-4p+\rho<N$; \item or from a term in some $u_i^{\beta_i}$, $i=2,\dots,d$, with support in $[0,(\beta_i-1)s_i+p]$. The maximum of the support of such a term is then at most $mJ_k+\rho+p-s_i<N$ since $s_i>4p$. \end{itemize} Hence, we get \begin{gather*} B_w^Nu^\beta=y+\frac{\varepsilon^m \eta_2^{\beta_2}\cdots \eta_d^{\beta_d} w_1\cdots w_{mJ_k+\rho}}{w_1\cdots w_{3p}}e_{3p}, \end{gather*} which belongs to $V$ by (\ref{condconv:3}) if $k$ is big enough. It remains now to show that properties (\ref{condconv:1}), (\ref{condconv:2}) and (\ref{condconv:3}) hold true.
Let us first prove \eqref{condconv:1}. By property (c) and an easy induction, there exist $q\geq 1$ and $C>0$ (depending on $r$ and $m$) such that, for all $k\geq 1$ and all $l\in\{0,\dots,p\}$, \begin{align*}
\frac{\|e_{J_k-3p+l}\|_r^{\frac{1}{2(m-1)}}\cdot \|e_{J_k}\|_r}{\|e_{J_k}\|_r^{\frac 12}}
&= \frac{\left(\|e_{J_k-3p+l}\|_r\cdot \|e_{J_k}\|_r^{m-1}\cdot \|e_\rho\|_r\right)^{\frac 1{2(m-1)}}}
{ \|e_\rho\|_r^{\frac 1{2(m-1)}}}\\
&\leq C \|e_{mJ_k-3p+l+\rho}\|_q^{\frac 1{2(m-1)}}. \end{align*} Hence,
\[ \varepsilon \|e_{J_k}\|_r \leq C \max_{0\leq l\leq p} \left(\frac { \|e_{mJ_k-3p+l+\rho}\|_q}{w_1\cdots w_{mJ_k-3p+l+\rho}}\right)^{\frac 1{2(m-1)}} \] and this goes to zero as $k$ tends to $+\infty$.
Regarding \eqref{condconv:2}, we first write \begin{align*}
|d_j|\cdot \|e_{J_k-3p+j+\rho}\|_r &\leq C \frac{\|e_{J_k-3p+j+\rho}\|_r}{w_1\cdots w_{mJ_k-3p+j+\rho}}\times \min_{0\leq l\leq p} \left(\frac{w_1\cdots w_{mJ_k-3p+l+\rho}}{\|e_{J_k-3p+l}\|_r}\right)^{\frac12}\\ &\quad\quad\quad\times
\max\left(\|e_{J_k}\|_r,(w_1\cdots w_{mJ_k+\rho})^{\frac 1m}\right)^{\frac{m-1}{2}} \\
&\leq C \frac{\|e_\rho\|_r^{\frac 12}\cdot \|e_{J_k-3p+j+\rho}\|_r^\frac12}{(w_1\cdots w_{mJ_k-3p+j+\rho})^{\frac12}}\\
&\quad\quad\quad\times \max\left(\|e_{J_k}\|_r,(w_1\cdots w_{mJ_k+\rho})^{\frac 1m}\right)^{\frac{m-1}{2}}, \end{align*} where the last line comes from the continuity of the product, more precisely from
$$ \|e_{J_k-3p+l+\rho}\|_r \leq \|e_{J_k-3p+l}\|_r\cdot \|e_\rho\|_r.$$
Assume first that the maximum is attained for $\|e_{J_k}\|_r$. In that case, using (c) in a similar way we write
\[ \frac{\|e_{J_k-3p+j+\rho}\|_r}{ w_1\cdots w_{mJ_k-3p+j+\rho}} \|e_{J_k}\|_r^{m-1}\leq C \frac{\|e_{mJ_k-3p+j+\rho}\|_{q}}{w_1\cdots w_{mJ_k-3p+j+\rho}} \] and the last parcel goes to zero. If the maximum is attained for $(w_1\cdots w_{mJ_k+\rho})^{\frac 1m}$, we now write \begin{gather*}
\frac{\|e_{J_k-3p+j+\rho}\|_r (w_1\cdots w_{mJ_k+\rho})^{\frac{m-1}m}}{ w_1\cdots w_{mJ_k-3p+j+\rho}}
= \frac{\bigl(\|e_{J_k-3p+j+\rho}\|_r^m (w_1\cdots w_{mJ_k+\rho})^{m-1}\bigr)^{\frac{1}{m}}}{w_1\cdots w_{mJ_k-3p+j+\rho}} \\
\leq C_1 \frac{\left ( (w_1\cdots w_{mJ_k+\rho})^{m-1} \|e_{mJ_k-m(3p-j)+m\rho} \|_{q}\right)^{\frac 1m} }{w_1\cdots w_{mJ_k-3p+j+\rho}}. \end{gather*} Now, using Corollary \ref{cor:wscauchy} for $n=mJ_k$, $N=3p$ and $s=3p-j$, we get the existence of $C_2>0$ and $q'\geq q$ (which does not depend on $k$) such that \begin{align*}
(w_1\cdots w_{mJ_k+\rho} )^{m-1} \|e_{mJ_k-m(3p-j)+m\rho}\|_q &\leq C_2(w_1\cdots w_{mJ_k-3p+j+\rho})^{m-1}\\
&\quad\quad\quad\times \|e_{mJ_k-3p+j+\rho}\|_{q'} \end{align*} so that
\[ \frac{\|e_{J_k-3p+j+\rho}\|_r (w_1\cdots w_{mJ_k+\rho})^{\frac{m-1}m}}{ w_1\cdots w_{mJ_k-3p+j\rho}}
\leq C_1 C_2^{\frac 1m} \left(\frac{\|e_{mJ_k-3p+j+\rho}\|_{q'}}{ w_1\cdots w_{mJ_k-3p+j+\rho}}\right)^{\frac 1m} \] and this goes to zero.
Finally, let us prove \eqref{condconv:3}. The proof is very similar. Indeed, for all $l=0,\dots,p$, \begin{align*}
&\left(\frac{\|e_{J_k-3p+l+\rho}\|_r}{w_1\cdots w_{mJ_k-3p+l+\rho}}\right)^{\frac m{2(m-1)}} (w_1\cdots w_{mJ_k+\rho})^{\frac 12}\\
&\quad\quad\quad\quad\leq \frac{\left(\|e_{J_k-3p+l+\rho}\|_r^m (w_1\cdots w_{mJ_k+\rho})^{m-1}\right)^{\frac1{2(m-1)}}}{(w_1\cdots w_{mJ_k-3p+l+\rho})^{\frac m{2(m-1)}}}\\
&\quad\quad\quad\quad\leq C_1 \frac{\left(\|e_{mJ_k-m(3p-l)+m\rho}\|_q (w_1\cdots w_{mJ_k+\rho})^{m-1}\right)^{\frac1{2(m-1)}}} {(w_1\cdots w_{mJ_k-3p+l+\rho})^{\frac m{2(m-1)}}}\\
&\quad\quad\quad\quad\leq C_1 C_2^{\frac 1{2(m-1)}} \frac{\left( \|e_{mJ_k-3p+l+\rho}\|_{q'} (w_1\cdots w_{mJ_k-3p+l+\rho})^{m-1}\right)^{\frac 1{2(m-1)}}} {(w_1\cdots w_{mJ_k-3p+l+\rho})^{\frac m{2(m-1)}}} \\
&\quad\quad\quad\quad\leq C_1 C_2^{\frac 1{2(m-1)}}\left(\frac{\|e_{mJ_k-3p+l+\rho}\|_{q'}} {w_1\cdots w_{mJ_k-3p+l+\rho}}\right)^{\frac 1{2(m-1)}}, \end{align*} which achieves the proof of \eqref{condconv:3}.
We now sketch briefly the proof when $m=1$. We still set $\rho =\beta_2s_2+\dots +\beta_ds_d$ and we now consider a sequence $(J_k)$ satisfying $$ (w_1\cdots w_{J_k+l+\rho})^{-1}e_{J_k+l+\rho}\xrightarrow{k\to+\infty} 0$$ for all $l=0,\dots,p$. We define \begin{gather*} u_1=x_1+\sum_{j=0}^pd_je_{J_k+j+\rho},\\ u_i=x_i+\eta_ie_{s_i},\text{ for }i=2,...,d, \end{gather*} where $$ d_j=\frac{y_jw_1\dots w_j}{\eta_2^{s_2}\dots \eta_d^{s_d}w_1\dots w_{J_k+j+\rho}}. $$ Setting $N=J_k+\rho$ for $k$ sufficiently large, it is easy to check that $u_i\in U_i$, $i=1,\dots ,d$, $B_w^N(u^{\alpha})=0$ if $\alpha \in A\setminus \{ \beta \}$, and $B_w^N(u^{\beta})\in V$, which concludes the proof. \end{proof}
85
\section{Frequently and upper frequently hypercyclic algebras}
\subsection{How to get upper frequently hypercyclic algebras}\label{sec:fhc}
In \cite{BoGre18}, following the proof made in \cite{BAYRUZSA} that the set of upper frequently hypercyclic vectors is either empty or residual, the authors gave an analogue to Birkhoff's transitivity theorem for upper frequent hypercyclicity. We adapt it in order to get upper frequently hypercyclic algebras.
\begin{proposition}\label{prop:ufhccriterion} Let $T$ be a continuous operator on an $F$-algebra $X$ satisfying the following condition: for each $I\in\mathcal P_f(\mathbb N)\backslash \{\varnothing\}$, there exists $m_0\in I$ such that, for each non-empty open subset $V$ of $X$ and each neighbourhood $W$ of the origin, there is $\delta >0$ such that for each non-empty open subset $U$ and each $N_0 \in \mathbb{N}$, there is $u\in U$ and $N\geq N_0$ satisfying $$ \frac{1}{N+1}\textrm{card}\left\{p\leq N: T^p(u^m)\in W \,\, \mbox{for} \,\, m\in I\backslash\{m_0\}\,\, \mbox{and} \,\, T^p(u^{m_0})\in V\right\}>\delta. $$ Then $T$ admits an upper frequently hypercyclic algebra. \end{proposition}
\begin{proof} Let $(V_k)_{k}$ be a basis for the topology of $X$ and let $(W_j)_{j}$ be a basis of open neighbourhoods of the origin. By the assumption for each $I\in\mathcal P_f(\mathbb N)\backslash \{\varnothing\}$, there exists $m_0=m_0(I)$ such that, for each $k,j$, there is $\delta _{k,j,I}>0$ such that for each non-empty open subset $U$ and each $N_0 \in \mathbb{N}$, there is $u\in U$ and $N\geq N_0$ satisfying \begin{equation*} \frac{1}{N+1}\textrm{card}\left\{p\leq N: T^p(u^m)\in W_j \,\, \mbox{for} \,\, m\in I\backslash\{m_0\} \,\, \mbox{and} \,\, T^p(u^{m_0})\in V_k\right\}>\delta _{k,j,I}. \end{equation*}
We set \begin{align*} A=\bigcap_{k,j,I,N_0\geq 1}\bigcup_{N\geq N_0}\Big\{u\in X&: \frac{1}{N+1}\textrm{card}\big\{p\leq N: T^p(u^m)\in W_j \,\, \mbox{for} \,\, m\in I\backslash\{m_0\} \,\, \\ &\mbox{and} \,\, T^p(u^{m_0})\in V_k\big\}>\delta_{k,j,I} \Big\} \end{align*} and show that $A$ is residual.
For fixed $k,j,I,N_0$ the set
\begin{align*}
\bigcup_{N\geq N_0}\Bigg\{u\in X: &\ \frac{1}{N+1}\textrm{card}\big\{p\leq N: T^p(u^m)\in W_j \,\, \mbox{for} \,\, m\in I\backslash\{m_0\} \,\, \mbox{and}\\ & \,\, T^p(u^{m_0})\in V_k\big\}>\delta_{k,j,I}\Big\}
\end{align*} is clearly open, and the definition of $\delta_{k,j,I}$ implies that it is also dense. By the Baire category theorem $A$ is residual.
Next, we check that if $u$ belongs to $A$ and if $P\in\mathbb C[z]$ is not constant, with $P(0)=0$, then $P(u)\in UFHC(T)$. We write $P(z)=\sum_{m\in I}\hat P(m)z^m$ for some $I\in\mathcal P_f(\mathbb N)\backslash \{\varnothing\}$ and $\hat P(m)\neq 0$ for all $m\in I$. Since $UFHC(T)$ is invariant under multiplication by a scalar we may assume that $\hat P(m_0)=1$.
Let $V$ be a non-empty open set and find $k,j$ such that $V_k+ (\textrm{card}(I)-1)\| \hat P \|_{\infty}W_j\subset V$. For each $N_0 \in \mathbb{N}$ there is $N\geq N_0$ such that $$ \frac{1}{N+1}\textrm{card}\left \{p\leq N: T^p(u^m)\in W_j \,\, \mbox{for} \,\, m\in I\backslash\{m_0\} \,\, \mbox{and} \,\, T^p(u^{m_0})\in V_k\right\}>\delta_{k,j,I}. $$ But if $T^p(u^m) \in W_j$ for $m\in I\backslash\{m_0\}$ and $T^p(u^{m_0})\in V_k$, then $T^p(P(u))\in V$. Therefore, \begin{align*} \frac{1}{N+1}\textrm{card}\left\{p\leq N: T^p(P(u))\in V\right\}> \delta_{k,j,I} \end{align*} which yields that $\overline{\textrm{dens}}(\{p\in \mathbb{N}: T^p(P(u))\in V\})> \delta_{k,j,I}>0$. \end{proof}
We will apply this lemma either for $m_0(I)=\min(I)$ or $m_0=\max(I)$. The proposition gets the simpler forms: \begin{corollary}\label{cor:ufhccriterion}
Let $X$ be an $F$-algebra. If for each nonempty subset $V$ of $X$, for each neighbourhood $W$ of the origin,
for any positive integers $m_0<m_1$, there is $\delta >0$ such that for each nonempty open subset $U$ and each $N_0 \in \mathbb{N}$, there is $u\in U$ and $N\geq N_0$ satisfying $$ \frac{1}{N+1}\textrm{card}\left\{p\leq N: T^p(u^m)\in W \,\, \mbox{for} \,\, m\in \{m_0+1,\dots,m_1\}\,\, \mbox{and} \,\, T^p(u^{m_0})\in V\right\}>\delta $$ then $T$ admits an upper frequently hypercyclic algebra. \end{corollary}
\begin{corollary}\label{cor:ufhccriterionmax}
Let $X$ be an $F$-algebra. If for each nonempty subset $V$ of $X$, for each neighbourhood $W$ of the origin,
for any positive integer $m$, there is $\delta >0$ such that for each nonempty open subset $U$ and each $N_0 \in \mathbb{N}$, there is $u\in U$ and $N\geq N_0$ satisfying $$ \frac{1}{N+1}\textrm{card}\left\{p\leq N: T^p(u^n)\in W \,\, \mbox{for} \,\, n\in \{1,\dots,m-1\}\,\, \mbox{and} \,\, T^p(u^{m})\in V\right\}>\delta $$ then $T$ admits an upper frequently hypercyclic algebra. \end{corollary}
\subsection[Upper frequently hypercyclic algebras I]{Existence of upper frequently hypercyclic algebras for weighted backward shifts - coordinatewise products}
We intend to apply the previous method to backward shift operators and prove Theorem \ref{thm:ufhcws}. The unconditional convergence of the series $\sum_{n\geq 1}(w_1\cdots w_n)^{-1}e_n$ will be used throughout the following lemma. \begin{lemma}\label{lem:unconditionalfhc}
Let $(w_n)$ be a weight sequence such that $\sum_{n\geq 1}(w_1\cdots w_n)^{-1}e_n$ converges unconditionally. Then, for all $\varepsilon>0$, for all $p>0$, for all $M>0$, there exists $N\geq p$ such that, for each sequence of complex numbers $(y(n,l))_{n\geq N,\ 0\leq l\leq p}$ with $|y(n,l)|\leq M$ for all $n,l$, then
\[ \left\|\sum_{n\geq N}\sum_{l=0}^p \frac{y(n,l)}{w_{l+1}\cdots w_{n+l}}e_{n+l}\right\|\leq\varepsilon. \] \end{lemma} \begin{proof} We first observe that the convergence of the series involved follows from the unconditional convergence of each series $\sum_{n\geq 1}(w_{l+1}\cdots w_{n+l})^{-1}e_{n+l}$. Setting $z(n,l)=y(n,l)/M$ and using the triangle inequality, the (almost) homogeneity of the $F-$norm implies that
\[\left\|\sum_{n\geq N}\sum_{l=0}^p \frac{y(n,l)}{w_{l+1}\cdots w_{n+l}}e_{n+l}\right\|\leq (M+1)\sum_{l=0}^p \left\|\sum_{n\geq N}\frac{z(n,l)}{w_{l+1}\cdots w_{n+l}}e_{n+l}\right\|.\] The existence of an $N$ such that the last term is less than $\varepsilon$ now follows directly from the unconditional convergence of the series $\sum_{n}(w_{l+1}\cdots w_{n+l})^{-1}e_{n+l}$ (see the introduction). \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:ufhcws}]
Let $m_0<m_1$ be two positive integers. Let $V,W\subset X$ be open and non-empty with $0\in W$. Let $p\geq 0$, $\varepsilon>0$ and $v=\sum_{l=0}^p v_l e_l$ be such that $B(v,\varepsilon)\subset V$ and $B(0,2\varepsilon)\subset W$. We also set $M=\max(1,\|v\|_\infty)^{m_1/m_0}$. Let $N\in\mathbb N$ be given by Lemma \ref{lem:unconditionalfhc} for these values of $\varepsilon$, $p$ and $M$. Without loss of generality, we may assume $N>p$. We set $\delta=\frac1{2N}$. Let $U\subset X$ be open and non-empty and let $x\in U$ with finite support. We also define, for $k\geq 0$, $$v(k)=\sum_{l=0}^p \frac{v_l^{1/m_0}}{(w_{l+1}\cdots w_{k+l})^{1/m_0}}e_{k+l}.$$ Let $N_1$ be very large (precise conditions on it will be given later; for the moment we just assume that $N_1$ is bigger than the maximum of the support of $x$). We finally set $$u=x+\sum_{k\geq N_1} v(Nk).$$
The unconditional convergence of the series $\sum_k (w_{l+1}\cdots w_{k+l})^{-1/m_0}e_{k+l}$ ensures that $u$ is well-defined and that $\|u-x\|<\varepsilon$ provided $N_1$ is large enough. Let now $m\in\{m_0,\cdots,m_1\}$ and $j\geq N_1$. Then, since $x$ and the $v(kN)$, $k\geq N_1$, have pairwise disjoint support, \begin{eqnarray*} B_w^{Nj}u^m&=&\sum_{l=0}^p \frac{v_l^{m/m_0}}{(w_{l+1}\cdots w_{jN+l})^{\frac{m}{m_0}-1}}e_l\\ &&\quad\quad+\sum_{k>j}\sum_{l=0}^p \frac{v_l^{m/m_0}w_{(k-j)N+l+1}\cdots w_{kN+l}}{(w_{l+1}\cdots w_{kN+l})^{\frac m{m_0}}}e_{(k-j)N+l}\\ &=:&z(1,j,m)+z(2,j,m). \end{eqnarray*}
If $m=m_0$, then $z(1,j,m)=v$ whereas, if $m\in\{m_0+1,\dots,m_1\}$, then since $|v_l|^{m/m_0}\leq M$
and since the sequences $(w_{l+1}\cdots w_{l+n})_n$ go to $+\infty$ (recall that $X$ has a continuous norm), we may adjust $N_1$ so that $\|z(1,j,m)\|<\varepsilon$ for all $j\geq N_1$. On the other hand, we may write \[ z(2,j,m)=\sum_{n\geq N}\sum_{l=0}^p \frac{y(j,n,l,m)}{w_{l+1}\cdots w_{n+l}}e_{n+l}\] where, for $s=k-j\geq 1$, $l=0,\dots,p$, $$y(j,sN,l,m)=\frac{v_l^{m/m_0}}{(w_{l+1}\cdots w_{(s+j)N})^{\frac m{m_0}-1}}$$ and $y(j,n,l,m)=0$ if $n$ is not a multiple of $N$.
Again taking $N_1$ large enough guarantees that $|y(j,n,l,m)|\leq M$ for all $j\geq N_1$, all $n\geq N$, all $l=0,\dots,p$ and all $m=m_0,\dots,m_1$. By the choice of $N$, we get $\|z(2,j,m)\|<\varepsilon$ for all $j\geq N_1$. Summarizing, we have proved that, for all $j\geq N_1$, $B_w^{Nj}u^{m_0}\in V$ and $B_w^{Nj}u^m\in W$ for all $m\in\{m_0+1,\dots,m_1\}$. Hence we may apply Corollary \ref{cor:ufhccriterion} to prove that $B_w$ supports an upper frequently hypercyclic algebra. \end{proof}
The Theorem \ref{thm:ufhcws} can be applied to the following examples: $\lambda B$ on $\ell_p$ for $\lambda>1$, $\lambda D$ on $H(\mathbb C)$ for $\lambda>0$ or $B_w$ on $c_0$ with $w_n=1+\lambda/n$, $\lambda>0$. Regarding this last weight, on $\ell_p$, $B_w$ is upper frequently hypercyclic if and only if $\lambda>1/p$. However we do not know the answer to the following question because of the divergence of $\sum_n (w_1\cdots w_n)^{-1/m}e_n$ for $m>\lambda p$. \begin{question} Let $X=\ell_p$ and $w_n=1+\lambda/n$ for $\lambda>1/p$. Does $B_w$ supports an upper frequently hypercyclic algebra? \end{question}
On $\ell_p$, it is known that $B_w$ is (upper) frequently hypercyclic if and only if $\sum_n (w_1\cdots w_n)^{-p}$ is convergent (see \cite{BAYRUZSA}). \begin{question} Let $X=\ell_p$ endowed with the coordinatewise product and let $w=(w_n)$ be an admissible weight sequence. Assume that $B_w$ supports an upper frequently hypercyclic algebra. Does this imply that $\sum_n (w_1\cdots w_n)^{-\gamma}$ is convergent for all $\gamma>0$? \end{question}
\subsection[Upper frequently hypercyclic algebras II]{Existence of upper frequently hypercyclic algebras for weighted backward shifts - convolution product}
We now study the existence of an upper frequently hypercyclic algebra for weighted backward shifts when the underlying Fréchet algebra is endowed with the convolution product. We shall give a general statement encompassing the case of the multiples of the backward shift and of the derivation operator.
\begin{theorem}\label{thm:ufhcconvolution}
Let $X$ be a regular Fréchet sequence algebra for the Cauchy product and let $(w_n)$ be an admissible weight sequence. Assume that
\begin{enumerate}[(a)]
\item $\sum_{n\geq 1}(w_1\cdots w_n)^{-1}e_n$ converges unconditionally.
\item for all $m\geq 2$, there exists $c\in (0,1)$ such that
\[ \lim_{\sigma\to+\infty} \sup_{z\in c_{00}\cap B_{\ell_\infty}}
\left\|\sum_{n\geq c\sigma} \frac{z_n (w_1\cdots w_{m\sigma})^{(m-1)/m}}{w_1\cdots w_{(m-1)\sigma+n}}e_n\right\|=0.
\] Then $B_w$ admits an upper frequently hypercyclic algebra.
\end{enumerate} \end{theorem}
\begin{proof}
We shall prove that the assumptions of Corollary \ref{cor:ufhccriterionmax} are satisfied. For $m=1$, this follows from condition (a) which implies that $T$ admits a dense
set of (upper) frequently hypercyclic vectors. Thus, let us assume that $m\geq 2$ and let $c\in (0,1)$ be given by (b). We also consider $d\in(c,(1+c)/2)\subset (0,1)$. Let $V$ be a non-empty
open subset of $X$, $W$ a neighbourhood of $0$, $y=\sum_{l=0}^p y_l e_l\in V$ and $\eta>0$ such that $B(0,2\eta)+y\subset V$. Let finally $q>p$ be such that, for all $z\in\ell_\infty$ with $\|z\|_\infty\leq \|y\|_\infty\big(\max(1,w_1,\dots,w_p)\big)^{p+1}$,
\[ \left\|\sum_{n\geq q}\frac{z_n}{w_1\cdots w_n}e_n\right\|<\eta. \]
We intend to prove that, for each non-empty open subset $U$ and each $N_0\in\mathbb N$,
there is $u\in U$ and $N\geq N_0$ satisfying
\[ \frac1{N+1}\textrm{card}\left\{s\leq N:B_w^s(u^n)\in W\textrm{ for }n<m\textrm{ and }B_w^s(u^m)\in V\right\}\geq\frac{d-c}{2\big((m-1)q+qd\big)}. \]
More precisely, we shall prove that, for all $\sigma$ large enough, setting
\[ E_\sigma=\left\{(m-1)q\sigma+qj: c\sigma\leq j<d\sigma\right\} ,\]
there exists $u\in U$ such that, for all $s\in E_\sigma$, $B_w^s(u^n)=0$ for $n<m$ and $B_w^s(u^m)\in V$. Since
\[ \lim_{\sigma\to+\infty} \frac{\textrm{card}(E_\sigma)}{\max(E_\sigma)}=\frac{d-c}{(m-1)q+qd} ,\]
we will get the claimed result.
We thus fix $x\in U$ with finite support (we denote by $p'$ the maximum of the support of $x$) and let $\sigma>0$ be such that $p'<c\sigma$. Inspired
by the proof of Theorem \ref{thm:wscauchy}, we set
\[ u=x+\sum_{j=c\sigma}^{d\sigma-1} \sum_{l=0}^p d_{j,l}e_{qj+l}+\varepsilon e_{q\sigma} \]
where
\begin{align*}
\varepsilon &= \frac{1}{(w_1\cdots w_{mq\sigma})^{1/m}}\\
d_{j,l}& = \frac{y_l}{m\varepsilon^{m-1} w_{l+1}\cdots w_{(m-1)q\sigma+qj+l}}.
\end{align*}
Let us first prove that, provided that $\sigma$ is large enough, $u$ belongs to $U$.
Let $r\geq 1$. Since $X$ is regular, there exists $\rho\geq r$ and $C>0$ such that
\[ \left\| \varepsilon e_{q\sigma}\right\|_r\leq \frac{C}{(w_1\cdots w_{mq\sigma})^{1/m}}\left\|e_{mq\sigma}\right\|^{1/m}_\rho\xrightarrow{\sigma\to+\infty}0. \] Furthermore, \begin{align*}
\left\| \sum_{j=c\sigma}^{d\sigma-1} \sum_{l=0}^p d_{j,l}e_{qj+l} \right\| &=
\left\|\sum_{j=c\sigma}^{d\sigma-1}\sum_{l=0}^p \frac{y_l w_1\cdots w_l (w_1\cdots w_{mq\sigma})^{(m-1)/m}}{m w_1\cdots w_{(m-1)q\sigma+qj+l}}e_{qj+l}\right \|\\
&= \left\|\sum_{n\geq cq\sigma}\frac{z_n (w_1\cdots w_{mq\sigma})^{(m-1)/m}}{w_1\cdots w_{(m-1)q\sigma+n}}e_n\right\| \end{align*} for some eventually null sequence $(z_n)$ such that
$$\|z\|_\infty\leq \|y\|_\infty \big(\max(1,w_1,\dots,w_p)\big)^{p}/m.$$ Assumption (b) allows us to conclude that $\sum_{j=c\sigma}^{d\sigma-1} \sum_{l=0}^p d_{j,l}e_{qj+l}$ tends to zero as $\sigma$ goes to $+\infty$.
Observe now that, for $n<m$, the support of $u^n$ is contained in $[0,nq\sigma]$ so that, for $s\in E_\sigma$, $B_w^s(u^n)=0$. On the other hand, \[ u^m = z+m\varepsilon^{m-1} \sum_{j=c\sigma}^{d\sigma-1} \sum_{l=0}^p d_{j,l}e_{(m-1)q\sigma+qj+l}+\varepsilon^m e_{mq\sigma} \] with $\text{supp}(z)\subset [0,(m-2)q\sigma+2qd\sigma]\cup [0,(m-1)q\sigma+p']$. It is not difficult to see that, because $d<(1+c)/2$, $\max(\text{supp}(z))\leq (m-1)q\sigma+qc\sigma$ for $\sigma$ large enough. Thus, for any $s=(m-1)q\sigma+qk\in E_\sigma$, \[ B_w^s(u^m)=y+\sum_{j=k+1}^{d\sigma-1}\sum_{l=0}^p \frac{y_l}{w_{l+1}\cdots w_{q(j-k)+l}}e_{q(j-k)+l}+\frac{1}{w_1\cdots w_{q\sigma-qk}}e_{q\sigma-qk}. \] We handle the second term of the right hand side of the equality by writing \[ \sum_{j=k+1}^{d\sigma-1} \sum_{l=0}^p \frac{y_l}{w_{l+1}\cdots w_{q(j-k)+l}}e_{q(j-k)+l} =\sum_{n\geq q} \frac{z_n}{w_1\cdots w_n}e_n \]
for some sequence $(z_n)$ such that $\|z\|_\infty\leq \|y\|_\infty \big(\max(1,w_1,\dots,w_p)\big)^p$. By our choice of $q$, this has $F$-norm less than $\eta$. Finally, $\|e_{q\sigma-qk}\|/(w_1\cdots w_{q\sigma-qk})$ becomes also less than $\eta$ provided $\sigma$, and thus $q\sigma-qd\sigma$, becomes large enough. \end{proof}
Assumption (a) in Theorem \ref{thm:ufhcconvolution} is what we need to get an (upper) frequently hypercyclic vector. Assumption (b) is also an assumption around unconditional convergence of the series $\sum_{n\geq 1}(w_1\cdots w_n)^{-1}e_n$. This looks clearer by writing \[ \sum_{n\geq c\sigma} \frac{z_n(w_1\cdots w_{m\sigma})^{(m-1)/m}}{w_1\cdots w_{n+(m-1)\sigma}}e_n=\sum_{n\geq c\sigma} \frac{z_n(w_1\cdots w_{m\sigma})^{(m-1)/m}}{w_{n+1}\cdots w_{n+(m-1)\sigma}}\times\frac{e_n}{w_1\cdots w_n}. \] Although it looks quite technical, it is satisfied by three natural examples (where we always endow the $F$-algebra with the Cauchy product). \begin{example}Let $X=\ell_1$ and $w_n=\lambda>1$ for all $n\in\mathbb N$. Then $B_w$ supports an upper frequently hypercyclic algebra. \end{example} \begin{proof}
The situation is very simple here because, for all $n\geq 1$,
\[ \frac{(w_1\cdots w_{m\sigma})^{(m-1)/m}} {w_{n+1}\cdots w_{n+(m-1)\sigma}}=1, \]
so that (b) is clearly satisfied. \end{proof} \begin{example}
Let $X=\ell_1$ and $w_1\cdots w_n=\exp(n^\alpha)$ for all $n\in\mathbb N$ and some $\alpha\in(0,1)$. Then $B_w$ supports an upper frequently hypercyclic algebra. \end{example} \begin{proof}
That (b) is satisfied follows from the classical asymptotic behavior
\begin{equation}\label{eq:wsexpnalpha}
\sum_{n\geq N}\exp\left(-n^\alpha\right)\sim_{N\to+\infty}\frac 1\alpha N^{1-\alpha}\exp(-N^\alpha).
\end{equation}
Assuming \eqref{eq:wsexpnalpha} is true, we just write \begin{align*}
& \left\|\sum_{n\geq c\sigma} \frac{z_n (w_1\cdots w_{m\sigma})^{(m-1)/m}}{w_1\cdots w_{(m-1)\sigma+n}}e_n\right\| \\ &\quad\quad = \exp\left(\frac{m-1}m m^\alpha\sigma^\alpha\right)\sum_{n\geq c\sigma}\exp\left( -\left((m-1)\sigma+n\right)^\alpha\right)\\
&\quad\quad \sim_{+\infty}C\sigma^{1-\alpha} \exp\left( \left(\frac {m-1}m m^\alpha-(m-1+c)^\alpha\right)\sigma^\alpha\right). \end{align*} Assumption (b) is satisfied for $c$ close enough to $1$, since in that case
\[ (m-1+c)^\alpha > \frac{(m-1)m^{\alpha}}{m}. \]
For the sake of completeness, we just mention that \eqref{eq:wsexpnalpha} follows from the formula of integration by parts:
\[ \int_{N}^{+\infty}\exp(-x^\alpha)dx=\frac 1\alpha N^{1-\alpha}\exp(-N^\alpha)+\frac{1-\alpha}{\alpha}\int_{N}^{+\infty}x^{-\alpha}\exp(-x^\alpha)dx. \]
\end{proof}
\begin{example}The derivation operator $D$ supports an upper frequently hypercyclic algebra on $H(\mathbb C)$.
\end{example}
\begin{proof}
It is sufficient to verify (b) replacing $\|\cdot\|$ by any seminorm $\|\cdot\|_r$. Now,
\begin{align*}
\left\|\sum_{n\geq c\sigma} \frac{z_n (w_1\cdots w_{m\sigma})^{(m-1)/m}}{w_1\cdots w_{(m-1)\sigma
+n}}e_n\right\|_r & =
\sum_{n\geq c\sigma}\frac{(m\sigma)!^{(m-1)/m}}{((m-1)\sigma+n)!}r^n\\
&= \frac{(m\sigma)!^{(m-1)/m}}{r^{(m-1)\sigma}}\sum_{n\geq c\sigma}\frac{r^{(m-1)\sigma+n}}{((m-1)\sigma+n)!}\\
&\leq C r^{c\sigma}\frac{(m\sigma)!^{(m-1)/m}}{(m-1+c)\sigma}.
\end{align*} Since for all $\varepsilon>0$ Stirling's formula implies \begin{gather*} (m\sigma)!^{(m-1)/m}\leq C_\varepsilon \sigma^{(m-1+\varepsilon)\sigma}\\ ((m-1+c)\sigma)!\geq C_\varepsilon\sigma^{(m-1+c-\varepsilon)\sigma}, \end{gather*} choosing $\varepsilon<2c$ it follows that, for all $c\in(0,1)$ and all $r\geq1$, we have \begin{align*}
\left\|\sum_{n\geq c\sigma} \frac{z_n (w_1\cdots w_{m\sigma})^{(m-1)/m}}{w_1\cdots w_{(m-1)\sigma
+n}}e_n\right\|_r & \leq C\frac{r^{c\sigma}}{\sigma^{(c-2\varepsilon)\sigma}}\xrightarrow{\sigma\to+\infty}0. \end{align*} Hence, assumption (b) is verified. \end{proof} Another natural operator that could admit an upper frequently hypercyclic algebra is the backward shift $B_w$ with $w_n=\left(1+\frac \lambda n\right)$, $\lambda>1$, acting over $\ell_1$ with the convolution product. Unfortunately, for this weight, the assumptions in Theorem \ref{thm:ufhcconvolution} are not verified. \begin{question} Let $X=\ell_1$ endowed with the convolution product and $w_n=\left(1+\frac \lambda n\right)$, $\lambda>1$. Does $B_w$ admit an upper frequently hypercyclic algebra? \end{question}
We can ask a similar question for convolution operators $\phi(D)$ on $H(\mathbb C)$, $|\phi(0)|<1$, which are frequently hypercyclic and admit a hypercyclic algebra.
\begin{question}
Let $X=H(\mathbb C)$ and let $\phi:\mathbb C\to\mathbb C$ be a nonconstant entire function with exponential type, not a multiple of an exponential function, with $|\phi(0)|<1$. Does $\phi(D)$ supports an upper frequently hypercyclic algebra? \end{question}
\subsection{Weighted shifts with a frequently hypercyclic algebra on $\omega$}
Despite the result of Falc\'o and Grosse-Erdmann, it is not so difficult to exhibit operators supporting a frequently hypercyclic algebra if we work on the big space $\omega$.
\begin{theorem}\label{thm:fhcalgomega} Let $w=(w_n)_{n\geq 1}$ be a weight sequence such that $(w_{1}\cdots w_{n})$ either tends to $+\infty$ or to 0. Then $B_w$, acting on $\omega$ endowed with the coordinatewise product, supports a frequently hypercyclic algebra. \end{theorem} \begin{proof} We first assume that $(w_1\cdots w_n)$ tends to $+\infty$ and observe that this clearly implies that, for all $l\geq 0$, $(w_{l+1}\cdots w_{n+l})$ tends to $+\infty$. Let $(v(p),m(p))$ be a dense sequence in $\omega\times\mathbb N$, where each $v(p)$ has finite support contained in $[0,p]$. We then write $v(p)=\sum_{l=0}^p v_l(p)e_l$. For $(n,p)\in\mathbb N^2$, we define \[ y(n,p)=\sum_{l=0}^p \frac{v_l(p)^{1/m(p)}}{(w_{l+1}\cdots w_{n+l})^{1/m(p)}}e_{n+l} .\]
By \cite[Lemma 6.19]{BM09} (see the forthcoming Lemma \ref{lem:setsfhc}), there exists a sequence $(A(p))$ of pairwise disjoint subsets of $\mathbb N$, with positive lower density, and such that $|n-n'|\geq p+q+1$ whenever $n\neq n'$ and $(n,n')\in A(p)\times A(q)$. In particular, the vectors $y(n,p)$ for $p\in\mathbb N$ and $n\in A(p)$ have disjoint support. Hence, we may define $u=\sum_{p\in\mathbb N}\sum_{n\in A(p)}y(n,p)$ and we claim that $u$ generates a frequently hypercyclic algebra.
Indeed, let $P\in\mathbb C[z]$ be non-constant with $P(0)=0$, $P(z)=\sum_{m=m_0}^{m_1}\hat P(m) z^m$, $\hat P(m_0)\neq 0$, and let $V$ be a non-empty open subset of $\omega$. Let $p\in\mathbb N$, $\varepsilon>0$ be such that $m(p)=m_0$ and any vector $x\in\omega$ satisfying $|x_l-\hat P(m_0)v_l(p)|<\varepsilon$ for all $l=0,\dots,p$ belongs to $V$. Now, for $l=0,\dots,p$ and $n\in A(p)$, \[ \big(B_w^n P(u)\big)_l=\hat P(m_0) v_l(p)+\sum_{m=m_0+1}^{m_1}\frac{\hat P(m) v_l(p)^{\frac m{m_0}}}{(w_{l+1}\cdots w_{n+l})^{\frac m{m_0}-1}}. \] Since $(w_{l+1}\cdots w_{n+l})$ tends to $+\infty$ for all $l$, $B_w^n P(u)$ belongs to $V$ for all $n$ in a cofinite subset of $A(p)$. Hence, $P(u)$ is a frequently hypercyclic vector for $B_w$.
The proof is completely similar if we assume that $(w_1\cdots w_n)$ tends to $0$. The only difference is that the dominant term is now given by the term of highest degree of $P$, namely we choose $p$ such that $m(p)=m_1$ and we write \[ \big(B_w^n P(u)\big)_l=\hat P(m_1) v_l(p)+\sum_{m=m_0}^{m_1-1}\frac{\hat P(m) v_l(p)^{\frac m{m_1}}}{(w_{l+1}\cdots w_{n+l})^{\frac m{m_1}-1}}. \] We will conclude because, for $m<m_1$, $(w_{l+1}\cdots w_{n+l})^{\frac m{m_1}-1}$ tends to $+\infty$. Details are left to the reader. \end{proof}
The unweighted backward shift on $\omega$ (still endowed with the coordinatewise product) supports a frequently hypercyclic algebra. Indeed, more generally, let $T$ be a multiplicative operator on an $F$-algebra $X$ with the property that for every non-zero polynomial $P$ vanishing at the origin, the map $$ \tilde{P}:X\rightarrow X, x\mapsto P(x) $$ has dense range. Then if $T$ is frequently hypercyclic, it supports a frequently hypercyclic algebra. The reason for that is the simple observation that if $U$ is a non-empty open set of $X$ and $P$ a non-zero polynomial vanishing at the origin,
\[ \left\{n\in\mathbb N:\ T^n(P(x))\in U\right\}=\left\{n\in\mathbb N:\ T^n x\in \tilde{P}^{-1}(U)\right\}. \]
From the same observation we may conclude that the translation operators $T_a$ acting on $C^{\infty}(\mathbb{R},\mathbb{C})$ for $a\in \mathbb{R}, a\neq 0$, admit a frequently hypercyclic algebra. The fact that $C^{\infty}(\mathbb{R},\mathbb{C})$ has the above mentioned property is proven in \cite[Proposition 20]{BCP18}.
\subsection[Sequence of sets with positive lower density]{A sequence of sets with positive lower density which are very far away from each other}
The remaining part of this section is devoted to the proof of Theorem \ref{thm:afhcc0}. The starting point to exhibit frequently hypercyclic vectors is the following lemma on the existence of subsets of $\mathbb N$ with positive lower density which are sufficiently separated.
\begin{lemma}[Lemma 6.19 in \cite{BM09}]\label{lem:setsfhc} Let $(a(p))$ be any sequence of positive real numbers. Then one can find a sequence $(A(p))$ of pairwise disjoint subsets of $\mathbb N$ such that \begin{enumerate}[(i)] \item each set $A(p)$ has positive lower density;
\item $\min A(p)\geq a(p)$ and $|n-n'|\geq a(p)+a(q)$ whenever $n\neq n'$ and $(n,n')\in A(p)\times A(q)$. \end{enumerate} \end{lemma}
To produce a frequently hypercyclic algebra for a weighted shift on $c_0$, we will need a refined version of this lemma where we add new conditions of separation.
\begin{theorem}\label{thm:setsafhc}
Let $(a(p))$ be any sequence of positive real numbers. Then one can find a sequence $(A(p))$ of pairwise disjoint subsets of $\mathbb N$ such that \begin{enumerate}[(i)] \item each set $A(p)$ has positive lower density;
\item $\min A(p)\geq a(p)$ and $|n-n'|\geq a(p)+a(q)$ whenever $n\neq n'$ and $(n,n')\in A(p)\times A(q)$.
\item for all $C>0$, there exists $\kappa>0$ such that, for all $(n,n')\in A(p)\times A(q)$ with $p\neq q$ and $\max(n,n')\geq \kappa$, then $|n-n'|\geq C$. \end{enumerate} \end{theorem}
The proof of this theorem is rather long. The strategy is to construct a sequence of sets satisfying only (i) and (iii), and then to modify them to add (ii). We begin with two sets. \begin{lemma}\label{lem:setsafhc1}
Let $E\subset\mathbb N$ be a set with positive lower density. There exist $A,B\subset E$ disjoint, with positive lower density, and such that, for all $C>0$, there exists $\kappa>0$ such that, for all $n\in A$ and all $n'\in B$ with $\max(n,n')\geq \kappa$, then $|n-n'|\geq C$. \end{lemma}
\begin{proof} We write $E=\{n_j:\ j\in\mathbb N\}$ in an increasing order. We set, for $k\geq 1$, $u_k=k$, $v_k=\lfloor \sqrt k\rfloor$. We define sequences $(M_k)$, $(N_k)$, $(P_k)$ and $(Q_k)$ by setting $M_1=1$ and, for $k\geq 1$, \[ N_k=M_k+u_k,\ P_k=N_k+v_k,\ Q_k=P_k+u_k,\ M_{k+1}=Q_k+v_k.\] We then define \[I=\bigcup_k [M_k,N_k),\ \ J=\bigcup_k [P_k,Q_k), \] \begin{align*} A=\left\{n_j:\ j\in I\right\},\ \ B=\left\{n_j:\ j\in J\right\}. \end{align*} The sets $I$ and $J$ have positive lower density. Indeed, for $N\in\mathbb N$, let $k$ be such that $N\in [M_k,M_{k+1})$. Then \[ \frac{\textrm{card}(I\cap [1,\dots,N])}N\geq \frac{u_1+\cdots+u_{k-1}}{2(u_1+\cdots+u_k+v_1+\cdots+v_k)}\geq\frac 14 \] provided $k$ is large enough. The same is true for $J$. Since $E$ has positive lower density, this yields that $A$ and $B$ have positive lower density. Moreover, let $C>0$. There exists $k\geq 0$ such that $v_{k-1}\geq C$. We set $\kappa=n_{M_k}$. Let $(n,n')\in A\times B$ with $\max(n,n')\geq \kappa$.
Assume for instance that $n\geq n_{M_k}$ and write $n=n_j$, $n'=n_{j'}$. Then $j\geq M_k$ and the construction of the sets $I$ and $J$ ensure that $j'$ does not belong to $[j-v_{k-1},j+v_{k-1}]$. Thus, $|n-n'|\geq |j-j'|\geq C$. \end{proof}
It is not difficult to require that (ii) in Theorem \ref{thm:setsafhc} holds when we restrict ourselves to $p=q$. \begin{lemma}\label{lem:setsafhc3}
Let $A\subset\mathbb N$ with positive lower density and $a>0$. There exists $B\subset A$ with positive lower density, $\min(B)\geq a$ and $|n-n'|\geq a$ for all $n,n'\in B$, $n\neq n'$. \end{lemma} \begin{proof} Write $A=\{n_j:\ j\in\mathbb N\}$ in an increasing order and define $B=\{n_{ka}:\ k\in\mathbb N\}$. \end{proof}
We then go inductively from two sets to a sequence of sets.
\begin{lemma}\label{lem:setsafhc2} There exists a sequence $(A(p))$ of pairwise disjoint subsets of $\mathbb N$ such that \begin{enumerate}[(i)] \item each set $A(p)$ has positive lower density;
\item for all $C>0$, there exists $\kappa>0$ such that, for all $(n,n')\in A(p)\times A(q)$ with $p\neq q$ and $\max(n,n')\geq \kappa$, then $|n-n'|\geq C$. \end{enumerate} \end{lemma} \begin{proof} We shall construct by induction two sequences of sets $(A(p))$ and $(B(p))$ and a sequence of integers $(\kappa_k)$ such that, at each step $r$, \begin{enumerate}[(a)] \item for all $1\leq p\leq r$, $A(p)$ and $B(p)$ are disjoint and have positive lower density. \item for all $1\leq p<q\leq r$, $A(q)\subset B(p)$ and $B(q)\subset B(p)$.
\item for all $C>0$, there exists $\kappa>0$ such that, for all $1\leq p\leq q\leq r$, for all $n\in A(p)$ and $n'\in B(q)$, $\max(n,n')\geq\kappa\implies |n-n'|\geq C$.
\item for all $k\in\{1,\dots,r\}$, for all $1\leq p\leq q\leq r$, for all $n\in A(p)$ and $n'\in B(q)$, $\max(n,n')\geq \kappa_k\implies |n'-n|\geq k$. \end{enumerate} It is straightforward to check that the resulting sequence $(A(p))$ satisfies the conclusions of Lemma \ref{lem:setsafhc2}. Observe nevertheless that it is condition (d) together with the inclusion $A(q)\subset B(p)$ for $q>p$ which gives (ii) in this lemma (which is uniform with respect to $p$ and $q$). Condition (c) is only helpful for the induction hypothesis.
We initialize the construction by applying Lemma \ref{lem:setsafhc1} to $E=\mathbb N$. We set $A(1)=A$ and $B(1)=B$ which satisfy (a), (b) and (c). In particular, applying (c) for $C=1$ we find some $\kappa$ that we call $\kappa_1$.
Assume now that the construction has been done until step $r$ and let us perform it for step $r+1$. Let $E$ be a subset of $B(r)$ with positive lower density and $|n-n'|\geq r+1$ provided $n\neq n'$ are in $E$. We apply Lemma \ref{lem:setsafhc1} to this set $E$ and we set $A(r+1)=A$ and $B(r+1)=B$, so that (a) and (b) are clearly satisfied. Upon taking a maximum, (c) is also easily satisfied: indeed, the only case which is not settled by the induction hypothesis is $p=q=r+1$ (when $p<r+1$ and $q=r+1$, use $B(q)\subset B(r)$); this case is solved by the construction of $A(r+1)$ and $B(r+1)$.
The proof of (d) is slightly more delicate. For $k=1,\dots,r$, we have to verify that for $1\leq p\leq r+1$, $n\in A(p)$ and $n'\in B(r+1)$, $\max(n,n')\geq \kappa_k\implies |n-n'|\geq k$. When $p\leq r$, again this follows from $B(r+1)\subset B(r)$. For $p=r+1$, this follows from $A(r+1),B(r+1)\subset E$ and the fact that distinct elements of $E$ have distance greater than or equal to $r+1$. Finally, applying (c) for $C=k+1$, we define $\kappa_{k+1}$. \end{proof}
We need now to ensure property (ii) in Theorem \ref{thm:setsafhc}. This will be done again inductively, the main step being the following lemma. \begin{lemma}\label{lem:setsafhc4}
Let $(A(p))$ be a sequence of pairwise disjoint subsets of $\mathbb N$ with positive lower density and let $(a(p))$ be a sequence of positive real numbers. There exists a sequence $(B(p))$ of subsets of $\mathbb N$ with positive lower density such that each $B(p)$ is contained in $A(p)$ and, for all $n\in B(1)$, for all $n'\in B(p)$, $p\geq 2$, $|n-n'|\geq a(1)+a(p)$. \end{lemma} \begin{proof} Since $A(1)$ has positive lower density, there exists $N\in\mathbb N$ and $\delta>0$ such that, for all $n\geq N$, $$\frac{\textrm{card}\big(A(1)\cap[0,\dots,n]\big)}{n+1}\geq\delta.$$ For $p\geq 2$, let $B(p)$ be a subset of $A(p)$ such that, for all $n\in\mathbb N$, $$\frac{\textrm{card}\big((B(p)+[-a(1)-a(p),a(1)+a(p)])\cap [0,\dots,n]\big)}{n+1}<\frac{\delta}{2^p}$$
but $B(p)$ still has positive lower density. This is possible if, writing $A(p)=\{n_j;\ j\in\mathbb N\}$, we set $B(p)=\{n_{ka}; k\geq 1\}$ for some sufficiently large $a$. We then define $B(1)=A(1)\backslash \bigcup_{p\geq 2}\big(B(p)+[-a(1)-a(p),a(1)+a(p)]\big)$. Then, for all $n\in B(1)$ and all $n'\in B(p)$, $p\geq 2$, one clearly has $|n-n'|\geq a(1)+a(p)$ whereas, for all $n\geq N$, \[ \frac{\textrm{card}\big(B(1)\cap[0,\dots,n]\big)}{n+1}\geq \delta-\sum_{p\geq 2}\frac{\delta}{2^p}\geq\frac\delta2 \] so that $B(1)$ still has positive lower density. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:setsafhc}] Applying Lemmas \ref{lem:setsafhc2} and \ref{lem:setsafhc3}, we start from a sequence $(A(p))$ of pairwise disjoint subsets of $\mathbb N$, satisfying properties (i) and (iii) of Theorem \ref{thm:setsafhc} and property (ii) when $p=q$. We construct by induction on $r$ sets $B(1),\dots,B(r)$, $A_r(k)$ for $k\geq r+1$ such that \begin{itemize} \item $B(k)\subset A(k)$ for all $k\leq r$, $A_r(k)\subset A(k)$ for all $k\geq r+1$; \item $B(k)$ and $A_r(k)$ have positive lower density;
\item for all $p\in\{1,\dots,r\}$, for all $q\geq p+1$, for all $n\in A(p)$, for all $n'\in B(q)$ if $q\leq r$, for all $n'\in A_r(q)$ if $q\geq r+1$, $|n-n'|\geq a(p)+a(q)$. \end{itemize} The sequence $(B(p))$ that we get at the end will answer the problem. Now the construction is easily done by successive applications of Lemma \ref{lem:setsafhc4} first with the sequence $(A(p))_{p\geq 1}$, then with the sequence $(A_1(p))_{p\geq 2}$, and so on. \end{proof}
\subsection{A weighted shift with a frequently hypercyclic algebra on $c_0$}
Let us now define a weight $(w_n)$ such that $B_w$, acting on $c_0$ endowed with the coordinatewise product, supports a frequently hypercyclic algebra. We start with the sequence $(A(p))_{p\geq 1}$ given by Theorem \ref{thm:setsafhc} for $a_p=p$.
We then construct inductively a sequence of integers $(M_k)$ such that, for all $(n,n')\in A(p)\times A(q)$ with $p\neq q$, $\max(n,n')\geq M_{k+1}\implies |n-n'|\geq M_k$. This follows directly from property (iii) of Theorem \ref{thm:setsafhc}, applied successively with $C=M_1=1$ to get $M_2$, $C=M_2$ to get $M_3$, and so on. We may also assume that the sequence $(M_{k+1}-M_k)$ is non-decreasing.
We define the weight $(w_n)_{n\geq 1}$ by the following inductive formulas: \begin{itemize} \item $w_n=2$ for all $n\leq M_2$; \item for all $k\geq 2$, for all $n\in\{M_k+1,\dots,M_{k+1}\}$, $$w_n=\left(w_1\cdots w_{M_k}\right)^\frac1{k(M_{k+1}-M_k)}$$ so that, and this is the crucial point, $$w_{M_k+1}\cdots w_{M_{k+1}}=\left(w_1\cdots w_{M_k}\right)^{\frac 1k}.$$ \end{itemize}
Let us summarize the properties of the weight which will be useful later. \begin{lemma}\label{lem:afhcws1} The weight $(w_n)$ satisfies the following properties: \begin{itemize} \item for all $n\geq 1$, $w_n\geq 1$; \item $(w_n)$ is non-increasing; \item $(w_1\cdots w_n)$ tends to $+\infty$; \item for all $\alpha>0$, for all $l\geq 0$, $\displaystyle \frac{w_{M_{k-1}+l+1}\cdots w_{M_{k+1}+l}}{\left(w_{l+1}\cdots w_{M_{k+1}+l}\right)^\alpha}\xrightarrow{k\to+\infty}0. $ \end{itemize} \end{lemma} \begin{proof} The first property is clear. For the second one, it suffices to prove that if $n\in\{M_k+1,\dots,M_{k+1}\}$ and $n'\in \{M_{k+1}+1,\dots,M_{k+2}\}$ for some $k$, then $w_{n'}\leq w_n$. We now write \begin{eqnarray*} w_{n'}&=&\left(w_1\cdots w_{M_{k+1}}\right)^{\frac{1}{(k+1)(M_{k+2}-M_{k+1})}}\\ &=&\left(w_1\cdots w_{M_{k}}\right)^{\frac{1}{(k+1)(M_{k+2}-M_{k+1})}}\left(w_{M_k+1}\cdots w_{M_{k+1}}\right)^{\frac{1}{(k+1)(M_{k+2}-M_{k+1})}}\\ &=&\left(w_1\cdots w_{M_k}\right)^{\frac{1}{k(M_{k+2}-M_{k+1})}}\\ &\leq&w_n. \end{eqnarray*} To prove that $(w_1\cdots w_n)$ tends to $+\infty$, we just observe that, for all $k\geq 2$, $$w_1\cdots w_{M_k}=(w_1\cdots w_{M_2})^{\prod_{j=2}^{k-1}\left(1+\frac 1j\right)},$$ and this goes to $+\infty$ as $k$ tends to $+\infty$. Finally, since $(w_n)$ is bounded and bounded below, we need only to prove the last property for $l=0$. Now we write \begin{align*} w_{M_{k-1}+1}\cdots w_{M_{k+1}}&=w_{M_{k-1}+1}\cdots w_{M_k}\left(w_1\cdots w_{M_k}\right)^{1/k}\\ &=\left(w_{M_{k-1}+1}\cdots w_{M_k}\right)^{1+\frac 1k} \left(w_1\cdots w_{M_{k-1}}\right)^{\frac 1k} \\ &=\left(w_1\cdots w_{M_{k-1}}\right)^{\frac 1{k-1}\left(1+\frac 1k\right)+\frac 1k}\\ &=\left(w_1\cdots w_{M_{k-1}}\right)^{\frac 2{k-1}} \end{align*} so that
$$\frac{w_{M_{k-1}+1}\cdots w_{M_{k+1}}}{\left(w_1\cdots w_{M_{k+1}}\right)^\alpha} \leq \frac1{\left(w_1\cdots w_{M_{k-1}}\right)^{\alpha-\frac{2}{k-1}}}$$ which indeed tends to zero. \end{proof}
We now prove that the operator $B_w$ acting on $c_0$ endowed with the coordinatewise product supports a frequently hypercyclic algebra. Let $(v(p),m(p))$ be a sequence dense in $c_0\times\mathbb N$ such that each $v(p)$ has finite support contained in $[0,p]$. We shall need a last technical lemma involving all the objects we constructed until now.
\begin{lemma}\label{lem:afhcws2} There exists a sequence of integers $(N(r))_{r\geq 1}$ satisfying the following properties: \begin{enumerate}[(i)] \item for all $r\geq 1$,
$$\sup_{n\geq N(r),\ l=0,\dots,r}\left|\frac{v_l(r)}{\left(w_{l+1}\cdots w_{n+l}\right)^{\frac{1}{m(r)+1}}}\right|^{\frac 1{m(r)}}<\frac 1r.$$ \item for all $r\geq 2$, for all $s\in\{1,\dots,r-1\}$, for all $(j,j')\in A(r)\times A(s)$ with $j\geq N(r)$, for all $l\in \{0,\dots,r\}$, for all $\alpha\geq\min\left(\frac 1{m(r)},\frac 1{m(s)}\right)$, \begin{align*}
j>j'&\implies \left|\frac{w_{l+(j-j')+1}\cdots w_{j+l}v_l(r)^\alpha}{\left(w_{l+1}\cdots w_{j+l}\right)^\alpha}\right|<\frac1r\\
j'>j&\implies \left|\frac{w_{l+(j'-j)+1}\cdots w_{j'+l}v_l(s)^\alpha}{\left(w_{l+1}\cdots w_{j'+l}\right)^\alpha}\right|<\frac1r. \end{align*} \end{enumerate} \end{lemma} \begin{proof} Let $r\geq 1$ be fixed, We first observe that it is easy to ensure (i), just by assuming that $N(r)$ is large enough. Let us choose $N(r)$ to ensure (ii). Upon taking a supremum, we may fix $s$ and $l$ and to simplify the notations, we will assume $l=0$. Let $\alpha_0=
\min\left(\frac 1{m(r)},\frac 1{m(s)}\right)$ and $C=\max(1,|v_0(r)|,|v_0(s)|)$. We define three integers $N_0$, $k_0$ and $k_1$ satisfying the following three conditions: $$n\geq N_0\implies \frac{C^2}{w_1\cdots w_n}<\frac 1r$$ $$n\geq M_{k_0}\implies \frac{C^2}{w_1\cdots w_n}<\frac 1r$$ $$k\geq k_1\implies \frac{w_{M_{k-1}+1}\cdots w_{M_{k+1}}C}{\left(w_1\cdots w_{M_{k+1}}\right)^{\alpha_0}} < \frac 1{r}.$$ We set $N(r)=\max(N_0,M_{k_0+1},M_{k_1})$. Let $(j,j')\in A(r)\times A(s)$ with $j\geq N(r)$. To fix the ideas, we assume that $j>j'$. If $\alpha\geq 2$, then
$$\left|\frac{w_{(j-j')+1}\cdots w_{j}v_0(r)^\alpha}{\left(w_{1}\cdots w_{j}\right)^\alpha}\right|
\leq \frac{w_{(j-j')+1}\cdots w_{j}}{w_1\cdots w_j}\times \left|\frac{v_0(r)^2}{w_1\cdots w_j}\right|^{\alpha/2}<1\times\left(\frac 1r\right)^{\frac\alpha 2}\leq \frac 1r.$$
If $\alpha\in [1,2]$, then
$$\left|\frac{w_{(j-j')+1}\cdots w_{j}v_0(r)^\alpha}{\left(w_{1}\cdots w_{j}\right)^\alpha}\right| \leq \frac{C^2}{w_1\cdots w_{j-j'}}.
$$ Since $j\geq N(r)$, $j-j'\geq M_{k_0}$ so that the last term is less than $1/r$. For $\alpha<1$, since $j\geq N(r)\geq M_{k_1}$, there exists a single integer $k\geq k_1$ such that $j\in [M_k,M_{k+1})$. Then $j-j'\geq M_{k-1}$ and \begin{align*}
\left|\frac{w_{(j-j')+1}\cdots w_j v_0(r)^\alpha}{\left(w_1\cdots w_{j}\right)^{\alpha}}\right| &\leq \frac{w_{M_{k-1}+1}\cdots w_{j} C}{\left(w_1\cdots w_{j}\right)^{\alpha_0}}\\ &\leq \frac{w_{M_{k-1}+1}\cdots w_{M_{k+1}}C}{\left(w_1\cdots w_{M_{k+1}}\right)^{\alpha_0}}\\ &<\frac 1r. \end{align*} \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:afhcc0}] We are now ready for the proof that $B_w$ supports a frequently hypercyclic algebra.
By Lemma \ref{lem:setsafhc3}, for each $p\geq 1$, let $B(p)$ be a subset of $A(p)$ with positive lower density such that $\min(B(p))\geq N(p)$ and $|n-n'|\geq N(p)$ for all $n\neq n'\in B(p)$. We set $$u(p)=\sum_{n\in B(p)}\sum_{l=0}^p \frac1{\left(w_{l+1}\cdots w_{n+l}\right)^{1/m(p)}} v_l(p)^{1/m(p)} e_{n+l}.$$
Since $(w_1\cdots w_n)$ tends to $+\infty$, $u(p)$ belongs to $c_0$. Moreover, the choice of $N(p)$
(here, (i) of Lemma \ref{lem:afhcws2}) ensures that $\|u(p)\|<1/p$. We also observe that the $u(p)$ have pairwise disjoint support. Hence we may define $u=\sum_{p\geq 1}u(p)$ which still belongs to $c_0$. We claim that the following property is true: for all $p\geq 1$, for all $q\neq p$, for all $n\in B(p)$, \begin{equation}\label{eq:afhcws1}
\left\|B_w^n u(p)^{m(p)}-v(p)\right\|<\frac 1p, \end{equation} \begin{equation}\label{eq:afhcws2}
\forall m>m(p),\ \left\|B_w^n u(p)^{m}\right\|<\frac 1p, \end{equation} \begin{equation}\label{eq:afhcws3}
\forall m\geq m(p),\ \left\|B_w^n u(q)^{m}\right\|<\frac 1p. \end{equation} Assume that these properties have been proved. Let $P$ be a non-constant polynomial with $P(0)=0$ and write it $P(z)=\sum_{m=m_0}^{m_1} \hat P(m) z^m$ with $\hat P(m_0)\neq 0$. We aim to prove that $P(u)$ is a frequently hypercyclic vector for $B_w$. Without loss of generality, we can assume $\hat P(m_0)=1$.
Let $V$ be a non-empty open subset of $c_0$. There exists $p\geq 1$ such that $B\left(v(p),\left(2+\sum_{m=m_0+1}^{m_1}|\hat P(m)|\right)/p\right)\subset V$ and $m(p)=m_0$. Then, for all $n\in B(p)$, \begin{align*}
\left\|B_w^n P(u)-v(p)\right\|&\leq \left\|B_w^n u(p)^{m(p)}-v(p)\right\|+\left\|\sum_{q\neq p} B_w^n u(q)^{m(p)}\right\|\\
&\quad\quad\quad+\sum_{m=m_0+1}^{m_1}|\hat P(m)|\left\|\sum_{q\geq 1} B_w^n u(q)^m\right\|\\
&\leq \frac{2+\sum_{m=m_0+1}^{m_1}|\hat P(m)|}{p}, \end{align*} where the last inequality follows from \eqref{eq:afhcws1}, \eqref{eq:afhcws2}, \eqref{eq:afhcws3} and the fact that the $B_w^n u(q)$ have pairwise disjoint support. Therefore, for all $n$ in a set of positive lower density, $B_w^n P(u)$ belongs to $V$, showing that $P(u)$ is a frequently hypercyclic vector for $B_w$. Hence, it remains to prove \eqref{eq:afhcws1}, \eqref{eq:afhcws2} and \eqref{eq:afhcws3}. We first observe that \[ B_w^n u(p)^{m(p)}-v(p)=\sum_{\substack{n'\in B(p)\\ n'\geq n}}\sum_{l=0}^p \frac{v_l(p)}{w_{l+1}\cdots w_{(n'-n)+l}} e_{(n'-n)+l}. \] Since $n'-n>N(p)$ for all $n'>n$, $n'\in B(p)$, \eqref{eq:afhcws1} follows from (i) in Lemma \ref{lem:afhcws2}. Next, for $m>m(p)$, we may write $B_w^n u(p)^m$ as $$\sum_{\substack{n'\in B(p)\\ n'\geq n}}\sum_{l=0}^p \frac{v_l(p)^{\frac{m}{m(p)}}}{\left(w_{l+1}\cdots w_{(n'-n)+l}\right)^{\frac m{m(p)}}\left(w_{(n'-n)+l+1}\cdots w_{n'+l}\right) ^{\frac{m}{m(p)}-1}}e_{(n'-n)+l}. $$ There is an additional difficulty since now we may have $n'=n$. We overcome this difficulty by writing \begin{align*}
&\left|\frac{v_l(p)^{\frac{m}{m(p)}}}{\left(w_{l+1}\cdots w_{(n'-n)+l}\right)^{\frac m{m(p)}}\left(w_{(n'-n)+l+1}\cdots w_{n'+l}\right)^{\frac{m}{m(p)}-1}}\right|\\ &\quad\quad\leq
\left| \frac{v_l(p)^{\frac{m}{m(p)}}}{\left(w_{l+1}\cdots w_{n'+l}\right)^{\frac{m}{m(p)}-1}}\right|\\
&\quad\quad\leq \left| \frac{v_l(p)^{\frac{1}{m(p)}}}{\left(w_{l+1}\cdots w_{n'+l}\right)^{\frac{1}{m(p)}-\frac 1m}}\right|^m\\
&\quad\quad\leq \left| \frac{v_l(p)^{\frac{1}{m(p)}}}{\left(w_{l+1}\cdots w_{n'+l}\right)^{\frac{1}{m(p)(m(p)+1)}}}\right|^m\\
&\quad\quad<\frac 1p. \end{align*}
Finally, for $m\geq m(p)$ and $q\neq p$, we write \[ B_w^n u(q)^m=\sum_{\substack{n'\in B(q)\\ n'>n}}\sum_{l=0}^q \frac{w_{l+(n'-n)+1}\cdots w_{n'+l}}{\left(w_{l+1}\cdots w_{n'+l}\right)^{\frac m{m(q)}}}v_l(q)^{\frac m{m(q)}}e_{(n'-n)+l}. \]
For $q>p$, we apply (ii) of Lemma \ref{lem:afhcws2} with $r=q$, $s=p$, $j=n'$, $j'=n$ and $\alpha=m/m(q)$. For $q<p$, we apply (ii) of Lemma \ref{lem:afhcws2} with $r=p$, $s=q$, $j=n$, $j'=n'$ and $\alpha=m/m(q)$. In both cases, we immediately find that all the coefficients of $B_w^n u(q)^m$ are smaller than $1/p$, yielding
\[ \left\|B_w^n u(q)^m\right\|<\frac 1p. \] This closes the proof of Theorem \ref{thm:afhcc0}. \end{proof}
This technical construction leads to an example over the not so difficult space $c_0$, but the following question remains open.
\begin{question} Does there exist a weighted shift on $\ell_p$ endowed with the pointwise product admitting a frequently hypercyclic algebra? \end{question}
Of course, it would also be nice to get simpler examples! On the other hand, for sequence spaces endowed with the convolution product, we have neither positive nor negative examples. For instance, it would be very interesting to solve the following questions.
\begin{question}
Does $B$ on $\omega$ endowed with the convolution product support a frequently hypercyclic algebra? \end{question}
\begin{question} Does $2B$ on $\ell_1$ endowed with the convolution product support a frequently hypercyclic algebra? \end{question}
\section{Concluding remarks and open questions}
\subsection{Closed hypercyclic algebras}
As pointed out in the introduction, provided $T$ is hypercyclic, $HC(T)\cup\{0\}$ always contains a dense subspace. When moreover $T$ satisfies the hypercyclicity criterion, there is a necessary and sufficient condition to determine whether $HC(T)\cup\{0\}$ contains an infinite-dimensional closed subspace (see for instance \cite{BM09}). In our context, it is natural to ask whether, for some of our examples, $HC(T)\cup\{0\}$ contains a closed non-trivial algebra (we will say that $T$ supports a closed hypercyclic algebra).
The third author and K. Grosse-Erdmann have shown that it is the case if $T$ is a translation operator acting on the space $\mathcal C^{\infty}(\mathbb R,\mathbb C)$. The fact that $T$ is an algebra homomorphism plays an important role here. We now give several negative results. The first one solves a question of \cite{shkarin}.
\begin{proposition} No convolution operator $P(D)$ induced by a nonconstant polynomial $P\in\mathbb C[z]$ admits a closed hypercyclic algebra. \end{proposition} \begin{proof} We write $P(z)=\sum_{s=0}^t\hat{P}(s)z^s$, with $\hat{P}(t)\neq0$, and let $f\in HC(P(D))$. We shall prove that the closed algebra generated by $f$ contains a non-zero and non-hypercyclic vector. Write $f(z)=a_0+\sum_{n\geq p}a_n z^n$, with $a_p\neq 0$. Without loss of generality, we may assume that $a_p=1$. We shall construct by induction a sequence of complex numbers $(b_k)$ such that
\[ |b_k|\leq \left(\frac{|\hat P(0)|+1}{|\hat{P}(t)|}\right)^{kp}\times\frac 1{(ktp)!}\]
for all $k$ and, setting $P_k(z)=\sum_{l=1}^k b_l ( z-a_0)^{lt}$, then
\[ |P(D)^{lp}(P_k\circ f)(0)|\geq (|\hat P(0)|+1)^{lp} \]
for all $1\leq l\leq k$. The conclusion follows easily. In fact, $(P_k)$ converges uniformly on compact subsets of $\mathbb C$ to some entire function $g$. From the uniformity of the convergence, we conclude that the function $g\circ f$ satisfies
\[ \left|P(D)^{lp}(g\circ f)(0)\right|\geq (|\hat P(0)|+1)^{lp}\]
for all $l\geq 1$. Let us set $h=g-g(0)$. The function $h\circ f$, which is in the algebra generated by $f$, satisfies \begin{align*}
\left |P(D)^{lp}(h\circ f)(0)\right| & \geq \left|P(D)^{lp}(g\circ f)(0)\right| - \left| P(D)^{lp}(g(0))\right|\\
&\geq (|\hat P(0)|+1)^{lp} - |\hat P(0)|^{lp} |g(0)|\\ &\xrightarrow{l\to+\infty}+\infty. \end{align*} Hence, $h\circ f$ is nonzero and it cannot be hypercyclic for $P(D)^p$. In particular, since $HC(P(D))\subset \bigcap_{n\geq1}HC(P(D)^n)$ (see \cite[Theorem 1]{ansari}), $h\circ f$ cannot be hypercyclic for $P(D)$ as well.
For the proof we will use the formula \[ P(D)^q=\sum_{{\bf j}\in I_q}{\binom q {\bf j}}\hat{P}(0)^{j_0}\cdots\hat{P}(t)^{j_t}D^{j_1+2j_2+\cdots+tj_t}, \] where $I_q=\{{\bf j}=(j_0,...,j_t)\in \mathbb N_0^{t+1} : j_0+\cdots+j_t=q\}$ and $\binom q {\bf j}$ denote the multinomial coefficient \[\binom q {j_0,\dots,j_t}=\frac{q!}{j_0!\cdots j_t!}.\]
Let us set $P_0(z)=0$ and let us assume that the construction has been done until step $k-1$. Then \begin{align*} P_{k-1}\circ f+b(f-a_0)^{kt}&=P_{k-1}\circ f+b(z^p+a_{p+1}z^{p+1}+\cdots)^{kt}\\
&=P_{k-1}\circ f+b z^{ktp}+\sum_{j\geq ktp+1}c_j z^j. \end{align*}
Hence, for $1\leq l\leq k$, \begin{align*} P(D)^{lp}(P_{k-1}\circ f+b(f-a_0)^{kt})(0)&=P(D)^{lp}(P_{k-1}\circ f)(0)+P(D)^{lp}(b(f-a_0)^{kt})(0)\\ &=P(D)^{lp}(P_{k-1}\circ f)(0)+g_l(0), \end{align*} where \begin{align*} g_l(z) &=P(D)^{lp}\left(bz^{ktp}+\sum_{j\geq ktp+1}c_jz^j\right). \end{align*} If $l\leq k-1$ then $\deg P^{lp}\leq(k-1)tp<ktp$, hence $g_l(0)=0$. By the induction hypothesis it follows that \begin{align*}
|P(D)^{lp}(P_{k-1}\circ f+b(f-a_0)^{kt})(0)|&=|P(D)^{lp}(P_{k-1}\circ f)(0)+g_l(0)|\\
&=|P(D)^{lp}(P_{k-1}\circ f)(0)|\geq (|\hat P(0)|+1)^{lp} \end{align*} whatever the value of $b$ is. On the other hand, if $l=k$, then \begin{align*} g_k(z) &=b\hat{P}(t)^{kp}(ktp)!\\ &\quad\quad\quad+\sum_{{\bf j}\in I_{kp}\backslash\{(0,...,0,kp)\}}\binom{kp}{\bf j}\hat{P}(0)^{j_0}\cdots\hat{P}(t)^{j_t}D^{j_1+2j_2+\cdots+tj_t}\left(bz^{ktp}+\sum_{j\geq ktp+1}c_jz^j\right), \end{align*} hence $g_k(0)=b\hat{P}(t)^{kp}(ktp)!$, that is, \begin{align*}
|P(D)^{kp}(P_{k-1}\circ f+b(f-a_0)^{kt})(0)|&=|P(D)^{kp}(P_{k-1}\circ f)(0)+g_k(0)|\\
&= |P(D)^{kp}(P_{k-1}\circ f)(0)+b\hat{P}(t)^{kp}(ktp)!|, \end{align*} so we can find $b$ satisfying
\[ |b|\leq \left(\frac{|\hat P(0)|+1}{|\hat{P}(t)|}\right)^{kp}\times\frac 1{(ktp)!}\] such that
\[ |P(D)^{lp}(P_{k-1}\circ f+b(f-a_0)^{kt})(0)|\geq (|\hat P(0)|+1)^{lp}. \]
The proof is now done by taking $b_k=b$. \end{proof}
\begin{question} Does there exist an entire function $\phi$ of exponential type such that $\phi(D)$ supports a closed hypercyclic algebra? \end{question}
\begin{proposition} Let $X=\ell_p$, $X=c_0$ or $X=\omega$, endowed with the coordinatewise product. No backward shift on $X$ supports a closed hypercyclic algebra. \end{proposition}
\begin{proof} We first consider the case $X=c_0$. Let $x\in X$ be a non-zero sequence, and let $D\subset \mathbb{C}$ be a compact disc centered at the origin and omitting at least one of the terms of $x$. For each $n\in \mathbb N$, consider the compact set $K_n$ as in Figure \ref{polyfigure} and $f$ a holomorphic function defined on a neighbourhood of $K_n$ and satisfying that $f(z)=0$ if $z\in D$ and $f(z)=1$ if $z\in K_n\setminus D$.
\begin{figure}\label{polyfigure}
\end{figure}
By Runge's approximation theorem we get a polynomial $P_n$ such that $\| P_n-f\| _{K_n}<\frac{1}{n}$. We end up with a sequence of polynomials $(P_n)$, satisfying that $P_n(z)\rightarrow 0$ uniformly on $D$, and $P_n(z)\rightarrow 1$ pointwise on $\mathbb{C}\setminus D$. Redefining $P_n$ by $P_n-P_n(0)$, we may also assume that $P_n(0)=0$, for every $n\in \mathbb N$.
Since $x\in c_0$, it follows that eventually all the terms of $x$ belong to $D$ which yields that $P_n(x)\rightarrow y=(y_k)$ in $c_0$, where $y_k=0$, if $x_k\in D$ and $y_k=1$ otherwise. We conclude, that $y$ is a non-zero element in the closed algebra generated by $x$ which is not hypercyclic for any weighted backward shift on $c_0$.
Let now $X=\ell ^p, p\geq 1$. Consider an $x\in X, x\neq 0$, and $D$ and $(P_n)$ defined as above. Cauchy's formula ensures that $P_n'\rightarrow 0$ uniformly on $\frac{1}{2}D$. Let $$
C=\sup \left\{ |P_n'(z)|: n\in \mathbb N, z\in \frac{1}{2}D \right\}. $$ Fix $\varepsilon >0$ and let $k_0\in \mathbb N$ be such that, for $k>k_0$, $x_k\in\frac 12 D$ and $$
2^pC^p\sum_{k>k_0}|x_k|^p<\frac{\varepsilon}{2}. $$ Find $N\in \mathbb N$ such that for all $m,n\geq N$, $$
\sum_{k=1}^{k_0}|P_m(x_k)-P_n(x_k)|^p<\frac{\varepsilon}{2}. $$ We have \begin{align*}
\|P_m(x)-P_n(x)\|_p^p &\leq \sum_{k=1}^{k_0}|P_m(x_k)-P_n(x_k)|^p+\sum_{k>k_0}(|P_m(x_k)|+|P_n(x_k)|)^p \\
&\leq \sum_{k=1}^{k_0}|P_m(x_k)-P_n(x_k)|^p+ 2^pC^p\sum_{k>k_0}|x_k|^p<\varepsilon. \end{align*} That means that the sequence $(P_n(x))$ is Cauchy in $\ell_p$ and the conclusion follows exactly as in the previous case.
Finally, we consider the case $X=\omega$. Letting $D=\{0\}$ and $K_n$ be as above, and by using Runge's approximation theorem, we get a sequence of polynomials $(Q_n)$ such that $Q_n(0)=0$, for every $n\in \mathbb N$ and $Q_n(z)\rightarrow 1$ for each $z\in \mathbb{C}\setminus \{0\}$. If $x\in \omega$ is a non-zero sequence, then $Q_n(x)\rightarrow y=(y_k)$, where $y_k=0$ if $x_k=0$, and $y_k=1$ otherwise. It is immediate that $y$ is a non-zero element in the closed algebra generated by $x$ which fails to be hypercyclic for any weighted backward shift on $\omega$. \end{proof}
\begin{question}
Does there exist a weight $(w_n)$ such that $B_w$, acting on $\ell_1$ endowed with the Cauchy product, supports a closed hypercyclic algebra? \end{question}
\subsection{Hypercyclic algebras in the ideal of compact operators}
Beyond the examples given in that paper, there are other examples where the existence of a hypercyclic algebra would be natural. One of them is given by hypercyclic operators acting on separable ideals of operators. For instance, assume that $H$ is a separable Hilbert space and denote by $X=\mathcal K(H)$ the (non-commutative) algebra of compact operators in $H$, endowed with the norm topology.
For $T\in\mathcal L(H)$, denote by $L_T$ the operator of left multiplication by $T$, defined on $\mathcal K(H)$. It is known (see for instance \cite[Chapter 8]{BM09}) that if $T$ satisfies the hypercyclicity criterion, then $L_T$ is a hypercyclic operator on $\mathcal K(H)$. This latter space being an algebra, it is natural to study whether $L_T$ supports a hypercyclic algebra. We do not know the answer to this question, but we point out that a positive answer would require different techniques. Indeed, Theorem \ref{thm:generalcriterion} can never be applied to these operators.
\begin{proposition} Let $T\in\mathcal L(H)$. Then $L_T$, acting on $\mathcal K(H)$, does not satisfy the assumptions of Theorem \ref{thm:generalcriterion} even for $d=1$. \end{proposition}
\begin{proof}
We fix $x\in H$, $x^*\in H^*$ with $x^*(x)=1$, $\|x\|=1$, $\|x^*\|=1$. Using the notations of Theorem \ref{thm:generalcriterion}, let $A=\{1,2\}$, \begin{align*}
U=V&=\left\{u\in \mathcal L(H):\ \|u-x^*\otimes x\|<1/4\right\},\\
W&=\left\{u\in \mathcal L(H):\ \|u\|<1/8\right\}. \end{align*} Assume first that $\beta=1$ and that there exist $u\in U$, $N\in\mathbb N$ with $T^N u\in V$ and $T^N u^2\in W$. Then we know that
\[ \left\|T^N u^2(x)-x^*(u(x))x\right\|<\frac{\|u(x)\|}4 \]
since $T^N u\in V$. Now, $\|u(x)-x\|<1/4$ so that $\|u(x)\|<5/4$ and $|x^*(u(x))|> 3/4$. Hence,
\[ \left\|T^N u^2(x)\right\| > \frac 34-\frac 5{16}>\frac 18. \] This contradicts $T^N u^2\in W$.
If we assume that $\beta=2$ and that there exist $u\in U$, $N\in\mathbb N$ with $T^N u\in W$ and $T^N u^2\in V$, then we get successively \begin{align*}
\left\| T^N u^2(x)-x\right\|&<\frac 14\textrm{ (since $T^N u^2\in V$)}\\
\left\| T^N u^2(x)\right\|&<\frac 18 \|u(x)\|\textrm{ (since $T^N u\in W$)}\\
\left\|u(x)-x\right\|&<\frac 14. \end{align*} These three inequalities yield easily a contradiction. \end{proof}
\subsection{Further question and remark}
As independently shown by Ansari in \cite{ansariexistence} and later-on by Bernal-Gonzáles in \cite{gonzalez}, every separable Banach space supports a hypercyclic operator. In the context of algebras a natural question arises.
\begin{question} Is it true that every separable Banach algebra supports a hypercyclic operator admitting a hypercyclic algebra? \end{question}
In all known results, the set of generators for hypercyclic algebras is either empty or residual. We observe below that this set can be non-empty and meager.
\begin{remark} For every pair $(X,T)$ where $X$ is a Banach space and $T$ a hypercyclic operator with a non-hypercyclic vector, we may define a product on $X$ turning it into a commutative Banach algebra and such that the set of generators for a hypercyclic algebra for $T$ is non-empty and nowhere dense. \end{remark}
\begin{proof}
Let $x\in HC(T)$ and $y$ a non-hypercyclic vector for $T$ with $\|y\|=1$. Consider $f\in X^{\ast}$ with $\|f\|=1$ and such that $f(x)=0$ and $f(y)=1$. We define the product $$ z\cdot w=f(z)f(w)y, \,\, \mbox{with} \,\, z,w \in X $$ and observe that it turns $X$ into a commutative Banach algebra. Now, $x^2=0$ so $A(x)=span(x)$, and thus $x$ is a generator for a hypercyclic algebra for $T$. Moreover, it is easy to check that the following holds, $$ \{x\in X: A(x)\setminus \{0\}\subset HC(T)\}=HC(T)\cap Ker(f). $$ Since $Ker(f)$ is a proper, closed hyperplane of $X$, we conclude that the set of generators for a hypercyclic algebra for $T$ is non-empty and nowhere dense. \end{proof}
\end{document} | arXiv |
\begin{document}
\enlargethispage{10mm}
\title{A critical blow-up exponent for flux limitation\\ in a Keller-Segel system}
\author{ Michael Winkler\footnote{[email protected]}\\ {\small Institut f\"ur Mathematik, Universit\"at Paderborn,}\\ {\small 33098 Paderborn, Germany} }
\date{} \maketitle
\begin{abstract} \noindent
The parabolic-elliptic cross-diffusion system
\begin{eqnarray*}
\left\{ \begin{array}{l}
u_t = \Delta u - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm]
0 = \Delta v - \mu + u,
\qquad \int_\Omega v=0,
\qquad
\mu:=\frac{1}{|\Omega|} \int_\Omega u dx,
\end{array} \right.
\end{eqnarray*}
is considered along with homogeneous Neumann-type boundary conditions in a smoothly bounded domain $\Omega\subset\mathbb{R}^n$, $n\ge 1$,
where $f$ generalizes the prototype given by
\begin{eqnarray*}
f(\xi) = (1+\xi)^{-\alpha},
\qquad \xi\ge 0,
\qquad \mbox{for all } \xi\ge 0,
\end{eqnarray*}
with $\alpha\in\mathbb{R}$.\\[5pt]
In this framework, the main results assert that if $n\ge 2$, $\Omega$ is a ball and
\begin{eqnarray*}
\alpha<\frac{n-2}{2(n-1)},
\end{eqnarray*}
then throughout a considerably large set of radially symmetric initial data, an associated initial value problem
admits solutions blowing up in finite time with respect to the $L^\infty$ norm of their first components.\\[5pt]
This is complemented by a second statement which ensures that in general and not necessarily symmetric settings,
if either $n=1$ and $\alpha\in\mathbb{R}$ is arbitrary, or $n\ge 2$ and $\alpha>\frac{n-2}{2(n-1)}$,
then any explosion is ruled out in the sense that for arbitrary nonnegative and continuous initial data,
a global bounded classical solution exists.\\[5pt]
\noindent {\bf Key words:} chemotaxis; flux limitation; finite-time blow-up\\ {\bf MSC 2010:} 35B44 (primary); 35K65, 92C17 (secondary) \end{abstract}
\section{Introduction}\label{intro}
We consider nonnegative solutions of the parabolic-elliptic cross-diffusion system \begin{equation} \label{00}
\left\{ \begin{array}{l}
u_t = \Delta u - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm]
0 = \Delta v - \mu + u,
\qquad \int_\Omega v=0,
\qquad
\mu:=\frac{1}{|\Omega|} \int_\Omega u dx,
\end{array} \right. \end{equation} where the given function $f$ appropriately generalizes the prototype determined by \begin{equation} \label{pr}
f(\xi) = (1+\xi)^{-\alpha},
\qquad \xi\ge 0,
\qquad \mbox{for all } \xi\ge 0, \end{equation} with $\alpha\in\mathbb{R}$. In mathematical biology, systems of this form arise as refinements of the classical Keller-Segel model for chemotaxis processes, that is, for processes during which individuals within a population, represented through its density $u=u(x,t)$, partially orient their diffusive movement toward increasing concentrations $v=v(x,t)$ of a chemical substance produced by themselves (\cite{KS}).\\[5pt]
Deviating from the simplest choice $f\equiv 1$ that underlies the minimal and most thoroughly studied representative within the class of Keller-Segel systems (\cite{KS}, \cite{hillen_painter2009}), (\ref{00}) with more general $f$ accounts for refined approaches in the more recent modeling literature, according to which in several relevant application frameworks an adequate description of chemotactic motion should include certain gradient-dependent limitations of cross-diffusive fluxes in the style of (\ref{00})-(\ref{pr}) (\cite{bellomo_flim}, \cite{perthame}, \cite{bianchi_painter_sherratt}, \cite{bianchi_painter_sherratt2016}).\\[5pt]
In light of well-known results asserting the occurrence of finite-time blow-up in the case when $f\equiv 1$ (\cite{JL}, \cite{nagai1995} , \cite{nagai2001}, \cite{biler}, \cite{suzuki_book}), a natural question seems to be how far (\ref{00}) retains a certain ability to support singularity formation also in cases when $f$ reflects some saturation effects al large signal gradients in the sense that $f(\xi)\to 0$ as $\xi\to\infty$. Since it can readily be seen that no such explosion arises when $f(\xi)=(1+\xi)^{-\frac{1}{2}}$ for $\xi\ge 0$ (cf.~also Proposition \ref{prop19}), in the context of (\ref{00})-(\ref{pr}) and in a slightly more ambitious formulation, this amounts to locating the number \begin{equation} \label{ac}
\alpha_c:=\sup \Big\{ \alpha \in \mathbb{R} \ \Big| \
\mbox{(\ref{00})-(\ref{pr}) admits at least one solution blowing up in finite time} \Big\}, \end{equation} and hence to deciding, in dependence on whether or not $\alpha_c$ belongs to the open interval $(0,\frac{1}{2})$, if $\alpha_c$ plays the role of a critical exponent that corresponds to a genuinely critical nonlinearity in (\ref{00}).\\[5pt]
{\bf The challenge of detecting critical parameter settings in Keller-Segel systems.}\quad
Here we remark that in the context of Keller-Segel type systems, the identification of explosion-critical constellations has successfully been accomplished only in quite a small number of cases yet, which may be viewed as reflecting an apparent lack of appropriate methods for blow-up detection in such classes of cross-diffusion problems. Indeed, the literature provides a rich variety of techniques capable of discovering situations in which the respective dissipative ingredients overbalance cross-diffusive destabilization and hence blow-up is ruled out (cf.~\cite{horstmann_DMV} and \cite{BBTW} for an incomplete overview). Only in a relatively small number of cases, however, analytical approaches are available which allow for findings on singularity formation in parameter settings complementary to those for which results on global existence and boundedness are available. Comparatively well-understood in this respect are thus only certain particular relatives of (\ref{00}) which share some essential features with simpler classes of scalar parabolic problems.\\[5pt]
Quite far-reaching results are available, e.g., for variants of (\ref{00}) in which the respective chemotactic sensitivity function depends on the population density $u$ itself, rather than its gradient, such as in the framework of quasilinear systems with their corresponding first equation given by \begin{equation} \label{Q}
u_t= \nabla \cdot (D(u)\nabla u) - \nabla \cdot (uS(u)\nabla v). \end{equation} Namely, when supplemented either by elliptic equations of the form in (\ref{00}), or even by more complex fully parabolic equations for $v$, such modifications of (\ref{00}) admit favorable gradient-like structures that provide accessibility to energy-based arguments both in the development of global existence theories and in the derivation of blow-up results, and accordingly a fairly comprehensive knowledge concerning the emergence of singularities could be achieved: In the prototypical context determined by the choices $D(\xi)=(\xi+1)^{m-1}$ and $S(\xi)=(\xi+1)^{q-1}$, $\xi\ge 0$, for instance, associated Neumann problems in bounded domains $\Omega\subset\mathbb{R}^n$ admit global bounded solutions for widely arbitrary initial data if $m\in\mathbb{R}$ and $q\in\mathbb{R}$ satisfy $m-q>\frac{n-2}{n}$ (\cite{taowin_subcrit}, \cite{djie_win}, \cite{senba_suzuki_AAA}, \cite{kowalczyk_szymanska}), whereas unbounded solutions exist when $\Omega$ is a ball and $m-q<\frac{n-2}{n}$ (\cite{cieslak_stinner_JDE2012}, \cite{cieslak_stinner_JDE2015}, \cite{cieslak_laurencot_DCDS}, \cite{win_collapse}, \cite{djie_win}; cf.~also \cite{cieslak_laurencot_ANIHPC}). In some subcases of the latter, the use of Lyapunov functionals even allowed for the construction of certain global solutions which blow up in infinite time (\cite{cieslak_stinner_JDE2012}, \cite{cieslak_stinner_JDE2015}, \cite{lankeit_ITBU}, \cite{win_ITBU}).\\[5pt]
Beyond this, however, most classes of relevant chemotaxis systems appear to lack comparable energy structures, and accordingly the few further studies concerned with rigorous blow-up detection rely on adequately designed ad hoc methods. In consequence, the identification of explosion-critical parameter settings has so far been achieved only in a small number of additional cases, in most of which either the derivation of collapsing ordinary differential inequalities for moment-like functionals in the flavor of classical blow-up proofs for semilinear heat equations (\cite{kaplan}, \cite{deng_levine}), or even a reduction to single parabolic equations allowing for comparison with exploding subsolutions is possible; examples of this flavor address critical exponents in nonlinear signal production rates (\cite{liu_tao}, \cite{win_NON_ct_signal_critexp}), optimal conditions on zero-order degradation with respect to blow-up (\cite{fuest_logistic}, \cite{tello_win}), threshold mass levels for singularity formation (\cite{JL}, \cite{NSY}, \cite{taowin_JEMS}), or also critical interplay of several ingredients (\cite{bellomo_win_TRAN}).\\[5pt]
{\bf Main results.} \quad
The present work now intends to address the corresponding issue of criticality in (\ref{00}) by a {\em combination} of an essentially moment-based approach with a comparison argument in a crucial first step of an associated blow-up argument. To substantiate this in the context of a full initial-boundary value problem, let us henceforth consider \begin{equation} \label{0}
\left\{ \begin{array}{rcll}
u_t &=& \Delta u - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big),
\qquad & x\in\Omega, \ t>0, \\[1mm]
0 &=& \Delta v - \mu + u,
\qquad \int_\Omega v=0,
\qquad & x\in\Omega, \ t>0, \\[1mm]
& & \hspace*{-13mm}
\frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu}=0,
\qquad & x\in\partial\Omega, \ t>0, \\[1mm]
& & \hspace*{-13mm}
u(x,0)=u_0(x),
\qquad & x\in\Omega,
\end{array} \right. \end{equation} in a bounded domain $\Omega\subset\mathbb{R}^n$, $n\ge 1$, where \begin{equation} \label{f1}
f\in C^2([0,\infty)), \end{equation} and where \begin{equation} \label{init}
u_0 \in C^0(\overline{\Omega})
\quad \mbox{is nonnegative}
\qquad \mbox{with} \qquad
- \hspace*{-4mm} \int_\Omega u_0 dx=\mu>0. \end{equation}
Indeed, we shall see in Section \ref{sect2} that if $\Omega=B_R(0)\subset\mathbb{R}^n$ with some $R>0$, and if $u_0$ is radially symmetric, then in the resulting radial framework the evolution can equivalently be described by considering the accumulated densities
$w=w(s,t):=\frac{1}{n|B_1(0)|} \int_{B_r(0)} u(x,t) dx$, $s=r^n\in [0,R^n]$, $t\ge 0$, which namely solve a Dirichlet problem for \begin{equation} \label{w00}
w_t = n^2 s^{2-\frac{2}{n}} w_{ss}
+ n \cdot \Big(w-\frac{\mu}{n} s\Big) \cdot w_s \cdot f\Big(s^{\frac{2}{n}-2} \cdot (w-\frac{\mu}{n}s)^2 \Big) \end{equation} (cf.~(\ref{0w})). Here, to adequately quantify the destabilizing potential of the second summand on the right-hand side we shall, in a first and yet quite basic step, rely on a comparison argument to make sure that an appropriate monotonicity assumption on $u_0$ entails nonnegativity of the expression $w-\frac{\mu}{n}s$ that appears in two crucial places. This will then enable us to suitably estimate the Burgers-type and shock-supporting action of the nonlinearity in (\ref{w00}) from below; for nonlinearities which suitably generalize that in (\ref{pr}) with $\alpha<\frac{n-2}{2(n-1)}$ when $n\ge 3$, this will be accomplished in Section \ref{sect3} by analyzing the evolution of the moment-like functional \begin{equation} \label{phi0}
\int_0^{s_0} s^{-\gamma} (s_0-s) \Big(w(s,t)-\frac{\mu}{n}s\Big) ds \end{equation} along trajectories, and by thereby making sure that for smooth initial data sufficiently concentrated near the origin, this quantity satisfies a superlinearly forced ordinary differential inequality (ODI) if the free parameter $\gamma$ herein is appropriately adjusted (Lemma \ref{lem16}). In consequence, this will establish the following result which can be viewed as the main outcome of this study.
\begin{theo}\label{theo18}
Let $\Omega=B_R(0) \subset \mathbb{R}^n$ with $n\ge 3$ and $R>0$, and let $f$ satisfy (\ref{f1}) as well as
\begin{equation} \label{f2}
f(\xi) \ge k_f \cdot (1+\xi)^{-\alpha}
\qquad \mbox{for all } \xi\ge 0
\end{equation}
with some $k_f>0$ and $\alpha> 0$ fulfilling
\begin{equation} \label{18.1}
\alpha<\frac{n-2}{2(n-1)}.
\end{equation}
Then for any choice of $\mu>0$ one can find $R_0=R_0(\mu)\in (0,R)$ with the property that whenever
$u_0$ satisfies (\ref{init}) and is such that
\begin{equation} \label{i1}
u_0 \mbox{ is radially symmetric}
\qquad \mbox{with} \qquad
- \hspace*{-4mm} \int_{B_r(0)} u_0 dx \ge - \hspace*{-4mm} \int_\Omega u_0 dx
\quad \mbox{for all } r\in (0,R)
\end{equation}
as well as
\begin{equation} \label{18.2}
- \hspace*{-4mm} \int_{B_{R_0}(0)} u_0 dx \ge \frac{\mu}{2} \Big(\frac{R}{R_0}\Big)^n,
\end{equation}
the corresponding solution $(u,v)$ of (\ref{0})
blows up in finite time; that is, for the uniquely determined local classical solution, maximally extended
up to some time $T_{max} \in (0,\infty]$ according to
Proposition \ref{prop_loc} below, we then have $T_{max}<\infty$ and
\begin{equation} \label{18.3}
\limsup_{t\nearrowT_{max}} \|u(\cdot,t)\|_{L^\infty(\Omega)} =\infty.
\end{equation}
\end{theo}
{\bf Remark.} \quad In order to construct simple examples of initial data which the above statement asserts to enforce blow-up, we only need to observe that (\ref{i1}) is satisfied whenever $u_0$ is radial and nonincreasing with respect to $|x|$, and that (\ref{18.2}) is trivially implied if ${\rm supp} \, u_0 \subset \overline{B}_{R_0}(0)$, for instance. Beyond this, however, by means of an almost verbatim copy of the reasoning detailed in \cite[Proposition 1.2]{bellomo_win_TRAN} the set of initial data fulfilling (\ref{i1}) and (\ref{18.2}) can in fact be seen to contain an open subset of radial functions with respect to the topology in $L^\infty(\Omega)$, and to furthermore even be dense in the set of all radial functions fulfilling (\ref{init}) in the framework of the topology in $L^p(\Omega)$ for each $p\in (0,1)$.
\vspace*{1mm}
In order to secondly make sure that the above results cannot be substantially improved, let us next consider the case when besides (\ref{f1}), $f$ satisfies \begin{equation} \label{f3}
f(\xi) \le K_f \cdot (1+\xi)^{-\alpha} \qquad \mbox{for all } \xi\ge 0 \end{equation} with some $K_f>0$ and $\alpha\in\mathbb{R}$. Then the following statement, valid even in general not necessarily radial frameworks, provides an essentially exhausting complement to Theorem \ref{theo18}.
\begin{prop}\label{prop19}
Let $n\ge 1$ and $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary, and suppose that $f$ satisfies
(\ref{f1}) and (\ref{f3}) with some $K_f>0$ and
\begin{equation} \label{19.01}
\left\{ \begin{array}{ll}
\alpha \in \mathbb{R} \qquad & \mbox{if } n=1, \\[2mm]
\alpha > \frac{n-2}{2(n-1)} \qquad & \mbox{if } n\ge 2.
\end{array} \right.
\end{equation}
Then for any choice of $u_0$ coplying with (\ref{init}), the problem (\ref{0}) possesses a unique global classical
solution $(u,v) \in (C^0(\overline{\Omega}\times [0,\infty)) \cap C^{2,1}(\overline{\Omega}\times (0,\infty))) \cap C^{2,0}(\overline{\Omega}\times (0,\infty))$
which is bounded in the sense that
\begin{equation} \label{19.1}
\|u(\cdot,t)\|_{L^\infty(\Omega)} \le C
\qquad \mbox{for all } t>0
\end{equation}
with some $C>0$.
\end{prop}
A result similar to that of Proposition \ref{prop19} has been the objective of a previous study (\cite{NT}) in which boundedness statements have been derived for (\ref{0}) in the particular case when $f(\xi)=\xi^{-\alpha}$, $\xi> 0$, within the slightly restricted range determined by \begin{eqnarray*}
\left\{ \begin{array}{ll}
\alpha \in (-\infty,\frac{1}{2}) \qquad & \mbox{if } n=1, \\[2mm]
\alpha \in \Big( \frac{n-2}{2(n-1)}, \frac{1}{2}\Big) \qquad & \mbox{if } n\ge 2.
\end{array} \right. \end{eqnarray*} Our approach toward Proposition \ref{prop19}, essentially reducing to a one-page argument presented in Section \ref{sect4}, apparently provides a somewhat shorter reasoning which, as we remark here without pursuing details in this regard, can readily be extended so as to cover this result as well.\\[5pt]
We have to leave open here the question how far the number $\alpha=\frac{n-2}{2(n-1)}$, thus playing the role of a critical exponent in the style of the definition in (\ref{ac}), belongs to either the blow-up range or the opposite regime. In view of precedents concerned with, e.g., the case $f\equiv 1$ when $n=2$, it may well be conceivable that also for arbitrary $n\ge 3$, choosing $f$ as in (\ref{pr}) with $\alpha=\frac{n-2}{2(n-1)}$ may enforce critical mass phenomena with respect to finite-time blow-up, and possible even go along with some global unbounded solutions; a refined analysis in this direction, however, would go beyond the scope of this study.
\mysection{Local existence and transformation to a scalar problem}\label{sect2}
In a straightforward manner adopting arguments well-established in the context of parabolic-elliptic Keller-Segel type systems (see e.g.~\cite{nagai1995}, \cite{djie_win} and \cite{cieslak_win} for suitable precedents), one can readily establish the following statement on local existence and extensibility of solutions to (\ref{0}):
\begin{prop}\label{prop_loc}
Let $n\ge 1$ and $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary,
and assume that $f$ and $u_0$ satisfy (\ref{f1}) and (\ref{init}).
Then there exist $T_{max}\in (0,\infty]$ and a uniquely determined pair $(u,v)$ of functions
\begin{eqnarray*}
\left\{ \begin{array}{l}
u \in C^0(\overline{\Omega}\times [0,T_{max})) \cap C^{2,1}(\overline{\Omega}\times (0,T_{max})), \\[1mm]
v\in \bigcap_{q>n} L^\infty_{loc}([0,T_{max});W^{1,q}(\Omega)) \cap C^{2,0}(\overline{\Omega}\times (0,T_{max})),
\end{array} \right.
\end{eqnarray*}
with $u\ge 0$ and $v\ge 0$ in $\Omega\times (0,T_{max})$, such that
$(u,v)$ solves (\ref{0}) classically in $\Omega\times (0,T_{max})$, that
\begin{equation} \label{mass}
\int_\Omega u(\cdot,t) = \int_\Omega u_0
\qquad \mbox{for all } t\in (0,T_{max}),
\end{equation}
and that
\begin{equation} \label{ext}
\mbox{if $T_{max}<\infty$, \quad then \quad }
\limsup_{t\nearrowT_{max}} \|u(\cdot,t)\|_{L^\infty(\Omega)} = \infty.
\end{equation}
Moreover, if $\Omega=B_R(0)$ with some $R>0$ and $u_0$ is radially symmetric with respect to $x=0$, then also
$u(\cdot,t)$ and $v(\cdot,t)$ are radially symmetric for each $t\in (0,T_{max})$.
\end{prop}
\mysection{Blow-up of radial solutions when $\alpha<\frac{n-2}{2(n-1)}$. Proof of Theorem \ref{theo18}}\label{sect3}
\subsection{A basic differential inequality for a moment-like functional $\phi$}
Throughout this section assuming that $\Omega=B_R(0)\subset\mathbb{R}^n$ is a ball with some $n\ge 2$ and $R>0$, for arbitrary $u_0=u_0(r)$ fulfilling (\ref{init}) we let $T_{max}\in (0,\infty]$ and the corresponding radial local solution $(u,v)=(u(r,t),v(r,t))$ of (\ref{0}) be as provided by Proposition \ref{prop_loc}, and in the spirit of \cite{JL} we introduce \begin{equation} \label{w}
w(s,t):=\int_0^{s^\frac{1}{n}} \rho^{n-1} u(\rho,t) d\rho,
\qquad s\in [0,R^n], \ t\in [0,T_{max}). \end{equation} Then \begin{equation} \label{ws}
w_s(s,t)=\frac{1}{n} u(s^\frac{1}{n},t) \ge 0
\qquad \mbox{for all $s\in (0,R^n)$ and } t\in (0,T_{max}), \end{equation} and \begin{equation} \label{0w}
\left\{ \begin{array}{lcll}
w_t &=& n^2 s^{2-\frac{2}{n}} w_{ss}
+ n \cdot \Big(w-\frac{\mu}{n} s\Big) \cdot w_s \cdot f\Big(s^{\frac{2}{n}-2} \cdot (w-\frac{\mu}{n}s)^2 \Big),
\qquad & s\in (0,R^n), \ t\in (0,T_{max}), \\[2mm]
& & \hspace*{-14mm}
w(0,t)=0, \quad w(R^n,t)=\frac{\mu R^n}{n},
\qquad & t\in (0,T_{max}), \\[1mm]
& & \hspace*{-14mm}
w(s,0)=w_0(s):=\int_0^{s^\frac{1}{n}} \rho^{n-1} u_0(\rho) d\rho,
\qquad & s\in (0,R^n).
\end{array} \right. \end{equation}
The role of our extra assumption (\ref{i1}) in Theorem \ref{theo18} will then become clear through the following additional information.
\begin{lem}\label{lem1}
Assume (\ref{f1}), (\ref{init}) and (\ref{i1}). Then for $w$ as in (\ref{w}) we have
\begin{eqnarray*}
w(s,t) \ge \frac{\mu}{n} \cdot s
\qquad \mbox{for all $s\in (0,R^n)$ and } t\in (0,T_{max}).
\end{eqnarray*}
\end{lem}
{\sc Proof.} \quad
Since for $\underline{w}(s,t):=\frac{\mu}{n} \cdot s$, $s\in [0,R^n], \ t\ge 0$, we have
\begin{eqnarray*}
\underline{w}_t - n^2 s^{2-\frac{2}{n}} \underline{w}_{ss} - n \cdot
\Big(\underline{w}-\frac{\mu}{n} s\Big) \cdot \underline{w}_s \cdot f\Big(s^{\frac{2}{n}-2} \cdot (\underline{w}-\frac{\mu}{n}s)^2 \Big)
=0
\qquad \mbox{in } (0,R^n) \times (0,\infty)
\end{eqnarray*}
with $\underline{w}(0,t)=0$ and $\underline{w}(R^n,t)=\frac{\mu R^n}{n}$ for all $t>0$, observing that our assumption (\ref{i1}) precisely asserts that
\begin{eqnarray*}
w_0(s)
= \frac{s}{n} - \hspace*{-4mm} \int_{B_{s^{1/n}}(0)} u_0 dx
\ge \frac{s}{n} \cdot \mu = \underline{w}(s,0)
\qquad \mbox{for all } s\in (0,R^n),
\end{eqnarray*}
this follows by applying
a comparison principle
(cf., e.g., \cite[Lemma 5.1]{bellomo_win_TRAN} for a version covering the present degenerate setting)
to (\ref{0w}).
$\Box$ \vskip.2cm
Based on the latter, namely, we can make use of a presupposed additional lower estimate of the form (\ref{f2}) for $f$ in establishing the following starting point of our subsequent blow-up analysis:
\begin{lem}\label{lem3}
Suppose that (\ref{f1}) and (\ref{f2}) hold with some $k_f>0$ and $\alpha>0$, and let (\ref{init}) and (\ref{i1})
be valid.
Then with $w$ and $w_0$ taken from (\ref{w}) and (\ref{0w}),
\begin{equation} \label{z}
z(s,t):=w(s,t)-\frac{\mu}{n} \cdot s,
\qquad s\in [0,R^n], \ t\in [0,T_{max}),
\end{equation}
is nonnegative and satisfies
\begin{equation} \label{0z}
z_t \ge n^2 s^{2-\frac{2}{n}} z_{ss} + nk_f z \cdot \Big(1+s^{\frac{2}{n}-2} z^2 \Big)^{-\alpha} \cdot w_s
\qquad \mbox{for all $s\in (0,R^n)$ and } t\in (0,T_{max}).
\end{equation}
\end{lem}
{\sc Proof.} \quad
The nonnegativity of $z$ has precisely been asserted by Lemma \ref{lem1}.
Since moreover $w_s\ge 0$ by (\ref{ws}), in (\ref{0w}) we may use (\ref{f2}) to estimate
\begin{eqnarray*}
n \cdot \Big(w-\frac{\mu}{n}\cdot s\Big) \cdot w_s \cdot f\Big(
s^{\frac{2}{n}-2}
(w-\frac{\mu}{n}s)^2\Big)
&=& nzw_s \cdot f(s^{\frac{2}{n}-2} z^2) \\
&\ge& nzw_s \cdot k_f (1+ s^{\frac{2}{n}-2} z^2)^{-\alpha}
\end{eqnarray*}
for $s\in (0,R^n)$ and $t\in (0,T_{max})$, and thus to obtain (\ref{0z}) from (\ref{0w}).
$\Box$ \vskip.2cm
We can thereby state a basic evolution property of a moment-type functional which, unlike those considered in related precedents (\cite{biler}, \cite{win_NON_ct_signal_critexp}), explicitly involves the shifted variable $z$ instead of the accumulated density $w$ itself:
\begin{lem}\label{lem4}
Assume (\ref{init}), (\ref{i1}), (\ref{f1}) and (\ref{f2}) with some $k_f>0$ and $\alpha>0$, and for
$\gamma\in (0,1)$ and $s_0\in (0,R^n)$, let
\begin{equation} \label{phi}
\phi(t):=\int_0^{s_0} s^{-\gamma} (s_0-s) z(s,t) ds,
\qquad t\in [0,T_{max}),
\end{equation}
where $z$ is as given by (\ref{z}).
Then $\phi\in C^0([0,T_{max})) \cap C^1((0,T_{max}))$ with
\begin{eqnarray}\label{4.1}
\phi'(t)
&\ge& - n^2 \Big(2-\frac{2}{n}-\gamma) \Big(\gamma-1+\frac{2}{n}\Big)
\int_0^{s_0} s^{-\frac{2}{n}-\gamma} (s_0-s) z(s,t) ds \nonumber\\
& & - 2n^2 \Big(2-\frac{2}{n}-\gamma\Big) \int_0^{s_0} s^{1-\frac{2}{n}-\gamma} z(s,t) ds \nonumber\\[2mm]
& & + \psi(t)
\qquad \mbox{for all } t\in (0,T_{max}),
\end{eqnarray}
where
\begin{equation} \label{psi}
\psi(t):=nk_f
\int_0^{s_0} s^{-\gamma}(s_0-s) z(s,t) \cdot \Big(1+s^{\frac{2}{n}-2} z^2(s,t)\Big)^{-\alpha} \cdot w_s(s,t) ds,
\qquad t\in (0,T_{max}),
\end{equation}
with $w$ taken from (\ref{w}).
\end{lem}
{\sc Proof.} \quad
Since $u \in C^0(\overline{\Omega}\times [0,T_{max}))$ and $u_t\in C^0(\overline{\Omega}\times (0,T_{max}))$, a standard argument based on the dominated convergence
theorem ensures that indeed $\phi$ belongs to $C^0([0,T_{max}))$ and to $C^1((0,T_{max}))$, and that for all $t\in (0,T_{max})$,
\begin{eqnarray}\label{4.4}
\phi'(t)
&=& \int_0^{s_0} s^{-\gamma} (s_0-s) z(s,t) ds \nonumber\\
&\ge& n^2 \int_0^{s_0} s^{2-\frac{2}{n}-\gamma}(s_0-s) z_{ss} ds
+ nk_f \int_0^{s_0} s^{-\gamma}(s_0-s) z \cdot \Big(1+s^{\frac{2}{n}-2} z^2 \Big)^{-\alpha} \cdot w_s ds
\end{eqnarray}
according to (\ref{0z}). Here, two integrations by parts show that for all $t\in (0,T_{max})$ we have
\begin{eqnarray*}
& & \hspace*{-20mm}
n^2 \int_0^{s_0} s^{2-\frac{2}{n}-\gamma}(s_0-s) z_{ss} ds \\
&=& - n^2 \int_0^{s_0} \Big\{ \Big(2-\frac{2}{n}-\gamma\Big) s^{1-\frac{2}{n}-\gamma} (s_0-s) - s^{2-\frac{2}{n}-\gamma}
\Big\} z_s ds \\
&
\ge
& n^2 \int_0^{s_0} \Big\{ \Big(2-\frac{2}{n}-\gamma\Big) \Big(1-\frac{2}{n}-\gamma\Big) s^{-\frac{2}{n}-\gamma} (s_0-s)
- 2\Big(2-\frac{2}{n}-\gamma\Big) s^{1-\frac{2}{n}-\gamma} \Big\} z ds,
\end{eqnarray*}
because due to the fact that $\gamma<1 \le 2-\frac{2}{n}$ and $z(\cdot,t)\in C^1([0,R^n])$ with $z(0,t)=0$
for all $t\in (0,T_{max})$, and hence $s^{1-\frac{2}{n}-\gamma}z(s,t) \to 0$ as $s\nearrow 0$ for all $t\in (0,T_{max})$, we have
\begin{eqnarray*}
n^2 s^{2-\frac{2}{n}-\gamma} (s_0-s) z_s \bigg|_{s=0}^{s=s_0}
=0
\qquad \mbox{for all } t\in (0,T_{max})
\end{eqnarray*}
and
\begin{eqnarray*}
- n^2 \cdot \Big\{ \Big( 2-\frac{2}{n}-\gamma\Big) s^{1-\frac{2}{n}-\gamma} (s_0-s) - s^{2-\frac{2}{n}-\gamma} \Big\} z
\bigg|_{s=0}^{s=s_0}
&=& n^2 s_0^{2-\frac{2}{n}-\gamma} z(s_0,t) \\
&\ge& 0
\qquad \mbox{for all } t\in (0,T_{max}).
\end{eqnarray*}
Therefore, (\ref{4.1}) is a consequence of (\ref{4.4}).
$\Box$ \vskip.2cm
\subsection{A first lower estimate for the cross-diffusive contribution $\psi$}
Let us next approach the core of our analysis by quantifying the cross-diffusive contribution to (\ref{4.1}) through a first estimate for the rightmost summand therein from below. This will be achieved on the basis of the following elementary observation.
\begin{lem}\label{lem2}
Let $\alpha\in\mathbb{R}$ and $\beta\in (0,1]$. Then
\begin{equation} \label{2.1}
(1+\xi)^{-\alpha} \ge 1- \frac{\alpha_+}{\beta} \xi^\beta
\qquad \mbox{for all } \xi\ge 0,
\end{equation}
where $\sigma_+:=\max\{\sigma,0\}$ for $\sigma\in\mathbb{R}$.
\end{lem}
{\sc Proof.} \quad
If $\alpha\le 0$, this is obvious. In the case when $\alpha$ is positive, (\ref{2.1})
can be verified by observing that $\varphi(\xi):=(1+\xi)^{-\alpha} - 1 + \frac{\alpha}{\beta} \xi^\beta$,
$\xi\ge 0$, satisfies $\varphi(0)=0$ as well as $\varphi'(\xi)=-\alpha(1+\xi)^{-\alpha-1} + \alpha \xi^{\beta-1}$
and hence $\varphi'(\xi)\ge 0$ for all $\xi>0$, because if $\xi\in (0,1)$ then $(1+\xi)^{-\alpha-1} \le 1 \le \xi^{\beta-1}$,
while if $\xi\ge 1$ then $(1+\xi)^{-\alpha-1} \le \xi^{-1} \le \xi^{\beta-1}$.
$\Box$ \vskip.2cm
A straightforward application thereof shows that the function $\psi$ in (\ref{psi}), up to perturbation terms to be estimated later on after suitably fixing the artificial parameter $\beta$, essentially dominates an integral containing both $z_s$ and a certain power of $z$ as factors in the integrand.
\begin{lem}\label{lem5}
Suppose that (\ref{init}), (\ref{i1}), (\ref{f1}) and (\ref{f2}) hold with some $kf>0$ and $\alpha\in\mathbb{R}$,
and let $\gamma\in (0,1)$ and $s_0\in (0,R^n)$.
Then for any choice of $\beta\in (0,1]$, the function $\psi$ from (\ref{psi}) satisfies
\begin{eqnarray}\label{5.1}
\psi(t)
&\ge& nk_f \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} (s_0-s) z^{1-2\alpha}(s,t) z_s(s,t) ds \nonumber\\
& & - \frac{nk_f \alpha_+}{\beta}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} (s_0-s) z^{1-2(\alpha+\beta)}(s,t) z_s(s,t) ds \nonumber\\
& & - \frac{\muk_f \alpha_+}{\beta}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} (s_0-s) z^{1-2(\alpha+\beta)}(s,t) ds
\qquad \mbox{for all } t\in (0,T_{max}).
\end{eqnarray}
\end{lem}
{\sc Proof.} \quad
By means of Lemma \ref{lem2}, again thanks to the nonnegativity of $z$ and $w_s$ we can estimate
\begin{eqnarray*}
z \cdot \Big(1+ s^{\frac{2}{n}-2} z^2 \Big)^{-\alpha} \cdot w_s
&=&
s^{(2-\frac{2}{n})\alpha}
z^{1-2\alpha} \cdot \Big(1+s^{2-\frac{2}{n}} z^{-2} \Big)^{-\alpha} \cdot w_s \\
&\ge&
s^{(2-\frac{2}{n})\alpha}
z^{1-2\alpha} \cdot \Big\{ 1 - \frac{\alpha_+}{\beta} (s^{2-\frac{2}{n}} z^{-2})^\beta \Big\} \cdot w_s \\
&=&
s^{(2-\frac{2}{n})\alpha}
z^{1-2\alpha} w_s - \frac{\alpha_+}{\beta}
s^{(2-\frac{2}{n})(\alpha+\beta)}
z^{1-2(\alpha+\beta)} w_s
\qquad \mbox{in } (0,R^n) \times (0,T_{max}).
\end{eqnarray*}
As $w_s=z_s + \frac{\mu}{n} \ge z_s$, this entails that
\begin{eqnarray*}
z \cdot \Big(1+ s^{\frac{2}{n}-2} z^2 \Big)^{-\alpha} \cdot w_s
&\ge&
s^{(2-\frac{2}{n})\alpha}
z^{1-2\alpha} z_s \\
& & - \frac{\alpha_+}{\beta}
s^{(2-\frac{2}{n})(\alpha+\beta)}
z^{1-2(\alpha+\beta)} z_s
- \frac{\mu \alpha_+}{n\beta}
s^{(2-\frac{2}{n})(\alpha+\beta)}
z^{1-2(\alpha+\beta)}
\end{eqnarray*}
in $(0,R^n) \times (0,T_{max})$, so that (\ref{5.1}) results from the definition (\ref{psi}) of $\psi$.
$\Box$ \vskip.2cm
Here another integration by parts enables us to further control the first summand on the right of (\ref{5.1}) from below by integral expressions no longer containing $z_s$, provided that $\alpha$ satisfies a condition weaker than that in Theorem \ref{theo18}, and that the free parameter $\gamma$ lies above some threshold.
\begin{lem}\label{lem6}
Let (\ref{f1}) and (\ref{f2}) be valid with some $k_f>0$ and $\alpha\in (-\infty,\frac{n}{2(n-1)})$, and let $\gamma\in (0,1)$
be such that $\gamma>(2-\frac{2}{n})\alpha$.
Then there exists $k>0$ such that whenever (\ref{init}) and (\ref{i1}) hold, with $z$ taken from (\ref{z}) and
for any choice of $s_0\in (0,R^n)$ we have
\begin{eqnarray}\label{6.1}
nk_f \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} (s_0-s) z^{1-2\alpha}(s,t) z_s(s,t) ds
&\ge& k \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha}(s,t) ds \nonumber\\
& & + k \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha}(s,t) ds
\end{eqnarray}
for all $t\in (0,T_{max})$.
\end{lem}
{\sc Proof.} \quad
Using that $\alpha<1$, we may integrate by parts to see that for all $t\in (0,T_{max})$,
\begin{eqnarray}\label{6.2}
\hspace*{-4mm}
nk_f \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} (s_0-s) z^{1-2\alpha} z_s ds
&=& \frac{nk_f}{2-2\alpha} \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} (s_0-s) (z^{2-2\alpha})_s ds \nonumber\\
&=& - \frac{nk_f}{2-2\alpha} \int_0^{s_0} \partial_s \Big\{ s^{(2-\frac{2}{n})\alpha-\gamma}(s_0-s)\Big\}
\cdot z^{2-2-\alpha} ds,
\end{eqnarray}
where we note that the corresponding boundary terms vanish again due to the fact that for each fixed $t\in (0,T_{max})$,
$z(\cdot,t)$ belongs to $C^1([0,\mathbb{R}^n])$ with $z(0,t)=0$ by Proposition \ref{prop_loc}:
This, namely, implies that for any such $t$,
\begin{eqnarray*}
s^{(2-\frac{2}{n})\alpha-\gamma}(s_0-s) z^{2-2\alpha}(s,t)
\le s_0 \|z_s(\cdot,t)\|_{L^\infty((0,R^n))}^{2-2\alpha} s^{(2-\frac{2}{n})\alpha-\gamma+2-2\alpha}
\to 0
\qquad \mbox{as } (0,s_0)\ni s \searrow 0,
\end{eqnarray*}
because $(2-\frac{2}{n})\alpha-\gamma+2-2\alpha=2-\frac{2\alpha}{n}-\gamma \ge 2-\frac{2}{n}-\gamma>0$
as a consequence of the inequality
$\gamma<1\le 2-\frac{2}{n}$.
Furthermore computing
\begin{eqnarray*}
\partial_s \Big\{ s^{(2-\frac{2}{n})\alpha-\gamma}(s_0-s)\Big\}
= -\Big[ \gamma-\Big(2-\frac{2}{n}\Big)\alpha \Big] s^{(2-\frac{2}{n})\alpha-\gamma-1}(s_0-s)
- s^{(2-\frac{2}{n})\alpha-\gamma},
\qquad s\in (0,s_0),
\end{eqnarray*}
we readily infer (\ref{6.1}) from (\ref{6.2}) with
$k:=
\frac{nk_f}{2-2\alpha}
\cdot \min \{ \gamma-(2-\frac{2}{n})\alpha \ , \ 1\}$ being positive since
$\gamma>(2-\frac{2}{n})\alpha$.
$\Box$ \vskip.2cm
\subsection{Controlling the ill-signed summands in (\ref{5.1}). A refined lower estimate for $\psi$}
When next estimating the second and third summands on the right of (\ref{5.1}), we may evidently concentrate on the case when $\alpha$ is positive. By imposing a suitable smallness condition on the auxiliary parameter $\beta$, we may first rewrite the first of the respective integrals through another integration by parts.
\begin{lem}\label{lem8}
Let (\ref{f1}) and (\ref{f2}) be valid with some $k_f>0$ and $\alpha\in (0,\frac{n}{2(n-1)})$, and let $\gamma\in (0,1)$.
Then for any $\beta>0$ such that $\beta<1-\alpha$ and each $s_0\in (0,R^n)$, with $z$ as in (\ref{z}) we have
\begin{eqnarray}\label{8.1}
& & \hspace*{-28mm}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} (s_0-s) z^{1-2(\alpha+\beta)}(s,t) z_s(s,t) ds \nonumber\\
&=& \frac{\gamma-(2-\frac{2}{n})(\alpha+\beta)}{2-2(\alpha+\beta)}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma-1} (s_0-s) z^{2-2(\alpha+\beta)}(s,t) ds \nonumber\\
& & + \frac{1}{2-2(\alpha+\beta)}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} z^{2-2(\alpha+\beta)}(s,t) ds
\end{eqnarray}
for all $t\in (0,T_{max})$.
\end{lem}
{\sc Proof.} \quad
Since $\alpha+\beta<1$, once more integrating by parts we compute
\begin{equation} \label{8.2}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} (s_0-s) z^{1-2(\alpha+\beta)} z_s ds
= - \frac{1}{2-2(\alpha+\beta)}
\int_0^{s_0} \partial_s \Big\{ s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} (s_0-s)\Big\} \cdot z^{2-2(\alpha+\beta)} ds
\end{equation}
for all $t\in (0,T_{max})$, where again no additional boundary terms appear, because
\begin{eqnarray*}
s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} (s_0-s) z^{2-2(\alpha+\beta)} (s,t)
\le s_0 \|z_s(\cdot,t)\|_{L^\infty((0,R^n)}^{2-2(\alpha+\beta)} s^{2-\frac{2}{n}(\alpha+\beta)-\gamma} \to 0
\qquad \mbox{as } (0,s_0)\ni s\searrow 0
\end{eqnarray*}
due to the fact that $\gamma<1<2-\frac{2}{n}<2-\frac{2}{n}(\alpha+\beta)$.
Evaluating the right-hand side in (\ref{8.2}) directly yields (\ref{8.1}).
$\Box$ \vskip.2cm
By means of Young's inequality, we can make sure that both integrals on the right of (\ref{8.1}) can appropriately be absorbed by expressions of the form in (\ref{6.1}), up to addition of some error terms merely depending on the potentially small parameter $s_0$.
\begin{lem}\label{lem9}
Let (\ref{f1}) and (\ref{f2}) be valid with some $k_f>0$ and $\alpha\in (0,\frac{n}{2(n-1)})$, and let $\gamma\in (0,1)$.
Then given any $\beta>0$ such that $\beta<1-\alpha$, for each $\varepsilon>0$ one can find $C=C(\beta,\varepsilon)>0$
such that whenever (\ref{init}) and (\ref{i1}) hold and $s_0\in (0,R^n)$,
with $z$
taken from (\ref{z})
we have
\begin{eqnarray}\label{9.1}
& & \hspace*{-20mm}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma-1} (s_0-s) z^{2-2(\alpha+\beta)}(s,t) ds \nonumber\\
&\le& \varepsilon \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha}(s,t) ds
+ C s_0^{3-\frac{2}{n}-\gamma}
\qquad \mbox{for all } t\in (0,T_{max})
\end{eqnarray}
as well as
\begin{eqnarray}\label{9.2}
& & \hspace*{-20mm}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} z^{2-2(\alpha+\beta)}(s,t) ds \nonumber\\
&\le& \varepsilon \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha}(s,t) ds
+ C s_0^{3-\frac{2}{n}-\gamma}
\qquad \mbox{for all } t\in (0,T_{max})
\end{eqnarray}
\end{lem}
{\sc Proof.} \quad
According to our assumption that $\alpha+\beta<1$, we may use Young's inequality to find $c_1=c_1(\beta,\varepsilon)>0$ such that
\begin{eqnarray}\label{9.3}
& & \hspace*{-10mm}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma-1} (s_0-s) z^{2-2(\alpha+\beta)} ds \nonumber\\
&=& \int_0^{s_0} \Big\{ s^{(2-\frac{2}{n})\alpha-\gamma-1}(s_0-s) z^{2-2\alpha} \Big\}^\frac{1-\alpha-\beta}{1-\alpha}
\cdot \Big\{
s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma-1 + \frac{[\gamma+1-(2-\frac{2}{n})\alpha] \cdot (1-\alpha-\beta)}{1-\alpha}}
\cdot (s_0-s)^\frac{\beta}{1-\alpha} \Big\} ds \nonumber\\
&\le& \varepsilon \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha} ds \nonumber\\
& & + c_1 \int_0^{s_0}
s^\frac{[(2-\frac{2}{n})(\alpha+\beta)-\gamma-1] \cdot (1-\alpha)
+ [\gamma+1-(2-\frac{2}{n})\alpha] \cdot (1-\alpha-\beta)}{\beta}
\cdot (s_0-s) ds
\qquad \mbox{for all } t\in (0,T_{max}),
\end{eqnarray}
where simplifying in the last integrand shows that
\begin{eqnarray*}
\int_0^{s_0}
s^\frac{[(2-\frac{2}{n})(\alpha+\beta)-\gamma-1] \cdot (1-\alpha)
+ [\gamma+1-(2-\frac{2}{n})\alpha] \cdot (1-\alpha-\beta)}{\beta}
\cdot (s_0-s) ds
&=& \int_0^{s_0} s^{1-\frac{2}{n}-\gamma} (s_0-s) ds \\
&\le& s_0 \int_0^{s_0} s^{1-\frac{2}{n}-\gamma} ds
= \frac{s_0^{3-\frac{2}{n}-\gamma}}{2-\frac{2}{n}-\gamma},
\end{eqnarray*}
because $\gamma<1<2-\frac{2}{n}$.
Therefore, (\ref{9.3}) implies (\ref{9.1}), whereas (\ref{9.2}) can similarly be derived by once more using
Young's inequality to obtain $c_2=c_2(\beta,\varepsilon)>0$ fulfilling
\begin{eqnarray*}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} z^{2-2(\alpha+\beta)} ds
&=& \int_0^{s_0} \Big\{ s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha} \Big\}^\frac{1-\alpha-\beta}{1-\alpha}
\cdot \Big\{
s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma + \frac{[\gamma-(2-\frac{2}{n})\alpha] \cdot (1-\alpha-\beta)}{1-\alpha}}
\Big\} ds\\
&\le& \varepsilon \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha} ds \nonumber\\
& & + c_2 \int_0^{s_0}
s^\frac{[(2-\frac{2}{n})(\alpha+\beta)-\gamma] \cdot (1-\alpha)
+ [\gamma-(2-\frac{2}{n})\alpha] \cdot (1-\alpha-\beta)}{\beta} ds \\
&=& \varepsilon \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha} ds
+ c_2 \int_0^{s_0} s^{2-\frac{2}{n}-\gamma} ds \\
&=& \varepsilon \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha} ds
+ \frac{c_2 s_0^{3-\frac{2}{n}-\gamma}}{3-\frac{2}{n}-\gamma}
\end{eqnarray*}
for all $t\in (0,T_{max})$ and any choice of $s_0\in (0,R^n)$.
$\Box$ \vskip.2cm
The last integral on the right of (\ref{5.1}) can directly be estimated in quite a similar manner.
\begin{lem}\label{lem11}
Let (\ref{f1}) and (\ref{f2}) be valid with some $k_f>0$ and
$\alpha\in (0,\frac{1}{2})$,
and let $\gamma\in (0,1)$ and $\beta>0$ satisfy $2(\alpha+ \beta)<1$.
Then for all $\varepsilon>0$ there exists $C=C(\beta,\varepsilon)>0$
such that if (\ref{init}) and (\ref{i1}) are valid and $s_0\in (0,R^n)$,
then the function $z$ from (\ref{z}) has the property that
\begin{eqnarray}\label{11.1}
& & \hspace*{-20mm}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} (s_0-s) z^{1-2(\alpha+\beta)}(s,t) ds \nonumber\\
&\le& \varepsilon \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha}(s,t) ds
+ C s_0^\frac{3-\frac{2\alpha}{n}-\gamma+2\beta(3-\frac{2}{n}-\gamma)}{1+2\beta}
\qquad \mbox{for all } t\in (0,T_{max}).
\end{eqnarray}
\end{lem}
{\sc Proof.} \quad
Using that $1-2(\alpha+\beta)$ is positive, again by means of Young's inequality we can find $c_1=c_1(\beta,\varepsilon)>0$
such that
\begin{eqnarray}\label{11.2}
& & \hspace*{-10mm}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} (s_0-s) z^{1-2(\alpha+\beta)} ds \nonumber\\
&=& \int_0^{s_0} \Big\{ s^{(2-\frac{2}{n})\alpha-\gamma-1}(s_0-s) z^{2-2\alpha}
\Big\}^\frac{1-2(\alpha+\beta)}{2-2\alpha}
\cdot \Big\{
s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma + \frac{[\gamma+1-(2-\frac{2}{n})\alpha] \cdot [1-2(\alpha+\beta)]}
{2-2\alpha}}
\cdot (s_0-s)^\frac{1+2\beta}{2-2\alpha} \Big\} ds \nonumber\\
&\le& \varepsilon \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha} ds \nonumber\\
& & + c_1 \int_0^{s_0}
s^\frac{[(2-\frac{2}{n})(\alpha+\beta)-\gamma] \cdot (2-2\alpha)
+ [\gamma+1-(2-\frac{2}{n})\alpha] \cdot [1-2(\alpha+\beta)]}{1+2\beta}
\cdot (s_0-s) ds
\qquad \mbox{for all } t\in (0,T_{max}).
\end{eqnarray}
Here,
\begin{eqnarray*}
& & \hspace*{-40mm}
\int_0^{s_0}
s^\frac{[(2-\frac{2}{n})(\alpha+\beta)-\gamma] \cdot (2-2\alpha)
+ [\gamma+1-(2-\frac{2}{n})\alpha] \cdot [1-2(\alpha+\beta)]}{1+2\beta}
\cdot (s_0-s) ds \\
&=& \int_0^{s_0} s^\frac{1-\frac{2\alpha}{n}-\gamma+2\beta(1-\frac{2}{n}-\gamma)}{1+2\beta} \cdot (s_0-s) ds \\
&\le& s_0 \int_0^{s_0} s^\frac{1-\frac{2\alpha}{n}-\gamma+2\beta(1-\frac{2}{n}-\gamma)}{1+2\beta} ds\\
&=& \frac{1+2\beta}{2-\frac{2\alpha}{n}-\gamma+2\beta(2-\frac{2}{n}-\gamma)}
\cdot s_0^\frac{3-\frac{2\alpha}{n}-\gamma+2\beta(3-\frac{2}{n}-\gamma)}{1+2\beta}
\end{eqnarray*}
due to the fact that since $\beta>0$, $\alpha<1$ and $\gamma<1<2-\frac{2}{n}$,
\begin{eqnarray*}
2-\frac{2\alpha}{n}-\gamma+2\beta \Big(2-\frac{2}{n}-\gamma\Big)
> 2-\frac{2\alpha}{n}-\gamma>2-\frac{2}{n}-\gamma>0.
\end{eqnarray*}
Therefore, (\ref{11.1}) results from (\ref{11.2}).
$\Box$ \vskip.2cm
We now combine Lemma \ref{lem8} with Lemma \ref{lem9} and Lemma \ref{lem11} to see upon suitably fixing $\beta$ that Lemma \ref{lem6} entails the following refined lower estimate for the function $\psi$ in (\ref{psi}).
\begin{lem}\label{lem12}
Let (\ref{f1}) and (\ref{f2}) be valid with some $k_f>0$ and
$\alpha\in (-\infty,\frac{1}{2})$,
and let $\gamma\in (0,1)$ be such that $\gamma>(2-\frac{2}{n})\alpha$.
Then one can find $C>0$ with
the
property that whenever (\ref{init}) and (\ref{i1}) hold,
for any choice of $s_0\in (0,R^n)$ the functions $\psi$ and $z$ from (\ref{psi}) and (\ref{z}) satisfy
\begin{eqnarray}\label{12.1}
\psi(t)
&\ge& \frac{k}{2} \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha}(s,t) ds \nonumber\\
& & + \frac{k}{2} \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha}(s,t) ds \nonumber\\
& & - C s_0^{3-\frac{2}{n}-\gamma}
\qquad \mbox{for all } t\in (0,T_{max}),
\end{eqnarray}
where $k>0$ is as given by Lemma \ref{lem6}.
\end{lem}
{\sc Proof.} \quad
In view of Lemma \ref{lem5} and Lemma \ref{lem6}, we only need to consider the case when $\alpha$ is positive, in which
we use that the restriction $\alpha < \frac{n}{2(n-1)}$ particularly requires that $\alpha<\frac{1}{2}$,
whence we can fix $\beta>0$ such that $2(\alpha+\beta)<1$.
An application of Lemma \ref{lem9} to suitably small $\varepsilon>0$ then yields $c_1>0$ and $c_2>0$ such that
\begin{eqnarray*}
& & \hspace*{-20mm}
\frac{nk_f \alpha}{\beta} \cdot \frac{\gamma-(2-\frac{2}{n})(\alpha+\beta)}{2-2(\alpha+\beta)}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma-1}(s_0-s) z^{2-2(\alpha+\beta)} ds \\
&\le& \frac{k}{4} \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1}(s_0-s) z^{2-2\alpha} ds
+ c_1 s_0^{3-\frac{2}{n}-\gamma}
\qquad \mbox{for all } t\in (0,T_{max})
\end{eqnarray*}
and
\begin{eqnarray*}
& & \hspace*{-20mm}
\frac{nk_f \alpha}{\beta} \cdot \frac{1}{2-2(\alpha+\beta)}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma} z^{2-2(\alpha+\beta)} ds \\
&\le& \frac{k}{2} \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha} ds
+ c_2 s_0^{3-\frac{2}{n}-\gamma}
\qquad \mbox{for all } t\in (0,T_{max}),
\end{eqnarray*}
while using Lemma \ref{lem11} we similarly find $c_3>0$ fulfilling
\begin{eqnarray*}
& & \hspace*{-20mm}
\frac{\mu k_f \alpha}{\beta}
\int_0^{s_0} s^{(2-\frac{2}{n})(\alpha+\beta)-\gamma}(s_0-s) z^{1-2(\alpha+\beta)} ds \\
&\le& \frac{k}{4} \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1}(s_0-s) z^{2-2\alpha} ds
+ c_3 s_0^\frac{3-\frac{2\alpha}{n}-\gamma+2\beta(3-\frac{2}{n}-\gamma)}{1+2\beta}
\qquad \mbox{for all } t\in (0,T_{max}).
\end{eqnarray*}
In light of Lemma \ref{lem5}, Lemma \ref{lem6} and Lemma \ref{lem8}, combining these inequalities shows that
\begin{eqnarray}\label{12.2}
\psi(t)
&\ge& \Big(k-\frac{k}{4}-\frac{k}{4}\Big)
\int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1}(s_0-s) z^{2-2\alpha} ds
+\Big(k-\frac{k}{2}\Big)
\int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha} ds \nonumber\\
& & - (c_1+c_2) s_0^{3-\frac{2}{n}-\gamma}
- c_3 s_0^\frac{3-\frac{2\alpha}{n}-\gamma+2\beta(3-\frac{2}{n}-\gamma)}{1+2\beta}
\qquad \mbox{for all } t\in (0,T_{max}).
\end{eqnarray}
Since
\begin{eqnarray*}
\frac{3-\frac{2\alpha}{n}-\gamma+2\beta(3-\frac{2}{n}-\gamma)}{1+2\beta} - \Big(3-\frac{2}{n}-\gamma\Big)
=\frac{2(1-\alpha)}{n(1+2\beta)}
\end{eqnarray*}
is positive, and since thus
\begin{eqnarray*}
c_3 s_0^\frac{3-\frac{2\alpha}{n}-\gamma+2\beta(3-\frac{2}{n}-\gamma)}{1+2\beta}
\le c_3 R^\frac{2(1-\alpha)}{1+2\beta} s_0^{3-\frac{2}{n}-\gamma}
\end{eqnarray*}
for any $s_0\in (0,R^n)$, from (\ref{12.2}) we directly obtain (\ref{12.1}).
$\Box$ \vskip.2cm
\subsection{Estimating the first two integrals on the right of (\ref{4.1})}
Let us next examine how far also the two first integrals appearing on the right-hand side of (\ref{4.1}) can be controlled by $\psi$. Indeed, further applications of Young's inequality show that each of these integrals can essentially be estimated against one of the first two summands on the right of (\ref{12.1}), provided that $\gamma$ satisfies an additional smallness condition which can be fulfilled within a range of $\alpha$ which is yet larger than that indicated in Theorem \ref{theo18}.
\begin{lem}\label{lem13}
Suppose that (\ref{f1}) and (\ref{f2}) hold with some $k_f>0$ and
$\alpha\in (-\infty,\frac{n-2}{2n-3})$,
and assume that $\gamma\in (0,1)$ is such that
\begin{equation} \label{13.1}
(1-2\alpha)\gamma < 2-\frac{4}{n} - 4\alpha + \frac{6\alpha}{n}.
\end{equation}
Then for all $\varepsilon>0$ there exists $C=C(\varepsilon)>0$ such that if (\ref{init}) and (\ref{i1}) are valid, then
for arbitrary $s_0\in (0,R^n)$ the function $z$ in (\ref{z}) satisfies
\begin{eqnarray}\label{13.2}
\int_0^{s_0} s^{-\frac{2}{n}-\gamma} (s_0-s) z(s,t) ds
&\le& \varepsilon
\int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha}(s,t) ds \nonumber\\
& & + Cs_0^\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha}
\qquad \mbox{for all } t\in (0,T_{max})
\end{eqnarray}
and
\begin{eqnarray}\label{13.3}
\int_0^{s_0} s^{1-\frac{2}{n}-\gamma} z(s,t) ds
&\le& \varepsilon
\int_0^{s_0} s^{(2-\frac{2}{n})\alpha
-\gamma
} z^{2-2\alpha}(s,t) ds \nonumber\\
& & + Cs_0^\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha}
\qquad \mbox{for all } t\in (0,T_{max}).
\end{eqnarray}
\end{lem}
{\sc Proof.} \quad
Based on the fact that $2-2\alpha>1$, we once again invoke Young's inequality and thereby find $c_1=c_1(\varepsilon)>0$ such that
for all $t\in (0,T_{max})$,
\begin{eqnarray}\label{13.4}
& & \hspace*{-20mm}
\int_0^{s_0} s^{-\frac{2}{n}-\gamma} (s_0-s) z ds \nonumber\\
&=& \int_0^{s_0} \Big\{ s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha} \Big\}^\frac{1}{2-2\alpha} \cdot
\Big\{ s^{-\frac{2}{n}-\gamma + \frac{\gamma+1-(2-\frac{2}{n})\alpha}{2-2\alpha}}
(s_0-s)^\frac{1-2\alpha}{2-2\alpha} \Big\} ds \nonumber\\
&\le& \varepsilon
\int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha} ds
+ c_1 \int_0^{s_0} s^\frac{(-\frac{2}{n}-\gamma)(2-2\alpha) + \gamma+1-(2-\frac{2}{n})\alpha}{1-2\alpha} (s_0-s) ds,
\end{eqnarray}
and note that here the assumption (\ref{13.1}) warrants that the exponent in the latter integral satisfies
\begin{eqnarray*}
\kappa
&:=& \frac{(-\frac{2}{n}-\gamma)(2-2\alpha) + \gamma+1-(2-\frac{2}{n})\alpha}{1-2\alpha} \\
&=& \frac{1-\frac{4}{n}-2\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha} \\
&>& \frac{1-\frac{4}{n}-2\alpha+\frac{6\alpha}{n}-\Big\{ 2-\frac{4}{n}-4\alpha+\frac{6\alpha}{n}\Big\}}{1-2\alpha} \\
&=& -1.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\int_0^{s_0} s^\frac{(-\frac{2}{n}-\gamma)(2-2\alpha) + \gamma+1-(2-\frac{2}{n})\alpha}{1-2\alpha} (s_0-s) ds
&\le& s_0 \int_0^{s_0} s^\kappa ds \\
&=& \frac{1}{1+\kappa} \cdot s_0^{2+\kappa} \\
&=& \frac{1}{1+\kappa} \cdot s_0^\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha},
\end{eqnarray*}
whence (\ref{13.4}) implies (\ref{13.2}).\\
Likewise, Young's inequality provides $c_2=c_2(\varepsilon)>0$ such that
\begin{eqnarray*}
\int_0^{s_0} s^{1-\frac{2}{n}-\gamma} z ds
&=& \int_0^{s_0} \Big\{ s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha} \Big\}^\frac{1}{2-2\alpha} \cdot
\Big\{ s^{1-\frac{2}{n}-\gamma + \frac{\gamma-(2-\frac{2}{n})\alpha}{2-2\alpha}} \Big\} ds \\
&\le& \varepsilon
\int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha} ds
+ c_2 \int_0^{s_0} s^\frac{(1-\frac{2}{n}-\gamma)(2-2\alpha) + \gamma-(2-\frac{2}{n})\alpha}{1-2\alpha} ds \\
&=& \varepsilon
\int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha} ds
+ c_2 \int_0^{s_0} s^\frac{2-\frac{4}{n}-4\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha} ds \\
&=& \varepsilon
\int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha} ds \\
& & + c_2 \cdot \frac{1-2\alpha}{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma} \cdot
s_0^\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha}
\qquad \mbox{for all } t\in (0,T_{max}),
\end{eqnarray*}
and thus establishes (\ref{13.3}).
$\Box$ \vskip.2cm
\subsection{A superlinearly forced ODI for $\phi$. The role of the condition $\alpha<\frac{n-2}{2(n-1)}$}
Finally, if $\gamma$ satisfies one further smallness condition, then the first integral on the right of (\ref{12.1}) can be identified as a potentially explosion-enforcing contribution: Namely, for such $\gamma$ the following lemma reveals that the superlinear and actually convex growth of $0\le \widehat{z} \mapsto \widehat{z}^{2-2\alpha}$ allows for an estimation of said integral in terms of a superlinear expression of $\phi$:
\begin{lem}\label{lem14}
Assume (\ref{f1}) and (\ref{f2}) with constants $k_f>0$ and $\alpha\in (-\infty,\frac{1}{2})$,
and let $\gamma\in (0,1)$ satisfy
\begin{equation} \label{14.1}
(1-2\alpha)\gamma < 2- 4\alpha + \frac{2\alpha}{n}.
\end{equation}
Then there exists $C>0$ such that whenever (\ref{init}) and (\ref{i1}) hold
and $z$ and $\phi$ are as in (\ref{z}) and (\ref{phi}), we have
\begin{eqnarray}\label{14.2}
\hspace*{-4mm}
\int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha}(s,t) ds
\ge C s_0^{-3+6\alpha-\frac{2\alpha}{n}+(1-2\alpha)\gamma} \phi^{2-2\alpha}(t)
\quad \mbox{for all } t\in (0,T_{max}).
\end{eqnarray}
\end{lem}
{\sc Proof.} \quad
Again relying on the property that $2-2\alpha>1$, we may use the H\"older inequality to estimate
\begin{eqnarray}\label{14.3}
\phi(t)
&=& \int_0^{s_0}
\Big\{ s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha} \Big\}^\frac{1}{2-2\alpha}
\cdot
\Big\{ s^{-\gamma+\frac{\gamma+1-(2-\frac{2}{n})\alpha}{2-2\alpha}} (s_0-s)^\frac{1-2\alpha}{2-2\alpha} \Big\} ds
\nonumber\\
&\le& \bigg\{ \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha} ds \bigg\}^\frac{1}{2-2\alpha}
\cdot \bigg\{
\int_0^{s_0} s^\frac{-(2-2\alpha)\gamma+\gamma+1-(2-\frac{2}{n})\alpha}{1-2\alpha} (s_0-s) ds
\bigg\}^\frac{1-2\alpha}{2-2\alpha}
\end{eqnarray}
for all $t\in (0,T_{max})$,
where using that $2-4\alpha+\frac{2\alpha}{n}-(1-2\alpha)\gamma$ is positive by (\ref{14.1}), we see that
\begin{eqnarray*}
\int_0^{s_0} s^\frac{-(2-2\alpha)\gamma+\gamma+1-(2-\frac{2}{n})\alpha}{1-2\alpha} (s_0-s) ds
&\le& s_0 \int_0^{s_0} s^\frac{-(2-2\alpha)\gamma+\gamma+1-(2-\frac{2}{n})\alpha}{1-2\alpha} ds \\
&=& c_1 s_0^\frac{3-6\alpha+\frac{2\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha}
\end{eqnarray*}
with $c_1:=\frac{1-2\alpha}{2-4\alpha+\frac{2\alpha}{n}-(1-2\alpha)\gamma}>0$.
Rearranging (\ref{14.3}) thus shows that
\begin{eqnarray*}
\bigg\{ \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha} ds \bigg\}^\frac{1}{2-2\alpha}
\ge c_1^{-\frac{1-2\alpha}{2-2\alpha}} s_0^{-\frac{3-6\alpha+\frac{2\alpha}{n}-(1-2\alpha)\gamma}{2-2\alpha}} \phi(t)
\qquad \mbox{for all } t\in (0,T_{max}),
\end{eqnarray*}
from which (\ref{14.2}) immediately follows.
$\Box$ \vskip.2cm
Now a crucial issue consists of clarifying how far our above, and at first glance quite technical, conditions on $\gamma$ can simultaneously be fulfilled. That this can indeed be achieved under the assumption from Theorem \ref{theo18}, essentially optimal in view of Proposition \ref{prop19}, results from a simple consideration:
\begin{lem}\label{lem15}
Let $n\ge 2$ and $\alpha\in\mathbb{R}$ be such that $\alpha<\frac{n-2}{2(n-1)}$.
Then
\begin{equation} \label{15.2}
\Big(4-\frac{4}{n}\Big) \alpha^2 - \Big(6-\frac{8}{n}\Big) \alpha + 2-\frac{4}{n}>0,
\end{equation}
and there exists $\gamma \in (0,1)$ such that $\gamma>(2-\frac{2}{n})\alpha$ and that moreover
(\ref{13.1}) as well as (\ref{14.1}) are valid.
\end{lem}
{\sc Proof.} \quad
Since
\begin{eqnarray*}
\Big(4-\frac{4}{n}\Big) \alpha^2 - \Big(6-\frac{8}{n}\Big) \alpha + 2-\frac{4}{n}
=\Big(4-\frac{4}{n}\Big) \cdot \Big( \frac{n-2}{2(n-1)} - \alpha \Big) \cdot (1-\alpha),
\end{eqnarray*}
it is evident that our assumption on $\alpha$ implies (\ref{15.2}).
We can thereby ensure simultaneous availability of (\ref{13.1}) and the condition that $\gamma>(2-\frac{2}{n})\alpha$:
Namely, since
\begin{eqnarray*}
\Big\{ 2-\frac{4}{n}-4\alpha+\frac{6\alpha}{n}\Big\}
- \Big\{ \Big(2-\frac{2}{n}\Big)\alpha \cdot (1-2\alpha)\Big\}
= \Big(4-\frac{4}{n}\Big) \alpha^2 - \Big(6-\frac{8}{n}\Big) \alpha + 2-\frac{4}{n},
\end{eqnarray*}
from (\ref{15.2}) it follows that the interval
$J:=\Big((2-\frac{2}{n})\alpha \, , \, \frac{2-\frac{4}{n}-4\alpha+\frac{6\alpha}{n}}{1-2\alpha}\Big)$
is not empty, so that noting that moreover
$(2-\frac{2}{n})\alpha<(2-\frac{2}{n}) \cdot \frac{n-2}{2(n-1)} = \frac{n-2}{n}<1$, we see that
it is possible to fix $\gamma \in (0,1)$ such that $\gamma\in J$.
Since
\begin{eqnarray*}
2-\frac{4}{n}-4\alpha+\frac{6\alpha}{n}
= 2-4\alpha+\frac{2\alpha}{n} - \frac{4(1-\alpha)}{n}
< 2-4\alpha+\frac{2\alpha}{n}
\end{eqnarray*}
due to the fact that $\alpha<1$, this furthermore entails (\ref{14.1}).
$\Box$ \vskip.2cm
Upon choosing $\gamma$ as thus specified, we can indeed turn (\ref{4.1}) into an autonomous and superlinearly forced ODI for $\phi$ in the following style.
\begin{lem}\label{lem16}
Let $n\ge 2$, assume (\ref{f1}) and (\ref{f2}) with some $k_f>0$ and $\alpha\in\mathbb{R}$ fulfilling $\alpha<\frac{n-2}{2(n-1)}$,
and let $\gamma \in (0,1)$ be as provided by Lemma \ref{lem15}.
Then there exists $C>0$ such that if (\ref{init}) and (\ref{i1}) hold, for any choice of $s_0\in (0,R^n)$ the function
$\phi$ from (\ref{phi}) satisfies
\begin{equation} \label{16.1}
\phi'(t) \ge \frac{1}{C} s_0^{-3+6\alpha - \frac{2\alpha}{n}+(1-2\alpha)\gamma} \phi^{2-2\alpha}(t)
- Cs_0^\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha}
\qquad \mbox{for all } t\in (0,T_{max}).
\end{equation}
\end{lem}
{\sc Proof.} \quad
Since $\gamma$ satisfies (\ref{13.1}), we may employ Lemma \ref{lem13} to find $c_1>0$ such that with $k>0$
given by Lemma \ref{lem6} and $\lambda:=\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha}$,
\begin{eqnarray*}
n^2 \Big(2-\frac{2}{n}-\gamma\Big)\Big(\gamma-1+\frac{2}{n}\Big)
\int_0^{s_0} s^{-\frac{2}{n}-\gamma} (s_0-s) zds
\le \frac{k}{4} \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha} ds
+ c_1 s_0^\lambda
\end{eqnarray*}
and
\begin{eqnarray*}
n^2 \Big(2-\frac{2}{n}-\gamma\Big)
\int_0^{s_0} s^{1-\frac{2}{n}-\gamma} zds
\le \frac{k}{2} \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma} z^{2-2\alpha} ds
+ c_1 s_0^\lambda
\end{eqnarray*}
for all $t\in (0,T_{max})$.
As furthermore $\gamma>(2-\frac{2}{n})\alpha$, we may use this in conjunction with Lemma \ref{lem4} and Lemma \ref{lem12}
to see that with some $c_2>0$ we have
\begin{equation} \label{16.2}
\phi'(t)
\ge \frac{k}{4} \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha} ds
- 2c_1 s_0^\lambda - c_2 s_0^{3-\frac{2}{n}-\gamma}
\qquad \mbox{for all } t\in (0,T_{max}),
\end{equation}
where we note that
\begin{eqnarray*}
\Big(3-\frac{2}{n}-\gamma\Big) -\lambda
= \frac{2(1-\alpha)}{n(1-2\alpha)}
\end{eqnarray*}
is positive and hence
\begin{equation} \label{16.3}
c_2 s_0^{3-\frac{2}{n}-\gamma}
= c_2 s_0^\frac{2(1-\alpha)}{n(1-2\alpha)} \cdot s_0^\lambda
\le c_3 s_0^\lambda
\end{equation}
with $c_3:=c_2 R^\frac{2(1-\alpha)}{1-2\alpha}$.\\
Apart from that, we may rely on the fact that $\gamma$ also satisfies (\ref{14.1}) to infer from Lemma \ref{lem14}
the existence of $c_4>0$ such that
\begin{eqnarray*}
\frac{k}{4} \int_0^{s_0} s^{(2-\frac{2}{n})\alpha-\gamma-1} (s_0-s) z^{2-2\alpha} ds
\ge c_4 s_0^{-3+6\alpha-\frac{2\alpha}{n}+(1-2\alpha)\gamma} \phi^{2-2\alpha}(t)
\qquad \mbox{for all } t\in (0,T_{max}).
\end{eqnarray*}
Therefore, (\ref{16.2}) together with (\ref{16.3}) establishes (\ref{16.1}) if we let
$C:=\max\{\frac{1}{c_4} \, , \, 2c_1+c_3\}$.
$\Box$ \vskip.2cm
\subsection{Selection of concentrated initial data. Proof of Theorem \ref{theo18}}
In order to derive a blow-up result from Lemma \ref{lem16} by means of a contradiction argument, it remains to select the free parameter $s_0$, along with initial data which reflect appropriate mass concentration near the origin, in such a way that (\ref{16.1}) at the initial instant, and hence throughout evolution by comparison, can be turned into a genuine superlinear ODI without appearance of any negative summand on its right:
\begin{lem}\label{lem17}
Let $n\ge 2$, and suppose that (\ref{f1}) and (\ref{f2}) hold with some $k_f>0$ and $\alpha\in\mathbb{R}$ such that
$\alpha<\frac{n-2}{2(n-1)}$.
Then for each $\mu>0$ one can find $s_0=s_0(\mu)\in (0,\frac{R^n}{4})$ such that whenever
(\ref{init}) and (\ref{i1}) hold and $w_0$ from (\ref{0w}) satisfies
\begin{equation} \label{17.1}
w_0 \Big(\frac{s_0}{2}\Big) \ge \frac{\mu R^n}{2n},
\end{equation}
the solution of (\ref{0}) has the property that $T_{max}<\infty$.
\end{lem}
{\sc Proof.} \quad
Relying on our assumption $\alpha<\frac{n-2}{2(n-1)}$, we can fix $\gamma\in (0,1)$ as given by Lemma \ref{lem15},
and then invoke Lemma \ref{lem16} to find $c_1>0$ and $c_2>0$ with the property that if (\ref{init}) and (\ref{i1}) hold,
then for arbitrary $s_0\in (0,R^n)$, the function $\phi$ as accordingly defined through (\ref{phi}) satisfies
\begin{equation} \label{17.2}
\phi'(t)
\ge c_1 s_0^{-3+6\alpha - \frac{2\alpha}{n}+(1-2\alpha)\gamma} \phi^{2-2\alpha}(t)
- c_2 s_0^\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha}
\qquad \mbox{for all } t\in (0,T_{max}).
\end{equation}
We now use that $\gamma<1$ in introducing
\begin{eqnarray*}
c_3=c_3(\mu):=\frac{\mu R^n}{16n} \cdot \frac{(\frac{3}{4})^{1-\gamma} -(\frac{1}{2})^{1-\gamma}}{1-\gamma}>0
\end{eqnarray*}
and then choose $s_0=s_0(\mu)\in (0,\frac{R^n}{4})$ small enough such that
\begin{equation} \label{17.3}
\frac{1}{2} c_1 c_3^{2-2\alpha} s_0^{1+2\alpha-\frac{2\alpha}{n}
-\gamma
}
\ge c_2 s_0^\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha},
\end{equation}
noting that the latter is possible due to (\ref{15.2}), which indeed ensures that
\begin{eqnarray*}
\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha}
- \Big\{ 1+2\alpha-\frac{2\alpha}{n}
-\gamma
\Big\}
= \frac{(4-\frac{4}{n})\alpha^2 -(6-\frac{8}{n})\alpha + 2-\frac{4}{n}}{1-2\alpha}
\end{eqnarray*}
is positive, and that hence (\ref{17.3}) holds for all suitably small $s_0>0$.
Keeping this value of $s_0$ fixed, we now assume to be given initial data which besides (\ref{init}) and (\ref{i1})
fulfill (\ref{17.1}).
Then by monotonicity of $w_0$, we have $w_0(s) \ge \frac{\mu R^n}{2n}$ for all $s\in (\frac{s_0}{2},\frac{3s_0}{4})$
and hence, by (\ref{z}) and our definition of $c_3$,
\begin{eqnarray*}
\phi(0)
&\ge& \frac{s_0}{4} \int_{\frac{s_0}{2}}^{\frac{3s_0}{4}} s^{-\gamma} z_0(s) ds \\
&=& \frac{s_0}{4} \int_{\frac{s_0}{2}}^{\frac{3s_0}{4}} s^{-\gamma} \Big( w_0(s)-\frac{\mu}{n} \cdot s\Big) ds \\
&\ge& \frac{\mu s_0}{4n} \int_{\frac{s_0}{2}}^{\frac{3s_0}{4}} s^{-\gamma} \Big( \frac{R^n}{2}-s\Big) ds \\
&\ge& \frac{\mu R^n s_0}{16n} \int_{\frac{s_0}{2}}^{\frac{3s_0}{4}} s^{-\gamma} ds \\
&=& c_3 s_0^{2-\gamma},
\end{eqnarray*}
because $s_0\le \frac{R^n}{4}$.
As a consequence of (\ref{17.3}), we thus obtain that
\begin{eqnarray}\label{17.99}
& & \hspace*{-20mm}
c_1 s_0^{-3+6\alpha - \frac{2\alpha}{n}+(1-2\alpha)\gamma} \phi^{2-2\alpha}(0)
- c_2 s_0^\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha} \nonumber\\
&\ge& \frac{1}{2} c_1 s_0^{-3+6\alpha - \frac{2\alpha}{n}+(1-2\alpha)\gamma} \phi^{2-2\alpha}(0),
\end{eqnarray}
whence an ODE comparison argument shows that thanks to (\ref{17.2}),
\begin{equation} \label{17.999}
\phi'(t) \ge \frac{1}{2} c_1 s_0^{-3+6\alpha - \frac{2\alpha}{n}+(1-2\alpha)\gamma} \phi^{2-2\alpha}(t)
\qquad \mbox{for all } t\in (0,T_{max}),
\end{equation}
for writing
\begin{eqnarray*}
\underline{\phi}(t):=\Bigg(
\frac{2c_2 s_0^\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha}}
{c_1 s_0^{-3+6\alpha - \frac{2\alpha}{n}+(1-2\alpha)\gamma}}
\Bigg)^\frac{1}{2-2\alpha},
\qquad t\ge 0,
\end{eqnarray*}
we directly observe that
\begin{eqnarray*}
\underline{\phi}_t - c_1 s_0^{-3+6\alpha - \frac{2\alpha}{n}+(1-2\alpha)\gamma} \underline{\phi}^{2-2\alpha}(t)
+ c_2 s_0^\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha}
&=& - c_2 s_0^\frac{3-\frac{4}{n}-6\alpha+\frac{6\alpha}{n}-(1-2\alpha)\gamma}{1-2\alpha} \\
&\le& 0
\qquad \mbox{for all } t>0,
\end{eqnarray*}
whereas (\ref{17.99}) ensures that $\underline{\phi}(0) \le \phi(0)$.
Since $2-2\alpha>1$, by positivity of $\phi(0)$
the inequality in (\ref{17.999}), however implies
that $\phi$, and thus clearly also $(u,v)$, must cease to
exist in finite time.
$\Box$ \vskip.2cm
Our main result on the occurrence of blow-up in (\ref{0}) can now be obtained by simply transforming the above back to the original variables:\\[5pt]
{\sc Proof} \ of Theorem \ref{theo18}. \quad
With $s_0=s_0(\mu)\in (0,\frac{R^n}{4})$ as provided by Lemma \ref{lem17}, we let $R_0=R_0(\mu)
:=(\frac{s_0}{2})^\frac{1}{n} \in (0,R)$, and assume
$u_0$ to satisfy (\ref{init}), (\ref{i1}) as well as (\ref{18.2}).
We then only need to observe that when rewritten in terms of the function $w_0$ defined in (\ref{0w}),
(\ref{18.2}) says that
\begin{eqnarray*}
w_0\Big(\frac{s_0}{2}\Big)
= \frac{R_0^n}{n} - \hspace*{-4mm} \int_{B_{R_0}(0)} u_0 dx
\ge \frac{R_0^n}{n} \cdot \frac{\mu}{2}\Big(\frac{R}{R_0}\Big)^n
= \frac{\mu R^n}{2n}.
\end{eqnarray*}
Therefore, namely, the claim becomes a direct consequence of Lemma \ref{lem17}.
$\Box$ \vskip.2cm
\mysection{Global boundedness for supercritical $\alpha$. Proof of Proposition \ref{prop19}}\label{sect4}
Let us finally make sure that our blow-up result indeed is essentially optimal with respect to the parameter range therein. Our brief reasoning in this direction will rely on a basic integrability feature of the taxis gradient which immediately results from the $L^1$ bound for $u$ implied by (\ref{mass}) due to standard elliptic regularity theory:
\begin{lem}\label{lem20}
Under the assumptions of Proposition \ref{prop19}, for all $q\in [1,\frac{n}{n-1})$ one can find $C=C(q)>0$ such that
\begin{eqnarray*}
\|\nabla v(\cdot,t)\|_{L^q(\Omega)} \le C
\qquad \mbox{for all } t\in (0,T_{max}).
\end{eqnarray*}
\end{lem}
{\sc Proof.} \quad
In view of (\ref{mass}),
this is a direct consequence of the second equation in (\ref{0}) when combined with
well-known regularity theory for elliptic problems with $L^1$ inhomogeneities
(\cite{brezis_strauss}).
$\Box$ \vskip.2cm
We can thereby easily derive the claimed statement on global existence and boundedness by means of a suitably designed $L^\infty$ estimation procedure based on smoothing properties of the Neumann heat semigroup:\\[5pt]
{\sc Proof} \ of Proposition \ref{prop19}. \quad
Without loss of generality assuming that $\alpha<\frac{1}{2}$, we note that since $2(n-1)\alpha>n-2$ and hence
$(1-2\alpha)n <\frac{n}{n-1}$, we can fix $q\in [1,\frac{n}{n-1})$ such that $q>(1-2\alpha)n$, whereupon it becomes
possible to pick $r>n$ such that still $q>(1-2\alpha)r$.\\
Then according to known smoothing properties of the Neumann heat semigroup $(e^{t\Delta})_{t\ge 0}$ in $\Omega$
(\cite{FIWY}), there exist $c_1>0$ and $\theta>0$ such that for all $\varphi\in C^1(\overline{\Omega};\mathbb{R}^n)$ such that
$\varphi\cdot\nu=0$ on $\partial\Omega$,
\begin{eqnarray*}
\|e^{t\Delta} \nabla\cdot\varphi\|_{L^\infty(\Omega)} \le c_1 t^{-\frac{1}{2}-\frac{n}{2r}} e^{-\theta t}
\|\varphi\|_{L^r(\Omega)}
\qquad \mbox{for all } t>0.
\end{eqnarray*}
Therefore, using
that $0\le e^{t\Delta} u_0 \le \|u_0\|_{L^\infty(\Omega)}$ for all $t>0$ by the maximum principle,
we infer that for all $t\in (0,T_{max})$,
\begin{eqnarray}\label{19.2}
\hspace*{-7mm}
\|u(\cdot,t)\|_{L^\infty(\Omega)}
&=& \Bigg\| e^{t\Delta} u_0
- \int_0^t e^{(t-s)\Delta}
\nabla \cdot \bigg\{ u(\cdot,s) f\Big( |\nabla v(\cdot,s)|^2 \Big) \nabla v(\cdot,s)
\bigg\} ds \Bigg\|_{L^\infty(\Omega)} \nonumber\\
&\le& \|e^{t\Delta} u_0\|_{L^\infty(\Omega)}
+ c_1 \int_0^t \bigg\| e^{(t-s)\Delta}
\nabla \cdot \bigg\{ u(\cdot,s) f\Big( |\nabla v(\cdot,s)|^2 \Big) \nabla v(\cdot,s)
\bigg\} \bigg\|_{L^\infty(\Omega)} ds \nonumber\\
&\le& \|u_0\|_{L^\infty(\Omega)}
+ c_1 \int_0^t (t-s)^{-\frac{1}{2}-\frac{n}{2r}} e^{-\theta(t-s)}
\bigg\| u(\cdot,s) f\Big( |\nabla v(\cdot,s)|^2 \Big) \nabla v(\cdot,s) \bigg\|_{L^r(\Omega)} ds,
\end{eqnarray}
where by (\ref{f3}) and the H\"older inequality, for all $s\in (0,T_{max})$ we see that
\begin{eqnarray*}
\bigg\| u(\cdot,s) f\Big( |\nabla v(\cdot,s)|^2 \Big) \nabla v(\cdot,s) \bigg\|_{L^r(\Omega)}
&\le& K_f \Big\| u(\cdot,s) \Big(1+|\nabla v(\cdot,s)|^2 \Big)^{-\alpha} \nabla v(\cdot,s) \Big\|_{L^r(\Omega)} \\
&\le& K_f \Big\| u(\cdot,s) |\nabla v(\cdot,s)|^{1-2\alpha} \Big\|_{L^r(\Omega)} \\
&\le& K_f \|u(\cdot,s)\|_{L^\frac{qr}{q-(1-2\alpha)r}(\Omega)} \|\nabla v(\cdot,s)\|_{L^q(\Omega)}^{1-2\alpha} \\
&\le& K_f \|u(\cdot,s)\|_{L^\infty(\Omega)}^a \|u(\cdot,s)\|_{L^1(\Omega)}^{1-a}
\|\nabla v(\cdot,s)\|_{L^q(\Omega)}^{1-2\alpha}
\end{eqnarray*}
with $a:=1-\frac{q-(1-2\alpha)r}{qr} \in (0,1)$.
As $\|u(\cdot,s)\|_{L^1(\Omega)}=\int_\Omega u_0$ for all $s\in (0,T_{max})$,
by means of Lemma \ref{lem20} we thus find $c_2>0$ such that writing
$M(T):=\sup_{t\in (0,T)} \|u(\cdot,t)\|_{L^\infty(\Omega)}$ for $T\in (0,T_{max})$, we have
\begin{eqnarray*}
\bigg\| u(\cdot,s) f\Big( |\nabla v(\cdot,s)|^2 \Big) \nabla v(\cdot,s) \bigg\|_{L^r(\Omega)}
\le c_2\|u(\cdot,s)\|_{L^\infty(\Omega)}^a
\le c_2 M^a(T)
\qquad \mbox{for all } s\in (0,T),
\end{eqnarray*}
whence (\ref{19.2}) shows that
\begin{eqnarray}\label{19.3}
\|u(\cdot,t)\|_{L^\infty(\Omega)}
&\le& \|u_0\|_{L^\infty(\Omega)}
+ c_1 c_2 M^a(T) \int_0^t (t-s)^{-\frac{1}{2} - \frac{n}{2r}} e^{-\theta(t-s)} ds \nonumber\\
&\le& c_3 + c_3 M^a(T)
\qquad \mbox{for all } t\in (0,T)
\end{eqnarray}
if we let
$c_3:=\max\left\{ \|u_0\|_{L^\infty(\Omega)} \, , \,
c_1 c_2 \int_0^\infty \sigma^{-\frac{1}{2}-\frac{n}{2r}} e^{-\theta\sigma} \right\}$ and note that $c_3$ is finite due to the
restriction that $r>n$.
In consequence, (\ref{19.3}) entails that
\begin{eqnarray*}
M(T) \le c_3 + c_3 M^a(T)
\qquad \mbox{for all } T\in (0,T_{max})
\end{eqnarray*}
and thereby asserts that $\|u(\cdot,t)\|_{L^\infty(\Omega)} \le \max\{1 \, , \, (2c_3)^\frac{1}{1-a} \}$
for all $t\in (0,T_{max})$.
Thanks to (\ref{ext}), this firstly ensures that $T_{max}=\infty$, and that secondly moreover (\ref{19.1}) holds.
$\Box$ \vskip.2cm
\vspace*{5mm}
{\bf Acknowledgement.} \quad
The author warmly thanks the anonymous reviewer for numerous helpful remarks and suggestions.
He furthermore
acknowledges support of the {\em Deutsche Forschungsgemeinschaft}
in the context of the project {\em Emergence of structures and advantages in cross-diffusion systems}
(Project No.~411007140, GZ: WI 3707/5-1).
\end{document} | arXiv |
Viscosity solution
In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in dynamic programming (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems,[1][2] as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games.
The classical concept was that a PDE
$F(x,u,Du,D^{2}u)=0$
over a domain $x\in \Omega $ has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that $x$, $u$, $Du$, $D^{2}u$ satisfy the above equation at every point.
If a scalar equation is degenerate elliptic (defined below), one can define a type of weak solution called viscosity solution. Under the viscosity solution concept, u does not need to be everywhere differentiable. There may be points where either $Du$ or $D^{2}u$ does not exist and yet u satisfies the equation in an appropriate generalized sense. The definition allows only for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations.
Definition
There are several equivalent ways to phrase the definition of viscosity solutions. See for example the section II.4 of Fleming and Soner's book[3] or the definition using semi-jets in the Users Guide.[4]
Degenerate elliptic
An equation $F(x,u,Du,D^{2}u)=0$ in a domain $\Omega $ is defined to be degenerate elliptic if for any two symmetric matrices $X$ and $Y$ such that $Y-X$ is positive definite, and any values of $x\in \Omega $, $u\in \mathbb {R} $ and $p\in \mathbb {R} ^{n}$, we have the inequality $F(x,u,p,X)\geq F(x,u,p,Y)$. For example, $-\Delta u=0$ (where $\Delta $ denotes the Laplacian) is degenerate elliptic since in this case, $F(x,u,p,X)=-{\text{trace}}(X)$, and the trace of $X$ is the sum of its eigenvalues. Any real first- order equation is degenerate elliptic.
Viscosity subsolution
An upper semicontinuous function $u$ in $\Omega $ is defined to be a subsolution of the above degenerate elliptic equation in the viscosity sense if for any point $x_{0}\in \Omega $ and any $C^{2}$ function $\phi $ such that $\phi (x_{0})=u(x_{0})$ and $\phi \geq u$ in a neighborhood of $x_{0}$, we have $F(x_{0},\phi (x_{0}),D\phi (x_{0}),D^{2}\phi (x_{0}))\leq 0$.
Viscosity supersolution
A lower semicontinuous function $u$ in $\Omega $ is defined to be a supersolution of the above degenerate elliptic equation in the viscosity sense if for any point $x_{0}\in \Omega $ and any $C^{2}$ function $\phi $ such that $\phi (x_{0})=u(x_{0})$ and $\phi \leq u$ in a neighborhood of $x_{0}$, we have $F(x_{0},\phi (x_{0}),D\phi (x_{0}),D^{2}\phi (x_{0}))\geq 0$.
Viscosity solution
A continuous function u is a viscosity solution of the PDE $F(x,u,Du,D^{2}u)=0$ in $\Omega $ if it is both a supersolution and a subsolution. Note that the boundary condition in the viscosity sense has not been discussed here.
Example
Consider the boundary value problem $|u'(x)|=1$, or $F(u')=|u'|-1=0$, on $(-1,1)$ with boundary conditions $u(-1)=u(1)=0$. Then, the function $u(x)=1-|x|$ is a viscosity solution.
Indeed, note that the boundary conditions are satisfied classically, and $|u'(x)|=1$ is well-defined in the interior except at $x=0$. Thus, it remains to show that the conditions for viscosity subsolution and viscosity supersolution hold at $x=0$. Suppose that $\phi (x)$ is any function differentiable at $x=0$ with $\phi (0)=u(0)=1$ and $\phi (x)\geq u(x)$ near $x=0$. From these assumptions, it follows that $\phi (x)-\phi (0)\geq -|x|$. For positive $x$, this inequality implies $\lim _{x\to 0^{+}}{\frac {\phi (x)-\phi (0)}{x}}\geq -1$, using that $|x|/x=sgn(x)=1$ for $x>0$. On the other hand, for $x<0$, we have that $\lim _{x\to 0^{-}}{\frac {\phi (x)-\phi (0)}{x}}\leq 1$. Because $\phi $ is differentiable, the left and right limits agree and are equal to $\phi '(0)$, and we therefore conclude that $|\phi '(0)|\leq 1$, i.e., $F(\phi '(0))\leq 0$. Thus, $u$ is a viscosity subsolution. Moreover, the fact that $u$ is a supersolution holds vacuously, since there is no function $\phi (x)$ differentiable at $x=0$ with $\phi (0)=u(0)=1$ and $\phi (x)\leq u(x)$ near $x=0$. This implies that $u$ is a viscosity solution.
In fact, one may prove that $u$ is the unique viscosity solution for such problem. The uniqueness part involves a more refined argument.
Discussion
The previous boundary value problem is an eikonal equation in a single spatial dimension with $f=1$, where the solution is known to be the signed distance function to the boundary of the domain. Note also in the previous example, the importance of the sign of $F$. In particular, the viscosity solution to the PDE $-F=0$ with the same boundary conditions is $u(x)=|x|-1$. This can be explained by observing that the solution $u(x)=1-|x|$ is the limiting solution of the vanishing viscosity problem $F(u')=[u']^{2}-1=\epsilon u''$ as $\epsilon $ goes to zero, while $u(x)=|x|-1$ is the limit solution of the vanishing viscosity problem $-F(u')=1-[u']^{2}=\epsilon u''$.[5] One can readily confirm that $u_{\epsilon }(x)=\epsilon [\ln(\cosh(1/\epsilon ))-\ln(\cosh(x/\epsilon ))]$ solves the PDE $F(u')=[u']^{2}-1=\epsilon u''$ for each $\epsilon >0$. Further, the family of solutions $u_{\epsilon }$ converges toward the solution $u=1-|x|$ as $\epsilon $ vanishes (see Figure).
Basic properties
The three basic properties of viscosity solutions are existence, uniqueness and stability.
• The uniqueness of solutions requires some extra structural assumptions on the equation. Yet it can be shown for a very large class of degenerate elliptic equations.[4] It is a direct consequence of the comparison principle. Some simple examples where comparison principle holds are
1. $u+H(x,\nabla u)=0$ with H uniformly continuous in both variables.
2. (Uniformly elliptic case) $F(D^{2}u,Du,u)=0$ so that $F$ is Lipschitz with respect to all variables and for every $r\leq s$ and $X\geq Y$, $F(Y,p,s)\geq F(X,p,r)+\lambda ||X-Y||$ for some $\lambda >0$.
• The existence of solutions holds in all cases where the comparison principle holds and the boundary conditions can be enforced in some way (through barrier functions in the case of a Dirichlet boundary condition). For first order equations, it can be obtained using the vanishing viscosity method[6][2] or for most equations using Perron's method.[7][8][2] There is a generalized notion of boundary condition, in the viscosity sense. The solution to a boundary problem with generalized boundary conditions is solvable whenever the comparison principle holds.[4]
• The stability of solutions in $L^{\infty }$ holds as follows: a locally uniform limit of a sequence of solutions (or subsolutions, or supersolutions) is a solution (or subsolution, or supersolution). More generally, the notions of viscosity sub- and supersolution are also conserved by half-relaxed limits.[4]
History
The term viscosity solutions first appear in the work of Michael G. Crandall and Pierre-Louis Lions in 1983 regarding the Hamilton–Jacobi equation.[6] The name is justified by the fact that the existence of solutions was obtained by the vanishing viscosity method. The definition of solution had actually been given earlier by Lawrence C. Evans in 1980.[9] Subsequently the definition and properties of viscosity solutions for the Hamilton–Jacobi equation were refined in a joint work by Crandall, Evans and Lions in 1984.[10]
For a few years the work on viscosity solutions concentrated on first order equations because it was not known whether second order elliptic equations would have a unique viscosity solution except in very particular cases. The breakthrough result came with the method introduced by Robert Jensen in 1988 to prove the comparison principle using a regularized approximation of the solution which has a second derivative almost everywhere (in modern versions of the proof this is achieved with sup-convolutions and Alexandrov theorem).[11]
In subsequent years the concept of viscosity solution has become increasingly prevalent in analysis of degenerate elliptic PDE. Based on their stability properties, Barles and Souganidis obtained a very simple and general proof of convergence of finite difference schemes.[12] Further regularity properties of viscosity solutions were obtained, especially in the uniformly elliptic case with the work of Luis Caffarelli.[13] Viscosity solutions have become a central concept in the study of elliptic PDE. In particular, Viscosity solutions are essential in the study of the infinity Laplacian.[14]
In the modern approach, the existence of solutions is obtained most often through the Perron method.[4] The vanishing viscosity method is not practical for second order equations in general since the addition of artificial viscosity does not guarantee the existence of a classical solution. Moreover, the definition of viscosity solutions does not generally involve physical viscosity. Nevertheless, while the theory of viscosity solutions is sometimes considered unrelated to viscous fluids, irrotational fluids can indeed be described by a Hamilton-Jacobi equation.[15] In this case, viscosity corresponds to the bulk viscosity of an irrotational, incompressible fluid. Other names that were suggested were Crandall–Lions solutions, in honor to their pioneers, $L^{\infty }$-weak solutions, referring to their stability properties, or comparison solutions, referring to their most characteristic property.
References
1. Dolcetta, I.; Lions, P., eds. (1995). Viscosity Solutions and Applications. Berlin: Springer. ISBN 3-540-62910-6.
2. Tran, Hung V. (2021). Hamilton-Jacobi Equations : Theory and Applications. Providence, Rhode Island. ISBN 978-1-4704-6511-7. OCLC 1240263322.{{cite book}}: CS1 maint: location missing publisher (link)
3. Wendell H. Fleming, H. M . Soner, (2006), Controlled Markov Processes and Viscosity Solutions. Springer, ISBN 978-0-387-26045-7.
4. Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis (1992), "User's guide to viscosity solutions of second order partial differential equations", Bulletin of the American Mathematical Society, New Series, 27 (1): 1–67, arXiv:math/9207212, Bibcode:1992math......7212C, doi:10.1090/S0273-0979-1992-00266-5, ISSN 0002-9904, S2CID 119623818
5. Barles, Guy (2013). "An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications". Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Lecture Notes in Mathematics. Vol. 2074. Berlin: Springer. pp. 49–109. doi:10.1007/978-3-642-36433-4_2. ISBN 978-3-642-36432-7. S2CID 55804130.
6. Crandall, Michael G.; Lions, Pierre-Louis (1983), "Viscosity solutions of Hamilton-Jacobi equations", Transactions of the American Mathematical Society, 277 (1): 1–42, doi:10.2307/1999343, ISSN 0002-9947, JSTOR 1999343
7. Ishii, Hitoshi (1987), "Perron's method for Hamilton-Jacobi equations", Duke Mathematical Journal, 55 (2): 369–384, doi:10.1215/S0012-7094-87-05521-9, ISSN 0012-7094
8. Ishii, Hitoshi (1989), "On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs", Communications on Pure and Applied Mathematics, 42 (1): 15–45, doi:10.1002/cpa.3160420103, ISSN 0010-3640
9. Evans, Lawrence C. (1980), "On solving certain nonlinear partial differential equations by accretive operator methods", Israel Journal of Mathematics, 36 (3): 225–247, doi:10.1007/BF02762047, ISSN 0021-2172
10. Crandall, Michael G.; Evans, Lawrence C.; Lions, Pierre-Louis (1984), "Some properties of viscosity solutions of Hamilton–Jacobi equations", Transactions of the American Mathematical Society, 282 (2): 487–502, doi:10.2307/1999247, ISSN 0002-9947, JSTOR 1999247
11. Jensen, Robert (1988), "The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations", Archive for Rational Mechanics and Analysis, 101 (1): 1–27, Bibcode:1988ArRMA.101....1J, doi:10.1007/BF00281780, ISSN 0003-9527, S2CID 5776251
12. Barles, G.; Souganidis, P. E. (1991), "Convergence of approximation schemes for fully nonlinear second order equations", Asymptotic Analysis, 4 (3): 271–283, doi:10.3233/ASY-1991-4305, ISSN 0921-7134
13. Caffarelli, Luis A.; Cabré, Xavier (1995), Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0437-7
14. Crandall, Michael G.; Evans, Lawrence C.; Gariepy, Ronald F. (2001), "Optimal Lipschitz extensions and the infinity Laplacian", Calculus of Variations and Partial Differential Equations, 13 (2): 123–129, doi:10.1007/s005260000065, S2CID 1529607
15. Westernacher-Schneider, John Ryan; Markakis, Charalampos; Tsao, Bing Jyun (2020). "Hamilton-Jacobi hydrodynamics of pulsating relativistic stars". Classical and Quantum Gravity. 37 (15): 155005. arXiv:1912.03701. Bibcode:2020CQGra..37o5005W. doi:10.1088/1361-6382/ab93e9. S2CID 208909879.
| Wikipedia |
Show that the Möbius group is isomorphic to the quotient of $GL_2(\Bbb C)$ by its centre.
Each element $\begin{pmatrix} a & b \\ c & d\\ \end{pmatrix}$ of $GL_2(\Bbb C)$ gives rise to a Möbius transformation $z \rightarrow \frac{az+b}{cz+d}$.
The first part of the question was to show that these transformations form a group under composition of functions. I managed to do this, but I got stuck on the second part. We have to show that this group is isomorphic to the quotient of $GL_2(\Bbb C)$ by its centre.
I wanted to use the first Isomorphism Theorem. But since we want to show that it's an isomorphism to $GL_2(\Bbb C) / centre$, we would have to find an $f : GL_2(\Bbb C) \rightarrow \text{Möbius group}$ where the Kernel = Centre.
I don't really know how I could come up with an $f$ that satisfies this. So far I only tried $f: \begin{pmatrix} a & b \\ c & d\\ \end{pmatrix} \rightarrow \frac{az+b}{cz+d}$ but then you don't get a homomorphism.
Tips on how to find a suitable $f$ are very welcome!
group-theory group-isomorphism group-homomorphism mobius-transformation
129k77 gold badges4343 silver badges121121 bronze badges
$\begingroup$ You should use $$f: \pmatrix{a&b\cr c&d\cr}\mapsto [z\mapsto \frac{az+b}{cz+d}].$$ The kernel consists of those matrices $A$ such that $f(A)=f(I_2)$. $\endgroup$ – Jyrki Lahtonen Dec 28 '17 at 11:49
$\begingroup$ @JyrkiLahtonen I never saw a function notated like this. I don't know how I could show that this is a homomorphism. Could you please explain that aswel? $\endgroup$ – user423841 Dec 28 '17 at 11:57
$\begingroup$ You can use whatever notation you want. The point is that $\dfrac{az+b}{cz+d}$ is a complex number or infinity, but a Möbius transformation is a function that sends a complex number to another $\endgroup$ – Jyrki Lahtonen Dec 28 '17 at 12:00
$\begingroup$ The operations that you need to prove to be compatible with $f$ are 1) the product of matrices, 2) the composition of two Möbius transformations. $\endgroup$ – Jyrki Lahtonen Dec 28 '17 at 12:03
Hint 1: two Möbius transformations are "equal" (ie, they should be in the same equivalence class) if $A=\lambda A'$ ($A$ and $A'$ the matrices)
Hint 2: the center of $GL_2(\mathbb{C})$ are the scalar matrices $\lambda I_2$ with $\lambda\neq0$
Martín Vacas VignoloMartín Vacas Vignolo
Show that each of the following are homomorphisms
Writing down elements using cycle notation.
Proof that Möbius transformations are group under composition - finding inverse element
Finding what elements are in the group centre and proving to which group it's isomorphic
Show that $\varphi:\mathbb{R}→Gl_2 (\mathbb{R})$ defined by $\varphi(a)=\begin{pmatrix} 1 & a \\ 0 & 1\end{pmatrix}$ is not an isomorphism
To complete the proof that $\operatorname{PSL}(2,\Bbb F_5)\cong A_5$
Prove that the kernel of $GL_2(\mathbb{R})$ acting on $\mathbb{R}^2$ is $\{I\}$
Show that $R^n/Im(\rho)=R^{n-1}\bigoplus R/2R$, where $R$ is an abelian group and $\rho$ is the following function.
Find group isomorphic to the quotient group
Show that a function from $\Bbb Z / 3600\mathbb{Z}$ to $\mathbb{Z} / 1200\mathbb{Z}$ is a well defined group homomorphism of additive groups | CommonCrawl |
Express Letter
Near-ridge-axis volcanism affected by hotspot: insights from effective elastic thickness and topography of the Ojin Rise Seamounts, east of Shatsky Rise in the northwest Pacific Ocean
Shoka Shimizu ORCID: orcid.org/0000-0002-5160-45391,
Masao Nakanishi2 &
Takashi Sano3
Earth, Planets and Space volume 72, Article number: 11 (2020) Cite this article
We used recently collected bathymetric data and published gravity data to examine the effective elastic thickness of the lithosphere and the crustal thickness beneath the Ojin Rise Seamounts, located east of Shatsky Rise in the northwest Pacific Ocean. An admittance analysis of the bathymetric and gravity data indicates that the effective elastic thickness of the Pacific plate under the Ojin Rise Seamounts is 2.7 ± 0.1 km, implying that the seamounts were formed on or near the spreading ridge between the Pacific and Farallon plates. The mean crustal thickness beneath the seamounts estimated from the mantle Bouguer anomaly is 10.1 ± 1.7 km, which is thicker than the surrounding crust. The thick crust was probably formed by the interaction between the Pacific–Farallon ridge and a hotspot forming Shatsky Rise. Our results indicate that late-stage volcanism after the formation of the main edifices of Shatsky Rise spread widely beyond the eastern side of the rise, forming the Ojin Rise Seamounts.
The formation of oceanic plateaus is an important unresolved problem in Earth science. Oceanic plateaus are thought to be emplaced by rapid, voluminous eruptions over a surfacing mantle plume head (e.g., Duncan and Richards 1991). However, secondary (late-stage or post-plateau) volcanism also plays an important role in the formation of oceanic plateaus (e.g., Ito and Clift 1998; Pietsch and Uenzelmann-Neben 2015). Late-stage volcanism includes volcanism along the hotspot track and rejuvenated-stage volcanism. According to the plume model, hotspot-track volcanism represents activity of the plume tail (e.g., Hawaiian–Emperor volcanic chain; Clague and Dalrymple 1987). Rejuvenated-stage volcanism may occur as a result of melting of lithospheric mantle beneath the plateau (e.g., Tejada et al. 2015). Late-stage volcanism produced volcanic cones on the Manihiki Plateau (Pietsch and Uenzelmann-Neben 2015) and may have thickened the crust of the Ontong Java Plateau (Ito and Clift 1998). Thus, a comprehensive understanding of the formation of oceanic plateaus requires adequate knowledge of late-stage volcanism.
Shatsky Rise is proposed to have formed by the emergence of a mantle plume head at the ridge–ridge–ridge triple junction among the Pacific, Izanagi, and Farallon plates (Nakanishi et al. 1999). It consists of three major volcanic edifices, the Tamu, Ori, and Shirshov massifs, and the bathymetric high of Papanin Ridge (Fig. 1a). Drilled samples have yielded radiometric dates of ~ 145 Ma for the Tamu Massif and ~ 134 Ma for the Ori Massif (Mahoney et al. 2005; Geldmacher et al. 2014; Heaton and Koppers 2014). The maximum crustal thickness of the Tamu Massif is estimated to be 30 km on the basis of seismic refraction surveys (Korenaga and Sager 2012). Several studies have proposed that the thick crust of the massif was produced by deeper melting, with a higher degree of partial melting, than is typical at normal mid-ocean ridges (e.g., Sano et al. 2012). Previous studies have also suggested that late-stage volcanism occurred on Shatsky Rise (e.g., Sager et al. 2016; Shimizu et al. 2013; Tejada et al. 2016).
Bathymetry of the study area. a Bathymetric map around the Ojin Rise Seamounts and Shatsky Rise. Bathymetric data are from Smith and Sandwell (1997); contour interval is 500 m. Shatsky Rise is outlined in red at the 5000 m bathymetric contour. Thin black lines represent magnetic anomaly lineations (Nakanishi et al. 1999). The white-edged black line at lower center is the seismic reflection survey line of Ohira et al. (2017). The two dashed rectangles are the areas shown in b and c. The inset shows the locations of Shatsky Rise, Japan, and present plate boundaries (dashed lines). b Bathymetric map based on data collected during cruise KR14-07. c Bathymetric map of the study area
The Ojin Rise Seamounts (ORS) lies east of Shatsky Rise and consists of approximately 80 volcanic edifices. Torsvik et al. (2019) devised an absolute Late Jurassic–Cretaceous Pacific plate model and showed that the ORS is a hotspot track connected to Shatsky Rise. Nakanishi et al. (1999) speculated that part of the ORS (around 37° 00′ N and 165° 30′ E) was formed near a spreading ridge because the elongation of the seamounts and the magnetic anomaly lineations have similar orientations (Fig. 1a). The crustal structure under the ORS is unknown. Tejada et al. (2016) indicated that the ORS was formed by late-stage volcanism related to the formation of Shatsky Rise and proposed two models of formation based on the geochemical data. One is that the ORS was formed by the interaction between volcanism of a spreading ridge and a hotspot. The other is that the ORS was formed only by deformation-related shallow-mantle volcanism. Age information for the ORS is indispensable for testing these models.
The effective elastic thickness (Te) value of the lithosphere under a seamount is an indicator of its tectonic setting (Watts 1978). The Te value, the thickness of the lithosphere behaving as an elastic body, can be estimated from bathymetric and gravity data. The relationship between Te and the age of the lithosphere at the time of loading was first proposed by Watts (1978). Small Te values (0–8 km) suggest that the seamounts were formed on or near a spreading ridge. For example, Te value of the lithosphere under the Foundation Seamounts near the Pacific–Antarctic Ridge is 0–5 km, suggesting an age difference of 0–10 Myr between the seamounts and the lithosphere under them (Maia and Arkani-Hamed 2002), and Te of the lithosphere under the Hawaiian Islands is 30–40 km, indicating an age difference of 70–80 Myr between the islands and the adjacent seafloor (Watts 1978). In this study, we determined the crustal thickness and the Te value of the lithosphere under the ORS to estimate its formation age, tectonic setting, and crustal structure.
Theory and method
We estimated the Te value of the lithosphere under the ORS by an admittance analysis, a method of spectral analysis of the bathymetry and free-air gravity anomaly (FGA) in the wavenumber domain. We used the admittance function (McKenzie and Bowin 1976),
$$ \begin{array}{*{20}c} {Z\left( k \right) = \frac{{C_{\text{C}} \left( k \right)}}{{E_{\text{t}} \left( k \right)}},} \\ \end{array} $$
where k is the wavenumber and Z(k) is the gravitational admittance between the bathymetry and the FGA. \( C_{\text{C}} \left( k \right) \) is a cross spectrum of FGA and bathymetry, and \( E_{\text{t}} \left( k \right) \) is a power spectrum of bathymetry:
$$ \begin{array}{*{20}c} {C_{\text{C}} \left( k \right) = \Delta g\left( k \right)H^{ *} \left( k \right),} \\ \end{array} $$
$$ \begin{array}{*{20}c} {E_{\text{t}} \left( k \right) = H\left( k \right)H^{*} \left( k \right),} \\ \end{array} $$
where ∆g(k) and H(k) are the Fourier transforms of the observed FGA and bathymetry, respectively, and the asterisk denotes the complex conjugate.
The Te value can be estimated by comparing the admittance function of an elastic plate model with observational admittances (Watts 2001). The elastic plate model used in this study is one in which the load of a seamount is compensated by flexure of a plate. The admittance function of this model is defined by
$$ \begin{array}{*{20}c} {Z\left( k \right) = 2\pi G\left( {\rho_{\text{c}} {-} \rho_{\text{w}} } \right){\text{e}}^{ - kd} \left( {1 - \varPhi_{\text{e}} \left( k \right){\text{e}}^{ - kt} } \right),} \\ \end{array} $$
where d is the mean water depth, G is the universal gravitational constant, t is the mean crustal thickness, and \( \rho_{\text{w}} \) and \( \rho_{\text{c}} \) are the densities of seawater and crust, respectively. The flexural response function of the lithosphere \( \varPhi_{\text{e}} \left( k \right) \) is (Walcott 1976)
$$ \begin{array}{*{20}c} { \varPhi_{\text{e}} \left( k \right) = \left[ {\frac{{Dk^{4} }}{{\left( {\rho_{\text{m}} - \rho_{\text{w}} } \right)g}} + 1} \right]^{ - 1} ,} \\ \end{array} $$
where g is the average gravitational acceleration, \( \rho_{\text{m}} \) is the density of the mantle, and D is the flexural rigidity of the lithosphere defined by
$$ \begin{array}{*{20}c} {D = \frac{{ET_{\text{e}}^{3} }}{{12\left( {1 - \nu^{2} } \right)}},} \\ \end{array} $$
where E is Young's modulus and \( \nu \) is Poisson's ratio. These and other parameters used in this study are listed in Table 1.
Table 1 Parameters used in this study
The most suitable Te value was determined by minimizing the root-mean-square (RMS) misfit between theoretical and observational admittances for various Te values (Kalnins and Watts 2009; Hu et al. 2015). Kalnins and Watts (2009) noted that use of a very small spatial window (≤ 400 km) produces spuriously low Te values by truncating the signals of features on stronger lithosphere. Our study area encompassing the ORS consisted of a 550 km × 550 km window that did not include any major topographic features of the Shatsky Rise and Emperor Seamount Chain (Fig. 1c; 35° N to 39° 56.98′ N and 163° 30′ E to 169° 47.16′ E). We extended the grid by line symmetry to avoid the Gibbs phenomenon.
The mean crustal thickness, which is required to calculate admittances with Eq. 4, can be estimated from the mantle Bouguer anomaly (MBA; Kuo and Forsyth 1988). The MBA can be calculated by subtracting the predicted attractions of the seawater–sediment, sediment–crust, and crust–mantle interfaces from the FGA under the assumption of a constant crustal thickness. When the crustal density is assumed to be constant, the MBA represents the spatial variation of crustal thickness, that is, undulations of the Moho. To calculate that spatial variation, we determined the constant reference crustal thickness for the ORS. To find a reasonable ORS reference crustal thickness, we calculated the crustal thickness within the area bounded by 27° N and 45° N and 152° E and 175° E (Fig. 1a), which is much larger than our study area. We chose the larger area because the crustal thickness in its southern part is known from a seismic reflection survey to range from 6.8 to 7.3 km (Ohira et al. 2017). We then calculated reference crustal thicknesses in steps of 0.1 km between 1 and 15 km and chose the value that minimized the RMS difference between the crustal thicknesses determined by Ohira et al. (2017) and our MBA calculation.
To determine the range of wavenumbers used to fit the observed and theoretical admittances, we used the coherence γ2(k), a measure of the portion of the FGA that is caused by bathymetry. McKenzie and Bowin (1976) defined coherence as
$$ \begin{array}{*{20}c} {\gamma^{2} \left( k \right) = \frac{{C_{\text{C}} \left( k \right)C_{\text{c}}^{*} \left( k \right)}}{{E_{\Delta g} \left( k \right)E_{\text{t}} \left( k \right)}},} \\ \end{array} $$
where \( E_{\Delta g} \left( k \right) \) is a power spectrum of the FGA. The coherence can range from 0 to 1, and a high coherence indicates that most of the FGA is caused by bathymetry. Kalnins and Watts (2009) suggested that wavenumbers with coherence greater than a threshold value of 0.4 are suitable for admittance analysis, and we adopted that threshold for our admittance analysis.
Bathymetric and gravity data
Research cruise KR14-07 of R/V Kairei was carried out in summer 2014 to investigate the origin of the ORS and its relationship with Shatsky Rise (Sano 2014). Bathymetric data were collected using the vessel's SeaBeam 3012 multibeam echo sounder (Fig. 1b). Because the survey covered only a small fraction of the study area, we filled the gaps using gridded bathymetric data with a 1′ × 1′ grid spacing made by Sager et al. (1999) using the spline-under-tension gridding algorithm of Smith and Wessel (1990), shown in Fig. 1c.
Because the coverage of shipboard gravity data by cruise KR14-07 and previous research cruises was insufficient for admittance analysis, we used the satellite-derived FGA gridded data of Sandwell et al. (2014), with a grid spacing of 1′ × 1′ (Fig. 2). The sediment thickness was obtained from the gridded data of sediment thickness of Straume et al. (2019).
Free-air gravity anomaly map of the study area (location in Fig. 1c). The data are from Sandwell et al. (2014); contour interval is 5 mGal
The seamounts of the ORS, distributed within the area defined by latitude 36° 30′ N and 39° N and longitude 164° E and 168° E, are ~ 1000–2000 m high and ~ 10–20 km in diameter (Fig. 1c). An elongated ridge, parallel to Hawaiian magnetic lineations, is situated between 36° 40′ N and 37° 20′ N and 165° E and 166° E and is 126 km long, 67 km wide, and 1800 m high (Fig. 1b). Four seamounts on top of this ridge have diameters of 16–18 km and elevations of ~ 800–1400 m above the ridge.
The FGA is positive over the seamounts of ORS, ~ 30 mGal, and negative over the abyssal plain around the seamounts, less than –20 mGal (Fig. 2). The wavelengths of these gravity features are ~ 20–30 km.
The optimum reference crustal thickness determined from our MBA analysis is 8.5 ± 0.1 km; at this thickness the RMS difference between the result of Ohira et al. (2017; see Fig. 3c) and the best-fit crustal thickness reaches a minimum of 0.26 km (Fig. 3a). The crustal thickness variations along the survey line of Ohira et al. (2017) is shown with the best-fit crustal thickness in Fig. 3b, and the crustal thickness variations in the area of the survey line based on the 8.5 km reference thickness is shown in Fig. 3c. The crust beneath the ORS study area is everywhere thicker than 7 km, reaching 14 km at 37° 25′ N, 166° 50′ E, and it has a mean thickness of 10.1 ± 1.7 km (Fig. 3d). We used this value in Eq. 4 to calculate the theoretical admittances. The mean water depth d and crustal density \( \rho_{\text{c}} \) used in this calculation were derived from the slope of a plot of log10 admittance vs wavenumber and its intersection at k = 0 (Fig. 4a; Watts 2001). The best-fit straight line for an admittance of 0.08 < k < 0.4 km−1 gave estimates of d = 5838 m and \( \rho_{\text{c}} \) = 2811 kg m−3.
Crustal thickness results. a RMS misfit between crustal thicknesses by our analysis and by Ohira et al. (2017) for an area of normal oceanic crust (location in Fig. 1a). The dashed line represents the crustal thickness at the RMS minimum. b Crustal thickness along the survey line of Ohira et al. (2017) as determined by Ohira et al. (2017) (dashed line) and by our calculations using a constant reference crustal thickness of 8.5 km (solid line). c Map of crustal thickness around the survey line of Ohira et al. (2017) calculated for a reference crustal thickness of 8.5 km; contour interval 1 km. The thick white-edged line is the survey line of Ohira et al. (2017); the red portion was used for fitting crustal thickness (see the text). d Map of crustal thickness in the ORS study area for a reference crustal thickness of 8.5 km; contour interval 1 km
Results of admittance analysis. a Admittance amplitude vs wavenumber k (red line) with dashed line indicating the best fit to the observed admittance for 0.08 < k < 0.4. b Coherence of the observation admittance values (red line). The gray area is the range of wavenumbers where coherence exceeds the threshold of 0.4. c Observational admittances (black circles with standard deviations) and the theoretical admittance for Te = 2.7 km (red line). d RMS misfit between observational and theoretical admittances (red line). The vertical dotted line represents the Te value at the minimum RMS. Dashed lines represent RMS misfits between observational and theoretical admittances for crustal thicknesses of 8.4 and 11.8 km
The coherence of bathymetry and gravity exceeds the threshold of 0.4 for wavenumbers between 0.04 and 0.22 km−1 (Fig. 4b). The coherence for the wavenumbers greater than 0.4 km−1 is almost zero. We therefore used only the data for the wavenumber band of 0.04 to 0.22 km−1 for the admittance analysis.
Figure 4c compares the observational admittances and the best-fitting admittance model. The Te value of this model is 2.7 ± 0.1 km, and the RMS is 4.4 mGal km−1 (Fig. 4d). Watts et al. (2006) showed that seamounts with Te values less than 8 km formed on young lithosphere, that is, on or near a spreading ridge. On this basis, we propose that the ORS was formed on or near a spreading ridge.
We determined Te values for several different cases to confirm the robustness of our analysis. To estimate the influence of the error in crustal thickness, we calculated Te values for crustal thicknesses between 8.4 and 11.8 km. These ranged from 2.3 to 3.2 km (Fig. 4d), implying that Te value is fairly constant within the tolerance interval of crustal thickness. The Te value for a crustal thickness of 8 km, typical of the normal oceanic crust around Shatsky Rise (e.g., Ohira et al. 2017; Zhang et al. 2016), was 3.6 ± 0.1 km. Given a window size of 600 km × 600 km, Te was 2.8 ± 0.1 km, which is within the error range of Te value with our 550 km × 550 km window. In all cases, the Te values around the ORS were less than 4 km, implying that our analysis is appropriate to address the formation of the ORS.
Te values less than 8 km for the underlying lithosphere imply that the seamounts of the ORS formed on a young and weak plate (Watts et al. 2006). Thus, the Te value of 2.7 ± 0.1 km for the seafloor around the ORS indicates that the seamounts formed on or near the spreading ridge between the Pacific and Farallon plates (Pacific–Farallon ridge). Similarly, with Te value less than 8 km, the age of the lithosphere at the time of seamount loading is less than 30 Myr (Watts et al. 2006), suggesting that the age of the seafloor in the western (older) margin of the ORS is about 134 Ma (Nakanishi et al. 1999) and that the ORS formed between 104 and 134 Ma. Our result is consistent with the ~ 120–124 Ma 40Ar/39Ar ages of the seamounts on top of the ridge (Sano et al. 2016; around 36° 45′ N and 166° 00′ E).
Tejada et al. (2016) raised the possibility that the ORS was formed by late-stage volcanism of Shatsky Rise and is a hotspot track representing the "plume head to tail" stage of hotspot development. The plate reconstruction model of Torsvik et al. (2019) also suggests that the ORS represents the track of the hotspot that formed Shatsky Rise from 135 to ~ 120 Ma. This age range is close to that of the seafloor around the ORS determined from magnetic anomaly lineations (125–134 Ma) and the radiometric ages of the seamounts. These lines of evidence support our inference that the ORS was formed on or near the Pacific–Farallon ridge.
The mean crustal thickness beneath the ORS is 10.1 ± 1.7 km, which is ~ 3 km thicker than that of the oceanic crust around Shatsky Rise. Some of the Foundation Seamounts near the East Pacific Rise, which are similar to the ORS in topography, size, and shape, are parallel or subparallel to the Pacific–Antarctic Ridge (O'Connor et al. 2004). The Foundation Seamounts, which were formed by the interaction between a spreading ridge and a hotspot (Maia and Arkani-Hamed 2002; O'Connor et al. 2001), have Te values of 0–5 km (Maia and Arkani-Hamed 2002), and the crust beneath them is 6 km thicker than the surrounding normal oceanic crust (Maia et al. 2000). These similarities between the ORS and Foundation Seamounts suggest that they were formed by the same mechanism. Thus, we conclude that the ORS was formed by the interaction between the Pacific–Farallon ridge and the hotspot forming Shatsky Rise, consistent with the model proposed by Tejada et al. (2016). The ORS does not show a clear hotspot track like the Hawaiian–Emperor volcanic chain, perhaps because interactions with the spreading ridge produced scattered near-ridge volcanoes and led to a diffuse hotspot track.
Tejada et al. (2016) proposed that the volcanic edifices on and around Shatsky Rise, such as Toronto Ridge and Cooperation Seamount, formed by late-stage volcanism related to the formation of Shatsky Rise that took place far from plate boundaries (Fig. 1a). The ages of Toronto Ridge and Cooperation Seamount, 129.4 ± 0.3 Ma and ~ 122 Ma, respectively (Tejada et al. 2016), are close to the age of the ORS estimated by our study. Several other volcanic cones on Shatsky Rise are also thought to have formed by late-stage volcanism (Zhang et al. 2017). Ohira et al. (2017) proposed late-stage volcanism beneath the abyssal basin southeast of Shatsky Rise to explain the thick crust–mantle transition layer there. These studies as well as our results indicate that late-stage volcanism related to the formation of Shatsky Rise not only occurred within the rise itself, but also extended to the abyssal basin to the east. Our results also indicate that the late-stage hotspot volcanism occurred not only within the Pacific plate, but also near a plate boundary, the Pacific–Farallon ridge.
Our analysis of bathymetric and gravity data provides new insight into the tectonic setting of the ORS, east of Shatsky Rise. Our study indicates that the effective elastic thickness of the lithosphere and the mean thickness of the crust beneath the ORS are 2.7 ± 0.1 km and 10.1 ± 1.7 km, respectively. The small effective elastic thickness suggests that the ORS formed on or near the Pacific–Farallon ridge, and the thick crust indicates that the ORS formed by the interaction between the Pacific–Farallon ridge and the hotspot forming Shatsky Rise. The volcanism forming the ORS was part of the late-stage volcanism associated with Shatsky Rise, which occurred on or near a spreading ridge. This late-stage volcanism extended beyond the rise itself to the abyssal basin to the east. Our conclusion should be tested by further sampling from the wide area of the ORS.
The bathymetric dataset of cruise KR14-07 is available in the Data and Sample Research System of the Japan Agency for Marine-Earth Science and Technology, http://www.godac.jamstec.go.jp/darwin/e.
FGA:
Free-air gravity anomaly
MBA:
Mantle Bouguer anomaly
ORS:
Ojin Rise Seamounts
RMS:
Root-mean-square
T e :
Effective elastic thickness
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We thank the crew and technical staff of cruise KR14-07 for collection of the geophysical data used in this study and M.L.G. Tejada for discussion and comments on this manuscript. We are grateful for the constructive reviews of the editor and two anonymous reviewers that significantly improved this manuscript. Funding was provided by the Japan Society for the Promotion of Science under KAKENHI Grants no. 26302010, 18H03746, and 18K03772.
Funding was provided by the Japan Society for the Promotion of Science under KAKENHI Grants no. 26302010, 18H03746, and 18K03772.
Graduate School of Science and Engineering, Chiba University, Chiba, 2638522, Japan
Shoka Shimizu
Graduate School of Science, Chiba University, Chiba, 2638522, Japan
Masao Nakanishi
Department of Geology and Paleontology, National Museum of Nature and Science, Ibaraki, 3050005, Japan
The KR14-07 cruise was planned by TS and MN. Analyses were performed by SS and MN, and the manuscript was written by SS with extensive input and discussion from MN and TS. All authors read and approved the final manuscript.
Correspondence to Shoka Shimizu.
Shimizu, S., Nakanishi, M. & Sano, T. Near-ridge-axis volcanism affected by hotspot: insights from effective elastic thickness and topography of the Ojin Rise Seamounts, east of Shatsky Rise in the northwest Pacific Ocean. Earth Planets Space 72, 11 (2020). https://doi.org/10.1186/s40623-020-1140-5
Accepted: 20 January 2020
Shatsky Rise | CommonCrawl |
Anh Tuan Duong 1, , Phuong Le 2,, and Nhu Thang Nguyen 3,
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
* Corresponding author: Phuong Le ([email protected])
Received August 2019 Published July 2020
$ u $
be a nonnegative solution to the equation
$ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * |x|^a u^p \right) |x|^a u^{p-1} \quad\text{ in } \mathbb{R}^n \setminus \{0\}, $
$ n \ge 2 $
$ 0 < \alpha < 2 $
$ 0 < \beta < n $
$ a>\max\{ -\alpha, -\frac{\alpha+\beta}{2} \} $
. By exploiting the method of scaling spheres and moving planes in integral forms, we show that
must be zero if
$ 1\le p<\frac{n+\beta+2a}{n-\alpha} $
and must be radially symmetric about the origin if
$ a<0 $
$ \frac{n+\beta+2a}{n-\alpha} \le p \le \frac{n+\beta+a}{n-\alpha} $
Keywords: Fractional Choquard equations, nonexistence of solutions, symmetry of solutions, method of moving planes, method of scaling spheres.
Mathematics Subject Classification: 35R11, 35B06, 35B53, 45G10.
Citation: Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265
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Anh Tuan Duong Phuong Le Nhu Thang Nguyen | CommonCrawl |
\begin{definition}[Definition:Borel Sigma-Algebra/Topological Space]
Let $\struct {S, \tau}$ be a topological space
The '''Borel sigma-algebra''' $\map \BB {S, \tau}$ of $\struct {S, \tau}$ is the $\sigma$-algebra generated by $\tau$.
That is, it is the $\sigma$-algebra generated by the set of open sets in $S$.
\end{definition} | ProofWiki |
The association of body mass index with difficult tracheal intubation management by direct laryngoscopy: a meta-analysis
Tingting Wang1 na1,
Shen Sun1 na1 &
Shaoqiang Huang ORCID: orcid.org/0000-0002-0515-80991
Obesity is a serious disorder and may bring about many difficulties of perioperative management. A systematic review was conducted to assess the association between obesity and difficult intubation.
We searched electronic databases for related reviews and references of meta-analyses on August 14, 2017. The databases of PubMed, Embase, and the Cochrane controlled trials register were searched compared obese with non-obese patients in which difficult intubation rate of the adult population were retrieved. Patients with a BMI ≥ 30 kg·m− 2 were considered obese. The primary outcome was difficult tracheal intubation; secondary outcomes were the rates of difficult laryngoscopy and Mallampati score ≥ 3. This review included papers published from 1998 to 2015.
This review included 204,303 participants in 16 studies. There was a statistically significant association between obesity and risk of difficult tracheal intubation (pooled RR = 2.04, 95% CI: 1.16–3.59, p = 0.01; I2 = 71%, p = 0.008, Power = 1.0). It also showed significantly association between obesity and risk of difficult laryngoscopy (pooled RR = 1.54, 95% CI: 1.25–1.89, p < 0.0001; I2 = 45%, p = 0.07, Power = 1.0), obesity and risk of Mallampati score ≥ 3 (pooled RR = 1.83, 95% CI: 1.24–2.69, p = 0.002; I2 = 81%, p < 0.00001, Power = 0.93). However, there were no association of obesity and risks of difficult intubation compared with non-obesity in the cohort studies (pooled RR = 3.41, 95% CI: 0.88–13.23, p = 0.08; I2 = 50%, p = 0.14) and the elective tracheal intubation (pooled RR = 2.31, 95% CI: 0.76–6.99, p = 0.14; I2 = 73%, p = 0.01), no associated with an increased risk of difficult laryngoscopy in the sniffing position (pooled RR = 2.00, 95% CI: 0.97–4.15, p = 0.06; I2 = 67%, p = 0.03).
Obesity was associated with an increased risk of difficult intubation, difficult laryngoscopy and Mallampati score ≥ 3 in adults patients undergoing general surgical procedures. However, there were no association of obesity and risks of difficult intubation compared with non-obesity in the cohort studies and the elective tracheal intubation, no associated with an increased risk of difficult laryngoscopy in the sniffing position. Future analyses should explore the association of BMI and difficult airway.
Obesity is a public health issue that leads to serious social, psychological and physical problems [1]. According to the World Health Organization survey, obesity rates have almost doubled worldwide since 1980 [2]. With the growing number of obese adults, increasing attention is being paid to difficult intubation (DI).
Although several tools (such as video laryngoscope, fibre-optic tracheal airway devices) can facilitate intubation or increase success rates, a DI can still be challenging for anaesthetists. The higher cost and uncomfortable nature of awake intubation compared with traditional laryngoscopy are common contributing causes to the difficulty [3, 4]. Accordingly, a direct laryngoscope (DL) remains the most widely used device for tracheal intubation [5].
Furthermore, there is no consensus about whether obesity is associated with the occurrence of a DI. For instance, Shiga et al. found the rate of DI in obese patients (body mass index > 30) to be more than three times that in normal patients [6]. Conversely, some studies after 2005 reached a different conclusion, indicating that a high body mass index (BMI) was not associated with DI [7, 8]. In addition, although most anaesthesiologists recommend the sniffing position and consider it to be essential for improving tracheal intubation [9], the superiority of this position has been questioned during the last decade [10].
Considering the points raised above, we performed a review to evaluate the association between BMI and DI using meta-analysis and furthermore stratified by study design (cohort or case-control) and position (sniffing or supine). The primary outcome was to compare the rate of difficult tracheal intubation in high BMI vs. low BMI patients with a DL. The secondary outcomes were the rate of difficult laryngoscopy and a Mallampati score ≥ 3.
This systematic review was performed in accordance with the Preferred Reporting Items for Systematic Reviews and Meta-analyses (PRISMA) guidelines.
The protocol was registered with PROSPERO under number CRD42017058340 on August 14, 2017.
We searched electronic databases for related reviews and references of meta-analyses on August 14, 2017. To identify relevant articles, searches in PubMed, Embase, and the Cochrane controlled trials register (CENTRAL) were performed using the keywords "failed tracheal intubation", "difficult tracheal intubation", "difficult laryngoscopy", "Cormack Lehane" , "Mallampati", "BMI" and "obesity" as MeSH components and text words. There were no limitations regarding language, time of publication or article type. To reduce publication bias, ongoing studies at ClinicalTrials.gov and proceedings from the American Society of Anesthesiologists (ASA) annual meetings over the last 5 years (from July 2012 to August 2017) were also retrieved.
Studies meeting the following criteria were included: (1) reference to humans; (2) DI rate as an outcome using an adult population; (3) comparison of obese with non-obese patients according to BMI and (4) results reported or obtained via calculation of effect estimates of the relative risk (RR), hazard ratio (HR) or odds ratio (OR) with 95% confidence intervals (CIs). BMI was calculated by dividing the patient's body weight in kilograms by the square of their height in metres. Patients with a BMI ≥ 30 kg·m− 2 were considered obese. All types of surgery were considered.
The exclusion criteria were as follows: (1) case reports, cross-section studies, editorials, reviews and abstracts; (2) known risk factors for difficult airway (traumatic facial abnormalities, airway malformation and pathology, cervical spine fractures, those with a history of airway or intubation difficulty); and (3) pre-hospital tracheal intubation, that is, studies of pre-hospital tracheal intubation were excluded because the airway management setting differs between out-of-hospital and in-hospital. Accordingly, a non-planned endotracheal intubation in a hospital was defined as an emergency tracheal intubation.
Primary and secondary outcomes
The primary outcome was rate of difficult tracheal intubation. The secondary outcomes were 1) rate of difficult laryngoscopy and 2) a Mallampati score ≥ 3. In 1993, the ASA has defined difficult endotracheal intubation as 3 attempts at endotracheal intubation when an average laryngoscope is used or when endotracheal intubation takes 10 min or more [11]. Then these Practice Guidelines were update in "Practice Guidelines for Management of the Difficult Airway" in 2013, and difficult tracheal intubation was defined as requiring multiple attempts in the presence or absence of tracheal pathology [12]. However, the concept was subjective and ambiguous. The Intubation Difficulty Scale (IDS) score, an objective scoring system that consists of numerical expressions of parameters and has been validated in many studies, was proposed to assess intubation difficulty in a standardized manner [13]. An IDS score of 0 means easy intubation, 1 to 5 means slight difficulty, and > 5 means moderate to major difficulty [14]. Thus, difficult intubation has been defined as requiring multiple attempts to place the tracheal tube into the trachea, lasting > 10 min using conventional laryngoscopy, or both, or and ISD > 5. Difficult laryngoscopy was assessed using the Cormack and Lehane Grades, classified into 4 grades: 1) visible vocal cords; 2) visible posterior commissure and epiglottis; 3) only epiglottis visible; and 4) no visible glottal structures. Grades 3 and 4 are considered a difficult laryngoscopy [15]. Mallampati scores were classified into 4 grades: 1) the tonsils, uvula and soft palate fully visible; 2) the soft palate and uvula visible; 3) the soft palate and base of uvula visible; and 4) only the hard palate visible [16]. Mallampati grades III or IV may be associated with difficult tracheal intubation.
Data were retrieved independently by two researchers (T. W. and S. S.); disagreements were considered by a third researcher (S. H.) and discussed until a consensus was reached. The discussion focused on whether the data conformed to the included criterion. One researcher (T. W.) designed a standard data form, and the other researchers (S. H. and S. S.) amended and validated the design before it was implemented. The authors of the retrieved studies were contacted (by S. H.) and asked to provide missing data that had not been reported or obtained by calculating the effect estimates of the RR, HR or OR with a 95% CI. If a response was not provided, the article was excluded. All of the studies were screened. The dataset included the name of the first author, year and country of publication, group situation (specified BMI to define obesity), number of participants, participant characteristics, study design and outcomes.
Risk of bias assessment was performed by two reviewers independently (S. H., T. W.) using the Risk of Bias in Non-randomized Studies – of Interventions (ROBINS-I) tool for observational studies [17]. The ROBINS-I tool assesses bias across six domains: confounding, participant selection, intervention classification, departure from intended interventions, missing data, measurement of outcomes and selection of reported results. For each domain, an outcome of low, moderate, serious, critical and no information for risk of bias is recorded. The overall risk of bias judgement is then determined through a combination of the six domains.
Review Manager (RevMan version 5.2.5; The Nordic Cochrane Centre, The Cochrane Collaboration, Copenhagen, Denmark) was utilized for data analysis. As the outcome of the study was rare among all populations, ORs and HRs were directly considered as RRs in this study [18]. A p value ≤0.05 was considered statistically significant. The I2 statistic was utilized for heterogeneity assessment, and I2 > 50% was considered to indicate significant heterogeneity. A random-effects model was accepted for data analysis in the case of heterogeneity, and a fixed-effect model was adopted when heterogeneity was not found. Sources of heterogeneity were investigated by analysis of prespecified subgroups, as defined according to the study design (cohort or case-control) and position (sniffing or supine). The sniffing position was defined as patients with pillows or towels under their shoulders, with the head elevated and neck extended [19]. The supine position was defined as patients lying supine or not specifically in the sniffing position. To control the Type I error rate for multiple hypothesis testing, we used the Bonferroni correction as follows:
$$ {\upalpha}^{\ast}\kern0.5em =\kern0.5em \frac{\alpha }{c} $$
where α* is our new alpha level, α is our a priori significance level of 0.05 for the family of comparisons, and c is the number of comparisons [20]. We calculated the power for the primary outcome using post hoc power analysis with G*Power 3.1 software [21, 22]. Sensitivity analyses were conducted to assess the robustness of the data by removing each study sequentially and excluding those with emergency tracheal intubation, those defining obesity as a BMI cut-off > 30, and those in which parturients were participants. Potential publication bias was evaluated with a funnel plot; in the absence of bias, these plots resemble a symmetrical inverted funnel.
Study selection
A total of 1,533 related studies were obtained from the database search, and 2 citations were retrieved from a manual reference list search of the eligible studies. In total, 108 studies that were removed because they were duplicates. We then excluded 968 studies after the initial review of the title and 341 after the abstract was reviewed. Overall, 118 studies were considered relevant and were read in full. Then, 102 articles were excluded for reasons such as obesity was not defined as BMI ≥ 30, difficult intubation or laryngoscopy was not mentioned, the study did not provide or obtain the effect estimates of RR/HR/OR by calculation, tracheal intubation was not performed in a hospital. After reviewing the full texts, 16 studies (published between 1998 and 2015) were selected for inclusion (Fig. 1). No unpublished study in clinicaltrial.gov met the inclusion criteria.
A total of 1533 related studies were obtained from the database search, and 2 citations were retrieved from the manual reference list search of the eligible studies. There were 108 studies that were removed because they were duplicates. Then, we excluded 968 studies after the initial review of the title and 341 after the abstract was reviewed due to they are not endotracheal intubation, letters, reviews, cross-sectional studies and case reports. There were 118 studies that were considered relevant and were read in full. Then 102 articles were excluded for reasons such as obesity was not defined as BMI = 30, not mentioned difficult intubation or laryngoscopy, not provided or obtained by calculation the effect estimates of RR/HR/OR, not tracheal intubation in hospital. After reviewing the full texts, 16 studies (published between 1998 and 2015) were selected for inclusion
Study characteristics
A total of 204,303 subjects were included in this meta-analysis; 12,757 were assigned to the obese group, and 191,546 were assigned to the non-obese group. Five studies involving 100,974 patients were included in the analysis of the association between obesity and risk of difficult tracheal intubation [7, 23, 24]. Nine studies with a total of 112,388 patients were included in the obesity and difficult laryngoscopy group [7, 23,24,25,26,27,28,29]. Twelve studies with a total of 5678 patients were analysed the association between obesity and risk of Mallampati score ≥ 3 [7, 14, 23, 24, 26, 28,29,30,31,32,33,34]. There were five case-control studies (two in the USA [23, 28], one [27] in Germany, one in Denmark [35], and one in Ireland [32]) and 11 cohort studies (two in France [7, 24], two in Turkey [25, 31], one in Brazil [30], one in Israel [14], two in the USA [26], one in Greece [29] one in Italy [34] and one in the UK [33]). Two studies included parturients [25, 31], and 14 recruited non-parturients [7, 14, 23, 24, 26,27,28,29,30, 32,33,34]. Of the included studies, emergent tracheal intubation was used in two [23, 25] and non-emergent tracheal intubation in the other 14 [7, 14, 24, 26,27,28,29,30,31,32,33,34]. The sniffing position was employed in five studies [7, 14, 24, 26, 28] and the supine position in 11 [23, 25, 27, 29,30,31,32,33,34]. The definition of obesity as a BMI cut-off = 30 was applied in nine studies [7, 14, 23, 25, 28,29,30,31,32], whereas a BMI cut-off > 30 was used in seven studies [24, 26, 27, 33, 34]. The general characteristics of the published articles included in this meta-analysis are shown in Table 1.
Table 1 Characteristics of eligible trials
The results of the quality assessment are presented in Table 2. The ROBINS I tool indicated an overall low to moderate risk of bias, which for the majority of studies originated from the selection of the reported results as well as from the presence of possible confounding factors.
Table 2 Risk of bias in non-randomised studies -of interventions (ROBINS-I) tool
Association between obesity and rate of difficult tracheal intubation
Five studies with a total of 100,974 patients were included in this analysis [7, 8,23, 24, 35]. There was a significant association (pooled RR = 2.04, 95% CI: 1.16–3.59, p = 0.01; I2 = 71%, p = 0.008, Power = 1.0) between obesity and risk of DI (Fig. 2). Subgroup analysis of case-control studies showed that obesity was associated with an increased risk of DI (pooled RR = 1.50, 95% CI: 1.03–2.18, p = 0.03; I2 = 67%, p = 0.08). Similarly, obesity was associated with an increased risk of DI in both the sniffing (pooled RR = 5.77, 95% CI: 2.29–14.58, p = 0.0002; I2 = 0%, p = 0.63) and supine (pooled RR = 1.30, 95% CI: 1.17–1.44, p = 0.04; I2 = 64%, p = 0.02) positions. However, subgroup analysis of cohort studies revealed no trend of obese patients having a higher risk of DI compared with non-obese patients (RR = 3.41, 95% CI: 0.88–13.23, p = 0.08; I2 = 50, p = 0.14) (Table 3). We sequentially removed each study and then reanalysed the remaining dataset; removal of the studies defined obesity as a BMI cut-off > 30 [24], reducing the heterogeneity without significantly affecting the RR (RR = 2.12, 95% CI: 1.30–3.47, p = 0.003; I2 = 6, p = 0.30). Nonetheless, there was no significant difference in the estimates after we excluded studies with emergency tracheal intubation (RR = 2.31, 95% CI: 0.76–6.99, p = 0.14; I2 = 73, p = 0.01). No obvious asymmetry was detected in funnel plots (Fig. 3).
The result is the pooled estimate of the 5 included studies by random effect model. RR: rate ratios. There was a statistically significant association pooled (RR = 2.04, 95% CI: 1.16–3.59, p = 0.01; I2 = 71%, p = 0.008, Power = 1.0) between obesity and risk of DI
Table 3 Subgroup analysis of the outcomes
No obvious asymmetry was detected in the funnel plots
Association between obesity and the rate of difficult laryngoscopy
Nine studies including 112,388 patients were evaluated [7, 23,24,25,26,27,28,29]. There was a statistically significant association (pooled RR = 1.54, 95% CI: 1.25–1.89, p < 0.0001; I2 = 45%, p = 0.07, Power = 1.0) between obesity and the risk of difficult laryngoscopy (Fig. 4). The results of subgroup analyses are presented in Table 3. Obesity was associated with an increased risk of difficult laryngoscopy in both cohort studies (pooled RR = 1.85, 95% CI: 1.31–2.63, p = 0.0005; I2 = 49%, p = 0.08) and in case–control studies (pooled RR = 1.34, 95% CI: 1.22–1.48, p < 0.00001; I2 = 0%, p = 0.87). Similarly, obesity was associated with an increased risk of difficult laryngoscopy in the supine position (pooled RR = 1.47, 95% CI: 1.23–1.76, p < 0.0001; I2 = 45%, p = 0.07). However, subgroup analysis showed that compared with non-obesity, there was no association with the risk of difficult laryngoscopy in obese patients in the sniffing position (pooled RR = 2.00, 95% CI: 0.97–4.15, p = 0.06; I2 = 67%, p = 0.03). We sequentially removed each study as well as studies with parturients and emergency tracheal intubation and then reanalysed the remaining dataset; there were no major changes in the direction or magnitude of the statistical findings. However, removal of studies defining obesity as a BMI cut-off > 30 [24, 26, 27] reduced heterogeneity without significantly affecting the RR (RR = 1.64, 95% CI: 1.26–2.14, p = 0.002; I2 = 9, p = 0.36). No evidence of publication bias was evident by visual inspection of a funnel plot (Fig. 5).
The result is the pooled estimate of the 9 included studies by random effect model. RR: rate ratios. There was a statistically significant association (pooled RR = 1.54, 95% CI: 1.25–1.89, p < 0.0001; I2 = 45%, p = 0.07, Power = 1.0) between obesity and risk of difficult laryngoscopy
Association between obesity and Mallampati score ≥ 3
Twelve studies with a total of 5678 patients were analysed [7, 14, 23, 24, 26, 28,29,30,31,32,33,34]. There was a significant association (pooled RR = 1.83, 95% CI: 1.24–2.69, p = 0.002; I2 = 81%, p < 0.00001, Power = 0.93) between obesity and a Mallampati score ≥ 3 (Fig. 6). The results of subgroup analyses are presented in Table 3. Obesity was associated with an increased rate of a Mallampati score ≥ 3 in both cohort (pooled RR = 1.90, 95% CI: 1.12–3.21, p = 0.02; I2 = 85%, p < 0.00001) and case-control (pooled RR = 1.62, 95% CI: 1.27–2.08, p = 0.0001; I2 = 0%, p = 0.61) studies. We sequentially removed each study and those that included emergency tracheal intubation or parturients and then reanalysed the remaining dataset. Although there were no major changes in the direction or magnitude of the statistical findings, removal of studies defining obesity as a BMI cut-off > 30 decreased heterogeneity without significantly affecting the RR (RR = 2.14, 95% CI: 1.55–2.96, p < 0.00001; I2 = 41, p = 0.10). No evidence of publication bias was evident by visual inspection of the funnel plot (Fig. 7).
The result is the pooled estimate of the 12 included studies by random effect model. RR: rate ratios. There was a statistically significant association (pooled RR = 1.83, 95% CI: 1.24–2.69, p = 0.002; I2 = 81%, p < 0.00001, Power = 0.93) between obesity and risk of Mallampati score > 3
To the best of our knowledge, this is the first meta-analysis focusing on evaluating the association between obesity with risk of DI in recent decades. The findings revealed a significant association between obesity and the rate of difficult tracheal intubation, difficult laryngoscopy and Mallampati score ≥ 3. However, subgroup analysis showed no trend of a higher association of obesity with a risk of DI compared with non-obesity in cohort studies and no association with an increased risk of difficult laryngoscopy in the sniffing position. In addition, no trend of obese patients having a higher association with DI risk compared with non-obese patients when removing emergency tracheal intubation was found in sensitivity analyses.
There are conflicting reports regarding the correlation of BMI with DI [7, 8, 35]. Two prospective studies found no correlation between BMI and DI [7, 8], whereas a retrospective study of large samples concluded that the correlation was weak but statistically significant [35]. These discrepancies may be explained by methodological differences and varying study designs. As a consequence, large-sample, sufficient-power and high-level evidence studies are essential. The powers of our outcomes were 1.00, 1.00 and 0.93, and we believe that these were sufficient because most researchers assess power using 0.80 as a standard for adequacy. Furthermore, to control for the Type I error rate, we used Bonferroni adjustment to control the significance criterion. As the overall qualities of the studies were satisfactory, we consider our results to be convincing.
Importantly, we chose difficult tracheal intubation as our primary outcome because clinicians want to know whether this procedure is more difficult in obese than in non-obese patients. However, difficult tracheal intubation has many influencing factors. Indeed, DI represents a complex interaction between patient factors, the clinical setting, and the skill of the practitioner. As a consequence, we chose the rates of difficult laryngoscopy and a Mallampati score ≥ 3 to be our secondary outcomes. The direct reason for difficult tracheal intubation by DL is the difficult laryngoscopy procedure, and the extent of laryngoscopy is an important prediction method for intubation that is widely used in clinical practice. In addition, the Mallampati score is a preoperative assessment widely applied due to its better alignment with Cormack-Lehane grades [36]. As a result, we believe that the three outcomes complement each other and are indispensable for evaluating difficult intubation.
It is worth mentioning that for the outcome of DI, the CIs obtained varied widely, and the heterogeneity was high. To identify sources of heterogeneity, we utilized analysis of prespecified subgroups and sensitivity analyses. However, subgroup analysis showed no significant association of obesity with DI risk compared with non-obesity in cohort studies. In obese patients, emergency tracheal intubation can be particularly challenging because of the increased risk of impaired respiratory mechanics [37, 38], and poor tolerance of apnea [39,40,41]. Therefore, we conducted sensitivity analyses by removing emergency tracheal intubation and found no significant differences in the estimates of elective tracheal intubation (RR = 2.31, 95% CI: 0.76–6.99, p = 0.14; I2 = 73, p = 0.01). Moreover, heterogeneity was reduced without significantly affecting the RR by removal of studies defining obesity as a BMI cut-off > 30 (RR = 2.12, 95% CI: 1.30–3.47, p = 0.003; I2 = 6, p = 0.30) [8, 24, 35], suggesting that this result was unstable.
The sniffing position has been commonly advocated as the standard head position for DL. In this position, the neck must be flexed on the chest, typically by elevating the head with a cushion under the occiput and extending the head on the atlanto-occipital joint [42, 43]. Regardless, the anatomic explanation of the advantage of the sniffing position has been called into question [44–46]. In subgroup analysis, obesity was not associated with an increased risk of difficult laryngoscopy in the sniffing position compared with non-obesity. This was a believable result, as it is based on a large sample of high-quality research, and it confirms the effect of the sniffing position in improving the laryngeal view in obese patients.
According to the Cochrane Collaboration common scheme for bias and the ROBINS-I tool, the studies demonstrated low/moderate risk of bias. For the majority of the studies, this bias originated from the selection of the reported results as well as from the presence of possible confounding factors. These studies had higher levels of evidence.
There were some limitations to this meta-analysis. First, methodologic limitations with regard to the studies and statistical heterogeneities among the studies were significant. Some biases were unavoidable. For example, it was not possible to blind either anaesthesiologists or patients regarding non-obese or obese patients. Thus, it is accepted that observational studies were essentially included. Second, we only explored difficult tracheal intubation by direct laryngoscopy and not by difficult airway, therefore lacking facemask data. As the risk factors for difficult mask ventilation and DI are quite different [47], future analyses should explore the association between BMI and difficult airway.
Current meta-analysis indicated that obesity was associated with an increased risk of DI, difficult laryngoscopy and a Mallampati score ≥ 3 in adults patients undergoing general surgical procedures. However, there was no association between obesity and risk of DI compared with non-obesity in cohort studies and elective tracheal intubation and no association between an increased risk of difficult laryngoscopy in the sniffing position. Nonetheless, high heterogeneity among the studies included in this analysis limits the generalizability of our findings. Future analyses should explore the association of BMI with difficult airway.
American Society of Anesthesiology
BMI:
Difficult intubation
DL:
Direct laryngoscope
Risk ratios
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Aslani A, Husarova V, Ecimovic P, Loughrey J, McCaul C. Anaesthetic outcomes in obese parturients: the effect of assessment in the high-risk clinic. Ir J Med Sci. 2012;181:93–7.
Combes X, Sauvat S, Leroux B, Dumerat M, Sherrer E, Motamed C, Brain A, D'Honneur G. Intubating laryngeal mask airway in morbidly obese and lean patients: a comparative study. Anesthesiology. 2005;102:1106–9.
Yildiz TS, Ozdamar D, Arslan I, Solak M, Toker K. The LMA CTrach™ in morbidly obese and lean patients undergoing gynecological procedures: a comparative study. J Anesth. 2010;24:849–53.
Lundstrøm LH, Møller AM, Rosenstock C, Astrup G, Wetterslev J. High body mass index is a weak predictor for difficult and failed tracheal intubation: a cohort study of 91,332 consecutive patients scheduled for direct laryngoscopy registered in the Danish Anesthesia Database. Anesthesiology. 2009;110:266–74.
Samsoon GL, Young JR. Difficult tracheal intubation: a retrospective study. Anaesthesia. 1987;42:487–90.
Koenig S. Pulmonary complications of obesity. Am J Med Sci. 2001;321:249–79.
Dixon B, Dixon J, Carden J, Burn A, Schachter L, Playfair J, Laurie C, O'Brien P. Preoxygenation is more effective in the 25 degrees head-up position than in the supine position in severely obese patients: a randomized controlled study. Anesthesiology. 2005;102:1110–5.
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Farmery A, Roe P. A model to describe the rate of oxyhaemoglobin desaturation during apnoea. Brit J Anaesth. 1996;76:284–91.
Benumof J, Dagg R, Benumof R. Critical hemoglobin desaturation will occur before return to an unparalyzed state following 1 mg/kg intravenous succinylcholine. Anesthesiology. 1997;87:979–82.
Miller RD: Endotracheal intubation, Anesthesia, 5th edition. Edited by Miller RD. Philadelphia: Churchill Livingstone, 2000, pp 1426–36.
Benumof JL: Conventional (laryngoscopic) orotracheal and nasotracheal intubation (single-lumen tube), Airway Management, Principles and Practices. Edited by Benumof JL. St Louis, Mosby, 1996, pp 261–76.
Adnet F, Borron SW, Lapostolle F, Lapandry C: The three axis alignment theory and the "sniffing position": Perpetuation of an anatomical myth? Anesthesiology. 1999;91:1964–5.
Adnet F, Borron SW, Dumas JL, Lapostolle F, Cupa M, Lapandry C: Study of the "sniffing position" by magnetic resonance imaging. Anesthesiology. 2001;94:83–6.
Bannister FB, Macbeth RG: Direct laryngoscopy and tracheal intubation. Lancet. 1944;2:651–4.
Jain D. Can positioning alter the success of endotracheal intubation in obese? Braz J Anesthesiol. 2014;64:216.
All the data supporting these findings is contained within this manuscript.
Tingting Wang and Shen Sun contributed equally to this work.
Department of Anaesthesia, Obstetrics & Gynecology Hospital, Fudan University, 128# Shenyang road, Shanghai, 200090, China
Tingting Wang, Shen Sun & Shaoqiang Huang
Tingting Wang
Shen Sun
Shaoqiang Huang
TTW: study design, data collection, data analysis and interpretation of the results, and writing of the paper; SS: data collection, data analysis, and writing of the paper; SQH: study design, data analysis, interpretation of the results, and writing of the paper. All authors read and approved the final manuscript.
Correspondence to Shaoqiang Huang.
Wang, T., Sun, S. & Huang, S. The association of body mass index with difficult tracheal intubation management by direct laryngoscopy: a meta-analysis. BMC Anesthesiol 18, 79 (2018). https://doi.org/10.1186/s12871-018-0534-4
Intratracheal | CommonCrawl |
Symmetric cone
In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional real Euclidean Jordan algebras, originally studied and classified by Jordan, von Neumann & Wigner (1934). The tube domain associated with a symmetric cone is a noncompact Hermitian symmetric space of tube type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other irreducible Hermitian symmetric spaces of noncompact type correspond to Siegel domains of the second kind. These can be described in terms of more complicated structures called Jordan triple systems, which generalize Jordan algebras without identity.[1]
Definitions
A convex cone C in a finite-dimensional real inner product space V is a convex set invariant under multiplication by positive scalars. It spans the subspace C – C and the largest subspace it contains is C ∩ (−C). It spans the whole space if and only if it contains a basis. Since the convex hull of the basis is a polytope with non-empty interior, this happens if and only if C has non-empty interior. The interior in this case is also a convex cone. Moreover, an open convex cone coincides with the interior of its closure, since any interior point in the closure must lie in the interior of some polytope in the original cone. A convex cone is said to be proper if its closure, also a cone, contains no subspaces.
Let C be an open convex cone. Its dual is defined as
$\displaystyle {C^{*}=\{X:(X,Y)>0\,\,\mathrm {for} \,\,Y\in {\overline {C}}\}.}$
It is also an open convex cone and C** = C.[2] An open convex cone C is said to be self-dual if C* = C. It is necessarily proper, since it does not contain 0, so cannot contain both X and −X.
The automorphism group of an open convex cone is defined by
$\displaystyle {\mathrm {Aut} \,C=\{g\in \mathrm {GL} (V)|gC=C\}.}$
Clearly g lies in Aut C if and only if g takes the closure of C onto itself. So Aut C is a closed subgroup of GL(V) and hence a Lie group. Moreover, Aut C* = (Aut C)*, where g* is the adjoint of g. C is said to be homogeneous if Aut C acts transitively on C.
The open convex cone C is called a symmetric cone if it is self-dual and homogeneous.
Group theoretic properties
• If C is a symmetric cone, then Aut C is closed under taking adjoints.
• The identity component Aut0 C acts transitively on C.
• The stabilizers of points are maximal compact subgroups, all conjugate, and exhaust the maximal compact subgroups of Aut C.
• In Aut0 C the stabilizers of points are maximal compact subgroups, all conjugate, and exhaust the maximal compact subgroups of Aut0 C.
• The maximal compact subgroups of Aut0 C are connected.
• The component group of Aut C is isomorphic to the component group of a maximal compact subgroup and therefore finite.
• Aut C ∩ O(V) and Aut0 C ∩ O(V) are maximal compact subgroups in Aut C and Aut0 C.
• C is naturally a Riemannian symmetric space isomorphic to G / K where G = Aut0 C. The Cartan involution is defined by σ(g)=(g*)−1, so that K = G ∩ O(V).
Spectral decomposition in a Euclidean Jordan algebra
See also: Formally real Jordan algebra
In their classic paper, Jordan, von Neumann & Wigner (1934) studied and completely classified a class of finite-dimensional Jordan algebras, that are now called either Euclidean Jordan algebras or formally real Jordan algebras.
Definition
Let E be a finite-dimensional real vector space with a symmetric bilinear product operation
$\displaystyle {E\times E\rightarrow E,\,\,\,a,b\mapsto ab=ba,}$
with an identity element 1 such that a1 = a for a in A and a real inner product (a,b) for which the multiplication operators L(a) defined by L(a)b = ab on E are self-adjoint and satisfy the Jordan relation
$\displaystyle {L(a)L(a^{2})=L(a^{2})L(a).}$
As will turn out below, the condition on adjoints can be replaced by the equivalent condition that the trace form Tr L(ab) defines an inner product. The trace form has the advantage of being manifestly invariant under automorphisms of the Jordan algebra, which is thus a closed subgroup of O(E) and thus a compact Lie group. In practical examples, however, it is often easier to produce an inner product for which the L(a) are self-adjoint than verify directly positive-definiteness of the trace form. (The equivalent original condition of Jordan, von Neumann and Wigner was that if a sum of squares of elements vanishes then each of those elements has to vanish.[3])
Power associativity
From the Jordan condition it follows that the Jordan algebra is power associative, i.e. the Jordan subalgebra generated by any single element a in E is actually an associative commutative algebra. Thus, defining an inductively by an = a (an−1), the following associativity relation holds:
$\displaystyle {a^{m}a^{n}=a^{m+n},}$
so the subalgebra can be identified with R[a], polynomials in a. In fact polarizing of the Jordan relation—replacing a by a + tb and taking the coefficient of t—yields
$\displaystyle {2L(ab)L(a)+L(a^{2})L(b)=2L(a)L(b)L(a)+L(a^{2}b).}$
This identity implies that L(am) is a polynomial in L(a) and L(a2) for all m. In fact, assuming the result for lower exponents than m,
$\displaystyle {a^{2}a^{m-1}=a^{m-1}(a^{2})=L(a^{m-1})L(a)a=L(a)L(a^{m-1})a=L(a)a^{m}=a^{m+1}.}$
Setting b = am – 1 in the polarized Jordan identity gives:
$\displaystyle {L(a^{m+1})=2L(a^{m})L(a)+L(a^{2})L(a^{m-1})-2L(a)^{2}L(a^{m-1}),}$
a recurrence relation showing inductively that L(am + 1) is a polynomial in L(a) and L(a2).
Consequently, if power-associativity holds when the first exponent is ≤ m, then it also holds for m+1 since
$\displaystyle {L(a^{m+1})a^{n}=2L(a)L(a^{m})a^{n}+L(a^{2})L(a^{m-1})a^{n}-2L(a)^{2}L(a^{m-1})a^{n}=a^{m+n+1}.}$
Idempotents and rank
An element e in E is called an idempotent if e2 = e. Two idempotents are said to be orthogonal if ef = 0. This is equivalent to orthogonality with respect to the inner product, since (ef,ef) = (e,f). In this case g = e + f is also an idempotent. An idempotent g is called primitive or minimal if it cannot be written as a sum of non-zero orthogonal idempotents. If e1, ..., em are pairwise orthogonal idempotents then their sum is also an idempotent and the algebra they generate consists of all linear combinations of the ei. It is an associative algebra. If e is an idempotent, then 1 − e is an orthogonal idempotent. An orthogonal set of idempotents with sum 1 is said to be a complete set or a partition of 1. If each idempotent in the set is minimal it is called a Jordan frame. Since the number of elements in any orthogonal set of idempotents is bounded by dim E, Jordan frames exist. The maximal number of elements in a Jordan frame is called the rank r of E.
Spectral decomposition
The spectral theorem states that any element a can be uniquely written as
$\displaystyle {a=\sum \lambda _{i}e_{i},}$
where the idempotents ei's are a partition of 1 and the λi, the eigenvalues of a, are real and distinct. In fact let E0 = R[a] and let T be the restriction of L(a) to E0. T is self-adjoint and has 1 as a cyclic vector. So the commutant of T consists of polynomials in T (or a). By the spectral theorem for self-adjoint operators,
$\displaystyle {T=\sum \lambda _{i}P_{i}}$
where the Pi are orthogonal projections on E0 with sum I and the λi's are the distinct real eigenvalues of T. Since the Pi's commute with T and are self-adjoint, they are given by multiplication elements ei of R[a] and thus form a partition of 1. Uniqueness follows because if fi is a partition of 1 and a = Σ μi fi, then with p(t)=Π (t - μj) and pi = p/(t − μi), fi = pi(a)/pi(μi). So the fi's are polynomials in a and uniqueness follows from uniqueness of the spectral decomposition of T.
The spectral theorem implies that the rank is independent of the Jordan frame. For a Jordan frame with k minimal idempotents can be used to construct an element a with k distinct eigenvalues. As above the minimal polynomial p of a has degree k and R[a] has dimension k. Its dimension is also the largest k such that Fk(a) ≠ 0 where Fk(a) is the determinant of a Gram matrix:
$\displaystyle {F_{k}(a)=\det _{0\leq m,n<k}(a^{m},a^{n}).}$
So the rank r is the largest integer k for which Fk is not identically zero on E. In this case, as a non-vanishing polynomial, Fr is non-zero on an open dense subset of E. the regular elements. Any other a is a limit of regular elements a(n). Since the operator norm of L(x) gives an equivalent norm on E, a standard compactness argument shows that, passing to a subsequence if necessary, the spectral idempotents of the a(n) and their corresponding eigenvalues are convergent. The limit of Jordan frames is a Jordan frame, since a limit of non-zero idempotents yields a non-zero idempotent by continuity of the operator norm. It follows that every Jordan frame is made up of r minimal idempotents.
If e and f are orthogonal idempotents, the spectral theorem shows that e and f are polynomials in a = e − f, so that L(e) and L(f) commute. This can be seen directly from the polarized Jordan identity which implies L(e)L(f) = 2 L(e)L(f)L(e). Commutativity follows by taking adjoints.
Spectral decomposition for an idempotent
If e is a non-zero idempotent then the eigenvalues of L(e) can only be 0, 1/2 and 1, since taking a = b = e in the polarized Jordan identity yields
$\displaystyle {2L(e)^{3}-3L(e)^{2}+L(e)=0.}$
In particular the operator norm of L(e) is 1 and its trace is strictly positive.
There is a corresponding orthogonal eigenspace decomposition of E
$\displaystyle {E=E_{0}(e)\oplus E_{1/2}(e)\oplus E_{1}(e),}$
where, for a in E, Eλ(a) denotes the λ-eigenspace of L(a). In this decomposition E1(e) and E0(e) are Jordan algebras with identity elements e and 1 − e. Their sum E1(e) ⊕ E0(e) is a direct sum of Jordan algebras in that any product between them is zero. It is the centralizer subalgebra of e and consists of all a such that L(a) commutes with L(e). The subspace E1/2(e) is a module for the centralizer of e, the centralizer module, and the product of any two elements in it lies in the centralizer subalgebra. On the other hand, if
$\displaystyle {U=8L(e)^{2}-8L(e)+I,}$
then U is self-adjoint equal to 1 on the centralizer algebra and −1 on the centralizer module. So U2 = I and the properties above show that
$\displaystyle {\sigma (x)=Ux}$
defines an involutive Jordan algebra automorphism σ of E.
In fact the Jordan algebra and module properties follow by replacing a and b in the polarized Jordan identity by e and a. If ea = 0, this gives L(e)L(a) = 2L(e)L(a)L(e). Taking adjoints it follows that L(a) commutes with L(e). Similarly if (1 − e)a = 0, L(a) commutes with I − L(e) and hence L(e). This implies the Jordan algebra and module properties. To check that a product of elements in the module lies in the algebra, it is enough to check this for squares: but if L(e)a = 1/2 a, then ea = 1/2 a, so L(a)2 + L(a2)L(e) = 2L(a)L(e)L(a) + L(a2e). Taking adjoints it follows that L(a2) commutes with L(e), which implies the property for squares.
Trace form
The trace form is defined by
$\displaystyle {\tau (a,b)=\mathrm {Tr} \,L(ab).}$
It is an inner product since, for non-zero a = Σ λi ei,
$\displaystyle {\tau (a,a)=\sum \lambda _{i}^{2}\mathrm {Tr} \,L(e_{i})>0.}$
The polarized Jordan identity can be polarized again by replacing a by a + tc and taking the coefficient of t. A further anyisymmetrization in a and c yields:
$\displaystyle {L(a(bc)-(ab)c)=[[L(a),L(b)],L(c)].}$
Applying the trace to both sides
$\displaystyle {\tau (a,bc)=\tau (ba,c),}$
so that L(b) is self-adjoint for the trace form.
Simple Euclidean Jordan algebras
The classification of simple Euclidean Jordan algebras was accomplished by Jordan, von Neumann & Wigner (1934), with details of the one exceptional algebra provided in the article immediately following theirs by Albert (1934). Using the Peirce decomposition, they reduced the problem to an algebraic problem involving multiplicative quadratic forms already solved by Hurwitz. The presentation here, following Faraut & Koranyi (1994), using composition algebras or Euclidean Hurwitz algebras, is a shorter version of the original derivation.
Central decomposition
If E is a Euclidean Jordan algebra an ideal F in E is a linear subspace closed under multiplication by elements of E, i.e. F is invariant under the operators L(a) for a in E. If P is the orthogonal projection onto F it commutes with the operators L(a), In particular F⊥ = (I − P)E is also an ideal and E = F ⊕ F⊥. Furthermore, if e = P(1), then P = L(e). In fact for a in E
$\displaystyle {ea=ae=L(a)P(1)=P(L(a)1)=P(a),}$
so that ea = a for a in F and 0 for a in F⊥. In particular e and 1 − e are orthogonal idempotents with L(e) = P and L(1 − e) = I − P. e and 1 − e are the identities in the Euclidean Jordan algebras F and F⊥. The idempotent e is central in E, where the center of E is defined to be the set of all z such that L(z) commutes with L(a) for all a. It forms a commutative associative subalgebra.
Continuing in this way E can be written as a direct sum of minimal ideals
$\displaystyle {E=\oplus E_{i}.}$
If Pi is the projection onto Ei and ei = Pi(1) then Pi = L(ei). The ei's are orthogonal with sum 1 and are the identities in Ei. Minimality forces Ei to be simple, i.e. to have no non-trivial ideals. For since L(ei) commutes with all L(a)'s, any ideal F ⊂ Ei would be invariant under E since F = eiF. Such a decomposition into a direct sum of simple Euclidean algebras is unique. If E = ⊕ Fj is another decomposition, then Fj=⊕ eiFj. By minimality only one of the terms here is non-zero so equals Fj. By minimality the corresponding Ei equals Fj, proving uniqueness.
In this way the classification of Euclidean Jordan algebras is reduced to that of simple ones. For a simple algebra E all inner products for which the operators L(a) are self adjoint are proportional. Indeed, any other product has the form (Ta, b) for some positive self-adjoint operator commuting with the L(a)'s. Any non-zero eigenspace of T is an ideal in A and therefore by simplicity T must act on the whole of E as a positive scalar.
List of all simple Euclidean Jordan algebras
• Let Hn(R) be the space of real symmetric n by n matrices with inner product (a,b) = Tr ab and Jordan product a ∘ b = 1/2(ab + ba). Then Hn(R) is a simple Euclidean Jordan algebra of rank n for n ≥ 3.
• Let Hn(C) be the space of complex self-adjoint n by n matrices with inner product (a,b) = Re Tr ab* and Jordan product a ∘ b = 1/2(ab + ba). Then Hn(C) is a simple Euclidean Jordan algebra of rank n for n ≥ 3.
• Let Hn(H) be the space of self-adjoint n by n matrices with entries in the quaternions, inner product (a,b) = Re Tr ab* and Jordan product a ∘ b = 1/2(ab + ba). Then Hn(H) is a simple Euclidean Jordan algebra of rank n for n ≥ 3.
• Let V be a finite dimensional real inner product space and set E = V ⊕ R with inner product (u⊕λ,v⊕μ) =(u,v) + λμ and product (u⊕λ)∘(v⊕μ)=( μu + λv) ⊕ [(u,v) + λμ]. This is a Euclidean Jordan algebra of rank 2, called a spin factor.
• The above examples in fact give all the simple Euclidean Jordan algebras, except for one exceptional case H3(O), the self-adjoint matrices over the octonions or Cayley numbers, another rank 3 simple Euclidean Jordan algebra of dimension 27 (see below).
The Jordan algebras H2(R), H2(C), H2(H) and H2(O) are isomorphic to spin factors V ⊕ R where V has dimension 2, 3, 5 and 9, respectively: that is, one more than the dimension of the relevant division algebra.
Peirce decomposition
See also: Peirce decomposition
Let E be a simple Euclidean Jordan algebra with inner product given by the trace form τ(a)= Tr L(a). The proof that E has the above form rests on constructing an analogue of matrix units for a Jordan frame in E. The following properties of idempotents hold in E.
• An idempotent e is minimal in E if and only if E1(e) has dimension one (so equals Re). Moreover E1/2(e) ≠ (0). In fact the spectral projections of any element of E1(e) lie in E so if non-zero must equal e. If the 1/2 eigenspace vanished then E1(e) = Re would be an ideal.
• If e and f are non-orthogonal minimal idempotents, then there is a period 2 automorphism σ of E such that σe=f, so that e and f have the same trace.
• If e and f are orthogonal minimal idempotents then E1/2(e) ∩ E1/2(f) ≠ (0). Moreover, there is a period 2 automorphism σ of E such that σe=f, so that e and f have the same trace, and for any a in this intersection, a2 = 1/2 τ(e) |a|2 (e + f).
• All minimal idempotents in E are in the same orbit of the automorphism group so have the same trace τ0.
• If e, f, g are three minimal orthogonal idempotents, then for a in E1/2(e) ∩ E1/2(f) and b in E1/2(f) ∩ E1/2(g), L(a)2 b = 1/8 τ0 |a|2 b and |ab|2 = 1/8 τ0 |a|2|b|2. Moreover, E1/2(e) ∩ E1/2(f) ∩ E1/2(g) = (0).
• If e1, ..., er and f1, ..., fr are Jordan frames in E, then there is an automorphism α such that αei = fi.
• If (ei) is a Jordan frame and Eii = E1(ei) and Eij = E1/2(ei) ∩ E1/2(ej), then E is the orthogonal direct sum the Eii's and Eij's. Since E is simple, the Eii's are one-dimensional and the subspaces Eij are all non-zero for i ≠ j.
• If a = Σ αi ei for some Jordan frame (ei), then L(a) acts as αi on Eii and (αi + αi)/2 on Eij.
Reduction to Euclidean Hurwitz algebras
Main article: Euclidean Hurwitz algebra
Let E be a simple Euclidean Jordan algebra. From the properties of the Peirce decomposition it follows that:
• If E has rank 2, then it has the form V ⊕ R for some inner product space V with Jordan product as described above.
• If E has rank r > 2, then there is a non-associative unital algebra A, associative if r > 3, equipped with an inner product satisfying (ab,ab)= (a,a)(b,b) and such that E = Hr(A). (Conjugation in A is defined by a* = −a + 2(a,1)1.)
Such an algebra A is called a Euclidean Hurwitz algebra. In A if λ(a)b = ab and ρ(a)b = ba, then:
• the involution is an antiautomorphism, i.e. (a b)*=b* a*
• a a* = ‖ a ‖2 1 = a* a
• λ(a*) = λ(a)*, ρ(a*) = ρ(a)*, so that the involution on the algebra corresponds to taking adjoints
• Re(a b) = Re(b a) if Re x = (x + x*)/2 = (x, 1)1
• Re(a b) c = Re a(b c)
• λ(a2) = λ(a)2, ρ(a2) = ρ(a)2, so that A is an alternative algebra.
By Hurwitz's theorem A must be isomorphic to R, C, H or O. The first three are associative division algebras. The octonions do not form an associative algebra, so Hr(O) can only give a Jordan algebra for r = 3. Because A is associative when A = R, C or H, it is immediate that Hr(A) is a Jordan algebra for r ≥ 3. A separate argument, given originally by Albert (1934), is required to show that H3(O) with Jordan product a∘b = 1/2(ab + ba) satisfies the Jordan identity [L(a),L(a2)] = 0. There is a later more direct proof using the Freudenthal diagonalization theorem due to Freudenthal (1951): he proved that given any matrix in the algebra Hr(A) there is an algebra automorphism carrying the matrix onto a diagonal matrix with real entries; it is then straightforward to check that [L(a),L(b)] = 0 for real diagonal matrices.[4]
Exceptional and special Euclidean Jordan algebras
The exceptional Euclidean Jordan algebra E= H3(O) is called the Albert algebra. The Cohn–Shirshov theorem implies that it cannot be generated by two elements (and the identity). This can be seen directly. For by Freudenthal's diagonalization theorem one element X can be taken to be a diagonal matrix with real entries and the other Y to be orthogonal to the Jordan subalgebra generated by X. If all the diagonal entries of X are distinct, the Jordan subalgebra generated by X and Y is generated by the diagonal matrices and three elements
$\displaystyle {Y_{1}={\begin{pmatrix}0&0&0\\0&0&y_{1}\\0&y_{1}^{*}&0\end{pmatrix}},\,\,\,Y_{2}={\begin{pmatrix}0&0&y_{2}^{*}\\0&0&0\\y_{2}&0&0\end{pmatrix}},\,\,\,Y_{3}={\begin{pmatrix}0&y_{3}&0\\y_{3}^{*}&0&0\\0&0&0\end{pmatrix}}.}$
It is straightforward to verify that the real linear span of the diagonal matrices, these matrices and similar matrices with real entries form a unital Jordan subalgebra. If the diagonal entries of X are not distinct, X can be taken to be the primitive idempotent e1 with diagonal entries 1, 0 and 0. The analysis in Springer & Veldkamp (2000) then shows that the unital Jordan subalgebra generated by X and Y is proper. Indeed, if 1 − e1 is the sum of two primitive idempotents in the subalgebra, then, after applying an automorphism of E if necessary, the subalgebra will be generated by the diagonal matrices and a matrix orthogonal to the diagonal matrices. By the previous argument it will be proper. If 1 - e1 is a primitive idempotent, the subalgebra must be proper, by the properties of the rank in E.
A Euclidean algebra is said to be special if its central decomposition contains no copies of the Albert algebra. Since the Albert algebra cannot be generated by two elements, it follows that a Euclidean Jordan algebra generated by two elements is special. This is the Shirshov–Cohn theorem for Euclidean Jordan algebras.[5]
The classification shows that each non-exceptional simple Euclidean Jordan algebra is a subalgebra of some Hn(R). The same is therefore true of any special algebra.
On the other hand, as Albert (1934) showed, the Albert algebra H3(O) cannot be realized as a subalgebra of Hn(R) for any n.[6]
Indeed, let π is a real-linear map of E = H3(O) into the self-adjoint operators on V = Rn with π(ab) = 1/2(π(a)π(b) + π(b)π(a)) and π(1) = I. If e1, e2, e3 are the diagonal minimal idempotents then Pi = π(ei are mutually orthogonal projections on V onto orthogonal subspaces Vi. If i ≠ j, the elements eij of E with 1 in the (i,j) and (j,i) entries and 0 elsewhere satisfy eij2 = ei + ej. Moreover, eijejk = 1/2 eik if i, j and k are distinct. The operators Tij are zero on Vk (k ≠ i, j) and restrict to involutions on Vi ⊕ Vj interchanging Vi and Vj. Letting Pij = Pi Tij Pj and setting Pii = Pi, the (Pij) form a system of matrix units on V, i.e. Pij* = Pji, Σ Pii = I and PijPkm = δjk Pim. Let Ei and Eij be the subspaces of the Peirce decomposition of E. For x in O, set πij = Pij π(xeij), regarded as an operator on Vi. This does not depend on j and for x, y in O
$\displaystyle {\pi _{ij}(xy)=\pi _{ij}(x)\pi _{ij}(y),\,\,\,\pi _{ij}(1)=I.}$
Since every x in O has a right inverse y with xy = 1, the map πij is injective. On the other hand, it is an algebra homomorphism from the nonassociative algebra O into the associative algebra End Vi, a contradiction.[7]
Positive cone in a Euclidean Jordan algebra
Definition
When (ei) is a partition of 1 in a Euclidean Jordan algebra E, the self-adjoint operators L(ei) commute and there is a decomposition into simultaneous eigenspaces. If a = Σ λi ei the eigenvalues of L(a) have the form Σ εi λi is 0, 1/2 or 1. The ei themselves give the eigenvalues λi. In particular an element a has non-negative spectrum if and only if L(a) has non-negative spectrum. Moreover, a has positive spectrum if and only if L(a) has positive spectrum. For if a has positive spectrum, a - ε1 has non-negative spectrum for some ε > 0.
The positive cone C in E is defined to be the set of elements a such that a has positive spectrum. This condition is equivalent to the operator L(a) being a positive self-adjoint operator on E.
• C is a convex cone in E because positivity of a self-adjoint operator T— the property that its eigenvalues be strictly positive—is equivalent to (Tv,v) > 0 for all v ≠ 0.
• C is an open because the positive matrices are open in the self-adjoint matrices and L is a continuous map: in fact, if the lowest eigenvalue of T is ε > 0, then T + S is positive whenever ||S|| < ε.
• The closure of C consists of all a such that L(a) is non-negative or equivalently a has non-negative spectrum. From the elementary properties of convex cones, C is the interior of its closure and is a proper cone. The elements in the closure of C are precisely the square of elements in E.
• C is self-dual. In fact the elements of the closure of C are just set of all squares x2 in E, the dual cone is given by all a such that (a,x2) > 0. On the other hand, (a,x2) = (L(a)x,x), so this is equivalent to the positivity of L(a).[8]
Quadratic representation
To show that the positive cone C is homogeneous, i.e. has a transitive group of automorphisms, a generalization of the quadratic action of self-adjoint matrices on themselves given by X ↦ YXY has to be defined. If Y is invertible and self-adjoint, this map is invertible and carries positive operators onto positive operators.
For a in E, define an endomorphism of E, called the quadratic representation, by[9]
$\displaystyle {Q(a)=2L(a)^{2}-L\left(a^{2}\right).}$
Note that for self-adjoint matrices L(X)Y = 1/2(XY + YX), so that Q(X)Y = XYX.
An element a in E is called invertible if it is invertible in R[a]. If b denotes the inverse, then the spectral decomposition of a shows that L(a) and L(b) commute.
In fact a is invertible if and only if Q(a) is invertible. In that case
$\displaystyle {Q(a)^{-1}a=a^{-1},\,\,\,Q\left(a^{-1}\right)=Q(a)^{-1}.}$
Indeed, if Q(a) is invertible it carries R[a] onto itself. On the other hand, Q(a)1 = a2, so
$\displaystyle {\left(Q(a)^{-1}a\right)a=aQ(a)^{-1}a=L(a)Q(a)^{-1}a=Q(a)^{-1}a^{2}=1.}$
Taking b = a−1 in the polarized Jordan identity, yields
$\displaystyle {Q(a)L\left(a^{-1}\right)=L(a).}$
Replacing a by its inverse, the relation follows if L(a) and L(a−1) are invertible. If not it holds for a + ε1 with ε arbitrarily small and hence also in the limit.
• If a and b are invertible then so is Q(a)b and it satisfies the inverse identity:
$\displaystyle {(Q(a)b)^{-1}=Q\left(a^{-1}\right)b^{-1}.}$
• The quadratic representation satisfies the following fundamental identity:
$\displaystyle {Q(Q(a)b)=Q(a)Q(b)Q(a).}$
• In particular, taking b to be non-negative powers of a, it follows by induction that
$\displaystyle {Q\left(a^{m}\right)=Q(a)^{m}.}$
These identities are easy to prove in a finite-dimensional (Euclidean) Jordan algebra (see below) or in a special Jordan algebra, i.e. the Jordan algebra defined by a unital associative algebra.[10] They are valid in any Jordan algebra. This was conjectured by Jacobson and proved in Macdonald (1960): Macdonald showed that if a polynomial identity in three variables, linear in the third, is valid in any special Jordan algebra, then it holds in all Jordan algebras.[11]
In fact for c in A and F(a) a function on A with values in End A, let DcF(a) be the derivative at t = 0 of F(a + tc). Then
$\displaystyle {c=D_{c}\left(Q(a)a^{-1}\right)=2\left[(L(a)L(c)+L(c)L(a)-L(ac))a^{-1}\right]+Q(a)D_{c}\left(a^{-1}\right)=2c+Q(a)D_{c}\left(a^{-1}\right).}$
The expression in square brackets simplifies to c because L(a) commutes with L(a−1).
Thus
$\displaystyle {D_{c}\left(a^{-1}\right)=-Q(a)^{-1}c.}$
Applying Dc to L(a−1)Q(a) = L(a) and acting on b = c−1 yields
$\displaystyle {(Q(a)b)\left(Q\left(a^{-1}\right)b^{-1}\right)=1.}$
On the other hand, L(Q(a)b) is invertible on an open dense set where Q(a)b must also be invertible with
$\displaystyle {(Q(a)b)^{-1}=Q\left(a^{-1}\right)b^{-1}.}$
Taking the derivative Dc in the variable b in the expression above gives
$\displaystyle {-Q(Q(a)b)^{-1}Q(a)c=-Q(a)^{-1}Q(b)^{-1}c.}$
This yields the fundamental identity for a dense set of invertible elements, so it follows in general by continuity. The fundamental identity implies that c = Q(a)b is invertible if a and b are invertible and gives a formula for the inverse of Q(c). Applying it to c gives the inverse identity in full generality.
Finally it can be verified immediately from the definitions that, if u = 1 − 2e for some idempotent e, then Q(u) is the period 2 automorphism constructed above for the centralizer algebra and module of e.
Homogeneity of positive cone
If a is an invertible operator and b is in the positive cone C, then so is Q(a)b.
The proof of this relies on elementary continuity properties of eigenvalues of self-adjoint operators.[12]
Let T(t) (α ≤ t ≤ β) be a continuous family of self-adjoint operators on E with T(α) positive and T(β) having a negative eigenvalue. Set S(t)= –T(t) + M with M > 0 chosen so large that S(t) is positive for all t. The operator norm ||S(t)|| is continuous. It is less than M for t = α and greater than M for t = β. So for some α < s < β, ||S(s)|| = M and there is a vector v ≠ 0 such that S(s)v = Mv. In particular T(s)v = 0, so that T(s) is not invertible.
Suppose that x = Q(a)b does not lie in C. Let b(t) = (1 − t) + tb with 0 ≤ t ≤ 1. By convexity b(t) lies in C. Let x(t) = Q(a)b(t) and X(t) = L(x(t)). If X(t) is invertible for all t with 0 ≤ t ≤ 1, the eigenvalue argument gives a contradiction since it is positive at t = 0 and has negative eigenvalues at t = 1. So X(s) has a zero eigenvalue for some s with 0 < s ≤ 1: X(s)w = 0 with w ≠ 0. By the properties of the quadratic representation, x(t) is invertible for all t. Let Y(t) = L(x(t)2). This is a positive operator since x(t)2 lies in C. Let T(t) = Q(x(t)), an invertible self-adjoint operator by the invertibility of x(t). On the other hand, T(t) = 2X(t)2 - Y(t). So (T(s)w,w) < 0 since Y(s) is positive and X(s)w = 0. In particular T(s) has some negative eigenvalues. On the other hand, the operator T(0) = Q(a2) = Q(a)2 is positive. By the eigenvalue argument, T(t) has eigenvalue 0 for some t with 0 < t < s, a contradiction.
It follows that the linear operators Q(a) with a invertible, and their inverses, take the cone C onto itself. Indeed, the inverse of Q(a) is just Q(a−1). Since Q(a)1 = a2, there is thus a transitive group of symmetries:
C is a symmetric cone.
Euclidean Jordan algebra of a symmetric cone
Construction
Let C be a symmetric cone in the Euclidean space E. As above, Aut C denotes the closed subgroup of GL(E) taking C (or equivalently its closure) onto itself. Let G = Aut0 C be its identity component. K = G ∩ O(E). It is a maximal compact subgroup of G and the stabilizer of a point e in C. It is connected. The group G is invariant under taking adjoints. Let σg =(g*)−1, period 2 automorphism. Thus K is the fixed point subgroup of σ. Let ${\mathfrak {g}}$ be the Lie algebra of G. Thus σ induces an involution of ${\mathfrak {g}}$ and hence a ±1 eigenspace decomposition
$\displaystyle {{\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}},}$
where ${\mathfrak {k}}$, the +1 eigenspace, is the Lie algebra of K and ${\mathfrak {p}}$ is the −1 eigenspace. Thus ${\mathfrak {p}}$⋅e is an affine subspace of dimension dim ${\mathfrak {p}}$. Since C = G/K is an open subspace of E, it follows that dim E = dim ${\mathfrak {p}}$ and hence ${\mathfrak {p}}$⋅e = E. For a in E let L(a) be the unique element of ${\mathfrak {p}}$ such that L(a)e = a. Define a ∘ b = L(a)b. Then E with its Euclidean structure and this bilinear product is a Euclidean Jordan algebra with identity 1 = e. The convex cone coincides C with the positive cone of E.[13]
Since the elements of ${\mathfrak {p}}$ are self-adjoint, L(a)* = L(a). The product is commutative since [${\mathfrak {p}}$, ${\mathfrak {p}}$] ⊆ ${\mathfrak {k}}$ annihilates e, so that ab = L(a)L(b)e = L(b)L(a)e = ba. It remains to check the Jordan identity [L(a),L(a2)] = 0.
The associator is given by [a,b,c] = [L(a),L(c)]b. Since [L(a),L(c)] lies in ${\mathfrak {k}}$ it follows that [[L(a),L(c)],L(b)] = L([a,b,c]). Making both sides act on c yields
$\displaystyle {[a,b^{2},c]=2[a,b,c]b.}$
On the other hand,
$\displaystyle {([b^{2},a,b],c)=(b^{2}(ba)-b(b^{2}a),c)=-(b^{2},[a,b,c])}$
and likewise
$\displaystyle {([b^{2},a,b],c)=(b,[a,b^{2},c]).}$
Combining these expressions gives
$\displaystyle {([b^{2},a,b],c)=0,}$
which implies the Jordan identity.
Finally the positive cone of E coincides with C. This depends on the fact that in any Euclidean Jordan algebra E
$\displaystyle {Q(e^{a})=e^{2L(a)}.}$
In fact Q(ea) is a positive operator, Q(eta) is a one-parameter group of positive operators: this follows by continuity for rational t, where it is a consequence of the behaviour of powers So it has the form exp tX for some self-adjoint operator X. Taking the derivative at 0 gives X = 2L(a).
Hence the positive cone is given by all elements
$\displaystyle {e^{2a}=Q(e^{a})1=e^{2L(a)}1=e^{X}\cdot 1,}$
with X in ${\mathfrak {p}}$. Thus the positive cone of E lies inside C. Since both are self-dual, they must coincide.
Automorphism groups and trace form
Let C be the positive cone in a simple Euclidean Jordan algebra E. Aut C is the closed subgroup of GL(E) taking C (or its closure) onto itself. Let G = Aut0 C be the identity component of Aut C and let K be the closed subgroup of G fixing 1. From the group theoretic properties of cones, K is a connected compact subgroup of G and equals the identity component of the compact Lie group Aut E. Let ${\mathfrak {g}}$ and ${\mathfrak {k}}$ be the Lie algebras of G and K. G is closed under taking adjoints and K is the fixed point subgroup of the period 2 automorphism σ(g) = (g*)−1. Thus K = G ∩ SO(E). Let ${\mathfrak {p}}$ be the −1 eigenspace of σ.
• ${\mathfrak {k}}$ consists of derivations of E that are skew-adjoint for the inner product defined by the trace form.
• [[L(a),L(c)],L(b)] = L([a,b,c]).
• If a and b are in E, then D = [L(a),L(b)] is a derivation of E, so lies in ${\mathfrak {k}}$. These derivations span ${\mathfrak {k}}$.
• If a is in C, then Q(a) lies in G.
• C is the connected component of the open set of invertible elements of E containing 1. It consists of exponentials of elements of E and the exponential map gives a diffeomorphism of E onto C.
• The map a ↦ L(a) gives an isomorphism of E onto ${\mathfrak {p}}$ and eL(a) = Q(ea/2). This space of such exponentials coincides with P the positive self-adjoint elements in G.
• For g in G and a in E, Q(g(a)) = g Q(a) g*.
Cartan decomposition
• G = P ⋅ K = K ⋅ P and the decomposition g = pk corresponds to the polar decomposition in GL(E).
• If (ei) is a Jordan frame in E, then the subspace ${\mathfrak {a}}$ of ${\mathfrak {p}}$ spanned by L(ei) is maximal Abelian in ${\mathfrak {p}}$. A = exp ${\mathfrak {a}}$ is the Abelian subgroup of operators Q(a) where a = Σ λi ei with λi > 0. A is closed in P and hence G. If b =Σ μi ei with μi > 0, then Q(ab)=Q(a)Q(b).
• ${\mathfrak {p}}$ and P are the union of the K translates of ${\mathfrak {a}}$ and A.
Iwasawa decomposition for cone
If E has Peirce decomposition relative to the Jordan frame (ei)
$\displaystyle {E=\bigoplus _{i\leq j}E_{ij},}$
then ${\mathfrak {a}}$ is diagonalized by this decomposition with L(a) acting as (αi + αj)/2 on Eij, where a = Σ αi ei.
Define the closed subgroup S of G by
$\displaystyle {S=\{g\in G|gE_{ij}\subseteq \bigoplus _{(p,q)\geq (i,j)}E_{pq}\},}$
where the ordering on pairs p ≤ q is lexicographic. S contains the group A, since it acts as scalars on Eij. If N is the closed subgroup of S such that nx = x modulo ⊕(p,q) > (i,j) Epq, then S = AN = NA, a semidirect product with A normalizing N. Moreover, G has the following Iwasawa decomposition:
$\displaystyle {G=KAN.}$
For i ≠ j let
$\displaystyle {{\mathfrak {g}}_{ij}=\{X\in {\mathfrak {g}}:[L(a),X]={1 \over 2}(\alpha _{i}-\alpha _{j})X,\,\,\,\mathrm {for} \,\,\,a=\sum \alpha _{i}e_{i}\}.}$
Then the Lie algebra of N is
$\displaystyle {{\mathfrak {n}}=\bigoplus _{i<j}{\mathfrak {g}}_{ij},\,\,\,\,{\mathfrak {g}}_{ij}=\{L(a)+2[L(a),L(e_{i})]:a\in E_{ij}\}.}$
Taking ordered orthonormal bases of the Eij gives a basis of E, using the lexicographic order on pairs (i,j). The group N is lower unitriangular and its Lie algebra lower triangular. In particular the exponential map is a polynomial mapping of ${\mathfrak {n}}$ onto N, with polynomial inverse given by the logarithm.
Complexification of a Euclidean Jordan algebra
Definition of complexification
Let E be a Euclidean Jordan algebra. The complexification EC = E ⊕ iE has a natural conjugation operation (a + ib)* = a − ib and a natural complex inner product and norm. The Jordan product on E extends bilinearly to EC, so that (a + ib)(c + id) = (ac − bd) + i(ad + bc). If multiplication is defined by L(a)b = ab then the Jordan axiom
$\displaystyle {[L(a),L(a^{2})]=0}$
still holds by analytic continuation. Indeed, the identity above holds when a is replaced by a + tb for t real; and since the left side is then a polynomial with values in End EC vanishing for real t, it vanishes also t complex. Analytic continuation also shows that all for the formulas involving power-associativity for a single element a in E, including recursion formulas for L(am), also hold in EC. Since for b in E, L(b) is still self-adjoint on EC, the adjoint relation L(a*) = L(a)* holds for a in EC. Similarly the symmetric bilinear form β(a,b) = (a,b*) satisfies β(ab,c) = β(b,ac). If the inner product comes from the trace form, then β(a,b) = Tr L(ab).
For a in EC, the quadratic representation is defined as before by Q(a)=2L(a)2 − L(a2). By analytic continuation the fundamental identity still holds:
$\displaystyle {Q(Q(a)b)=Q(a)Q(b)Q(a),\,\,\,Q(a^{m})=Q(a)^{m}\,\,(m\geq 0).}$
An element a in E is called invertible if it is invertible in C[a]. Power associativity shows that L(a) and L(a−1) commute. Moreover, a−1 is invertible with inverse a.
As in E, a is invertible if and only if Q(a) is invertible. In that case
$\displaystyle {Q(a)^{-1}a=a^{-1},\,\,\,Q(a^{-1})=Q(a)^{-1}.}$
Indeed, as for E, if Q(a) is invertible it carries C[a] onto itself, while Q(a)1 = a2, so
$\displaystyle {(Q(a)^{-1}a)a=aQ(a)^{-1}a=L(a)Q(a)^{-1}a=Q(a)^{-1}a^{2}=1,}$
so a is invertible. Conversely if a is invertible, taking b = a−2 in the fundamental identity shows that Q(a) is invertible. Replacing a by a−1 and b by a then shows that its inverse is Q(a−1). Finally if a and b are invertible then so is c = Q(a)b and it satisfies the inverse identity:
$\displaystyle {(Q(a)b)^{-1}=Q(a^{-1})b^{-1}.}$
Invertibility of c follows from the fundamental formula which gives Q(c) = Q(a)Q(b)Q(a). Hence
$\displaystyle {c^{-1}=Q(c)^{-1}c=Q(a)^{-1}Q(b)^{-1}b=Q(a)^{-1}b^{-1}.}$
The formula
$\displaystyle {Q(e^{a})=e^{2L(a)}}$
also follows by analytic continuation.
Complexification of automorphism group
Aut EC is the complexification of the compact Lie group Aut E in GL(EC). This follows because the Lie algebras of Aut EC and Aut E consist of derivations of the complex and real Jordan algebras EC and E. Under the isomorphism identifying End EC with the complexification of End E, the complex derivations is identified with the complexification of the real derivations.[14]
Structure groups
The Jordan operator L(a) are symmetric with respect to the trace form, so that L(a)t = L(a) for a in EC. The automorphism groups of E and EC consist of invertible real and complex linear operators g such that L(ga) = gL(a)g−1 and g1 = 1. Aut EC is the complexification of Aut E. Since an automorphism g preserves the trace form, g−1 = gt.
The structure groups of E and EC consist of invertible real and complex linear operators g such that
$\displaystyle {Q(ga)=gQ(a)g^{t}.}$
They form groups Γ(E) and Γ(EC) with Γ(E) ⊂ Γ(EC).
• The structure group is closed under taking transposes g ↦ gt and adjoints g ↦ g*.
• The structure group contains the automorphism group. The automorphism group can be identified with the stabilizer of 1 in the structure group.
• If a is invertible, Q(a) lies in the structure group.
• If g is in the structure group and a is invertible, ga is also invertible with (ga)−1 = (gt)−1a−1.
• If E is simple, Γ(E) = Aut C × {±1}, Γ(E) ∩ O(E) = Aut E × {±1} and the identity component of Γ(E) acts transitively on C.
• Γ(EC) is the complexification of Γ(E), which has Lie algebra ${\mathfrak {k}}\oplus {\mathfrak {p}}$.
• The structure group Γ(EC) acts transitively on the set of invertible elements in EC.
• Every g in Γ(EC) has the form g = h Q(a) with h an automorphism and a invertible.
The unitary structure group Γu(EC) is the subgroup of Γ(EC) consisting of unitary operators, so that Γu(EC) = Γ(EC) ∩ U(EC).
• The stabilizer of 1 in Γu(EC) is Aut E.
• Every g in Γu(EC) has the form g = h Q(u) with h in Aut E and u invertible in EC with u* = u−1.
• Γ(EC) is the complexification of Γu(EC), which has Lie algebra ${\mathfrak {k}}\oplus i{\mathfrak {p}}$.
• The set S of invertible elements u such that u* = u−1 can be characterized equivalently either as those u for which L(u) is a normal operator with uu* = 1 or as those u of the form exp ia for some a in E. In particular S is connected.
• The identity component of Γu(EC) acts transitively on S
• g in GL(EC) is in the unitary structure group if and only if gS = S
• Given a Jordan frame (ei) and v in EC, there is an operator u in the identity component of Γu(EC) such that uv = Σ αi ei with αi ≥ 0. If v is invertible, then αi > 0.
Given a frame (ei) in a Euclidean Jordan algebra E, the restricted Weyl group can be identified with the group of operators on ⊕ R ei arising from elements in the identity component of Γu(EC) that leave ⊕ R ei invariant.
Spectral norm
Let E be a Euclidean Jordan algebra with the inner product given by the trace form. Let (ei) be a fixed Jordan frame in E. For given a in EC choose u in Γu(EC) such that ua = Σ αi ei with αi ≥ 0. Then the spectral norm ||a|| = max αi is independent of all choices. It is a norm on EC with
$\displaystyle {\|a^{*}\|=\|a\|,\,\,\,\|\{a,a^{*},a\}\|=\|a\|^{3}.}$
In addition ||a||2 is given by the operator norm of Q(a) on the inner product space EC. The fundamental identity for the quadratic representation implies that ||Q(a)b|| ≤ ||a||2||b||. The spectral norm of an element a is defined in terms of C[a] so depends only on a and not the particular Euclidean Jordan algebra in which it is calculated.[15]
The compact set S is the set of extreme points of the closed unit ball ||x|| ≤ 1. Each u in S has norm one. Moreover, if u = eia and v = eib, then ||uv|| ≤ 1. Indeed, by the Cohn–Shirshov theorem the unital Jordan subalgebra of E generated by a and b is special. The inequality is easy to establish in non-exceptional simple Euclidean Jordan algebras, since each such Jordan algebra and its complexification can be realized as a subalgebra of some Hn(R) and its complexification Hn(C) ⊂ Mn(C). The spectral norm in Hn(C) is the usual operator norm. In that case, for unitary matrices U and V in Mn(C), clearly ||1/2(UV + VU)|| ≤ 1. The inequality therefore follows in any special Euclidean Jordan algebra and hence in general.[16]
On the other hand, by the Krein–Milman theorem, the closed unit ball is the (closed) convex span of S.[17] It follows that ||L(u)|| = 1, in the operator norm corresponding to either the inner product norm or spectral norm. Hence ||L(a)|| ≤ ||a|| for all a, so that the spectral norm satisfies
$\displaystyle {\|ab\|\leq \|a\|\cdot \|b\|.}$
It follows that EC is a Jordan C* algebra.[18]
Complex simple Jordan algebras
The complexification of a simple Euclidean Jordan algebra is a simple complex Jordan algebra which is also separable, i.e. its trace form is non-degenerate. Conversely, using the existence of a real form of the Lie algebra of the structure group, it can be shown that every complex separable simple Jordan algebra is the complexification of a simple Euclidean Jordan algebra.[19]
To verify that the complexification of a simple Euclidean Jordan algebra E has no ideals, note that if F is an ideal in EC then so too is F⊥, the orthogonal complement for the trace norm. As in the real case, J = F⊥ ∩ F must equal (0). For the associativity property of the trace form shows that F⊥ is an ideal and that ab = 0 if a and b lie in J. Hence J is an ideal. But if z is in J, L(z) takes EC into J and J into (0). Hence Tr L(z) = 0. Since J is an ideal and the trace form degenerate, this forces z = 0. It follows that EC = F ⊕ F⊥. If P is the corresponding projection onto F, it commutes with the operators L(a) and F⊥ = (I − P)EC. is also an ideal and E = F ⊕ F⊥. Furthermore, if e = P(1), then P = L(e). In fact for a in E
$\displaystyle {ea=ae=L(a)P(1)=P(L(a)1)=P(a),}$
so that ea = a for a in F and 0 for a in F⊥. In particular e and 1 − e are orthogonal central idempotents with L(e) = P and L(1 − e) = I − P.
So simplicity follows from the fact that the center of EC is the complexification of the center of E.
Symmetry groups of bounded domain and tube domain
According to the "elementary approach" to bounded symmetric space of Koecher,[20] Hermitian symmetric spaces of noncompact type can be realized in the complexification of a Euclidean Jordan algebra E as either the open unit ball for the spectral norm, a bounded domain, or as the open tube domain T = E + iC, where C is the positive open cone in E. In the simplest case where E = R, the complexification of E is just C, the bounded domain corresponds to the open unit disk and the tube domain to the upper half plane. Both these spaces have transitive groups of biholomorphisms given by Möbius transformations, corresponding to matrices in SU(1,1) or SL(2,R). They both lie in the Riemann sphere C ∪ {∞}, the standard one-point compactification of C. Moreover, the symmetry groups are all particular cases of Möbius transformations corresponding to matrices in SL(2,C). This complex Lie group and its maximal compact subgroup SU(2) act transitively on the Riemann sphere. The groups are also algebraic. They have distinguished generating subgroups and have an explicit description in terms of generators and relations. Moreover, the Cayley transform gives an explicit Möbius transformation from the open disk onto the upper half plane. All these features generalize to arbitrary Euclidean Jordan algebras.[21] The compactification and complex Lie group are described in the next section and correspond to the dual Hermitian symmetric space of compact type. In this section only the symmetries of and between the bounded domain and tube domain are described.
Jordan frames provide one of the main Jordan algebraic techniques to describe the symmetry groups. Each Jordan frame gives rise to a product of copies of R and C. The symmetry groups of the corresponding open domains and the compactification—polydisks and polyspheres—can be deduced from the case of the unit disk, the upper halfplane and Riemann sphere. All these symmetries extend to the larger Jordan algebra and its compactification. The analysis can also be reduced to this case because all points in the complex algebra (or its compactification) lie in an image of the polydisk (or polysphere) under the unitary structure group.
Definitions
Let E be a Euclidean Jordan algebra with complexification A = EC = E + iE.
The unit ball or disk D in A is just the convex bounded open set of elements a such the ||a|| < 1, i.e. the unit ball for the spectral norm.
The tube domain T in A is the unbounded convex open set T = E + iC, where C is the open positive cone in E.
Möbius transformations
The group SL(2,C) acts by Möbius transformations on the Riemann sphere C ∪ {∞}, the one-point compactification of C. If g in SL(2,C) is given by the matrix
$\displaystyle {g={\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}},}$
then
$\displaystyle {g(z)=(\alpha z+\beta )(\gamma z+\delta )^{-1}.}$
Similarly the group SL(2,R) acts by Möbius transformations on the circle R ∪ {∞}, the one-point compactification of R.
Let k = R or C. Then SL(2,k) is generated by the three subgroups of lower and upper unitriangular matrices, L and U', and the diagonal matrices D. It is also generated by the lower (or upper) unitriangular matrices, the diagonal matrices and the matrix
$\displaystyle {J={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}.}$
The matrix J corresponds to the Möbius transformation j(z) = −z−1 and can be written
$\displaystyle {J={\begin{pmatrix}1&0\\-1&1\end{pmatrix}}{\begin{pmatrix}1&1\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\-1&1\end{pmatrix}}.}$
The Möbius transformations fixing ∞ are just the upper triangular matrices B = UD = DU. If g does not fix ∞, it sends ∞ to a finite point a. But then g can be composed with an upper unitriangular matrix to send a to 0 and then with J to send 0 to infinity. This argument gives the one of the simplest examples of the Bruhat decomposition:
$\displaystyle {\mathbf {SL} (2,k)=\mathbf {B} \cup \mathbf {B} \cdot J\cdot \mathbf {B} ,}$
the double coset decomposition of SL(2,k). In fact the union is disjoint and can be written more precisely as
$\displaystyle {\mathbf {SL} (2,k)=\mathbf {B} \cup \mathbf {B} \cdot J\cdot \mathbf {U} ,}$
where the product occurring in the second term is direct.
Now let
$\displaystyle {T(\beta )={\begin{pmatrix}1&\beta \\0&1\end{pmatrix}}.}$
Then
$\displaystyle {{\begin{pmatrix}\alpha &0\\0&\alpha ^{-1}\end{pmatrix}}=JT(\alpha ^{-1})JT(\alpha )JT(\alpha ^{-1}).}$
It follows SL(2,k) is generated by the group of operators T(β) and J subject to the following relations:
• β ↦ T(β) is an additive homomorphism
• α ↦ D(α) = JT(α−1)JT(α)JT(α−1) is a multiplicative homomorphism
• D(−1) = J
• D(α)T(β)D(α)−1 = T(α2β)
• JD(α)J−1 = D(α)−1
The last relation follows from the definition of D(α). The generator and relations above is fact gives a presentation of SL(2,k). Indeed, consider the free group Φ generated by J and T(β) with J of order 4 and its square central. This consists of all products T(β1)JT(β2)JT(β3)J ... T(βm)J for m ≥ 0. There is a natural homomorphism of Φ onto SL(2,k). Its kernel contain the normal subgroup Δ generated by the relations above. So there is a natural homomorphism of Φ/Δ onto SL(2,k). To show that it is injective it suffices to show that the Bruhat decomposition also holds in Φ/Δ. It is enough to prove the first version, since the more precise version follows from the commutation relations between J and D(α). The set B ∪ B J B is invariant under inversion, contains operators T(β) and J, so it is enough to show it is invariant under multiplication. By construction it is invariant under multiplication by B. It is invariant under multiplication by J because of the defining equation for D(α).[22]
In particular the center of SL(2,k) consists of the scalar matrices ±I and it is the only non-trivial normal subgroup of SL(2,k), so that PSL(2,k) = SL(2,k)/{±I} is simple.[23] In fact if K is a normal subgroup, then the Bruhat decomposition implies that B is a maximal subgroup, so that either K is contained in B or KB = SL(2,k). In the first case K fixes one point and hence every point of k ∪ {∞}, so lies in the center. In the second case, the commutator subgroup of SL(2,k) is the whole group, since it is the group generated by lower and upper unitriangular matrices and the fourth relation shows that all such matrices are commutators since [T(β),D(α)] = T(β − α2β). Writing J = kb with k in K and b in B, it follows that L = k U k−1. Since U and L generate the whole group, SL(2,k) = KU. But then SL(2,k)/K ≅ U/U ∩ K. The right hand side here is Abelian while the left hand side is its own commutator subgroup. Hence this must be the trivial group and K = SL(2,k).
Given an element a in the complex Jordan algebra A = EC, the unital Jordan subalgebra C[a] is associative and commutative. Multiplication by a defines an operator on C[a] which has a spectrum, namely its set of complex eigenvalues. If p(t) is a complex polynomial, then p(a) is defined in C[a]. It is invertible in A if and only if it is invertible in C[a], which happen precisely when p does not vanish on the spectrum of a. This permits rational functions of a to be defined whenever the function is defined on the spectrum of a. If F and G are rational functions with G and F∘G defined on a, then F is defined on G(a) and F(G(a)) = (F∘G)(a). This applies in particular to complex Möbius transformations which can be defined by g(a) = (αa + β1)(γa + δ1)−1. They leave C[a] invariant and, when defined, the group composition law holds. (In the next section complex Möbius transformations will be defined on the compactification of A.)[24]
Given a primitive idempotent e in E with Peirce decomposition
$\displaystyle {E=E_{1}(e)\oplus E_{1/2}(e)\oplus E_{0}(e),\,\,\,\,A=A_{1}(e)\oplus A_{1/2}(e)\oplus A_{0}(e).}$
the action of SL(2,C) by Möbius transformations on E1(e) = C e can be extended to an action on A so that the action leaves invariant the components Ai(e) and in particular acts trivially on E0(e).[25] If P0 is the projection onto A0(e), the action is given be the formula
$\displaystyle {g(ze\oplus x_{1/2}\oplus x_{0})={\alpha z+\beta \over \gamma z+\delta }\cdot e\oplus (\gamma z+\delta )^{-1}x_{1/2}\oplus x_{0}-(\gamma z+\delta )^{-1}P_{0}(x_{1/2}^{2}).}$
For a Jordan frame of primitive idempotents e1, ..., em, the actions of SL(2,C) associated with different ei commute, thus giving an action of SL(2,C)m. The diagonal copy of SL(2,C) gives again the action by Möbius transformations on A.
Cayley transform
See also: Cayley transform
The Möbius transformation defined by
$\displaystyle {C(z)=i{1+z \over 1-z}=-i+{2i \over 1-z}}$
is called the Cayley transform. Its inverse is given by
$\displaystyle {P(w)={w-i \over w+i}=1-{2i \over w+i}.}$
The inverse Cayley transform carries the real line onto the circle with the point 1 omitted. It carries the upper halfplane onto the unit disk and the lower halfplane onto the complement of the closed unit disk. In operator theory the mapping T ↦ P(T) takes self-adjoint operators T onto unitary operators U not containing 1 in their spectrum. For matrices this follows because unitary and self-adjoint matrices can be diagonalized and their eigenvalues lie on the unit circle or real line. In this finite-dimensional setting the Cayley transform and its inverse establish a bijection between the matrices of operator norm less than one and operators with imaginary part a positive operator. This is the special case for A = Mn(C) of the Jordan algebraic result, explained below, which asserts that the Cayley transform and its inverse establish a bijection between the bounded domain D and the tube domain T.
In the case of matrices, the bijection follows from resolvant formulas.[26] In fact if the imaginary part of T is positive, then T + iI is invertible since
$\displaystyle {\|(T+iI)x\|^{2}=\|(T-iI)x\|^{2}+4(\mathrm {Im} (T)x,x).}$
In particular, setting y = (T + iI)x,
$\displaystyle {\|y\|^{2}=\|P(T)y\|^{2}+4(\mathrm {Im} (T)x,x).}$
Equivalently
$\displaystyle {I-P(T)^{*}P(T)=4(T^{*}-iI)^{-1}[\mathrm {Im} \,T](T+iI)^{-1}}$
is a positive operator, so that ||P(T)|| < 1. Conversely if ||U|| < 1 then I − U is invertible and
$\displaystyle {\mathrm {Im} \,C(U)=(2i)^{-1}[C(U)-C(U)^{*}]=(1-U^{*})^{-1}[I-U^{*}U](I-U)^{-1}.}$
Since the Cayley transform and its inverse commute with the transpose, they also establish a bijection for symmetric matrices. This corresponds to the Jordan algebra of symmetric complex matrices, the complexification of Hn(R).
In A = EC the above resolvant identities take the following form:[27]
$\displaystyle {Q(1-u^{*})Q(C(u)+C(u^{*}))Q(1-u)=-4B(u^{*},u)}$
and equivalently
$\displaystyle {4Q(\mathrm {Im} \,a)=Q(a^{*}-i)B(P(a)^{*},P(a))Q(a+i),}$
where the Bergman operator B(x,y) is defined by B(x,y) = I − 2R(x,y) + Q(x)Q(y) with R(x,y) = [L(x),L(y)] + L(xy). The inverses here are well defined. In fact in one direction 1 − u is invertible for ||u|| < 1: this follows either using the fact that the norm satisfies ||ab|| ≤ ||a|| ||b||; or using the resolvant identity and the invertibility of B(u*,u) (see below). In the other direction if the imaginary part of a is in C then the imaginary part of L(a) is positive definite so that a is invertible. This argument can be applied to a + i, so it also invertible.
To establish the correspondence, it is enough to check it when E is simple. In that case it follows from the connectivity of T and D and because:
• For x in E, Q(x) is a positive operator if and only if x or −x lies in C
• B(a*,a) is a positive operator if and only if a or its inverse (if invertible) lies in D
The first criterion follows from the fact that the eigenvalues of Q(x) are exactly λiλj if the eigenvalues of x are λi. So the λi are either all positive or all negative. The second criterion follows from the fact that if a = u Σ αi ei = ux with αi ≥ 0 and u in Γu(EC), then B(a*,a) = u*Q(1 − x2)u has eigenvalues (1 − αi2)(1 − αj2). So the αi are either all less than one or all greater than one.
The resolvant identity is a consequence of the following identity for a and b invertible
$\displaystyle {Q(a)Q(a^{-1}+b^{-1})Q(b)=Q(a+b).}$
In fact in this case the relations for a quadratic Jordan algebra imply
$\displaystyle {R(a,b)=2Q(a)Q(a^{-1},b)=2Q(a,b^{-1})Q(b)}$
so that
$\displaystyle {B(a,b)=Q(a)Q(a^{-1}-b)=Q(b^{-1}-a)Q(b).}$
The equality of the last two terms implies the identity, replacing b by −b−1.
Now set a = 1 − x and b = 1 − y. The resolvant identity is a special case of the more following more general identity:
$\displaystyle {Q(1-x)Q(C(x)+C(y))Q(1-y)=-4B(x,y).}$
In fact
$\displaystyle {C(x)+C(y)=-2i(1-a^{-1}-b^{-1}),}$
so the identity is equivalent to
$\displaystyle {Q(a)Q(1-a^{-1}-b^{-1})Q(b)=B(1-a,1-b).}$
Using the identity above together with Q(c)L(c−1) = L(c), the left hand side equals Q(a)Q(b) + Q(a + b) − 2L(a)Q(b) − 2Q(a)L(b). The right hand side equals 2L(a)L(b) + 2L(b)L(a) − 2L(ab) − 2L(a)Q(b) − 2Q(a)L(b) + Q(a)Q(b) + Q(a) + Q(b). These are equal because of the formula 1/2[Q(a + b) − Q(a) − Q(b)] = L(a)L(b) + L(b)L(a) − L(ab).
Automorphism group of bounded domain
The Möbius transformations in SU(1,1) carry the bounded domain D onto itself.
If a lies in the bounded domain D, then a − 1 is invertible. Since D is invariant under multiplication by scalars of modulus ≤ 1, it follows that a − λ is invertible for |λ| ≥ 1. Hence for ||a|| ≤ 1, a − λ is invertible for |λ| > 1. It follows that the Möbius transformation ga is defined for ||a|| ≤ 1 and g in SU(1,1). Where defined it is injective. It is holomorphic on D. By the maximum modulus principle, to show that g maps D onto D it suffices to show it maps S onto itself. For in that case g and its inverse preserve D so must be surjective. If u = eix with x = Σ ξiei in E, then gu lies in ⊕ C ei. This is a commutative associative algebra and the spectral norm is the supremum norm. Since u = Σ ςiei with |ςi| = 1, it follows that gu = Σ g(ςi)ei where |g(ςi)| = 1. So gu lies in S.
The unitary structure group of EC carries D onto itself.
This is a direct consequence of the definition of the spectral norm.
The group of transformations SU(1,1)m corresponding to a Jordan frame carries D onto itself.
This is already known for the Möbius transformations, i.e. the diagonal in SU(1,1)m. It follows for diagonal matrices in a fixed component in SU(1,1)m because they correspond to transformations in the unitary structure group. Conjugating by a Möbius transformation is equivalent to conjugation by a matrix in that component. Since the only non-trivial normal subgroup of SU(1,1) is its center, every matrix in a fixed component carries D onto itself.
D is a bounded symmetric domain.
Given an element in D an transformation in the identity component of the unitary structure group carries it in an element in ⊕ C ei with supremum norm less than 1. An transformation in SU(1,1)m the carries it onto zero. Thus there is a transitive group of biholomorphic transformations of D. The symmetry z ↦ −z is a biholomorphic Möbius transformation fixing only 0.
The biholomorphic mappings of D onto itself that fix the origin are given by the unitary structure group.
If f is a biholomorphic self-mapping of D with f(0) = 0 and derivative I at 0, then f must be the identity.[28] If not, f has Taylor series expansion f(z) = z + fk + fk + 1(z) + ⋅⋅⋅ with fi homogeneous of degree iand fk ≠ 0. But then fn(z) = z + n fk(z). Let ψ be a functional in A* of norm one. Then for fixed z in D, the holomorphic functions of a complex variable w given by hn(w) = ψ(fn(wz)) must have modulus less than 1 for |w| < 1. By Cauchy's inequality, the coefficients of wk must be uniformly bounded independent of n, which is not possible if fk ≠ 0.
If g is a biholomorphic mapping of D onto itself just fixing 0 then if h(z) = eiα z, the mapping f = g ∘ h ∘ g−1 ∘ h−α fixes 0 and has derivative I there. It is therefore the identity map. So g(eiα z) = eiαg(z) for any α. This implies g is a linear mapping. Since it maps D onto itself it maps the closure onto itself. In particular it must map the Shilov boundary S onto itself. This forces g to be in the unitary structure group.
The group GD of biholomorphic automorphisms of D is generated by the unitary structure group KD and the Möbius transformations associated to a Jordan frame. If AD denotes the subgroup of such Möbius transformations fixing ±1, then the Cartan decomposition formula holds: GD = KD AD KD.
The orbit of 0 under AD is the set of all points Σ αi ei with −1 < αi < 1. The orbit of these points under the unitary structure group is the whole of D. The Cartan decomposition follows because KD is the stabilizer of 0 in GD.
The center of GD is trivial.
In fact the only point fixed by (the identity component of) KD in D is 0. Uniqueness implies that the center of GD must fix 0. It follows that the center of GD lies in KD. The center of KD is isomorphic to the circle group: a rotation through θ corresponds to multiplication by eiθ on D so lies in SU(1,1)/{±1}. Since this group has trivial center, the center of GD is trivial.[29]
KD is a maximal compact subgroup of GD.
In fact any larger compact subgroup would intersect AD non-trivially and it has no non-trivial compact subgroups.
Note that GD is a Lie group (see below), so that the above three statements hold with GD and KD replaced by their identity components, i.e. the subgroups generated by their one-parameter cubgroups. Uniqueness of the maximal compact subgroup up to conjugacy follows from a general argument or can be deduced for classical domains directly using Sylvester's law of inertia following Sugiura (1982).[30] For the example of Hermitian matrices over C, this reduces to proving that U(n) × U(n) is up to conjugacy the unique maximal compact subgroup in U(n,n). In fact if W = Cn ⊕ (0), then U(n) × U(n) is the subgroup of U(n,n) preserving W. The restriction of the hermitian form given by the inner product on W minus the inner product on (0) ⊕ Cn. On the other hand, if K is a compact subgroup of U(n,n), there is a K-invariant inner product on C2n obtained by averaging any inner product with respect to Haar measure on K. The Hermitian form corresponds to an orthogonal decomposition into two subspaces of dimension n both invariant under K with the form positive definite on one and negative definite on the other. By Sylvester's law of inertia, given two subspaces of dimension n on which the Hermitian form is positive definite, one is carried onto the other by an element of U(n,n). Hence there is an element g of U(n,n) such that the positive definite subspace is given by gW. So gKg−1 leaves W invariant and gKg−1 ⊆ U(n) × U(n).
A similar argument, with quaternions replacing the complex numbers, shows uniqueness for the symplectic group, which corresponds to Hermitian matrices over R. This can also be seen more directly by using complex structures. A complex structure is an invertible operator J with J2 = −I preserving the symplectic form B and such that −B(Jx,y) is a real inner product. The symplectic group acts transitively on complex structures by conjugation. Moreover, the subgroup commuting with J is naturally identified with the unitary group for the corresponding complex inner product space. Uniqueness follows by showing that any compact subgroup K commutes with some complex structure J. In fact, averaging over Haar measure, there is a K-invariant inner product on the underlying space. The symplectic form yields an invertible skew-adjoint operator T commuting with K. The operator S = −T2 is positive, so has a unique positive square root, which commutes with K. So J = S−1/2T, the phase of T, has square −I and commutes with K.
Automorphism group of tube domain
There is a Cartan decomposition for GT corresponding to the action on the tube T = E + iC:
$\displaystyle {G_{T}=K_{T}A_{T}K_{T}.}$
• KT is the stabilizer of i in iC ⊂ T, so a maximal compact subgroup of GT. Under the Cayley transform, KT corresponds to KD, the stabilizer of 0 in the bounded symmetric domain, where it acts linearly. Since GT is semisimple, every maximal compact subgroup is conjugate to KT.
• The center of GT or GD is trivial. In fact the only point fixed by KD in D is 0. Uniqueness implies that the center of GD must fix 0. It follows that the center of GD lies in KD and hence that the center of GT lies in KT. The center of KD is isomorphic to the circle group: a rotation through θ corresponds to multiplication by eiθ on D. In Cayley transform it corresponds to the Möbius transformation z ↦ (cz + s)(−sz + c)−1 where c = cos θ/2 and s = sin θ/2. (In particular, when θ = π, this gives the symmetry j(z) = −z−1.) In fact all Möbius transformations z ↦ (αz + β)(−γz + δ)−1 with αδ − βγ = 1 lie in GT. Since PSL(2,R) has trivial center, the center of GT is trivial.[31]
• AT is given by the linear operators Q(a) with a = Σ αi ei with αi > 0.
In fact the Cartan decomposition for GT follows from the decomposition for GD. Given z in D, there is an element u in KD, the identity component of Γu(EC), such that z = u Σ αjej with αj ≥ 0. Since ||z|| < 1, it follows that αj < 1. Taking the Cayley transform of z, it follows that every w in T can be written w = k∘ C Σ αjej, with C the Cayley transform and k in KT. Since C Σ αiei = Σ βjej i with βj = (1 + αj)(1 − αj)−1, the point w is of the form w =ka(i) with a in A. Hence GT = KTATKT.
Iwasawa decomposition
There is an Iwasawa decomposition for GT corresponding to the action on the tube T = E + iC:[32]
$\displaystyle {G_{T}=K_{T}A_{T}N_{T}.}$
• KT is the stabilizer of i in iC ⊂ T.
• AT is given by the linear operators Q(a) where a = Σ αi ei with αi > 0.
• NT is a lower unitriangular group on EC. It is the semidirect product of the unipotent triangular group N appearing in the Iwasawa decomposition of G (the symmetry group of C) and N0 = E, group of translations x ↦ x + b.
The group S = AN acts on E linearly and conjugation on N0 reproduces this action. Since the group S acts simply transitively on C, it follows that ANT=S⋅N0 acts simply transitively on T = E + iC. Let HT be the group of biholomorphisms of the tube T. The Cayley transform shows that is isomorphic to the group HD of biholomorphisms of the bounded domain D. Since ANT acts simply transitively on the tube T while KT fixes ic, they have trivial intersection.
Given g in HT, take s in ANT such that g−1(i)=s−1(i). then gs−1 fixes i and therefore lies in KT. Hence HT = KT ⋅A⋅NT. So the product is a group.
Lie group structure
By a result of Henri Cartan, HD is a Lie group. Cartan's original proof is presented in Narasimhan (1971). It can also be deduced from the fact the D is complete for the Bergman metric, for which the isometries form a Lie group; by Montel's theorem, the group of biholomorphisms is a closed subgroup.[33]
That HT is a Lie group can be seen directly in this case. In fact there is a finite-dimensional 3-graded Lie algebra ${\mathfrak {g}}_{T}$ of vector fields with an involution σ. The Killing form is negative definite on the +1 eigenspace of σ and positive definite on the −1 eigenspace. As a group HT normalizes ${\mathfrak {g}}_{T}$ since the two subgroups KT and ANT do. The +1 eigenspace corresponds to the Lie algebra of KT. Similarly the Lie algebras of the linear group AN and the affine group N0 lie in ${\mathfrak {g}}_{T}$. Since the group GT has trivial center, the map into GL(${\mathfrak {g}}_{T}$) is injective. Since KT is compact, its image in GL(${\mathfrak {g}}_{T}$) is compact. Since the Lie algebra ${\mathfrak {g}}_{T}$ is compatible with that of ANT, the image of ANT is closed. Hence the image of the product is closed, since the image of KT is compact. Since it is a closed subgroup, it follows that HT is a Lie group.
Generalizations
Euclidean Jordan algebras can be used to construct Hermitian symmetric spaces of tube type. The remaining Hermitian symmetric spaces are Siegel domains of the second kind. They can be constructed using Euclidean Jordan triple systems, a generalization of Euclidean Jordan algebras. In fact for a Euclidean Jordan algebra E let
$\displaystyle {L(a,b)=2([L(a),L(b)]+L(ab)).}$
Then L(a,b) gives a bilinear map into End E such that
$\displaystyle {L(a,b)^{*}=L(b,a)},\,\,\,L(a,b)c=L(c,b)a$
and
$\displaystyle {[L(a,b),L(c,d)]=L(L(a,b)c,d)-L(c,L(b,a)d).}$
Any such bilinear system is called a Euclidean Jordan triple system. By definition the operators L(a,b) form a Lie subalgebra of End E.
The Kantor–Koecher–Tits construction gives a one-one correspondence between Jordan triple systems and 3-graded Lie algebras
$\displaystyle {{\mathfrak {g}}={\mathfrak {g}}_{-1}\oplus {\mathfrak {g}}_{0}\oplus {\mathfrak {g}}_{1},}$
satisfying
$\displaystyle {[{\mathfrak {g}}_{p},{\mathfrak {g}}_{q}]\subseteq {\mathfrak {g}}_{p+q}}$
and equipped with an involutive automorphism σ reversing the grading. In this case
$\displaystyle {L(a,b)=\mathrm {ad} \,[a,\sigma (b)]}$
defines a Jordan triple system on ${\mathfrak {g}}_{-1}$. In the case of Euclidean Jordan algebras or triple systems the Kantor–Koecher–Tits construction can be identified with the Lie algebra of the Lie group of all homomorphic automorphisms of the corresponding bounded symmetric domain. The Lie algebra is constructed by taking ${\mathfrak {g}}_{0}$ to be the Lie subalgebra ${\mathfrak {h}}$ of End E generated by the L(a,b) and ${\mathfrak {g}}_{\pm 1}$ to be copies of E. The Lie bracket is given by
$\displaystyle {[(a_{1},T_{1},b_{1}),(a_{2},T_{2},b_{2})]=(T_{1}a_{2}-T_{2}a_{1},[T_{1},T_{2}]+L(a_{1},b_{2})-L(a_{2},b_{1}),T_{2}^{*}b_{1}-T_{1}^{*}b_{2})}$
and the involution by
$\displaystyle {\sigma (a,T,b)=(b,-T^{*},a).}$
The Killing form is given by
$\displaystyle {B((a_{1},T_{1},b_{1}),(a_{2},T_{2},b_{2}))=(a_{1},b_{2})+(b_{1},a_{2})+\beta (T_{1},T_{2}),}$
where β(T1,T2) is the symmetric bilinear form defined by
$\displaystyle {\beta (L(a,b),L(c,d))=(L(a,b)c,d)=(L(c,d)a,b).}$
These formulas, originally derived for Jordan algebras, work equally well for Jordan triple systems.[34] The account in Koecher (1969) develops the theory of bounded symmetric domains starting from the standpoint of 3-graded Lie algebras. For a given finite-dimensional vector space E, Koecher considers finite-dimensional Lie algebras ${\mathfrak {g}}$ of vector fields on E with polynomial coefficients of degree ≤ 2. ${\mathfrak {g}}_{-1}$ consists of the constant vector fields ∂i and ${\mathfrak {g}}_{0}$ must contain the Euler operator H = Σ xi⋅∂i as a central element. Requiring the existence of an involution σ leads directly to a Jordan triple structure on V as above. As for all Jordan triple structures, fixing c in E, the operators Lc(a) = L(a,c) give E a Jordan algebra structure, determined by e. The operators L(a,b) themselves come from a Jordan algebra structure as above if and only if there are additional operators E± in ${\mathfrak {g}}_{\pm 1}$ so that H, E± give a copy of ${\mathfrak {sl}}_{2}$. The corresponding Weyl group element implements the involution σ. This case corresponds to that of Euclidean Jordan algebras.
The remaining cases are constructed uniformly by Koecher using involutions of simple Euclidean Jordan algebras.[35] Let E be a simple Euclidean Jordan algebra and τ a Jordan algebra automorphism of E of period 2. Thus E = E+1 ⊕ E−1 has an eigenspace decomposition for τ with E+1 a Jordan subalgebra and E−1 a module. Moreover, a product of two elements in E−1 lies in E+1. For a, b, c in E−1, set
$\displaystyle {L_{0}(a,b)c=L(a,b)c}$
and (a,b)= Tr L(ab). Then F = E−1 is a simple Euclidean Jordan triple system, obtained by restricting the triple system on E to F. Koecher exhibits explicit involutions of simple Euclidean Jordan algebras directly (see below). These Jordan triple systems correspond to irreducible Hermitian symmetric spaces given by Siegel domains of the second kind. In Cartan's listing, their compact duals are SU(p + q)/S(U(p) × U(q)) with p ≠ q (AIII), SO(2n)/U(n) with n odd (DIII) and E6/SO(10) × U(1) (EIII).
Examples
• F is the space of p by q matrices over R with p ≠ q. In this case L(a,b)c= abtc + cbta with inner product (a,b) = Tr abt. This is Koecher's construction for the involution on E = Hp + q(R) given by conjugating by the diagonal matrix with p digonal entries equal to 1 and q to −1.
• F is the space of real skew-symmetric m by m matrices. In this case L(a,b)c = abc + cba with inner product (a,b) = −Tr ab. After removing a factor of √(-1), this is Koecher's construction applied to complex conjugation on E = Hn(C).
• F is the direct sum of two copies of the Cayley numbers, regarded as 1 by 2 matrices. This triple system is obtained by Koecher's construction for the canonical involution defined by any minimal idempotent in E = H3(O).
The classification of Euclidean Jordan triple systems has been achieved by generalizing the methods of Jordan, von Neumann and Wigner, but the proofs are more involved.[36] Prior differential geometric methods of Kobayashi & Nagano (1964), invoking a 3-graded Lie algebra, and of Loos (1971), Loos (1985) lead to a more rapid classification.
Notes
1. This article uses as its main sources Jordan, von Neumann & Wigner (1934), Koecher (1999) and Faraut & Koranyi (1994), adopting the terminology and some simplifications from the latter.
2. Faraut & Koranyi 1994, pp. 2–4
3. For a proof of equivalence see:
• Koecher 1999, p. 118, Theorem 12
• Faraut & Koranyi 1994, pp. 42, 153–154
4. See:
• Freudenthal 1985
• Postnikov 1986
• Faraut & Koranyi 1994
• Springer & Veldkamp 2000
5. See:
• Freudenthal 1985
• Jacobson 1968
• Zhevlakov et al. 1982
• Hanche-Olsen & Størmer 1984
• Springer & Veldkamp 2000, pp. 117–141
6. See:
• Hanche-Olsen & Størmer 1984, pp. 58–59
• Faraut & Koranyi 1994, pp. 74–75
• Jacobson 1968
• Clerc 1992, pp. 49–52
7. Clerc 1992, pp. 49–52
8. Faraut & Koranyi 1994, pp. 46–49
9. Faraut & Koranyi 1994, pp. 32–35
10. See:
• Koecher 1999, pp. 72–76
• Faraut & Koranyi 1994, pp. 32–34
11. See:
• Jacobson 1968, pp. 40–47, 52
• Hanche-Olsen & Størmer 1984, pp. 36–44
12. See:
• Koecher 1999, p. 111
• Hanche-Olsen & Størmer 1984, p. 83
• Faraut & Koranyi 1994, p. 48
13. Faraut & Koranyi 1994, pp. 49–50
14. Faraut & Koranyi 1994, pp. 145–146
15. Loos 1977, p. 3.15-3.16
16. Wright 1977, pp. 296–297
17. See Faraut & Koranyi (1994, pp. 73, 202–203) and Rudin (1973, pp. 270–273). By finite-dimensionality, every point in the convex span of S is the convex combination of n + 1 points, where n = 2 dim E. So the convex span of S is already compact and equals the closed unit ball.
18. Wright 1977, pp. 296–297
19. Faraut & Koranyi 1994, pp. 154–158
20. See:
• Koecher 1999
• Koecher 1969
21. See:
• Loos 1977
• Faraut & Koranyi 1994
22. Lang 1985, pp. 209–210
23. Bourbaki 1981, pp. 30–32
24. See:
• Koecher 1999
• Faraut & Koranyi 1994, pp. 150–153
25. Loos 1977, pp. 9.4–9.5
26. Folland 1989, pp. 203–204
27. See:
• Koecher 1999
• Faraut & Koranyi 1994, pp. 200–201
28. Faraut & Koranyi 1994, pp. 204–205
29. Faraut & Koranyi 1994, p. 208
30. Note that the elementary argument in Igusa (1972, p. 23) cited in Folland (1989) is incomplete.
31. Faraut & Koranyi 1994, p. 208
32. Faraut & Koranyi 1994, p. 334
33. See:
• Cartan 1935
• Helgason 1978
• Kobayashi & Nomizu 1963
• Faraut & Koranyi 1994
34. See:
• Koecher 1967
• Koecher 1968
• Koecher 1969
• Faraut & Koranyi 1994, pp. 218–219
35. Koecher 1969, p. 85
36. See:
• Loos 1977
• Neher 1979
• Neher 1980
• Neher 1981
• Neher 1987
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| Wikipedia |
\begin{document}
\title{Scanning integer points with lex-inequalities: A finite cutting plane algorithm for integer programming with linear objective}
\begin{abstract} We consider the integer points in a unimodular cone $K$ ordered by a lexicographic rule defined by a lattice basis. To each integer point $x$ in $K$ we associate a family of inequalities (lex-inequalities) that defines the convex hull of the integer points in $K$ that are not lexicographically smaller than $x$. The family of lex-inequalities contains the Chv\'atal--Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program $\min \{cx: x\in S\cap \mathbb{Z}^n\}$, where $S\subset \mathbb{R}^n$ is a compact set and $c\in \mathbb{Z}^n$. We analyze the number of iterations of our algorithm. \end{abstract}
\section{Introduction} The area of integer nonlinear programming is rich in applications but quite challenging from a computational point of view. We refer to the articles~\cite{belotti2013mixed,burer-letchford} for comprehensive surveys on these topics. The tools that are mainly used are sophisticated techniques that exploit relaxations, constraint enforcement (e.g.,~cutting planes) and convexification of the feasible set. Reformulations in an extended space and cutting planes for integer nonlinear programs have been investigated and proposed for some time, see e.g.~\cite{ceria1999convex,frangioni2006,stubbs:1999}. This line of research mostly provides a nontrivial extension of the theory of disjunctive programming to the nonlinear case. To the best of our knowledge, these results are obtained under some restrictive conditions: typically, the feasible set is assumed to be convex or to contain 0/1 points only (these cases cover some important areas of application).
In this paper we present a finite cutting plane algorithm for problems of the form \begin{equation}\label{eq:problem} \min \{cx: x\in S\cap \mathbb{Z}^n\}, \end{equation} where $S$ is a compact subset of $\mathbb{R}^n$ (not necessarily convex or connected) and $c\in \mathbb{Z}^n$. Our algorithm uses a new family of cutting planes which
do not make any use of a description of the set $S$. The cutting planes employed in our algorithm are obtained as follows. We consider the integer points in a unimodular cone $K$, ordered by a lexicographic rule, associated with a lattice basis. To each integer point $x$ in $K$, we associate a family of inequalities (lex-inequalities) that, in a sense, is best possible, as it defines the convex hull of the integer points in $K$ that are not lexicographically smaller than $x$. Our family of cuts includes the Chv\'atal--Gomory cuts, but it does not contain, nor is it contained in, the family of split cuts.
Our algorithm recursively solves optimization problems of the form $\min\{cx:x\in S\cap P\}$, where $P$ is a polyhedron, and we assume that an algorithm for problems of this type is available as a black box. We remark that as long as this black box is available, no assumption on $S$ other than compactness is required by our algorithm. To the best of our knowledge, this is in contrast to the rest of the literature where convexity (or even polyhedrality) is a common assumption.
The cuts we introduce are {\em linear} inequalites. As the convex hull of the integer points in a bounded subset of $\mathbb{R}^n$ is a polytope, a finite number of linear inequalities suffices for its characterization, and only $n$ such inequalities determine an optimal point.
Furthermore a finite number of linear inequalities suffices to describe some relevant relaxations of the convex hull of the integer points in a bounded set: most notably, Dadush, Dey and Vielma~\cite{dadush:2014} proved that the Chv\'atal--Gomory closure of a compact convex set is a polytope (whereas this is not the case for the split closure of the set).
However, nonlinear inequalities have also been successfully used to provide elegant convex hull characterizations. We mention the work by Andersen and Jensen~\cite{andersen2013intersection} where a formula to describe the convex hull of a split disjunction applied to a second-order cone is provided. Their work is related to the paper by Modaresi et al.\ \cite{modaresi2016intersection}, where the authors derive split cuts for convex sets described by a single conic quadratic inequality and extend general intersection cuts to a wide variety of quadratic sets. Belotti et al.~\cite{belotti2013families,belotti:2017} introduce the so called disjunctive conic cuts studying families of quadratic surfaces intersected with two given hyperplanes. Burer and K{\i}l{\i}n\c{c}-Karzan~\cite{burer:2017} extend the works cited above and show that the convex hull of the intersection of a second-order-cone representable set and a single homogeneous quadratic inequality can be described by adding a single nonlinear inequality, defining an additional second-order-cone representable set.
From an algorithmic perspective, deriving a finite cutting plane procedure that uses a well defined family of inequalities does not seem to be straightforward. The oldest and most notable example is Gomory's finite cutting plane algorithm for integer linear programming over bounded sets based on fractional cuts~\cite{gomory:1958,gomory:1963}.
Other finite cutting plane algorithms (again for bounded sets) can be found in~\cite{armstrong1979page,bell1973cutting,bowman1970finiteness,he2017another} for integer linear programming and in~\cite{Lee2017} for mixed integer linear programming.
While in all the papers cited above the correctness of the algorithms is based on a specific procedure for solving the continuous relaxation, there are methods that only assume that an optimal solution of the continuous relaxation is given by a black box. This is the case for the lift-and-project method of Balas, Ceria and Cornu\'ejols~\cite{balas1993lift} for mixed 0/1 linear problems, the procedure described by Orlin~\cite{orlin1985finitely} for 0/1 integer linear programming, and the algorithm presented by Neto~\cite{neto2012simple} for integer linear programming over bounded sets.
The family of cuts used in Neto's algorithm is related to ours. As it will be clarified later, the inequalities introduced in~\cite{neto2012simple} are weaker than the lex-inequalities and, in particular, they are derived under the assumption that a box containing the set $S$ is known.
We notice that a common feature of the above papers is the (explicit or implicit) use of some lexicographic rule for the choice of an optimal solution of the continuous relaxation or the selection of the cut. This seems to be a key tool to prove finite convergence of this type of algorithms.
The paper is organized as follows. In Section~\ref{sec:mainres}, we introduce the lex-inequalities with their properties. In Section~\ref{sec:ip}, we present the cutting plane algorithm and we show that it terminates in a finite number of iterations. In Section~\ref{sec:exp}, an instance where the algorithm stops after an exponential number of iterations is provided. Furthermore, we compare the performance of our algorithm with a natural enumeration approach. A comparison with Chv\'atal--Gomory cuts and split cuts is presented in Section~\ref{sec:comp}. Section~\ref{sec:conc} concludes the paper.
\section{Lexicographic orderings and lex-inequalities}\label{sec:mainres}
A {\em lattice basis} of $\mathbb{Z}^n$ is a set of $n$ linearly independent vectors $c^1,\dots,c^n\in\mathbb{Z}^n$ such that for every $v\in\mathbb{Z}^n$ we have that $\lambda_1,\dots,\lambda_n\in\mathbb{Z}$ in the unique expression $v=\sum_{i=1}^n\lambda_ic^i$.
The lex-inequalities that we introduce in this paper are defined for a given lattice basis of $\mathbb{Z}^n$. To simplify the presentation, we first work with the standard basis and then extend the results to general lattice bases.
We will use standard notions in the theory of polyhedra, for which we refer the reader to \cite{SchrijverBOOK}.
\subsection{Standard basis}
We consider the {\em lexicographic ordering $\prec$} associated with the standard basis $e^1,\dots,e^n$: given $x^1,\,x^2\in \mathbb{R}^n$, $x^1\prec x^2$ if and only if $x^1\ne x^2$ and $x_i^1< x_i^2$, where $i$ is the smallest index for which $x_i^1\ne x_i^2$. We use $\preceq$, $\succ$, $\succeq$ with the obvious meaning.
We consider the cone $K=\mathbb{R}^n_+=\{x\in \mathbb{R}^n:x_i\ge 0,\,i=1,\dots, n\}$.
Given $\bar{x}\in K\cap \mathbb{Z}^n$, we define \[Q(\bar{x}):=\operatorname{conv}\{x\in K\cap \mathbb{Z}^n: x\succeq \bar{x}\},\] where ``conv'' denotes the convex hull operator.
Given $\bar x\in K\setminus\{\mathbf0\}$, we define the {\em leading index} $\ell(\bar x)$ as the largest index $i$ such that $\bar{x}_i>0$.
\begin{lemma}\label{lem:extreme-pts} Fix $\bar x\in K\cap\mathbb{Z}^n$. The set $Q(\bar x)$ is a full-dimensional polyhedron. Its vertices are precisely the following points $v^1,\dots,v^{\ell(\bar x)}$:
for $k=1,\dots,\ell(\bar x)-1$, $v^k$ has entries \begin{alignat*}4 v_i^k &= \bar{x}_i,&\quad&i=1,\dots,k-1\\ v_k^k &= \bar{x}_k+1\\ v_i^k &=0, &&i=k+1,\dots,n, \end{alignat*} and $v^{\ell(\bar x)}=\bar x$. Furthermore, the recession cone of $Q(\bar x)$ is $K$. \end{lemma}
\begin{proof} Define $X:=\{x\in K\cap \mathbb{Z}^n:x\succeq\bar x\}$ and $X_i=v^i+(K\cap\mathbb{Z}^n)$ for every $i\in\{1,\dots,\ell(\bar x)\}$. Note that $X=\bigcup_{i=1}^{\ell(\bar x)}X_i$ and therefore $\operatorname{conv}(X) =\operatorname{conv}\left(\bigcup_{i=1}^{\ell(\bar x)}\operatorname{conv}(X_i)\right)$. As any rational polyhedral cone is an integral polyhedron, we have $\operatorname{conv}(K\cap\mathbb{Z}^n)=K$. As $v^i\in\mathbb{Z}^n$, this implies $\operatorname{conv}(X_i) = v^i +K$ for every $i$, and thus these sets are integer translates of $K$. Therefore $\operatorname{conv}(X)=\operatorname{conv}\{v^1,\dots,v^{\ell(\bar x)}\}+K$. This shows that $\operatorname{conv}(X)$ is a full-dimensional polyhedron with recession cone $K$ and its vertices are contained in $\{v^1,\dots,v^{\ell(\bar x)}\}$. It is easy to verify that $v^1,\dots,v^{\ell(\bar x)}$ are actually all vertices of $\operatorname{conv}(X)$. \end{proof}
Let $\bar {x}\in K$ be given. For every $k\in\{1,\dots,n\}$ and $i\in\{1,\dots,k\}$ we define \[d^k_i:=\begin{cases} 1 & \mbox{if $i=k$}\\ \bar{x}_k & \mbox{if $i=k-1$},\\ \bar{x}_k\prod_{j=i+1}^{k-1}(\bar{x}_j+1),&\mbox{if } i\le k-2. \end{cases}\] (Note that the $d^k_i$'s depend on the choice of $\bar x$, but we omit the dependence on $\bar x$ to keep notation simpler: this will never generate any ambiguity.)
For every $k\in\{1,\dots,n\}$, the {\em $k$-th lex-inequality} associated with $\bar {x}$ is the inequality \begin{equation}\label{eq:lex-ineq} \sum_{i=1}^{k}d^k_i x_i\ge \sum_{i=1}^kd^k_i\bar{x}_i. \end{equation} Note that when $\bar x_k=0$, \eqref{eq:lex-ineq} is the inequality $x_k\ge 0$.
\begin{theorem}
\label{thm:convex-hull} If $\bar x\in K\cap\mathbb{Z}^n$, then the lex-inequalities \eqref{eq:lex-ineq} for $k=1,\dots,n$
and the inequalities $x_i\ge0$ for $i=1,\dots,n$ provide a description of the polyhedron $Q(\bar x)$. \end{theorem}
\begin{proof} As $K$ is the recession cone of $Q(\bar{x})$ (Lemma \ref{lem:extreme-pts}) and $Q(\bar x)\subseteq K$, it follows that every facet inducing inequality for $Q(\bar{x})$ (indeed every valid inequality) is of the type \begin{equation}\label{eq:facet}
\sum_{i=1}^{n} a_ix_i\ge a_0 \end{equation} where $a_i\ge 0, i=0,\dots,n$.
Given $k\in\{1,\dots,n\}$, we let $Q_k(\bar{x})\subseteq \mathbb{R}^k$ denote the orthogonal projection of $Q(\bar{x})$ onto the first $k$ variables, and we define $\bar{x}_{[k]}:=(\bar{x}_1, \dots, \bar{x}_k)$. It follows from the definition of lexicographic ordering that $Q_k(\bar{x})=Q(\bar{x}_{[k]})$.
Therefore the facet inducing inequalities of $Q_k(\bar{x})$ are the facet inducing inequalities of $Q(\bar{x})$ such that $a_j=0$ for $j=k+1,\dots, n$. (This can be seen, e.g., as a consequence of the method of Fourier--Motzkin to compute projections.)
As the theorem trivially holds for $Q_1(\bar{x})$, to prove the result by induction on $n$ it suffices to characterize the facets with $a_n>0$. As the only facet inducing inequality with $a_n>0$ and $a_0=0$ is $x_n\ge 0$, from now on we consider a facet inducing inequality \eqref{eq:facet} with $a_n>0$ and $a_0>0$.
Assume first that $\bar{x}_n=0$. Then by Lemma \ref{lem:extreme-pts} we have that $Q(\bar{x})=Q_{n-1}(\bar{x})\times \{x_n\in \mathbb{R}:x_n\ge 0\}$ and we are done by induction. Therefore we assume $\bar{x}_n>0$. Recall that, by Lemma \ref{lem:extreme-pts}, $Q(\bar{x})$ has $n$ vertices, $v^1,\dots, v^n=\bar{x}$.
\noindent {\bf Claim 1:} {\em $\bar{x}$ satisfies \eqref{eq:facet} at equality.}
\noindent Since $v^k_n=0$ for $k=1,\dots,n-1$, if $\bar{x}$ does not satisfy \eqref{eq:facet} at equality, the inequality \begin{equation*} \sum_{i=1}^{n-1} a_ix_i+(a_n-\varepsilon)x_n\ge a_0 \end{equation*} is valid for $Q(\bar{x})$ for some $\varepsilon >0$. Since \eqref{eq:facet} is the sum of $\varepsilon x_n\ge 0$ and the above inequality, and these inequalities are not multiples of each other as $a_0>0$, \eqref{eq:facet} does not induce a facet of $Q(\bar{x})$. This proves Claim 1.
\noindent{\bf Claim 2:} {\em $a_k>0$ for $k=1,\dots,n$.}
\noindent By Claim 1 we have that
\begin{equation*}
\sum_{i=1}^{n} a_i\bar{x}_i= a_0. \end{equation*} Pick $k\in\{1,\dots,n-1\}$. Since $v^k$ satisfies \eqref{eq:facet}, we have that
\begin{equation*}
\sum_{i=1}^{k} a_i\bar{x}_i+a_k\ge a_0. \end{equation*} Subtracting the above equation from this inequality, we obtain \begin{equation*}
a_k\ge \sum_{i=k+1}^{n} a_i\bar{x}_i>0, \end{equation*} where the strict inequality follows because $a_i\ge0$ for $i=1,\dots,n$ and $a_n\bar{x}_n>0$. This proves Claim 2.
Claim 2 shows that if $x''\ne x'$, $x''\ge x'$ (componentwise) and $x'$ satisfies \eqref{eq:facet}, then $x''$ cannot satisfy \eqref{eq:facet} at equality. In particular, if $x'$ satisfies \eqref{eq:facet} at equality and $r$ is a nonzero ray of $Q(\bar x)$ then $x'+r$ cannot satisfy \eqref{eq:facet} at equality. Therefore, as $Q(\bar{x})$ is a full dimensional polyhedron and \eqref{eq:facet} induces a facet, this inequality must be satisfied at equality by $v^1,\dots, v^n$. By imposing these $n$ equations, it can be derived that, up to positive scaling, $a_i = d_i^n$, for all $i=1,\ldots,n$ and $a_0 = \sum_{i=1}^nd^n_i\bar{x}_i$. This implies that \eqref{eq:facet} is \[\sum_{i=1}^{n}d^n_ix_i\ge \sum_{i=1}^nd^n_i\bar{x}_i\] and the theorem is proven. \end{proof}
\begin{remark} In the description given by Theorem \ref{thm:convex-hull}, for every $k$ such that $\bar x_k=0$ the $k$-th lex-inequality is redundant, as it is the inequality $x_k\ge0$. Furthermore, if $\bar x_1>0$ then also the inequality $x_1\ge0$ is redundant, as it is dominated by the first lex-inequality (which is $x_1\ge \bar x_1$). It can be verified that the remaining inequalities provide an irredundant description of $Q(\bar x)$. \end{remark}
\subsection{General lattice bases}
Let $\{c^1,\dots,c^n\}$ be a lattice basis of $\mathbb{Z}^n$. Then the $n\times n$ matrix $C$ whose rows are $c^1,\dots,c^n$ is unimodular, i.e., it is an integer matrix with determinant 1 or $-1$. The unimodular transformation $x\mapsto Cx$ and its inverse map integer points to integer points. By applying the transformation $x\mapsto Cx$, the results of the previous subsection can be immediately extended to the lattice basis $\{c^1,\dots,c^n\}$.
In particular, the lexicographic ordering defined by the lattice basis is as follows: given $x^1,\,x^2\in \mathbb{R}^n$, we have $x^1\prec x^2$ if and only if $x_1\ne x_2$ and $c^ix^1< c^ix^2$, where $i$ is the smallest index for which $c^ix^1\ne c^ix^2$.
The unimodular cone $K$ is defined as $K:=\{x\in\mathbb{R}^n:c^ix\ge0,\,i=1,\dots,n\}$ and, for $\bar x\in K\cap\mathbb{Z}^n$, $Q(\bar x):=\operatorname{conv}\{x\in K\cap\mathbb{Z}^n:x\succeq\bar x\}$.
The leading index $\ell(\bar x)$, for $\bar x\in K\setminus\{\mathbf0\}$, is the largest index $i$ such that $c^i\bar{x}>0$. Lemma \ref{lem:extreme-pts} now reads as follows:
\begin{lemma}\label{lem:extreme-pts-gen} Fix $\bar x\in K\cap\mathbb{Z}^n$. The convex set $Q(\bar x)$ is a full-dimensional polyhedron. Its vertices are precisely the following points $v^1,\dots,v^{\ell(\bar x)}$:
for $k=1,\dots,\ell(\bar x)-1$, $v^k$ is the unique point satisfying \begin{alignat*}4 c^iv^k &= c^i\bar{x},&\quad&i=1,\dots,k-1\\ c^kv^k &= c^k\bar{x}+1\\ c^iv^k &=0, &&i=k+1,\dots,n, \end{alignat*} and $v^{\ell(\bar x)}=\bar x$. Furthermore, the recession cone of $Q(\bar x)$ is $K$. \end{lemma}
For $\bar {x}\in K$, $k\in\{1,\dots,n\}$ and $i\in\{1,\dots,k\}$, the definition of the $d^k_i$'s is as follows: \begin{equation}\label{eq:d} d^k_i:=\begin{cases} 1 & \mbox{if $i=k$}\\ c^k\bar{x} & \mbox{if $i=k-1$},\\ c^k\bar{x}\prod_{j=i+1}^{k-1}(c^j\bar{x}+1),&\mbox{if $i\le k-2$}. \end{cases} \end{equation} The {\em $k$-th lex-inequality} associated with $\bar {x}$ is the following: \begin{equation}\label{eq:lex-cut-gen} \sum_{i=1}^{k}d^k_i c^ix\ge \sum_{i=1}^kd^k_ic^i\bar{x}. \end{equation}
Theorem \ref{thm:convex-hull} now reads as follows:
\begin{proposition}\label{prop:valIn} If $\bar x\in K\cap\mathbb{Z}^n$, then the lex-inequalities \eqref{eq:lex-cut-gen} for $k=1,\dots,n$ and the inequalities $c^ix\ge0$ for $i=1,\dots,n$ provide a description of the polyhedron $Q(\bar x)$. \end{proposition}
Neto \cite{neto2012simple} describes a family of inequalities that, although presented in a different setting, can be seen to be valid for $Q(\bar x)$ when the lattice basis $\{c^1,\dots,c^n\}$ is the standard basis. However, those inequalities in general do not induce facets of $Q(\bar x)$ and are therefore weaker than the lex-inequalities. In particular, the inequalities in \cite{neto2012simple} are derived under the assumption that a box containing the continuous set $S$ is known, and their coefficients depend on the size of the box. In contrast, our inequalities only depend on the current fractional solution $\bar x$. As a consequence, we obtain inequalities with smaller dynamism (i.e., with smaller ratio between the largest and the smallest absolute value of the coefficients), which is a desirable property in practice.
In order to compare Neto's inequalities with ours, let $n=2$, $\bar x=(1,1)$, and consider the box $[0,3]\times[0,3]$. Neto's inequalities are in this case $x_1\ge 1$ and $3x_1+x_2\ge 4$,\footnote{We note that Neto presents his inequalities in a different form, as he considers the integer points that are lexicographically {\em smaller} than $\bar x$.} while the lex-inequalities are $x_1\ge 1$ and $x_1+x_2\ge 2$. Since the inequality $3x_1+x_2\ge 4$ is a proper conic combination of the two lex-inequalities, it cannot be facet inducing for $Q(\bar x)$.
It should also be noted that in \cite{neto2012simple} the inequalities are described only for the case in which the continuous set $S$ is a bounded polyhedron, although it is not difficult to extend them to the case of a compact set.
Furthermore, we remark that a linear-inequality description of the set $Q(\bar x)$ can be inferred from a result proved by Gupte \cite[Theorem 2]{gupte2016} in the context of super-increasing knapsack problems: one needs to apply a change of variables and observe that by removing the lower bounds appearing in \cite{gupte2016} the remaining facet inducing inequalities are unaffected.
\section{The cutting plane algorithm}\label{sec:ip}
Let $\mathcal S$ be a family of compact (not necessarily connected or convex) subsets of $\mathbb{R}^n$ with the following property:
{\centerline{\em if $S\in\mathcal S$ and $H$ is a closed halfspace in $\mathbb{R}^n$, then $S\cap H\in\mathcal S$.}}
{\em Linear optimization} over $\mathcal S$ is the following problem:
given $S\in\mathcal S$ and $c\in\mathbb{Z}^n$, determine an optimal solution to the problem $\min\{cx:x\in S\}$ or certify that $S=\emptyset$. (Since $S$ is compact, either $S=\emptyset$ or the minimum is well defined.)
{\em Integer linear optimization} over $\mathcal S$ is defined similarly, but the feasible region is $S\cap \mathbb{Z}^n$, the set of integer points in $S$.
We prove that an oracle for solving linear optimization over $\mathcal S$ suffices to design a finite cutting plane algorithm that solves integer linear optimization over $\mathcal S$.
We now make this statement more precise. Given a compact subset $S$ of $\mathbb{R}^n$ and $c\in \mathbb{Z}^n$, let $ \bar x\in S$ be an optimal solution of the program $\min\{cx:x\in S\}$. A {\em cutting plane} is a linear inequality that is valid for $S\cap \mathbb{Z}^n$ and is violated by $\bar x$. Note that a cutting plane exists if and only if $\bar x\notin\operatorname{conv}(S\cap\mathbb{Z}^n)$. In particular, this is certainly the case if $\bar x$ is a non-integral extreme point of $\operatorname{conv}(S)$.
A (pure) {\em cutting plane algorithm} for integer linear optimization over $\mathcal S$ is an iterative procedure of the following type: \begin{itemize} \item[-]Let $S\in\mathcal S$ and $c\in\mathbb{Z}^n$ be given. \item[-]If $S=\emptyset$, then $S\cap \mathbb{Z}^n=\emptyset$. Otherwise, find an optimal solution $\bar x$ of $\min\{cx:x\in S\}$. \item[-]If $\bar x\in S\cap \mathbb{Z}^n$, stop: $\bar x$ is an optimal solution to $\min\{cx:x\in S\cap\mathbb{Z}^n\}$. Otherwise, detect a cutting plane and let $H$ denote the corresponding half-space. Replace $S$ with $S\cap H$ and iterate. \end{itemize}
Assume without loss of generality that the objective function vector $c$ is nonzero and has relatively prime entries. Then there exists a lattice basis $\{c^1,\dots,c^n\}$ of $\mathbb{Z}^n$ such that $c^1=c$. The optimal solution $\bar x$ of $\min\{cx:x\in S\}$ found by our algorithm will be a {\em lexicographically minimum} or {\em lex-min} solution in $S$ with respect to the lattice basis: i.e., $\bar{x}\prec x$ for every $x\in S\setminus \{\bar{x}\}$. The lex-min vector $\bar{x}$ in $S$ satisfies the following conditions: \begin{itemize} \item $c^1\bar x=\min\{c^1x:x\in S\}$; \item $c^2\bar x=\min\{c^2x:x\in S,\, c^1x=c^1\bar {x}\}$; \item $c^3\bar x=\min\{c^3x:x\in S,\, c^1x=c^1\bar {x},\, c^2x=c^2\bar {x}\}$; \item \dots \item $c^n\bar x=\min\{c^nx:x\in S,\, c^1x=c^1\bar {x},\dots,\, c^{n-1}x=c^{n-1}\bar {x}\}$. \end{itemize} Since $S$ is nonempty and compact, the above minima are well-defined and can be computed by applying the oracle $n$ times. Furthermore these conditions uniquely define $\bar x$. One verifies that $\bar x$ is an extreme point of $\operatorname{conv}(S)$.
\begin{algorithm}[!ht] \label{alg:weak} \caption{Resolution of integer linear optimization over $\mathcal S$} \SetKwInput{Input}{Input} \SetKwInput{Output}{Output}
\Input{$S\in\mathcal S$ with $S\ne\emptyset$, $c\in\mathbb{Z}^n\setminus\{\mathbf0\}$ with relatively prime entries, and a lattice basis $\{c^1,\dots,c^n\}$ of $\mathbb{Z}^n$ with $c^1=c$.}
\Output{an optimal integer solution $\bar x$ for the problem $\min\{cx:x\in S\}$ or a certificate that $S\cap \mathbb{Z}^n=\emptyset$.}
Compute $\ell^*_i:=\min\{c^ix:x\in S\}$ and $\ell_i:=\ceil {\ell^*_i}$ for $1\le i\le n$, and apply a translation so that $\ell_i=0$ for $1\le i \le n$.
Let $K:=\{x\in \mathbb{R}^n:c^ix\ge 0,\,i=1,\dots, n\}$ and replace $S$ with $S\cap K$.\label{step:preprocessing}\\
If $S=\emptyset$, stop: the given problem is infeasible.\label{step:empty}\\
Else, compute the lex-min solution $\bar x$ in $S$ with respect to $\{c^1,\dots,c^n\}$.\label{step:lex-min}\\
If $\bar x\in\mathbb{Z}^n$, return $\bar x$.\\
Else, let $k$ be the smallest index such that $c^k\bar x\notin\mathbb{Z}$\label{step:k} and compute
\[d^k_i:=\begin{cases} 1 & \mbox{if $i=k$}\\ \ceil{c^k\bar{x}} & \mbox{if $i=k-1$},\\ \ceil{c^k\bar{x}}\prod_{j=i+1}^{k-1}(c^j\bar{x}+1),&\mbox{if $i\le k-2$}. \end{cases} \]
Replace $S$ with $S\cap H$, where $H$ is the halfspace defined by the inequality \eqref{eq:cut}
\begin{equation*}
\sum_{i=1}^{k} d_i^kc^ix\ge \sum_{i=1}^{k-1} d_i^kc^i\bar x + d_k^k\ceil{c^k\bar x}
\end{equation*}
and go to step \ref{step:empty}. \label{step:cut} \end{algorithm}
Algorithm \ref{alg:weak} describes the procedure in detail.
Note that since $S$ is compact, numbers $\ell^*_1,\dots,\ell^*_n$ (as defined in Algorithm \ref{alg:weak}) exist and can be determined by querying the linear optimization oracle $n$ times. Moreover, as $\{c^1,\dots,c^n\}$ is a lattice basis of $\mathbb{Z}^n$, an index $k$ as in step \ref{step:k} always exists when $\bar x\notin\mathbb{Z}^n$.
Given $x\in K$, let $x^{\uparrow}$ be the lex-min vector in $K\cap \mathbb{Z}^n$ such that $x\preceq x^{\uparrow}$. Obviously $x=x^{\uparrow}$ if and only if $x\in \mathbb{Z}^n$. If $x\not\in \mathbb{Z}^n$, let $k$ be the smallest index such that $c^k x\not\in\mathbb{Z}$. It is easy to see that $x^{\uparrow}$ is the unique point satisfying the following conditions:
\begin{equation}\label{eq:uparrow}
c^ix^{\uparrow}=c^i x,\,i<k;\quad c^kx^{\uparrow}=\ceil{c^k x};\quad
c^ix^{\uparrow}=0,\,i>k.
\end{equation}
\begin{definition}\label{def:lexcut}
Let $\bar x\not\in S\cap \mathbb{Z}^n$ and let $k$ be the smallest index such that $c^k \bar x\not\in\mathbb{Z}$.
The \emph{$k$-th lex-cut} is the $k$-th
lex-inequality associated with $\bar x^{\uparrow}$:
\begin{equation}\label{eq:cut}
\sum_{i=1}^{k} d_i^kc^ix\ge \sum_{i=1}^{k-1} d_i^kc^i\bar x + d_k^k\ceil{c^k\bar x}
\end{equation}
(This is the cut introduced at step \ref{step:cut} of Algorithm \ref{alg:weak}.)
\end{definition}
\begin{proposition}\label{prop:cut} Inequality \eqref{eq:cut} defines a cutting plane. Algorithm \ref{alg:weak} terminates after a finite number of iterations. \end{proposition}
\begin{proof}
Since, after the preprocessing of step \ref{step:preprocessing}, $S\subseteq K$ and $\bar{x}$ is the lex-min point in $S$, $\bar{x}\preceq \bar{x}^{\uparrow}\prec x'$ for every $x'\in S\cap \mathbb{Z}^n\setminus\{\bar{x}^{\uparrow}\}$. Thus $S\cap \mathbb{Z}^n\subseteq Q(\bar{x}^{\uparrow})$ and by Proposition \ref{prop:valIn} inequality \eqref{eq:cut} is valid for $S\cap \mathbb{Z}^n$. As $c^k\bar{x}\not\in \mathbb{Z}$ and $d^k_k>0$, the inequality is violated by $\bar{x}$. This shows that \eqref{eq:cut} defines a cutting plane.
As different iterations of the algorithm use cuts \eqref{eq:cut} associated with lexicographically increasing vectors
in $S\cap \mathbb{Z}^n$, and $S$ is bounded, the number of iterations of the algorithm is finite.
\end{proof}
\iffalse \begin{rem}\label{rem:cuts} Inequality \eqref{eq:cut} is of the form $\sum_{i=1}^{k} d_ic^ix\ge \delta$ for some $k\in\{1,\dots, n\}$ and $\delta\in\mathbb{R}$. Therefore, Algorithm \ref{alg:weak} produces cuts with only $n$ predetermined normal vectors. In particular, if $c^1,\dots,c^n$ are nonnegative vectors, as e.g.\ in the standard basis, all cuts have nonnegative coefficients. \end{rem}
\begin{rem} We remark that the above algorithm requires an oracle that solves the continuous problem {\em exactly} and not just with arbitrary accuracy. (This is in theory the case for {\em any} cutting plane algorithm). However, from the above proofs we can observe that the assumption that the oracle finds an optimal solution for the continuous problem can be weakened. Indeed, it is sufficient that for any $c\in\mathbb{Z}^n$ the oracle finds a feasible solution $\bar x$ such that $\ceil{c\bar x}=\ceil{z^*}$, where $z^*$ is the optimal value of the continuous problem. However, when $z^*$ is integer this means that the oracle has to find an optimal solution. \end{rem} \fi
We mention that Akshay Gupte (personal communication) has elaborated an algorithm to solve $\min\{cx : x\in S\}$, assuming that a box $B$ containing $S$ is given. His algorithm iteratively constructs the convex hull of a set of the form $\{x\in B\cap\mathbb{Z}^n : x\succeq \hat x\}$ for some $\hat x\in \mathbb{Z}^n$, which can be seen as a truncated version of $Q(\hat x)$. However, while in Algorithm \ref{alg:weak} at each iteration we use $\hat x=\bar x^\uparrow$, where $\bar x$ is the optimal solution of the continuous relaxation, in Gupte's algorithm $\hat x$ is obtained by ``rounding'' the point optimizing an objective function with superincreasing coefficients that is different from the original objective function. As a consequence, in Gupte's algorithm one can have $\hat x \prec \bar x^\uparrow$, which makes $Q(\hat x)$ (or its truncated version) weaker.
\section{Lexicographic enumeration and the number of iterations}\label{sec:exp}
Recall the notation $x^\uparrow$ introduced in \eqref{eq:uparrow}. We extend that definition to sets as follows: given $S\subseteq \mathbb{R}^n$, let $S^{\uparrow}:=\{x^{\uparrow}:x\in S\}$. Since $S$ is bounded, $S^\uparrow$ is a finite set, as, given $y\in S^\uparrow$ and $i\in\{1,\dots,n\}$, $c^iy$ is an integer value satisfying $\min\{c^ix:x\in S\}\le c^iy\le\ceil{\max\{c^ix:x\in S\}}$.
\begin{obs}\label{obs:performance} Given a nonempty set $S\in \mathcal S$, let $(\bar{x})$ be the sequence of points in $S$ computed at step \ref{step:lex-min} of Algorithm \ref{alg:weak}. Then the sequence $(\bar{x}^{\uparrow})$ is the lex-ordering of some distinct points in $S^{\uparrow}$. \end{obs}
\begin{proof} If $\bar x$ is a point computed at step \ref{step:lex-min} of Algorithm \ref{alg:weak}, then clearly $\bar x^\uparrow\in S^\uparrow$, as $\bar x \in S$. Thus we only have to show that if $\bar x$ and $\tilde x$ are points computed at step \ref{step:lex-min} in two consecutive iterations (say iterations $q$ and $q+1$), then $\bar x^\uparrow\prec\tilde x^\uparrow$. Assume not. Then $\bar x^\uparrow=\tilde x^\uparrow$ and therefore the cuts introduced at these two iterations would be exactly the same. But then the cut generated at iteration $q$ would already cut off $\tilde x$, contradicting the fact that at iteration $q+1$ the point computed at step \ref{step:lex-min} is $\tilde x$. \end{proof}
\begin{corollary}\label{cor:sup}
$|S^{\uparrow}|$ is an upper bound on the number of cuts produced by Algorithm \ref{alg:weak}. \end{corollary}
We next construct a convex body containing no integer points for which the bound $|S^{\uparrow}|$ on the number of cuts is exponential and tight.
\begin{proposition}\label{prop:exp} For every $n\in\mathbb{N}$, there is a convex subset $S$ of $[0,1]^n$ (described by a single convex constraint plus variable bounds) on which Algorithm \ref{alg:weak}
computes $|S^{\uparrow}|=2^n-1$ cuts. \end{proposition}
\begin{proof}
We choose the standard basis $\{e^1,\dots,e^n\}$ as lattice basis of $\mathbb{Z}^n$. Let $\mathbf1$ be the point in $\mathbb{R}^n$ with all entries equal to 1, and let $\|\cdot\|$ denote the Euclidean norm. Define
\[S:=\left\{x\in[0,1]^n:\left\|x-\frac{\mathbf1}2\right\|^2\le \frac n4-\frac3{16}\right\}.\] Note that $S\cap\mathbb{Z}^n=\emptyset$ and $\ell_i=0$ for $i=1,\dots,n$. Furthermore, for every $x\in\{0,1\}^n\setminus\{\mathbf1\}$, $S$ contains the point $z(x)$ obtained from $x$ by setting to $\frac14$ the entry with largest index
that is 0. As $S\subseteq[0,1]^n$, this shows that $S^{\uparrow}=\{0,1\}^n\setminus \{\mathbf0\}$, and thus $|S^\uparrow|=2^n-1$.
We now show that every point in $S^{\uparrow}$ is of the form $\bar x^\uparrow$ for some point $\bar x$ found in step \ref{step:lex-min}. Let $\bar x$ be the point computed at some iteration of step \ref{step:lex-min} and assume $\bar x^\uparrow\ne\mathbf1$. By Theorem \ref{thm:convex-hull}, the lex-cut associated with $\bar x^\uparrow$ is satisfied by all $x\in \{0,1\}^n$ such that $x\succeq \bar x^\uparrow$. As the lex-cut associated with $\bar x^\uparrow$ is an inequality with nonnegative coefficients, it is also satisfied by the point $z(\bar x^\uparrow)$. This implies that, if we denote by $\tilde x$ the point computed in step \ref{step:lex-min} at the next iteration, $\tilde x^\uparrow$ is the lex-min point in $\{0,1\}^n$ that is lexicographically larger than $\bar x^\uparrow$. Thus every point in $S^{\uparrow}$ is of the form $\bar x^\uparrow$ for some point $\bar x$ found in step \ref{step:lex-min}. Together with Observation \ref{obs:performance},
this shows that precisely $|S^\uparrow|$ cuts are needed to discover that $S$ contains no integer points, which happens at step \ref{step:empty} immediately after the iteration in which $\bar x^\uparrow=\mathbf1$. \end{proof}
Corollary \ref{cor:sup} gives a guarantee on the maximum number of iterations of Algorithm \ref{alg:weak}. One may ask whether there exists an enumerative algorithm that achieves the same performance. We propose Algorithm \ref{alg:enum}, which we think is the best candidate.
\begin{algorithm}[ht] \label{alg:enum} \caption{Resolution of integer linear optimization over $\mathcal S$ via lex-enumeration} \SetKwInput{Input}{Input} \SetKwInput{Output}{Output}
\Input{$S\in\mathcal S$, $c\in\mathbb{Z}^n\setminus\{\mathbf0\}$ with relatively prime entries and a lattice basis $\{c^1,\dots,c^n\}$ of $\mathbb{Z}^n$, with $c^1=c$.}
\Output{an optimal integer solution $\bar x$ for the problem $\min\{cx:x\in S\}$ or a certificate that $S\cap \mathbb{Z}^n=\emptyset$.}
Translate $S$ so that $S\subseteq \{x\in \mathbb{R}^n:c^ix\ge 0,\,i=1,\dots, n\}$. Set $\alpha_1:=\dots :=\alpha_n:=0$ and $i^*:=1$.\\ Let $S^*:=S\cap\{x\in \mathbb{R}^n: c^ix=\alpha_i,\,i<i^*;\:c^ix\ge\alpha_i,\,i\ge i^*\}$\label{recursion}.\\
If $S^*=\emptyset$:\\
\qquad If $i^*=1$, stop: $S\cap\mathbb{Z}^n=\emptyset$.\label{step:i^*=1}\\
\qquad Else update {$i^*:= i^*-1$, $\alpha_{i^*}:=\alpha_{i^*}+1$, $\alpha_{i}:=0$ for $i>i^*$, and go to step \ref{recursion}\label{alpha1}}.\\
Else\\
\qquad Let $\bar{x}$ be the lex-min point in $S^*$\label{bar x}.\\
\qquad If $\bar{x}^{\uparrow}\in S^*$, stop: $\bar{x}^{\uparrow}$ is the lex-min point in $S\cap\mathbb{Z}^n$.\\
\qquad Else update $i^*:=n$, $\alpha_{i}:=c^i\bar{x}^{\uparrow}$ for $i=1,\dots, n$, and go to step \ref{recursion}\label{alpha2}. \end{algorithm}
The correctness of this algorithm is based on the following lemma (whose proof also explains how the algorithm works).
\begin{lemma}\label{lem:consecutive} In Algorithm \ref{alg:enum}, if $\bar x$ and $\tilde x$ denote the points computed at two consecutive executions of line \ref{bar x}, then $\tilde x$ is the lex-min point in $S$ that is lexicographically larger than $\bar x^\uparrow$. \end{lemma}
\begin{proof} Let $\bar x$ denote the point found at some execution of step \ref{bar x}. If $\bar x^\uparrow\notin S^*$, then, at steps \ref{alpha2} and \ref{recursion}, $S^*$ is defined as the set of points $x\in S$ satisfying $c^ix=c^i\bar{x}^{\uparrow}$ for all $i\le n-1$ and $c^nx\ge c^n\bar{x}^{\uparrow}$. If $S^*\ne\emptyset$ then the next execution of step \ref{bar x} yields the lex-min point in $S$ that is lexicographically larger than $\bar x^\uparrow$. Otherwise, step \ref{alpha1} is executed, which updates $S^*$ to the set of points $x\in S$ satisfying $c^ix=c^i\bar{x}^{\uparrow}$ for all $i\le n-2$, $c^{n-1}x\ge c^{n-1}\bar{x}^{\uparrow}+1$ and $c^nx\ge0$. Again, if $S^*\ne\emptyset$ then the next execution of step \ref{bar x} yields the lex-min point in $S$ that is lexicographically larger than $\bar x^\uparrow$. Otherwise, this point is found after some further executions of step \ref{alpha1} (unless the condition in step \ref{step:i^*=1} is satisfied). \end{proof}
To analyze the performance of Algorithm \ref{alg:enum}, we need the following definitions. Let $C$ be the $n\times n$ matrix whose rows are $c^1,\dots,c^n$ and let $S\in\mathcal S$ be given. For every $\bar x \in S^\uparrow$, let $V(\bar x)$ be the set of the following $n$ vectors $\alpha^1,\dots,\alpha^n$: for $k=1,\dots,n-1$, $\alpha^k$ is defined as \begin{alignat*}4 \alpha^k_i &= c^i\bar{x},&\quad&i=1,\dots,k-1\\ \alpha^k_k &= c^k\bar{x}+1\\ \alpha^k_i &=0, &&i=k+1,\dots,n, \end{alignat*} and $\alpha^n=C\bar x$. Notice that by Lemma \ref{lem:extreme-pts-gen}, $V(\bar x)$ contains all vectors of the form $Cx$ where $x$ is a vertex of $Q(\bar{x})$.
Let $V(S)=\bigcup_{\bar x\in S^\uparrow}V(\bar x)$. Notice that, given $\bar x,\bar y\in S^\uparrow$, the set $V(\bar x)\cap V(\bar y)$ may be nonempty.
\begin{proposition}\label{prop:bound-alg2} Given a set $S\in \mathcal S$, let $(\alpha)$ be the sequence of vectors used to define the sequence of sets $(S^*)$ in step \ref{recursion} of Algorithm \ref{alg:enum}. \begin{itemize} \item If $S\cap\mathbb{Z}^n=\emptyset$, then $(\alpha)$ is the lex-ordering of all points in $V(S)\cup\{\mathbf0\}$ with respect to the standard basis. \item If $S\cap\mathbb{Z}^n\ne \emptyset$, the sequence is truncated to the lex-min vector $\alpha$ (with respect to the standard basis) such that $C^{-1}\alpha\in S\cap\mathbb{Z}^n=S\cap S^{\uparrow}$. \end{itemize} \end{proposition}
\begin{proof} Clearly the sequence $(\alpha)$ starts with $\alpha=\mathbf0$ and is lexicographically increasing with respect to the standard basis.
Let $\alpha\ne0$ be a vector used in step \ref{recursion} at some iteration $q>1$ and let $\bar x$ be the last point computed at line \ref{bar x} before iteration $q$; say that $\bar x$ is computed at iteration $q'<q$. If $q'=q-1$, then $\alpha=C\bar x^\uparrow$ and therefore $\alpha\in V(S)$. If $q'=q-t$ for some $t>1$, then line \ref{alpha1} is executed $t-1$ times between iterations $q'$ and $q$. In this case, $\alpha$ is the vector defined by $\alpha_i=c^i\bar x^\uparrow$ for $i\le n-t$, $\alpha_{n-t+1}=c^{n-t+1}\bar x^\uparrow+1$, $\alpha_i=c^i\bar x^\uparrow$ for $i\ge n-t+2$, and therefore $\alpha \in V(S)$.
We now show that every point in $V(S)$ is in the sequence $(\alpha)$. By Lemma \ref{lem:consecutive}, the sequence $(\alpha)$ contains all points of the form $Cx$ for $x\in S^\uparrow$. Let now $\alpha\in V(S)$, where $\alpha$ is not of the form $Cx$ for any $x\in S^\uparrow$. Then there exist $\hat x\in S^\uparrow$ and an index $k<n$ such that $\alpha_i=c^i\hat x$ for $i<k$, $\alpha_kx=c^k\hat x+1$, and $\alpha_i=0$ for $i>k$. Consider the last iteration of line \ref{bar x} in which $\bar x^\uparrow$ satisfies $c^i\bar x^\uparrow=c^i\hat x$ for $i\le k$ (this definition makes sense because, as shown above, $\hat x=\bar x^\uparrow$ at some iteration of line \ref{bar x}). The algorithm now sets $\alpha=C\bar x^\uparrow$ and executes line \ref{alpha1} $k$ consecutive times. After this, we have $\alpha=Cx$. This shows that every point in $V(S)$ is in the sequence $(\alpha)$. \end{proof}
We remark that in the definition of $\alpha$ at line \ref{alpha2}, we could impose the stronger condition $\alpha_n:=c^n\bar x^\uparrow+1$. However, this would not change substantially the bounds on the number of iterations shown above. Moreover, when $S$ is convex the number of iterations is precisely the same in both cases.
By Observation \ref{obs:performance} and Proposition \ref{prop:bound-alg2}, the number of iterations of Algorithm \ref{alg:weak} and Algorithm \ref{alg:enum} is
upper-bounded by $|S^\uparrow|$ and $|V(S)|+1$, respectively. Note that the latter bound is always larger than the former. In particular, for the example in Proposition \ref{prop:exp} we have $|V(S)|=2^n+2^{n-1}-2$, thus in that case Algorithm \ref{alg:enum} executes roughly 50\% more iterations than Algorithm \ref{alg:weak}. However comparing the two algorithms by counting the number of iterations may not be ``fair'', as the computational effort varies from iteration to iteration: for instance, the computation of a lex-min solution (line \ref{step:lex-min} of Algorithm \ref{alg:weak} and line \ref{bar x} of Algorithm \ref{alg:enum}) requires up to $n$ oracle calls, while the iterations of Algorithm \ref{alg:enum} in which $S^*$ is empty only require a single oracle call. Nonetheless the results on the number of iterations at least indicate that, from the theoretical point of view, Algorithm \ref{alg:weak} tends to be more efficient than Algorithm \ref{alg:enum}.
\section{Comparison with Gomory and split cuts}\label{sec:comp}
Given a set $S$, a {\em Chv\'atal--Gomory inequality} for $S$ is a linear inequality of the form $gx\ge\ceil{\gamma}$ for some $g\in\mathbb{Z}^n$ and $\gamma\in\mathbb{R}$ such that the inequality $gx\ge\gamma$ is valid for $S$. We call $gx\ge\ceil{\gamma}$ a {\em proper} Chv\'atal--Gomory inequality if $gx\ge\ceil{\gamma}$ is violated by at least one point in $S$.
\begin{proposition} Given $S\in\mathcal S$, every proper Chv\'atal--Gomory inequality for $S$ is a lex-cut for some lattice basis $\{c^1,\dots,c^n\}$ of $\mathbb{Z}^n$. \end{proposition}
\begin{proof} Let $gx\ge\ceil{\gamma}$ be a proper Chv\'atal--Gomory inequality for $S$. Without loss of generality, we assume that the entries of $g$ are relatively prime integers. Let $\bar x$ be the lex-min solution found at the first iteration of Algorithm \ref{alg:weak} with respect to some lattice basis $\{c^1,\dots,c^n\}$, with $c^1=g$. Since $gx\ge\ceil{\gamma}$ is a proper Chv\'atal--Gomory inequality for $S$, we have $\gamma\le g\bar x<\ceil{\gamma}$. In particular, $g\bar x\notin\mathbb{Z}$. Then the corresponding lex-cut is (equivalent to) $gx\ge\ceil{g\bar x}=\ceil{\gamma}$. \end{proof}
The converse of the above proposition is false; this will follow from a stronger result.
A linear inequality is a {\em split cut} for $S$ if there exist $\pi\in\mathbb{Z}^n$ and $\pi_0\in\mathbb{Z}$ such that the inequality is valid for both $\{x\in S:\pi x\le\pi_0\}$ and $\{x\in S:\pi x\ge\pi_0+1\}$. It is known that every Chv\'atal--Gomory inequality is a split cut but not vice versa.
The next result shows that our family of cuts is not included in and does not include the family of split cuts. Combined with the previous proposition, this implies that our family of cuts strictly contains the Chv\'atal--Gomory inequalities.
\begin{proposition}\label{prop:split} There exist a bounded polyhedron $S$ and a split cut for $S$ that cannot be obtained as (and is not implied by) a lex-cut for any choice of the lattice basis $\{c^1,\dots,c^n\}$. Conversely, there exist a bounded polyhedron $S$ and a lex-cut that is not a split cut for $S$. \end{proposition}
\begin{proof} Let $S\subseteq\mathbb{R}^2$ be the triangle with vertices $(0,0)$, $(1,0)$ and $(1/2,-1)$. (See Figure \ref{fig:split1} to follow the proof.) The inequality $x_2\ge0$ is a split cut for $S$, as it is valid for both sets $\{x\in S:x_1\le0\}$ and $\{x\in S:x_1\ge1\}$. Note that after the application of the cut, the continuous relaxation becomes the segment with endpoints $(0,0)$ and $(1,0)$, which is the convex hull of the integer points in $S$.
Assume that the cut $x_2\ge0$ can be obtained via an iteration of Algorithm \ref{alg:weak} for some lattice basis $\{c^1,c^2\}$ and the corresponding bounds $\ell_1,\ell_2\in\mathbb{Z}$. In the following, we will write $c^1=(c^1_1,c^1_2)$ and $c^2=(c^2_1,c^2_2)$.
Recall that in Algorithm \ref{alg:weak} a translation is applied such that $\ell_i=0$ for every $i$. However, in this proof it is convenient to work without applying the translation. It is easy to see that in this case the form of the lex-cut is still \eqref{eq:cut}, but now the $d^k_i$ are defined as follows: \[d^k_i:=\begin{cases} 1 & \mbox{if $i=k$}\\ \ceil{c^k\bar{x}-\ell_k} & \mbox{if $i=k-1$},\\ \ceil{c^k\bar{x}-\ell_k}\prod_{j=i+1}^{k-1}(c^j\bar{x}+1-\ell_j),&\mbox{if $i\le k-2$}. \end{cases}\]
Since the point $(1/2,-1)$ is the only fractional vertex of $S$, we must have $\bar x=(1/2,-1)$, otherwise no cut is generated. Suppose $k=1$, i.e., $c^1\bar x\notin\mathbb{Z}$ (see step \ref{step:cut} of the algorithm). Then the inequality generated by the algorithm is equivalent to $c^1x\ge\ceil{c^1\bar x}$. Since this inequality must be equivalent to $x_2\ge0$ and the entries of $c^1$ are relatively prime integers, we necessarily have $c^1=(0,1)$. But then $c^1\bar x=-1$, a contradiction to the assumption $c^1\bar x\notin\mathbb{Z}$.
Suppose now $k=2$, i.e., $c^1\bar x\in\mathbb{Z}$ and $c^2\bar x\notin\mathbb{Z}$. Then the inequality given by the algorithm is \begin{equation}\label{eq:nonsplit} d^2_1\left(c^1x-c^1\bar x\right)+c^2x-\ceil{c^2\bar x}\ge0. \end{equation}
We claim that $c^1_1\ne0$. If this is not the case, then $c^1_1=0$ and $c^2_1\ne0$ (as $\{c^1,c^2\}$ is a basis), and inequality \eqref{eq:nonsplit} does not reduce to the desired cut $x_2\ge0$, as the coefficient of $x_1$ is $d^2_1c^1_1+c^2_1=c^2_1\ne0$. Thus $c^1_1\ne0$. This implies that either the point $(0,-1)$ or the point $(1,-1)$ satisfies the strict inequality $c^1x>c^1\bar x$. We assume that this holds for $\hat x:=(0,-1)$ (the other case is similar). Note that $c^1\hat x\ge c^1\bar x+1$, as $c^1\bar x\in\mathbb{Z}$ and $c^1,\hat x\in\mathbb{Z}^2$. Furthermore, the slope of the line defined by the equation $c^1x=c^1\bar x$ is positive.
If $c^2\hat x\ge \ell_2$, then $\hat x$ satisfies inequality \eqref{eq:nonsplit}, as $c^1\hat x-c^1\bar x\ge1$ and $c^2\hat x-c^2\bar x\ge \ell_2-c^2\bar x\ge-d^2_1$. Since the point $(1,0)$ also satisfies \eqref{eq:nonsplit} (as it is an integer point in $S$), the middle point of $\hat x$ and $(1,0)$ satisfies \eqref{eq:nonsplit}. However, the middle point is $(1/2,-1/2)$, which is in $S$. This shows that in this case \eqref{eq:nonsplit} is not equivalent to $x_2\ge0$.
Therefore we assume $c^2\hat x<\ell_2$. Since $c^2\bar x\ge \ell_2$, the line defined by the equation $c^2x=\ell_2$ intersects the line segments $[\hat x,\bar x]$ in a point distinct from $\hat x$. Then, because $(0,0)$ satisfies the inequality $c^2x\ge \ell_2$ (as it is in $S$), the slope of the line defined by the equation $c^2x=\ell_2$ is negative. Furthermore, since $c^2,\hat x\in\mathbb{Z}^2$, we have $c^2\hat x\le\floor{\ell_2}$, and thus the line defined by the equation $c^2x=\floor{\ell_2}$ intersects $[\hat x,\bar x]$ in some point $x^*$.
Now consider the system $c^1x=c^1\bar x$, $c^2x=\floor{\ell_2}$. Since the constraint matrix is unimodular (as $\{c^1,c^2\}$ is a lattice basis of $\mathbb{Z}^2$) and the right-hand sides are integer, the unique solution to this system is an integer point. However, the first equation defines a line with positive slope containing $\bar x$ and the second equation defines a line with negative slope containing $x^*$. From this we see that the intersection of the two lines is a point satisfying $0<x_1\le1/2$ and therefore cannot be an integer point, a contradiction. This concludes the proof that there is a split cut that cannot be obtained via an iteration of Algorithm \ref{alg:weak}.
\begin{figure}
\caption{Illustration of the first part of the proof of Proposition \ref{prop:split}. The inequality $x_2\ge0$ is a split cut for the shadowed triangle, but is not of the type \eqref{eq:cut}.}
\label{fig:split1}
\end{figure}
For the converse, let $S\subseteq\mathbb{R}^2$ be the triangle with vertices $(0,3/2)$, $(1/4,0)$ and $(1,0)$. If we take $c^1,c^2$ to be the vectors in the standard basis of $\mathbb{R}^2$, and $\ell_1=\ell_2=0$, then Algorithm \ref{alg:weak} yields the cut $2x_1+x_2\ge2$. Note that every point in $S$ other than $(1,0)$ is cut off by this inequality. Thus, if the inequality $2x_1+x_2\ge2$ is a split cut for $S$, then there exist $\pi\in\mathbb{Z}^2$ and $\pi_0\in\mathbb{Z}$ such that $S$ is contained in the ``strip'' $\{x\in\mathbb{R}^2:\pi_0\le\pi x\le\pi_0+1\}$. Since $S$ contains a horizontal and a vertical segment of length $3/4$, this is possible only if the Euclidean distance between the lines $\{x\in\mathbb{R}^2:\pi x=\pi_0\}$
and $\{x\in\mathbb{R}^2:\pi x=\pi_0+1\}$ is at least $\frac3{4\sqrt2}$. Therefore $\|\pi\|^2\le\left(\frac{4\sqrt2}3\right)^2=\frac{32}{9}<4$. Since $\pi$ is an integer vector, we deduce that $\pi_1,\pi_2\in\{0,1,-1\}$. It can be verified that if $|\pi_1|=|\pi_2|=1$ then $S$ is not contained in the strip. Therefore one entry of $\pi$ is 0 and the other is 1 or $-1$. It can be checked that the only strip of this type containing $S$ is $\{x\in\mathbb{R}^2:0\le x_1\le 1\}$. However, the inequality $2x_1+x_2\ge2$ is not valid for all the points in $\{x\in S:x_1\le0\}\cup\{x\in S:x_1\ge1\}$, as the point $(0,3/2)$ is in this set but violates the inequality. \end{proof}
\section{Concluding remarks}\label{sec:conc} An obvious variant of Algorithm \ref{alg:weak} is the following: instead of being computed only once at the beginning of the procedure, the lower bounds $\ell_i$ can be updated at every iteration or whenever it seems convenient. It can be verified that the bounds of Observation \ref{obs:performance} and Proposition \ref{prop:exp} also hold for this variant of the algorithm: the proofs are the same.
In view of Observation \ref{obs:performance} and Proposition \ref{prop:exp}, the cardinality of $S^{\uparrow}$ truncated to the lex-min point in $S^{\uparrow}\cap S$ plays a crucial role in the performance of Algorithm \ref{alg:weak}. This number is dependent on the choice of the lattice basis and its ordering. It is easy to see that different choices of the lattice basis (or different choices of the ordering of the elements of the same lattice basis) may result in a different number of iterations of the algorithm. However, this is not always the case: for instance, in the example in Proposition \ref{prop:exp} Algorithm \ref{alg:weak} would produce the same number of iterations regardless of the ordering of the standard basis.
A natural question is whether the approach described in this paper can be generalized to the mixed integer case, i.e., to problems of the form $\min \{cx+dy: (x,y)\in S\cap(\mathbb{Z}^n\times\mathbb{R}^p)\}$, where $S\subseteq\mathbb{R}^{n+p}$ is a compact set. However, it does not seem that our algorithm can be easily extended to deal with this case.
\end{document} | arXiv |
\begin{document}
\title[Picard groups]{Picard groups in $p$-adic Fourier theory}
\author[Tobias Schmidt]{Tobias Schmidt} \address{Mathematisches Institut\\ Westf\"alische Wilhelms-Universit\"at M\"unster\\ Einsteinstr. 62\\ D-48149 M\"unster, Germany} \email{[email protected]}
\maketitle \begin{abstract} Let $L\neq\mathbb{Q}_p$ be a proper finite field extension of ${\mathbb Q}_p$ and $o\subset L$ its ring of integers viewed as an abelian locally $L$-analytic group. Let $\hol$ be the rigid $L$-analytic group parametrizing the locally analytic characters of $o$ constructed by Schneider-Teitelbaum. Let $K/L$ be a finite extension field. We show that the base change $\hol_K$ has a Picard group $Pic(\hol_K)$ which is profinite and that the unit section in $\hol_K$ provides a divisor class of infinite order. In particular, the abelian group $Pic(\hol_K)$ is not finitely generated and is not a torsion group. On the way we show that $\hol_K$ is a nontrivial \'etale covering of the affine line over $K$ realized via the logarithm map of a Lubin-Tate formal group. We finally prove that rank and determinant mappings induce an isomorphism between $K_0(\hol_K) $ and ${\mathbb Z} \oplus Pic(\hol_K)$. \end{abstract} \section{Introduction} \footnote[0]{2000 Mathematics Subject Classification: 22E50 (primary), 14G22 (secondary).}
Let $L$ be a complete nonarchimedean field and ${\bf B}$ a rigid $L$-analytic open polydisc. A twisted form of ${\bf B}$ is a rigid analytic space $X$ over $L$ together with a complete nonarchimedean extension $L\subseteq L'$ and an isomorphism $X_{L'}\car {\bf B}_{L'}$. The study of the forms of ${\bf B}$ with respect to a given extension $L\subseteq L'$ is an important yet difficult problem which is still in its infancy. As a first result A. Ducros recently has shown \cite{Ducros} that, if $L\subseteq L'$ is finite and tamely ramified, any form $X$ is already trivial, in the sense that $X$ is itself isomorphic to an open polydisc. He also showed that there are finite wildly ramified extensions that support plenty of nontrivial forms, even in dimension one.
\vskip8pt
A coarser but more accessible problem concerns the description of basic invariants of such forms such as the Picard group or the Grothendieck group. The aim of this note is to make a first step in this direction and study the Picard group of an interesting form that comes from $p$-adic representation theory. To give more details, let $L\neq\mathbb{Q}_p$ be from now on a proper finite extension of ${\mathbb Q}_p$ and let ${\bf B}$ be the open unit disc of dimension one. In \cite{ST2} Schneider-Teitelbaum generalize the classical $p$-adic Fourier theory of Y. Amice \cite{Amice2} for the group ${\mathbb Z}_p$ to the additive group of integers $o$ in $L$. The key step is the construction of a certain nontrivial form $\hol$ of ${\bf B}$ ($\hol$ is even a group object) with respect to the transcendental extension $L\subset {\mathbb C}_p$ having the surprising property of admitting no trivializiation over any discretely valued complete subfield of ${\mathbb C}_p$. In particular, this forces its Picard group to be nontrivial. In \cite{Teitelbaum} J. Teitelbaum suggested to study this Picard group, partly for representation-theoretic reasons which we will describe below.
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Slightly more general, we will study the series of Picard groups $Pic(\hol_K)$ where $\hol_K$ denotes the base change of $\hol$ to finite extensions $K$ of $L$. Although we are not able to determine the structure of $Pic(\hol_K)$ explicitly, our main result shows that it is of enormous size. More precisely, we prove that $Pic(\hol_K)$ is a profinite group and that the unit section of $\hol_K$ supports a divisor class of infinite order. In particular, the abelian group $Pic(\hol_K)$ is not finitely generated and is not a torsion group. On the way we show that $\hol_K$ is a nontrivial \'etale covering of the affine line over $K$ (in the sense of \cite{deJongET}). We finally show that rank and determinant mappings induce an isomorphism $K_0(\hol_K) \car{\mathbb Z} \oplus Pic(\hol_K)$.
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As already indicated the variety $\hol_K$ has a representation-theoretic interpretation. Indeed the additive group $o$ is among the first examples of a compact abelian locally $L$-analytic group. The generalized Amice-Fourier isomorphism of Schneider-Teitelbaum \cite{ST2} induces an isomorphism of the ring of holomorphic functions on $\hol_K$ with the $K$-valued locally analytic distribution algebra of $o$. Since $\hol_K$ is a quasi-Stein space in the sense of R. Kiehl \cite{Kiehl} the group $Pic(\hol_K)$ controls the ideal structure of this distribution algebra and therefore the locally analytic representation theory of $o$. For example, the rational points of $\hol_K$ are in bijective correspondence with the locally analytic characters $o\rightarrow K^\times$.
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In the following we briefly outline the article. As with most nontrivial Picard groups there is by no means a straightforward way to compute the structure of $Pic(\hol_K)$. Our strategy is to first determine the local Picard groups corresponding to a suitable open affinoid covering of the quasi-Stein space $\hol_K$ and then 'glue' these informations. We begin by recalling some results on the divisor theory for Dedekind domains. The point is that the affinoid algebra of a twisted form of a nonarchimedean closed disc with respect to a finite extension is a Dedekind domain. In sect. \ref{section-forms} we use descent theory to prove a finiteness result for the Picard group of a certain class of such forms. In sect. \ref{section-fourier} we turn to the variety $\hol_K$. We prove that it admits an admissible affinoid covering $\hol_K=\cup_n \holnk$ where each $\holnk$ is a twisted form of a closed disc with respect to a finite Galois extension (depending on $n$). The Galois cocycle giving the descent datum comes out of a Lubin-Tate group $\G$ for $o$ and the logarithm $\log_\G$ identifies $\holnk$ with a finite \'etale covering of a closed disc whose degree equals $q^{en}.$ Here, $e$ and $q$ equal the ramification index of $L/\mathbb{Q}_p$ and the cardinality of the residue field of $L$ respectively. Passing to the limit in $n$ proves $\hol_K$ to be an \'etale covering of the affine line over $K$. The results of sect. \ref{section-forms} may be applied to $\holnk$ and yield the finiteness of $Pic(\holnk)$. Using a spectral sequence argument combined with a vanishing result of L. Gruson \cite{GrusonFi} we find $Pic(\hol_K)=\underleftarrow{\lim}_n Pic(\holnk)$. Building on ideas of A. de Jong \cite{deJongCrystalline} we show that the zero section of the group $\hol_K$ supports a divisor class of infinite order in $Pic(\hol_K)$. Finally, since the ring of global sections $\mathcal{O}(\hol_K)$ is a Pr\"uferian domain, Serre's theorem from algebraic $K$-theory \cite{Bass} implies the result on $K_0(\hol_K)$.
\vskip8pt
As explained above the points of $\hol$ parametrize the locally analytic characters of $o$. Generalizing the construction of $\hol$ M. Emerton has introduced such a character variety for any abelian locally $L$-analytic group which is topologically finitely generated \cite{EmertonA}. We conclude this work by briefly explaining how the problem of determining the Picard group of general character varieties can essentially be reduced to the case of (copies of) $\hol$.
~\\{\it Acknowledgements.} I am indebted to Peter Schneider and Jeremy Teitelbaum for their useful advice and comments during the preparation of this article and for generously providing me with some helpful private notes on their own work. Parts of this work were written during a stay of the author at the {\it Tata Institute of Fundamental Research}, Mumbai, supported by the Deutsche Forschungsgemeinschaft. The author is grateful for the support of both institutions.
\section{Preliminaries on Dedekind domains}\label{section-dedekind}
We recall some divisor theory for Dedekind domains, cf. \cite{B-CA}, VII.\S2, thereby fixing some notation. Recall that an integral domain is a {\it Dedekind domain} if it is noetherian, integrally closed and every nonzero prime ideal is maximal.
\vskip8pt
Let $B$ be a Dedekind domain with field of fractions $F$. The free abelian group $D(B)$ on the set $Sp(B)$ of maximal ideals of $B$ is called the {\it divisor group} of $B$. Given $P\in Sp(B)$ the localization $B_P$ of $B$ at $P$ is a discrete valuation ring. Let $v_P$ be the associated valuation. We have the well-defined group homomorphism \[div_B: F^\times\longrightarrow D(B)~,~~~ x\mapsto\sum_{P\in Sp(B)} v_P(x)P\] whose cokernel $$Cl(B):=D(B)/{\rm im~} div_B$$ is called the {\it divisor class group}.
On the other hand, let $Pic(B)$ denote the {\it Picard group} of $B$, i.e. the group of isomorphism classes of locally free $B$-modules of rank $1$ (with the tensor product of $B$-modules as group law). There is a commutative diagram of abelian groups with exact rows \begin{equation}\label{cartier}\xymatrix{ 1\ar[r]& B^\times \ar[r]^{\subseteq} \ar[d]^= & F^\times \ar[r] \ar[d]^= & Cart(B) \ar[r]\ar[d]^{\iota} &Pic(B)\ar[d]^{\bar{\iota}}\ar[r] &1\\ 1\ar[r]& B^\times \ar[r]^{\subseteq} & F^\times \ar[r]^{div_B} & D(B) \ar[r]& Cl(B) \ar[r] & 1. }\end{equation} Here, $Cart(B)$ refers to the group of invertible fractional ideals of $B$ and the map $F^\times\rightarrow Cart(B)$ is given by $x\mapsto xB$. The map $\iota: Cart(B)\rightarrow D(B)$ is given by $I\mapsto\sum_{P\in Sp(B)} v_P(I)P$ where $v_P(I)$ equals the order of the extended fractional ideal in the discretely valued field $Quot(B_P)$. Both $\iota$ and $\bar{\iota}$ are bijections, e.g. \cite{WeibelK}, Cor. I.3.8.1.
Now suppose that $A\subseteq B$ is a subring which is a Dedekind domain itself such that $A\rightarrow B$ is integral or flat. Given $P'\in Sp(A),~ P\in Sp(B)$ with $P\cap A=P'$ let $e(P/P')\in\mathbb{N}$ denote the ramification index of $P$ over $P'$. Then $j(P'):=\sum e(P/P')P$ induces a well-defined group homomorphism \[j: D(A)\rightarrow D(B)\] where the sum runs through all $P\in Sp(B), P\cap A=P'$. It factors into a group homomorphism \[\overline{\jmath}: Cl(A)\rightarrow Cl(B),\] cf. \cite{B-CA}, VII.\S1.10 Prop. 14.
Now suppose additionally that $A\subseteq B$ is a finite Galois extension with group $G$ and such that $Cl(B)=1$. The group $G$ acts on $Sp(B)$ and on $D(B)$. The map $j$ induces an isomorphism $D(A)\car D(B)^G$. Furthermore, $Quot(A)=F^G$. Taking $G$-invariants in the lower horizontal row of (\ref{cartier}) and using Hilbert 90 yields the exact sequence \[1\longrightarrow A^\times\longrightarrow Quot(A)^\times\longrightarrow D(A)\stackrel{\delta}{\longrightarrow} H^1(G,B^\times)\longrightarrow 1.\]
We obtain a canonical isomorphism $$\bar{\delta}: Cl(A)\stackrel{\cong}{\longrightarrow}H^1(G,B^\times).$$
\section{Class groups of twisted affinoid discs}\label{section-forms}
Let $L$ be a complete non-archimedean field, i.e. a field that is complete with respect to a specified nontrivial non-archimedean absolute value. We assume that the reader is familiar with the classical theory of affinoid spaces over such a field \cite{BGR}.
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For any $L$-affinoid algebra $B$ we denote by $\mathring{B}$ the subring of power-bounded elements, by $\check{B}$ the $\mathring{B}$-ideal of topologically nilpotent elements and by $\tilde{B}:=\mathring{B}/\check{B}$
the reduction of $B$. Passing to the reduction is a covariant functor from $L$-affinoid algebras to algebras over the residue field of $L$. If $|.|$ denotes the spectral seminorm on $B$ we have $\mathring{B}=\{b\in B: |b|\leq 1\}$ and $\check{B}=\{b\in B:|b|<1\}$. Moreover, if the ring $B$ is reduced, the spectral seminorm is a norm and defines the Banach topology of $B$.
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After these preliminaries let $K/L$ be a finite Galois extension and let $B$ be the one dimensional Tate algebra over $K$, i.e.
$$B=\{ \sum_{n\geq 0}a_nz^n, a_n\in K, |a_n|\rightarrow 0 {\rm~for~} n\rightarrow\infty \}.$$
Here, we denote the unique extension of the absolute value on $L$
to $K$ also by $|.|$. Suppose $A$ is a $L$-affinoid algebra equipped with an isomorphism $$A\otimes_L K \car B$$ of $K$-algebras. In other words, $A$ is a {\it twisted form} of $B$ with respect to the extension $K/L$, cf. \cite{KnusOjanguren}, II.\S8. Let $G:=Gal(K/L)$. The Galois group $G$ acts on $A\otimes_L K$ by $\sigma(a\otimes x)=a\otimes\sigma(x)$ and, via transport of structure, we obtain in this way a semilinear Galois action on the $K$-algebra $B$.
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Remark: The forms of the affinoid algebra $B$ with respect to the Galois extension $K/L$ are classified by the nonabelian Galois cohomology $H^1(G, {\rm Aut}_K(B))$ according to \cite{SchmidtDISC}. Here, ${\rm Aut}_K(B)$ refers to the automorphism group of the $K$-algebra $B$. Any automorphism of $B$ is completely determined by its value on the variable $z$ and induces, by functoriality, an automorphism of the algebra $\tilde{B}=\tilde{K}[z]$. Consequently, the map $f\mapsto f(z)$ induces a group isomorphism between ${\rm Aut}_K(B)$ and the group of formal power series
$$a_0+a_1z+a_2z^2+...$$ subject to the conditions $|a_0|\leq 1,
|a_1|=1, |a_i|<1 {\rm~for~all~}i>1$, cf. \cite{BGR}, Corollary 5.1.4/10. By the enormous size of this group the classification of forms of $B$ seems to be a difficult task. In \cite{SchmidtDISC} it is shown that any form of $B$ with respect to a finite tamely ramified extension is trivial, in the sense that it is itself isomorphic to a closed disc. Moreover, it is shown that there are plenty of wildly ramified forms. Apart from these results the author does not know of any results in this direction in the literature.
\begin{lem}\label{lem-dedekind} The ring $A$ is a Dedekind domain. \end{lem} \begin{proof} The ring $B$ is a principal ideal domain. Since the extension $A\subseteq B$ is integral the usual Going Up theorem shows $A$ to be of dimension $1$, e.g. \cite{Matsumura}, Ex. 9.2. Finally, let $a\in Quot(A)$ be integral over $A$ . Since $B$ is integrally closed we have $a\in B$ and thus, by Galois invariance, $a\in A$. \end{proof} We therefore have the canonical isomorphism $Cl(A)\car H^1(G,B^\times)$ at our disposal.
\begin{lem}\label{lem-sections} Suppose that the semilinear action of $G$ maps the principal ideal $(z)$ of $B$ to itself. Then $\mathring{B}^\times\subseteq B^\times$ induces a short split exact sequence
\[1\longrightarrow |K^\times|/|L^\times|\longrightarrow H^1(G,\mathring{B}^\times)\longrightarrow H^1(G,B^\times)\longrightarrow 1.\] \end{lem} \begin{proof} Let $U:=\{ b_0+b_1z+...\in B^\times: b_0=1\}$. Then $B^\times=UK^\times$ and $\mathring{B}^\times=U\mathring{K}^\times$ and the Galois action on $B$ respects these direct products by assumption. Since cohomology commutes with direct sums we obtain, again by Hilbert 90, a short exact sequence \[1\longrightarrow H^1(G,\mathring{K}^\times)\longrightarrow H^1(G,\mathring{B}^\times)\longrightarrow H^1(G,B^\times)\longrightarrow 1.\] The map $UK^\times\rightarrow U\mathring{K}^\times, (u,x)\mapsto (u,1)$ induces a splitting of this sequence. Finally, the long exact cohomology sequence induced by the short exact sequence
$$1\longrightarrow \mathring{K}^\times\longrightarrow K^\times\stackrel{|.|}{\longrightarrow} |K^\times|\longrightarrow 1$$ induces, again by Hilbert 90, the isomorphism $
|K^\times|/|L^\times|\car H^1(G,\mathring{K}^\times)$. \end{proof}
For any real number $0<m<1$ we let $$\mathring{B}^{(m)}:=\{b\in\mathring{B}: |b-1|\leq m\}.$$ It is a Galois stable subgroup of $\mathring{B}^\times$.
\begin{lem}\label{lem-sen} There is a real number $m=m(K)$ with $0<m<1$ such that the image of the map \[H^1(G, \mathring{B}^{(m)})\longrightarrow H^1(G, \mathring{B}^\times)\] induced by the inclusion $\mathring{B}^{(m)}\subseteq \mathring{B}^\times$ is the trivial group $\{1\}$. \end{lem} \begin{proof} This is a mild generalization of a lemma in \cite{ST2b}. Denote by $Tr$ the trace map of the field extension $K/L$. Since $K/L$ is separable there is an element $c\in K$ with $Tr(c)=1$. Then any
$0<m<1$ such that $m\,|c|<1$ will do. Indeed, let $\psi$ be a cocycle representing an element of $H^1(G, \mathring{B}^{(m)})$ and consider the element \[\phi:=\sum_{\sigma\in G}\psi(\sigma)\sigma(c)\in B.\] We have
\[|\phi-1|=|\sum_{\sigma\in G}(\psi(\sigma)-1)\sigma(c)|\leq m\,|c|<1\] and therefore $\phi\in \mathring{B}^\times$. Given $\tau\in G$ we compute $$\tau(\phi)=\sum_{\sigma\in G}\psi(\sigma)^\tau(\tau\sigma)(c)=\sum_{\sigma\in G}\psi(\tau)^{-1}\psi(\tau\sigma)(\tau\sigma)(c)= \psi(\tau)^{-1}\phi$$ and, thus, $\psi(\tau)=\phi^{1-\tau}$. Hence, the image of the class of $\psi$ in $H^1(G, \mathring{B}^\times)$ coincides with the class of the coboundary $\tau\mapsto \phi^{1-\tau}$. \end{proof} In the situation of the lemma the canonical homomorphism \begin{equation}\label{injective}H^1(G,\mathring{B}^\times)\longrightarrow H^1(G,\mathring{B}^\times/\mathring{B}^{(m)})\end{equation} is therefore {\it injective}.
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We fix an algebraic closure $K\subseteq \bar{K}$ of the field $K$ and we extend the absolute value from $K$ to $\bar{K}$. Let $r\in
|\bar{K}^\times |$ and consider the generalized Tate algebra $B_r$ of all affinoid functions on the closed disc of radius $r$ around zero. It is given by all formal series
$$a_0+a_1z+a_2z^2+...$$ subject to $a_n\in K,
|a_n|r^{n}\rightarrow 0$ for $n\rightarrow\infty$. Suppose $r'\in
|\bar{K}^\times |$ is a second radius with $r<r'$. It is well-known that, in case the field $K$ is locally compact, the canonical inclusion $$f: B_{r'}\stackrel{\subset}{\longrightarrow} B_r$$ is a compact continuous linear map between $K$-Banach spaces, e.g. \cite{NFA}, Example \S16. By loc.cit., Remark 16.3 this implies that the image of any bounded $\mathring{K}$-module in $B_{r'}$ has compact closure. We also remark that, since the inclusion $Sp(B_r)\rightarrow Sp(B_{r'})$ identifies the source with an affinoid subdomain in the target, the ring extension underlying $f$ is flat, cf. \cite{BGR}, Cor. 7.3.2/6.
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We know place ourselves in the following situation. Suppose
$r_1<r_2$ are two numbers in $|K^\times|$, the value group of $K$. The corresponding affinoid algebra on the closed disc with radius $r_i$ is denoted by $B_i$. We {\it suppose} that there is a $K$-semilinear $G$-action on $B_i$ stabilizing the principal ideal $(z)$ such that the inclusion map $f: B_2\rightarrow B_1$ is equivariant. We suppose furthermore, that the inclusion of the ring of invariants $A_i:=(B_i)^G$ (an affinoid $L$-algebra) into $B_i$ is a finite Galois extension with group $G$ and that the induced morphism between affinoids $Sp(A_1)\rightarrow Sp(A_2)$ is an inclusion making the source an affinoid subdomain in the target. In this situation we prove the \begin{prop}\label{prop-finite} If the field $K$ is locally compact, the class group $Cl(A_1)$ is a finite group. \end{prop} \begin{proof} First, since $r_i$ is in the value group of the ground field $K$ it is almost obvious that the results proved above for the ordinary Tate algebra $B$ hold verbatim for the algebras $B_i$.
To start with the ring extension $A_2\subseteq A_1$ is flat and so we have the homomomorphism $j:D(A_2)\rightarrow D(A_1)$ from the previous section. On the other hand, the map $Sp(A_1)\rightarrow Sp(A_2)$ induces a group homomorphism $\varphi: D(A_1)\rightarrow D(A_2)$. According to \cite{BGR}, Prop. 7.2.2/1 (iii) it is a splitting of $j$, i.e. $j\circ\varphi= id|_{D(A_1)}$. We have similar maps $j$ associated to the flat extensions $A_i\subseteq B_i$ and $f$. By multiplicativity of the ramification index they assemble to the commutative diagram \[\xymatrix{ D(A_2) \ar[d] \ar[r] & D(B_2) \ar[d] \\ D(A_1) \ar[r] & D(B_1). } \] It follows that the maps $\delta_i: D(A_i)\rightarrow H^1(G,B^\times_i)$ sit in the commutative diagram
\[\xymatrix{ D(A_2) \ar[d]^{j} \ar[r]^<<<<{\delta_2} & H^1(G,B^\times_2) \ar[d]^{H^1(f)} \\ D(A_1) \ar[r]^<<<<{\delta_1} &H^1(G,B^\times_1) } \] and, consequently, we have the identity \begin{equation}\label{commute}H^1(f)\circ\delta_2\circ\varphi=\delta_1\circ j\circ\varphi=\delta_1.\end{equation}
After this preliminary discussion we consider for $0<m<1$ the following diagram $(+)$ of abelian groups
\[\xymatrix{ D(A_1) \ar[d]^{\delta_2\circ\varphi} \ar[r]^{\delta_1} & H^1(G,B_1^\times) \ar[d]^{=} \\ H^1(G,B_2^\times) \ar[d]^{s_2} \ar[r]^{H^1(f)} & H^1(G,B_1^\times) \ar[d]^{s_1} \\ H^1(G,\mathring{B}_2^\times) \ar[d] \ar[r] & H^1(G,\mathring{B}_1^\times) \ar[d] \\ H^1(G, \mathring{B}_2^\times/(\mathring{B}_2^\times\cap\mathring{B}^{(m)}_1))\ar[r] & H^1(G, \mathring{B}_1^\times/\mathring{B}^{(m)}_1). } \] Here, the maps $s_i$ are the canonical sections from Lemma \ref{lem-sections} and the remaining maps are the obvious ones. We prove in a first step that all squares in this diagram are commutative. The identity (\ref{commute}) means that the upper square commutes. Consider the square in the middle. Since $r_i\in
|K^\times|$ we have direct product decompositions $B_i^\times=U_iK^\times$ and $\mathring{B}_i^\times=U_i\mathring{K}^\times$ as in the proof of Lemma \ref{lem-sections} where $U_i:=\{ b_0+b_1z+...\in B_i^\times: b_0=1\}$. These decompositions are respected by $f$. The middle square is therefore induced by the commutative diagram of Galois modules
\[\xymatrix{ B_2^\times \ar[d] \ar[r]^{f} & \ar[d] B_1^\times \\
\mathring{B}_2^\times \ar[r] & \mathring{B}_1^\times } \] where the vertical maps are given by $(u,x)\mapsto (u,1)$. It is therefore commutative. Finally, the commutativity of the lowest square is clear.
We now prove the assertion of the proposition. In $(+)$ the upper horizontal arrow $\delta_1$ factores into an isomorphism $\bar{\delta}_1: Cl(A_1)\car H^1(G,B_1^\times)$. By (\ref{injective}) we may adjust the number $m=m(K)$ so that the lowest vertical arrow on the right hand side becomes injective. Since the right-hand vertical maps are now all injective it suffices to show that the image of the lower horizontal map $$H^1(G,\mathring{B}_2^\times/(\mathring{B}_2^\times\cap\mathring{B}^{(m)}_1))\longrightarrow H^1(G, \mathring{B}_1^\times/\mathring{B}^{(m)}_1)$$ is finite. We claim that already the set $\mathring{B}_2^\times/(\mathring{B}_2^\times\cap\mathring{B}^{(m)}_1)$ is finite which yields the claim according to \cite{SerreL}, Cor. VIII.\S2.2. It is here where we use the local compactness of the field $K$. Indeed, consider the composite homomorphism $$h: \mathring{B}_2^\times\stackrel{h_1}{\longrightarrow}\mathring{B}_1^\times\stackrel{h_2}{\longrightarrow}\mathring{B}_1^\times/\mathring{B}^{(m)}_1$$ with kernel $\mathring{B}_2^\times\cap\mathring{B}^{(m)}_1$. Here, $h_1$ is induced from the compact map $f$ and $h_2$ is the canonical projection. Since $B_i$ is reduced the spectral norm induces its Banach topology and so $\mathring{B}_i$ is open and closed in $B_i$. We equip $\mathring{B}_i^\times$ with the induced topology from $\mathring{B}_i$. The maps $h_i$ are then continuous. We have $\overline{h_1(\mathring{B}_2^\times)}\subseteq \overline{f(\mathring{B}_2)}$ and the right hand side is a compact subset of $\mathring{B}_1$. Hence so is the left hand side. But the units $\mathring{B}_1^\times$ are a closed subset of the complete adic ring $\mathring{B}_1$ and therefore $\overline{h_1(\mathring{B}_2^\times)}\subseteq\mathring{B}_1^\times$. The image $h_2(\overline{h_1(\mathring{B}_2^\times)})$ is compact and contains the image of $h$. But $\mathring{B}^{(m)}_i$ is open in $\mathring{B}_i^\times$ and therefore the target of $h$ is a discrete space. Hence the image of $h$ must be finite. This proves the assertion. \end{proof}
\section{$p$-adic Fourier theory}\label{section-fourier} For the basic theory of locally analytic groups over $p$-adic fields we refer to P. Schneider's monograph \cite{SchneiderBookLie}.\vskip8pt
Let $|.|$ be the absolute value on $\mathbb{C}_p$ normalized via
$|p|=p^{-1}$. Let $$\mathbb{Q}_p\subseteq L\subseteq\mathbb{C}_p$$ be a finite extension field. Let $e=e(L/\mathbb{Q}_p)$ be the ramification index, $k$ be the residue field of $L$ and $q=\# k$ its cardinality. Let $o\subseteq L$ be the integers in $L$ and let $e_1,...,e_{[L:\mathbb{Q}_p]}$ be a $\mathbb{Z}_p$-basis of $o$. We always view $o$ as an abelian locally $L$-analytic group of dimension one.
We denote by ${\bf B}^{s}$ the rigid $L$-analytic open unit disc around zero of dimension $s\geq 1$ and by ${\bf B}^{s}(r)$ a closed subdisc of a real radius $r$. If $s=1$ we usually omit it from the notation. Let $z_1,...,z_{[L:\mathbb{Q}_p]}$ be a set of parameters on the disc ${\bf B}^{[L:\mathbb{Q}_p]}$.
\vskip8pt Let $\log (1+Z)=Z-Z^2/2+Z^3/3-...$ be the usual logarithm series. The central object of our investigations will be the closed analytic subvariety $$\hol\subseteq {\bf B}^{[L:\mathbb{Q}_p]}$$ defined by the equations \[ e_i\log(1+z_j)-e_j\log(1+z_i)=0\] for $i,j=1,...,[L:\mathbb{Q}_p]$. It is a connected smooth one dimensional rigid $L$-analytic variety, cf. \cite{ST2},\cite{Teitelbaum}. As explained in the introduction it is the central object of the $p$-adic Fourier theory developed in the article \cite{ST2}. A particular feature of $\hol$ is that for any intermediate complete field $L\subseteq K\subseteq \mathbb{C}_p$ the $K$-valued points $z\in\hol(K)$ are in natural bijection with the set of $K$-valued locally analytic characters $\kappa_z$ of $o$. This makes $\hol$ a group object.
As explained in the introduction we propose to study the Picard group and the Grothendieck group of $$\hol_K:=\hol\;\hat{\otimes}_L K$$ for any intermediate field $L\subseteq K\subseteq \mathbb{C}_p$ which is a finite extension of $L$. We remark straightaway that \cite{ST2}, Lem. 3.10 implies that the ideal sheaf corresponding to the zero section in the group $\hol_K$ is an invertible sheaf whose class in $Pic(\hol_K)$ is nontrivial if $L\neq\mathbb{Q}_p$. Since $\hol_K={\bf B}_K$ in case $L=\mathbb{Q}_p$ and $Pic({\bf B}_K)=1$, cf. \cite{Lazardzeros}, Thm. 7.2, this shows \[Pic(\hol_K)\neq 1\Longleftrightarrow L\neq\mathbb{Q}_p.\]
Our approach to the Picard group of $\hol_K$ rests upon the fact that $\hol$ is a twisted form of ${\bf B}$ with respect to the extension $L\subseteq\mathbb{C}_p$ and that the Galois cocycle giving the descent datum comes out of a Lubin-Tate group for $o$. We give more details in the next subsection.
\subsection{Lubin-Tate groups}\label{additivegroup} For a quick introduction to Lubin-Tate theory we suggest \cite{LangCyc}. Recall that a {\it Lubin-Tate formal group} for a fixed prime element $\pi\ino$ is a certain one dimensional commutative formal group $\G$ over $o$ of $p$-height $[L:\mathbb{Q}_p]$. It comes equipped with a unital ring homomorpism $[.]: o\rightarrow {\rm End}(\G)$. We will assume that $\G$ has the property $[\pi]=\pi X+X^q\ino[[X]]$. This is no essential restriction (loc.cit., Thm. 1.1/3.1).
Viewing $\G$ as a connected $p$-divisible group let $\G'$ be its Cartier dual and $T(\G')$ be the corresponding Tate module \cite{Tatediv}. The latter is a free rank one $o$-module carrying an action of the absolute Galois group \[G_L:=G(\bar{L}/L)\] of $L$ which is given by a continuous character \[\tau': G_L\longrightarrowo^\times.\] Denote by $\G_m$ the formal multiplicative group over $o$. There is a canonical Galois equivariant isomorphism of $o$-modules \begin{equation}\label{tatemodul} T(\G')\cong {\rm Hom}_{o_{\mathbb{C}_p}}(\G_{o_{\mathbb{C}_p}},\mathbb{G}_{m,o_{\mathbb{C}_p}})\end{equation} where on the right-hand side, $G_L$ acts coefficientwise on formal power series over $o_{\mathbb{C}_p}$ and the $o$-module structure comes by functoriality from the formal $o$-module $\G$. Choose once and for all an $o$-module generator $t'$ for $T(\G')$ and denote by \[F_{t'}(Z)=\omega Z+...\ino_{\mathbb{C}_p}[[Z]]Z\] the corresponding homomorphism of formal groups.
According to \cite{deJongCrystalline}, Lem. 7.3.4, we may identify $\G$ with the rigid $L$-analytic open unit disc ${\bf B}$ around zero and the latter becomes an $o$-module object in this way. The bijection between $\mathbb{C}_p$-valued points $z$ of ${\bf B}$ and $\mathbb{C}_p$-valued locally analytic characters $\kappa_z$ mentioned in the introduction to this section is then given by \[\kappa_z(g)=1+F_{t'}([g].z)\] for $g\ino$. This bijection comes in fact from an underlying rigid $\mathbb{C}_p$-analytic isomorphism \begin{equation}\label{keyiso}\kappa: {\bf B}_{\mathbb{C}_p}\car\hol_{\mathbb{C}_p}\end{equation} and the corresponding Galois cocycle is given by \begin{equation}\label{cocycle}\sigma\mapsto [\tau'(\sigma)^{-1}]\end{equation} for $\sigma\in G_L$. Here, $[\tau'(\sigma)^{-1}]$ is viewed as an element of the automorphism group of the algebra $\mathcal{O}({\bf B}_{\mathbb{C}_p})$ in the obvious way. We point out the trivial but useful identity \begin{equation}\label{explog} [\tau'(\sigma)^{-1}]=\exp_\G(\tau'(\sigma)^{-1}\log_\G(Z)) \end{equation} for any $\sigma\in G_L$ where $\exp_\G$ and $\log_\G$ are the formal exponential and logarithm series of $\G$ respectively. Note also that the linear coefficient $\omega$ of the power series $F_{t'}$ is a period (in the sense of $p$-adic Hodge theory) for the character $\tau'$ of the absolute Galois group $G_L$. Indeed, (\ref{tatemodul}) implies $\sigma.F_{t'}=\tau'(\sigma)F_{t'}$ and therefore \begin{equation}\label{period}\omega^\sigma=\tau'(\sigma)\omega\end{equation} for all $\sigma\in G_L$.
Let \[L\subseteq L_\infty\subseteq\bar{L}\] be the algebraic field extension of $L$ obtained by adjoining the $p^n$-torsion points of the $p$-divisible group $\G'$ to $L$ for all $n\geq 1$. By the main result of \cite{ST2}, Appendix, the period $\omega\in\mathbb{C}_p$ lies in the closure of $L_\infty$. By construction the isomorphism $\kappa$ therefore descends from $\mathbb{C}_p$ to this closure. If $L\neq\mathbb{Q}_p$ then, according to \cite{ST2}, Lemma 3.9, the twisted form $\hol$ of ${\bf B}$ cannot be trivialized over a discretely valued complete subfield of this closure. We also remark that (\ref{tatemodul}) implies that $L_\infty$ coincides with the field extension of $L$ obtained by adjoining all torsion points of $\G$ and all $p$-power roots of unity to $L$. Consequently, it contains wild ramification. It is interesting in this situation to recall from our introduction that A. Ducros has recently shown in \cite{Ducros} that any form of the open unit disc ${\bf B}$ with respect to a tamely ramified finite field extension is trivial in the sense that it is itself isomorphic to an open disc. He also showed that there are plenty of nontrivial wildly ramified forms.
After this review we begin our investigations by showing in a first step that, locally, $\hol_K$ is a twisted form of a rigid analytic group on a closed disc which admits trivializations over {\it finite} extensions of $K$ inside $KL_\infty$. To do this, define for $n\geq 0$ the increasing sequence of radii \[r_n:=r^{1/q^{en}}\] where $r:=p^{-q/e(q-1)}$ and consider \begin{equation}\label{quasistein1}\holn:=\hol\cap{\bf B}^{[L:\mathbb{Q}_p]}(r_n).\end{equation} If $n$ varies these affinoids form a countable increasing admissible open covering of $\hol$. On the other hand, each one dimensional disc ${\bf B}(r_n)$ is an $o$-module object with respect to the induced Lubin-Tate group structure coming from ${\bf B}(r_n)\subseteq{\bf B}$. In this situation the isomorphism (\ref{keyiso}) induces for each $n$ isomorphisms between affinoids over $\mathbb{C}_p$ \begin{equation}\label{quasistein2}{\bf B}(r_n)_{\mathbb{C}_p}\car\hol_{n,\mathbb{C}_p}\end{equation} according to \cite{ST2}, Thm. 3.6. Define a formal power series \[h_n(Z):=\exp_\G((\omega_n/\omega)\log_\G(Z))\in\mathbb{C}_p[[Z]]Z.\] Since $\omega$ lies in the closure of $L_\infty$ we may fix once and for all $\omega_n\in L_\infty$ such that
\begin{equation}\label{condition}|\omega_n/\omega-1|<p^{-n}\end{equation} for all $n\geq 0$.
\begin{lem}\label{rigidanalytic} For all $n\in\mathbb{N}$ the power series $h_n$ is a rigid analytic group automorphism of ${\bf B}(r_n)_{\mathbb{C}_p}$. \end{lem} \begin{pr} We give the details of a proof sketched in the unpublished note \cite{ST2b}. Let $\G(X,Y)\ino[[X,Y]]$ be the formal group law underlying $\G$. Using basic properties of $\exp_\G$ and $\log_\G$, one computes that $h_n(Z)$ equals \begin{equation}\label{group1} \exp_\G(\G_a(\log_\G(Z),(\omega_n/\omega-1)\log_\G(Z)))=\G(Z,\exp_\G((\omega_n/\omega-1)\log_\G(Z)))\end{equation} as formal power series over $\mathbb{C}_p$ where $\G_a$ denotes the formal additive group. By \cite{ST2}, Lem. 3.2 we have $[p^n].{\bf B}(r_n)={\bf B}(r)$ whence \[ p^n\log_\G({\bf B}(r_n))=\log_\G([p^n].{\bf B}(r_n))=\log_\G({\bf B}(r))={\bf B}(r)\] where the last identity follows from \cite{LangCyc}, Lem. \S8.6.4. Hence, on ${\bf B}(r_n)$ we have the composite of the rigid analytic functions \begin{equation}\label{group2} {\bf B}(r_n)_{\mathbb{C}_p}\stackrel{\log_\G}{\longrightarrow}p^{-n}{\bf B}(r)_{\mathbb{C}_p} \stackrel{(\omega_n/\omega)-1}{\longrightarrow}{\bf B}(r)_{\mathbb{C}_p} \stackrel{\exp_\G}{\longrightarrow}{\bf B}(r)_{\mathbb{C}_p}.\end{equation} Using that the group law $\G$ is defined over $o$ it follows that $h_n$ is a rigid analytic function on ${\bf B}(r_n)_{\mathbb{C}_p}$. Applying the same reasoning to the formal inverse $h_n^{-1}(Z)=\exp_\G((\omega/\omega_n)\log_\G(Z))$ shows that $h_n$ is a rigid automorphism of ${\bf B}(r_n)_{\mathbb{C}_p}$. It is clear from the definition of $h_n$ that it respects the Lubin-Tate group structure on ${\bf B}(r_n)_{\mathbb{C}_p}$.\end{pr}
We fix once and for all a chain of finite Galois extensions \[L:=L_0\subseteq L_1\subseteq...\subseteq L_\infty\] of $L$ with the property $\omega_n\in L_n$.
\begin{lem}\label{descent} The group isomorphism \[\kappa\circ h_n:{\bf B}(r_n)\,\hat{\otimes}_L\mathbb{C}_p\stackrel{\cong}{\longrightarrow}\holn\,\hat{\otimes}_L\mathbb{C}_p\] is already defined over the finite extension $L_n$ of $L$. \end{lem} \begin{pr} Denote by $B_n$ and $A_n$ the affinoid algebras of ${\bf B}(r_n)$ and $\holn$ respectively. We have obvious actions of the Galois group \[G_n:=Gal(\bar{L}/L_n)\] on ${\bf B}(r_n)_{\mathbb{C}_p}, B_n\,\hat{\otimes}_L\mathbb{C}_p$ and ${\rm Aut}_{\mathbb{C}_p}(B_n\,\hat{\otimes}_L\mathbb{C}_p)$. Here, the latter refers to the group of $\mathbb{C}_p$-algebra automorphisms of $B_n\,\hat{\otimes}\mathbb{C}_p$. Given $\sigma\in G_n$ and a $\mathbb{C}_p$-valued point $z$ of ${\bf B}(r_n)$ we find \[ h_n^{-1}(\sigma(h_n(\sigma^{-1}(z))))=\exp_\G((\omega/\omega^\sigma)\log_\G(z))=\exp_\G(\tau'(\sigma)^{-1}\log_\G(z)) =[\tau'(\sigma)^{-1}].z\] where the middle identity and the final identity come from (\ref{period}) and (\ref{explog}) respectively. Denoting by $h_n^\sharp$ the algebra automorphism associated to $h_n$ it follows for each $\sigma\in G_n$ that \[\sigma. h_n^\sharp=h_n^\sharp\circ[\tau'(\sigma)^{-1}]\] in ${\rm Aut}_{\mathbb{C}_p}(B_n\,\hat{\otimes}_L\mathbb{C}_p)$. By (\ref{cocycle}) the cocycle \[G_L\rightarrow {\rm Aut}_{\mathbb{C}_p}(B_n\,\hat{\otimes}_L\mathbb{C}_p),~\sigma\mapsto[\tau'(\sigma)^{-1}]\] gives the descent datum for the twisted form $\kappa^\sharp: A_n\,\hat{\otimes}_L\mathbb{C}_p\stackrel{\cong}{\longrightarrow}B_n\,\hat{\otimes}_L\mathbb{C}_p$. According to the usual formalism of Galois descent (cf. \cite{KnusOjanguren}, \S9) we may therefore conclude that the $\mathbb{C}_p$-algebra automorphism \[h_n^\sharp\circ\kappa^\sharp:A_n\,\hat{\otimes}_L\mathbb{C}_p\car B_n\,\hat{\otimes}_L\mathbb{C}_p\] is $G_n$-equivariant. By the existence of topological $L$-bases for $A_n$ and $B_n$, in the sense of \cite{NFA}, Prop. 10.1, taking $G_n$-invariants and applying Tate's theorem $\mathbb{C}_p^{G_n}=L_n$ (cf. \cite{Tatediv}, Prop. 3.1.8) yields the claim. \end{pr}
Remark: Using that $F_{t'}(Z)=\exp(\omega\log_\G(Z))$ as power series over $\mathbb{C}_p$ (cf. \cite{ST2}, \S4) the isomorphism $\kappa\circ h_n$ of the preceding proposition is given on $\mathbb{C}_p$-points $z$ of ${\bf B}(r_n)$ via \[ (\kappa\circ h_n)_z(g)=\exp(g\omega_n\log_\G(z)),~g\ino.\]
\begin{prop}\label{prop-cocycle2} The Galois cocycle of the twisted form $\hol_n$ of ${\bf B}(r_n)$ with respect to the finite extension $L_n/L$ is given by $$\sigma\mapsto \exp_\G((\omega_n/\omega_n^{\sigma})\log_\G(Z))$$ where $\sigma\in Gal(L_n/L)$. \end{prop} \begin{proof}
Let $\sigma\in Gal(L_n/L)$ and let $\tilde{\sigma}\in G_L$ be any extension to $\bar{L}$. The value of the cocycle on $\sigma$ equals the element
$$(h_n^\sharp\kappa^\sharp)\sigma (h_n^\sharp\kappa^\sharp)^{-1} \sigma^{-1}=h_n^\sharp\kappa^\sharp\sigma(\kappa^\sharp)^{-1}(h_n^\sharp)^{-1}\sigma^{-1} =h_n^\sharp(\kappa^\sharp\tilde{\sigma}(\kappa^\sharp)^{-1}\tilde{\sigma}^{-1}) \tilde{\sigma}(h_n^\sharp)^{-1}\sigma^{-1}$$ in the automorphism group of the $L_n$-affinoid algebra $\mathcal{O}({\bf B}(r_n))\otimes_L L_n$, cf. \cite{KnusOjanguren},\S9. As usual, we identify this automorphism with its value on the parameter and consequently, with the power series
\[ \begin{array}{cl} h_n^\sharp(\kappa^\sharp\tilde{\sigma}(\kappa^\sharp)^{-1}\tilde{\sigma}^{-1}) \tilde{\sigma}(h_n^\sharp)^{-1}(Z) & \stackrel{(\ref{cocycle})}{=}h_n^\sharp[\tau'(\sigma)^{-1}]\tilde{\sigma}(h_n^\sharp)^{-1}(Z) \\
& \\
&\stackrel{(\ref{explog}),(\ref{period})}{=}h_n^\sharp\exp_\G((\omega/\omega^{\tilde{\sigma}})\log_\G)\tilde{\sigma}(h_n^\sharp)^{-1}(Z) \\
& \\
&\stackrel{(*)}{=}\exp_\G((\omega_n/\omega)(\omega/\omega^{\tilde{\sigma}})(\omega^{\tilde{\sigma}}/\omega_n^{\sigma})\log_\G(Z))\\ & \\
&=\exp_\G((\omega_n/\omega_n^{\sigma})\log_\G(Z)).\\
\end{array} \] Here, we used the identity $$\tilde{\sigma}.(h_n^\sharp)^{-1}(Z)=\exp_\G((\omega^{\tilde{\sigma}}/\omega_n^\sigma)\log_\G(Z))\in Z\mathbb{C}_p[[Z]]$$ in $(*)$. \end{proof}
We now consider for each fixed $n$ the base extension $\holnk$ of $\holn$ to $K$. Again, if $n$ varies these affinoids give a countable increasing open admissible covering of $\hol_K$. As a result of our discussion each $\holnk$ is a twisted form of the Lubin-Tate group on ${\bf B}(r_n)_K$ trivialized over $K_n:=KL_n$ by $\kappa\circ h_n$ and the Galois cocycle giving the descent datum is given by the preceding proposition.
\subsection{An \'etale covering} We briefly like to indicate an alternative intrinsic characterization of the twisted form $\hol$ and its affinoid subdomains $\holn$. This builds on the analytic mapping properties of the logarithm $$\lambda:=\log_\G$$ associated to the Lubin-Tate group $\G$. It is best formulated in the language Berkovich analytic spaces \cite{BerkovichBook},\cite{BerkovichEtale}. It also involves the beginnings of A.J. de Jong's theory of \'etale covering maps for Berkovich spaces \cite{deJongET}.
Let ${\mathbb B}$ and ${\mathbb A}^1$ be the Berkovich analytic spaces over $L$ equal to the one dimensional open unit disc around zero and the affine line respectively. The logarithm $$\lambda: {\mathbb B}\longrightarrow{\mathbb A}^1$$ is an \'etale and surjective morphism, cf. \cite{RameroET}, Lem. 6.1.1. Let $\bar{L}_\infty$ be the closure of $L_\infty\subseteq\mathbb{C}_p$ and $G_\infty:=Gal(L_\infty/L)$. We endow the space ${\mathbb B}_{\bar{L}_\infty}$ with the semilinear Galois action associated to the cocycle (\ref{cocycle}). Similarly, we endow the space ${\mathbb A}^1_{\bar{L}_\infty}$ with the semilinear Galois action associated to the cocycle $$\sigma\mapsto \tau'(\sigma)^{-1}Z.$$ Using (\ref{explog}) it is elementary to see that with these definitions the map $\lambda_{\bar{L}_\infty}$ becomes equivariant. \begin{lem} The Galois descent of ${\mathbb A}^1_{\bar{L}_\infty}$ is canonically isomorphic to ${\mathbb A}^1_L$. \end{lem} \begin{proof} The semilinear Galois action on ${\mathbb A}^1_{\bar{L}_\infty}$ respects the natural increasing covering by closed discs around zero. Let
$s_n$ be a family of real numbers in $|\bar{L}^\times|$ tending towards infinity. Let $B_n$ be the affinoid algebra of ${\mathbb B}(s_n)_{\bar{L}_\infty}$. Let $z\in B_n$ be a parameter, $b:=\sum_{m\geq 0}a_mz^m\in B_n$ and $\sigma\in G_\infty$. By (\ref{period}) we have $$\sigma.(\sum_{m\geq 0}a_mz^m)=\sum_{m\geq 0} a_m^\sigma (\omega/\omega^\sigma)^mz^m$$ and thus $\sigma.b=b$ if and only if $a_m/\omega^m=(a_m/\omega^m)^\sigma$ for all $m\geq 0$. Consequently, $b$ is Galois invariant if and only if $a_m/\omega^m\in L$ for all $m\geq 0$. It follows that the subring of Galois invariants in $B_n$ is given by
$$B_n^{G_\infty}=\{\sum_{m\geq 0}c_m(\omega z)^m, c_m\in L, |c_m| (|\omega|s_n)^m\rightarrow 0
{\rm~for~} n\rightarrow\infty\}.$$ It is canonically isomorphic to the affinoid algebra of the closed disc of radius $|\omega|s_n$. For varying $s_n$ these isomorphism glue to a canonical isomorphism between the descent of ${\mathbb A}^1_{\bar{L}_\infty}$ and ${\mathbb A}^1_L$. \end{proof} We point out here that
$$|\omega|=p^\nu {\rm ~~~with~}\nu=\frac{1}{p-1}-\frac{1}{e(q-1)}$$ according to \cite{ST2}, Lemma 3.4b. Let $\hol^{an}$ be the Galois descent of ${\mathbb B}_{\bar{L}_{\infty}}$. The Galois descent of the map $\lambda_{\bar{L}_\infty}$ is a morphism $$\hol^{an}\longrightarrow{\mathbb A}^1_L$$ between Berkovich analytic spaces over $L$. In the following we show that it is \'etale and surjective and therefore an \'etale covering of the affine line over $L$.
\vskip8pt
To do this we make use of the arguments in proof of \cite{RameroET}, Lemma 6.1.1. For $n\geq 1$ we let
$$s_n:=r|\pi|^{-en}$$ and let ${\mathbb B}(s_n)\subset{\mathbb A}^{1}$ be the closed disc of radius $s_n$. We let $E(s_n)$ be the connected component containing zero of $\lambda^{-1}({\mathbb B}(s_n))$ so that the induced map $$\lambda: E(s_n)\longrightarrow {\mathbb B}(s_n)$$ is finite \'etale and surjective. As in loc.cit. one obtains that $E(s_n)$ is the connected component containing zero of the inverse image of ${\mathbb B}(r)$ by $[\pi^{en}]$. Indeed, since $r<p^{-1/e(q-1)}$, the maps $\log_\G$ and $\exp_\G$ induce mutually inverse isomorphisms of ${\mathbb B}(r)$ (\cite{LangCyc}, \S8.6 Lemma 4) so that $\exp_\G(\pi^{en}{\mathbb B}(s_n))={\mathbb B}(r)$. Next, we have $[\pi](Z)=\pi Z+Z^q$ and so, by the arguments given in the proof of \cite{ST2}, Lemma 3.2 one has $$E(s_n)=[\pi^{en}]^{-1}({\mathbb B}(r))={\mathbb B}(r^{1/q^{en}})={\mathbb B}(r_n).$$ As a result of this discussion we have a finite \'etale surjective morphism $$\lambda: {\mathbb B}(r_n)\longrightarrow{\mathbb B}(s_n)$$ for any $n\geq 1$. Using the compatibility with $\lambda$ and $[\pi]$ one finds that its degree equals $q^{en}$. The same properties hold for its base change from $L$ to the field $L_n$. We endow the space ${\mathbb B}(r_n)_{L_n}$ with the semilinear $Gal(L_n/L)$-action associated to the cocycle from Proposition \ref{prop-cocycle2}. Similarly, we endow the space ${\mathbb B}(s_n)_{L_n}$ with the semilinear $Gal(L_n/L)$-action associated to the cocycle $$\sigma\mapsto (\omega_n/\omega_n^\sigma)(Z).$$ This makes the map $\lambda_{L_n}: {\mathbb B}(r_n)_{L_n}\rightarrow {\mathbb B}(s_n)_{L_n}$ equivariant. The following lemma is proved in the same way as the preceding one. \begin{lem}
The Galois descent of ${\mathbb B}(s_n)_{L_n}$ is canonically isomorphic to the closed disc ${\mathbb B}(t_n)$ where $t_n:=|\omega_n|s_n$. \end{lem} We let $\holn^{an}$ be the Galois descent of ${\mathbb B}(r_n)_{L_n}$. The above discussion yields a finite \'etale surjective morphism $\holn^{an}\rightarrow{\mathbb B}(t_n)$ whose degree equals $q^{en}$. Taking the union of these maps over all $n$ we find that the morphism $$\hol^{an}\longrightarrow{\mathbb A}^1_L$$ is \'etale and surjective. \subsection{Cohomology and inverse limits}\label{cohomologylimits} To deal with the Picard group of the rigid analytic space $\hol_K$ we need to establish a general result on sheaf cohomology and inverse limits.
Recall that a rigid analytic space $X$ is called {\it quasi-Stein} if there is a countable increasing admissible open affinoid covering $\{X_n\}_{n\in\mathbb{N}}$ of $X$ such that the restriction maps $\mathcal{O}(X_{n+1})\longrightarrow\mathcal{O}(X_n)$ have dense image (cf. \cite{Kiehl}, Def. 2.3).
Let $X$ be a fixed quasi-Stein space and let $\{X_n\}$ be a defining covering. Let $Ab(X)$ be the category of abelian sheaves on $X$. \begin{prop}\label{universal} For any $\mathcal{F}\in Ab(X)$ there is a short exact sequence \[0\longrightarrow \underleftarrow{\lim}^{(1)}_n\,\mathcal{F}(X_n) \longrightarrow H^1(X,\mathcal{F})\longrightarrow\underleftarrow{\lim}_n H^1(X_n,\mathcal{F})\longrightarrow 0\] where the right surjection is induced by the inclusions $X_n\rightarrow X$ for each $n\in\mathbb{N}$ and $H^1$ denotes sheaf cohomology. \end{prop} \begin{pr} Our argument is based on the ideas in \cite{SS}, Prop. 2.4. Let $Proj_\mathbb{N}(Ab)$ be the category of $\mathbb{N}$-projective systems over the category of abelian groups $Ab$ and let $F:=\underleftarrow{\lim}_n$ be the projective limit viewed as an additive functor $Proj_\mathbb{N}(Ab)\rightarrow Ab$. Consider the additive functor $G: Ab(X)\rightarrow Proj_\mathbb{N}(Ab),~\mathcal{F}\mapsto (\mathcal{F}(X_n))_n.$ Both $F$ and $G$ are left exact. For any $\mathcal{F}\in Ab(X)$ there is a short exact sequence \[0\longrightarrow (R^1F)(G\mathcal{F})\longrightarrow R^1(FG)(\mathcal{F})\longrightarrow (FR^1G)(\mathcal{F})\longrightarrow 0.\] Indeed, an injective sheaf $\mathcal{I}\in Ab(X)$ is flasque
whence the system $G(\mathcal{I})$ is surjective and therefore $F$-acyclic. The five-term exact sequence associated to the Grothendieck spectral sequence (e.g. \cite{WeibelH} Thm. 5.8.3) \[E_2^{pq}:=(R^pF)(R^qG)(\mathcal{F})\Rightarrow R^{p+q}(FG)(\mathcal{F})\] yields the desired short exact sequence since $R^2F=0$ (cf. [loc.cit.], Cor. 3.5.4).
After this observation let $res_n: Ab(X)\rightarrow Ab(X_n)$ be the restriction functor. It is almost immediate that we have a cohomological $\delta$-functor $(T^{i})_{i\geq 0}: Ab(X)\rightarrow Proj_\mathbb{N}(Ab)$ where \[T^{i}:=(H^{i}(X_n, res_n\circ(.)))_n.\]
On any $G$-ringed space the restriction of an injective sheaf to an admissible open subset remains injective (cf. \cite{HartshorneO}, I.\S2). Hence each $res_n$ preserves injectives. Since $Ab(X)$ has enough injectives it follows that each $T^{i},~i>0$ is effaceable and therefore $(T^{i})_{i\geq 0}$ is universal.
It now follows that the right-derived functors of the functor $G$ are given by $(T^{i})_{i\geq 0}$ and it remains to observe that the sheaf property yields $F\circ G=\Gamma$, the global section functor. \end{pr} \subsection{Picard groups} In this subsection we prove the structure result on the Picard group $Pic(\hol_K)$. \begin{prop}\label{shortexact2} There is an exact sequence\footnote{We turn back to our earlier convention and denote the unit element in an abelian group by $1$.} \[1\longrightarrow \underleftarrow{\lim}^{(1)}_n \mathcal{O}(X_n)^\times \longrightarrow Pic (X)\longrightarrow\underleftarrow{\lim}_n Pic(X_n)\longrightarrow 1.\] \end{prop} \begin{pr} Let $\mathcal{O}_X$ be the structure sheaf of $X$. We apply the last result of the preceding subsection to $\mathcal{F}=\mathcal{O}_X^\times$ and use $H^1(X,\mathcal{O}_X^\times)=Pic(X)$ (cf. \cite{FresnelVanderPut}, Prop. 4.7.2). \end{pr}
Remark: The vanishing of the $\lim^{(1)}$-term in the above sequence depends heavily on the base field of $X$. To give an example let for a moment $\mathbb{Q}_p\subseteq K\subseteq \mathbb{C}_p$ by an arbitrary complete field. The open unit disc ${\bf B}_K$ over $K$ is quasi-Stein. A defining covering is provided by the closed subdiscs ${\bf B}(r)_K, r<1$. In this situation $\varprojlim_r^{(1)}\mathcal{O}({\bf B}(r)_K)^\times=1$ is equivalent to $K$ being spherically complete. This is a consequence of work of M. Lazard, cf. \cite{Lazardzeros}, Thm. 2 and Prop. 6.
\vskip8pt
Recall that the {\it dimension} of $X$ at a point $x\in X$ equals the dimension of the local ring at $x$. \begin{lem}\label{equivalence} Let $X$ be equidimensional with finitely many connected components. Taking global sections furnishes an equivalence of exact categories between vector bundles on $X$ and finitely generated projective $\mathcal{O}(X)$-modules. In particular, $K_0(X)\cong K_0(\mathcal{O}(X))$ and $Pic(X)\cong Pic(\mathcal{O}(X))$ canonically. \end{lem} \begin{pr} Following \cite{GrusonFi}, Remarque V.1. one may apply {\it mutatis mutandis} the arguments appearing in the proof of [loc.cit.], Thm. V.1 on each connected component of $X$. \end{pr}
The space $\hol_K$ is a quasi-Stein space with respect to the covering given by the affinoids $\holnk$. This follows from the definition (\ref{quasistein1}) and the fact that ${\bf B}^{[L:\mathbb{Q}_p]}$ is quasi-Stein with respect to the covering given by the ${\bf B}^{[L:\mathbb{Q}_p]}(r_n)$ (cf. remark above). \begin{prop}\label{inverselimits} The canonical homomorphism $$Pic(\hol_K)\car\underleftarrow{\lim}_n Pic(\holnk)$$ is bijective. \end{prop} \begin{proof} We let $A=\mathcal{O}(\hol_K)$ and $A_n:=\mathcal{O}(\holnk)$. By the preceding discussion it suffices to show that the natural homomorphism \[Pic(A)\longrightarrow \underleftarrow{\lim}_n Pic(A_n)\] is injective. To show this we use an adaption of the argument given in \cite{GrusonFi}, Prop. V.3.2. Let therefore $P$ be a projective rank $1$ module over $A$ whose class is in the kernel of this homomorphism. Let $P_n:=P\otimes_A A_n$. The natural restriction map $A_{n+1}\rightarrow A_n$ is injective since this is true for its base change to $\mathbb{C}_p$ (cf. (\ref{quasistein2})). Applying $P\otimes_A (\cdot)$ we have an injective map $ P_{n+1}\rightarrow P_n$ which we view as an inclusion. The first step now is to exhibit a well-chosen generator for each free $A_n$-module $P_n$. This is done by induction. Let $x_1$ be any generator for $P_1$. This starts the induction. To make the induction step suppose $x_n$ is a well-chosen generator of the free $A_n$-module $P_n$. Let $y_{n+1}$ be an arbitrary generator for $P_{n+1}$. Since the image of $y_{n+1}$ under the map $P_{n+1}\rightarrow P_n$ generates $P_n=P_{n+1}\otimes_{A_{n+1}}A_n$ we may write $y_{n+1}=a_nx_n$ with $a_n\in A_n^\times$. Let $$B_n:=\mathcal{O}({\bf B}(r_n)_{K_n}),~~~ U_n:=\{ b_0+b_1z+...\in B_n^\times: b_0=1\},~~~G_n:=G(K_n/K).$$ Passing to a finite extension of $K_n$ (if necessary) we have
$r_n\in |K_n^\times|$ and so, as in the proof of Lemma \ref{lem-sections}, a direct product decomposition $B_n^\times=U_nK_n^\times$. Since the $G_n$-action on ${\bf B}(r_n)_{K_n}$ preserves the origin taking invariants gives a direct product decomposition \begin{equation}\label{constant} A_n^\times=W_nK^\times.\end{equation} with $W_n:=(U_n)^{G_n}$. Let $a_n=wv$ with $w\in W_n, v\in K^\times$. We now let $x_{n+1}:=v^{-1}y_{n+1}$ which completes the induction step. Note that by construction $x_{n+1}\in W_nx_n$ for all $n$.
Now $U_n=1+U'_n$ with an $o_{K_n}$-module $U'_n$ and $W_n=1+(U'_n)^{G_n}$ with the $o_K$-module $(U'_n)^{G_n}$. We see that $$V_n:=W_nx_n\subseteq P_n$$ is a convex subset of the $L$-Banach space $P_n$. Moreover, $B_{n+1}\subseteq B_n$ implies $U_{n+1}\subseteq U_n$ and hence $W_{n+1}\subseteq W_n$. Thus, $V_{n+1}=W_{n+1}x_{n+1}\subseteq W_nx_n=V_n$ and so $$V_{n+1}\subseteq V_n$$ for all $n$. Now the map $P_{n+1}\rightarrow P_{n}$ is a compact linear map between $L$-Banach spaces (cf. \cite{ST5}, Lem. 6.1). This implies (\cite{NFA}, Remark 16.3) that the image of $V_{n+1}$ in $V_n$ is relatively compact. We may therefore argue as in \cite{GrusonFi}, Prop. V.3.2 to obtain that $\cap_n V_n\neq\emptyset$. By Theorem B for quasi-Stein spaces (\cite{Kiehl}) any nonzero element in this intersection generates $P$. Then $P$ is free on this generator since ${\rm Ann}_A(x)\subseteq {\rm Ann}_{A_n}(x)=0$. Hence the class of $P$ in $Pic(A)$ is trivial. \end{proof}
\begin{prop} The group $Pic(\holnk)=Cl(\holnk)$ is finite for all $n\geq 1$. \end{prop} \begin{proof}
We pass to a finite extension of $K_{n+1}$ (if necessary) to obtain $r_n,r_{n+1}\in |K_{n+1}^\times|$. Let $G=Gal(K_{n+1}/K)$. We equip ${\bf B}(r_n)_{K_{n+1}}$ and ${\bf B}(r_{n+1})_{K_{n+1}}$ with the semilinear Galois action coming from the cocycle $$\sigma\mapsto \exp_\G((\omega_{n+1}/\omega_{n+1}^{\sigma})\log_\G(Z))$$ for $\sigma\in G$ (Lemma \ref{prop-cocycle2}). By the Lemmas \ref{rigidanalytic} and \ref{descent} this is an action by group automorphisms with respect to the Lubin-Tate group structure on these discs. In particular, the origin of these discs remains fixed under this action. We let $B_i:=\mathcal{O}({\bf B}(r_i)_{K_{n+1}})$ and $A_i:=\mathcal{O}(\hol_{K,i})$ for $i=1,2$. The canonical inclusion $B_2\rightarrow B_1$ is thus equivariant. Since $\omega_{n+1}$ also satisfies the defining condition (\ref{condition}) for $\omega_n$ we have $A_n=(B_n)^G$ (and of course $A_{n+1}=(B_{n+1})^G$). Thus, taking $G$-invariants of $B_2\rightarrow B_1$ yields the restriction map $A_2\rightarrow A_1$. Moreover, by the very definition of $\hol_{K,n}$ the inclusion $\hol_{K,n}\subseteq\hol_{K,n+1}$ identifies the source with an affinoid subdomain in the target. Since $K_{n+1}$ is locally compact we may therefore apply Prop. \ref{prop-finite} which yields the assertion. \end{proof}
We now prove that, in case $L\neq\mathbb{Q}_p$, the group $Pic(\hol_K)$ is not a finite group. We need two auxiliary lemmas. \begin{lem}\label{lem-codim} Let $R$ and $R'$ be two normal $o_K$-algebras of topologically finite type which are flat over $o_K$. Suppose $R$ is an integral domain. Let $R\rightarrow R'$ be a finite ring extension such that $R\otimes_{o_K}K\rightarrow R'\otimes_{o_K}K$ is a finite free extension of degree, say, $m$. Let $f:Spec(R')\rightarrow Spec(R)$ be the associated morphism. There is a closed subset $V\subset Spec(R)$ with ${\rm codim}_{Spec(R)}V\geq 2$ such that the induced morphism $$Spec(R')\setminus f^{-1}(V)\longrightarrow Spec(R)\setminus V$$ is finite flat of degree equal to $m$. \end{lem}
\begin{proof} We argue along the lines of \cite{deJongCrystalline}, Lemma 7.3.2. Since $R$ is noetherian we may consider a finite presentation of the $R$-module $R'$ $$R^{m_1}\stackrel{\alpha}{\longrightarrow}R^{m_0}\longrightarrow R'\longrightarrow 0.$$ We may assume here that $0\leq m_0-m\leq m_1$. We consider the $m$-th Fitting ideal $F_m(R')$ of the $R$-module $R'$ (\cite{Eisenbud}, III.20.2), i.e. the ideal of $R$ generated by the $(m_0-m,m_0-m)$-minors of the matrix $\alpha$. After inverting a prime element of $o_K$ the $R$-module $R'$ is generated by $m$ elements. This implies $F_m(R')\neq 0$ (loc.cit. Cor. 20.5/Prop. 20.6). On the complement of $V:=Spec(R/F_m(R'))$ the ideal $F_m(R')$ becomes invertible. According to \cite{BoschLuetkeII}, Lemma 3.14 the $R$-module $R'$ is therefore locally free of rank $m$ over the open set $Spec(R)\setminus V$. Finally, using the normality of $R$ and $R'$ it follows as in the proof of \cite{deJongCrystalline}, Lemma 7.3.2 that ${\rm codim}_{Spec(R)} V\geq 2$. \end{proof}
For any rigid $K$-analytic space $X$ we denote by $\mathcal{O}(X)^0\subseteq\mathcal{O}(X)$ the $K$-subalgebra consisting of holomorphic functions which are bounded by $1$. \begin{lem} Let $f: X\rightarrow{\bf B}_K$ be a finite morphism of rigid $K$-analytic spaces.
Suppose the induced homomorphism $\mathcal{O}({\bf B}_K)^0\rightarrow\mathcal{O}(X)^0$ is an integral ring extension.
Then any function $F\in\mathcal{O}(X)^0$ has only finitely many zeroes on
$X$. \end{lem} \begin{proof} Let $z$ be a parameter on the disc ${\bf B}_K$. According to \cite{deJongCrystalline}, Lem. 7.3.4, we have $\mathcal{O}({\bf B}_K)^0=o_K[[z]]$, the ring of formal power series over $o_K$ in the variable $z$. Let $H\in \mathcal{O}(X)^0$ and consider an equation $$ H^m+b_{1}H^{m-1}+...+b_m=0$$ with $b_i\in\mathcal{O}({\bf B}_K)^0$. Since $\mathcal{O}({\bf B}_K)^0$ is an integral domain we may assume $b_m\neq 0$. If $H(x)=0$ for some $x\in X$ then $b_m(f(x))=0$. By the Weierstrass preparation theorem for $o_K[[X]]$, \cite{B-CA}, VII, \S3.8 Prop. 6, the power series $b_m$ has at most finitely many zeroes. Since $f$ has finite fibres according to \cite{BGR}, Cor. 9.6.3/6, we conclude that $H$ has at most finitely many zeroes on $X$. \end{proof}
\begin{prop} Let $L\neq\mathbb{Q}_p$. The ideal sheaf defining the zero section of the rigid group $\hol_K$ is not a torsion element in $Pic(\hol_K)$. \end{prop} \begin{pr} Abbreviate $A:=\mathcal{O}(\hol_K),~B:=\mathcal{O}({\bf B}_K)$ and $B_{\mathbb{C}_p}:=\mathcal{O}({\bf B}_{\mathbb{C}_p})$ and let $z\in B$ be a parameter. Assume for a contradiction that the ideal sheaf in question is a torsion element. If $I\subseteq A$ denotes the corresponding ideal of global sections there is $1<m<\infty$ such that $$I^m=(f)$$ with some $f\in A$. Since the trivialization $\kappa$ preserves the origin we have $I\mapsto (z)$ via $Pic(A)\rightarrow Pic(B_{\mathbb{C}_p})$ whence $(f)\mapsto (z^m)$. Now consider $f$ as a rigid $K$-analytic map $\hol_K\rightarrow\mathbb{A}^1_K$ into the affine line over $K$. The composite \[{\bf B}_{\mathbb{C}_p}\stackrel{\kappa}{\longrightarrow}\hol_{\mathbb{C}_p}\stackrel{f_{\mathbb{C}_p}}{\longrightarrow}\mathbb{A}^1_{\mathbb{C}_p}\] is then given by a power series $F$ generating the ideal $(z^m)$ of $B$ whence
\[F(z)=az^m(1+b_1z+b_2z^2+...)\] with $a\in\mathbb{C}_p^\times,~b_i\in o_{\mathbb{C}_p}$ according to \cite{HopkinsGrossLT}, Prop. 18.7. We may pass to a finite extension of $K$ (if necessary) and have an element $x\in K$ with $|x|=|a|$. Passing to $x^{-1}f$ we see that $F$ induces a rigid $\mathbb{C}_p$-analytic map ${\bf B}_{\mathbb{C}_p}\rightarrow{\bf B}_{\mathbb{C}_p}$ being the union over $\mathbb{C}_p$-affinoid maps ${\bf B}(r_n)_{\mathbb{C}_p}\rightarrow{\bf B}(r_n)_{\mathbb{C}_p}$. Hence, $f$ is in fact a rigid $K$-analytic map \[f: \hol_K\rightarrow{\bf B}_K\] equal to the union of $K$-affinoid maps $f_n: \holnk\rightarrow{\bf B}(r_n)_K$. The latter are induced by functions $f_n\in\mathcal{O}(\holnk)$ generating $I^m\mathcal{O}(\holnk)$, the $m$-th power of the ideal defining the zero section $Sp~K\rightarrow\holnk$.
Next, we show that the $\mathcal{O}_{{\bf B}_K}$-module $f_*(\mathcal{O}_{\hol_K})$ is locally free. Denote by $f^\sharp$ and $f_n^\sharp$ the corresponding ring homomorphisms on global sections. Fix $n$ and let $$A_n:=\mathcal{O}(\holnk),~~~~B_n:=\mathcal{O}({\bf B}(r_n)_{K_n}).$$ We pass to a finite extension of $K_n$ (if necessary) and have an element
$a_n\in K_n$ such that $|a_n|=r_n$. Identifying $A_n\otimes_K K_n\simeq B_n$ via the group isomorphism $\kappa\circ h_n$ the map $$f_n^\sharp\otimes_K K_n: B_n\longrightarrow B_n$$ is given by a power series in $B_n$ defining the $m$-th power of the zero section $Sp~K_n\rightarrow {\bf B}(r_n)_{K_n}$. Hence $f_n^\sharp\otimes_K K_n$ equals the map $(a_n^{-1}z)\mapsto
(a_n^{-1}z)^m\epsilon$ with suitable $\epsilon\in B_n^\times$. Since $|a_n^{-1}z|=1$ this is an isometry with associated graded map \[\gor (f_n^\sharp\otimes_K K_n): \sigma(a_n^{-1}z)\mapsto \sigma(a_n^{-1}z)^m\sigma(\epsilon)\] and $\sigma(\epsilon)\in (\gor K_n)^\times$. Here, $\gor$ denotes the reduction functor from $K_n$-affinoid algebras into graded $(\gor K_n)$-algebras introduced by M. Temkin \cite{TemkinII}. Clearly, the homomorphism $\gor (f_n^\sharp\otimes_K K_n)$ is finite free of rank $m$ on the homogeneous basis elements $\sigma(a_n^{-1}z)^0,...,\sigma(a_n^{-1}z)^{m-1}$. Hence, $f_n^\sharp\otimes_K K_n$ is finite free of rank $m$ according to \cite{LVO}, Lem. I.6.4. By faithfully flat descent $A_n$ is therefore a finitely generated projective $B_n$-module of rank $m$ via $f_n^\sharp$. This shows the $\mathcal{O}_{{\bf B}_K}$-module $f_*(\mathcal{O}_{\hol_K})$ to be a vector bundle of rank $m$.
By \cite{GrusonFi}, V.2 Remarque $3^o$ this vector bundle must be trivial and so $f^\sharp$ induces a finite free ring extension $B\rightarrow A$ of degree $m$. Let $A^0$ and $B^0$ be the holomorphic functions on $\hol_K$ and ${\bf B}_K$ respectively that are bounded above by $1$. We claim that the ring extension $f^\sharp: B^0\rightarrow A^0$ is integral. To see this we argue along the lines of \cite{deJongCrystalline}, Lem. 7.3.3. Let $H\in A^0$. It satisfies an integral equation $$T^m+b_1T^{m-1}+...+b_m=0$$ with $b_i\in B$. We consider the ring extension induced by $f^\sharp_n$ $$R:=\mathcal{O}({\bf B}(r_n)_K)^0\stackrel{\subseteq}{\longrightarrow} R':=\mathcal{O}(\holnk)^0.$$ It is a finite extension by \cite{BGR}, Cor. 6.4.1/6. Since ${\bf B}_K$ and $\hol_K$ are normal, $R$ and $R'$ are normal $o_K$-algebras of topologically finite type which are flat over $o_K$. Clearly, $R$ is an integral domain. Applying the Lemma \ref{lem-codim} we find a closed set $V\subset Spec(R)$ with ${\rm codim}_{Spec(R)}V\geq 2$ such that the extension $R\rightarrow R'$ is finite flat of degree $d$ over $Spec(R)\setminus V$. Consider $H$ as an element of $R'$ via the natural restriction map $A\rightarrow \mathcal{O}(\holnk)$. It then satisfies an equation $$T^m+b'_1T^{m-1}+...+b'_m=0$$ with $b_i'\in\Gamma(Spec(R)\setminus V, \mathcal{O}_{Spec(R)})=\Gamma(Spec(R),\mathcal{O}_{Spec(R)})=R$. Comparing these $b_i'$ to the $b_i$ above we see that they must be equal as elements of $\mathcal{O}({\bf B}(r_n)_K)$. Consequently, each $b_i$ is of norm $\leq 1$ on ${\bf B}(r_n)_K\subset{\bf B}_K$ for all $n$ which means $b_i\in B^0$. This shows $f^\sharp: B^0\rightarrow A^0$ to be an integral extension. By the preceding lemma we see that any element of $A^0$ has at most finitely many zeroes on $\hol_K$. But according to (the proof of) \cite{ST2}, Lem. 3.9 the nonzero holomorphic function on $\hol_K$ given on $\mathbb{C}_p$-valued points via $$\kappa_z\mapsto\kappa_z(1)-\kappa_z(0)$$ is bounded above by $1$ and has infinitely many zeroes. So we have arrived at a contradiction. \end{pr}
As a result of the above discussion we have the following theorem. \begin{theo}\label{pro-p} The group $Pic(\hol_K)$ is a profinite group. In case $L\neq\mathbb{Q}_p$ the isomorphism class of the ideal sheaf defining the zero section $Sp~K\rightarrow\hol_K$ is an element of infinite order in $Pic(\hol_K)$. \end{theo}
We briefly explain the relation of $Pic(\hol_K)$ to the Grothendieck group $K_0(\hol_K)$. By the Lemma \ref{equivalence} the latter coincides with $K_0(A)$ where $A=\mathcal{O}(\hol_K)$.
\vskip8pt
For the following basic notions from algebraic $K$-theory we refer to \cite{Bass}. Let $R$ be a commutative associative unital ring, $Pic(R)$ its Picard group and $K_0(R)$ its Grothendieck group. Mapping $1\mapsto [R]$ induces an injective group homomorphism $\mathbb{Z}\rightarrow K_0(R)$ which factores through the kernel of the determinant $\det: K_0(R)\rightarrow Pic(R)$.
Denoting by $H_0(R)$ the abelian group of continuous maps $Spec(R)\rightarrow\mathbb{Z}$ we have the rank mapping ${\rm rk}: K_0(R)\rightarrow H_0(R)$
and a surjection \begin{equation}\label{ss}{\rm rk}\oplus\det: K_0(R)\rightarrow H_0(R)\oplus Pic(R).\end{equation} The kernel $SK_0(R)$ consists of classes $[P]-[R^n]$ where $P$ has constant rank, say, $n$ and $\wedge^n P\cong R$. If $R$ is noetherian of dimension one or a Pr\"uferian domain (i.e. an integral domain such that any finitely generated ideal is invertible) then $SK_0(R)=0$. Indeed, in both cases Serre's theorem (loc.cit., Thm. IV.2.5) yields that any finitely generated projective module is isomorphic to a direct sum of an invertible and a free module (cf. also \cite{GrusonFi}, V.2 Remarque $3^{o}$).
By \cite{ST2}, final discussion in sect. 3, the algebra of global sections $\mathcal{O}(\hol_K)$ is a Pr\"uferian domain which implies the \begin{prop}\label{rkplusdet} The rank and determinant homomorphisms give a canonical isomorphism of abelian groups $${\rm rk}\oplus\det: K_0(\hol_K)\car \mathbb{Z}\oplus Pic(\hol_K).$$ \end{prop}
\vskip8pt
Remark: Let $\gor$ be the reduction functor for $K$-affinoids introduced by M. Temkin \cite{TemkinII}. In \cite{SchmidtDISC} the author develops a method to compute the Picard group of twisted forms of the closed unit disc which are '$\gor$-smooth', i.e. whose reduction is smooth over the 'graded field' $\gor K$. A large class of such forms is given by the tamely ramified ones. Since the field $L_\infty$ is generated by the torsion points of the $p$-divisible group $\G'$ it is not tamely ramified over $L$. Therefore, the form $\holn$ is generally not tamely ramified. Even worse, the descent datum of the form $\holn$ involves the logarithm series $\log_\G$ (Prop. \ref{prop-cocycle2}). To compute the reduction of $\holn$ seems to require therefore a detailed knowledge of the coefficients of $\log_\G$ which is not available. We have therefore not been able to prove that $\holn$ is $'\gor$-smooth'. In fact, we are rather sceptical about $\holn$ having this property.
\subsection{General character spaces} As explained above the space $\hol$ parametrizes the locally analytic characters of the additive group $o$. In the following we explain briefly why the problem of determining the Picard group of general character spaces essentially reduces to the case of (copies of) $\hol$.
\vskip8pt
So consider an arbitrary locally $L$-analytic group $Z$ which is abelian and topologically finitely generated. Let $d:={\rm dim}_L Z$. The following generalization of the rigid analytic character variety $\hol$ has been introduced by M. Emerton in \cite{EmertonA}, (6.4). Let $Rig(K)$ be the category of rigid analytic spaces over $K$. For each $X\in Rig(K)$ let $\hat{Z}(X)$ be the group of abstract group homomorphisms $Z\rightarrow\mathcal{O}(X)^\times$ with the property that for each admissible open affinoid subspace $U\subseteq X$ the map \[Z\rightarrow\mathcal{O}(X)^\times\stackrel{res}{\longrightarrow}\mathcal{O}(U)\] is a $\mathcal{O}(U)$-valued locally $L$-analytic function on $Z$. This defines a contravariant functor \[\hat{Z}_K: Rig(K)\rightarrow Ab\] which is in fact representable by a smooth rigid $K$-analytic group $\hat{Z}_K$ on a quasi-Stein space. Our character group $\hol_K$ corresponds to the case $Z=o$. The association $Z\mapsto\hat{Z}_K$ is a contravariant functor that converts direct products into fibre products over $K.$
Examples: Let $\Gm$ and $\mu_n$ be the rigid $K$-analytic multiplicative group and the rigid $K$-analytic group of roots of unity of order $n\geq 1$ respectively. Mapping a locally analytic character to its value on $1$ respectively on $1 {\rm~mod~} m$ induces group isomorphisms $$\hat{\mathbb{Z}}_K\car \Gm,~~~~~(\widehat{\mathbb{Z}/m\mathbb{Z}})_K\car \mu_m$$ (loc.cit.). \vskip8pt
To get a first impression of the space $\hat{Z}_K$ we look at dimension and number of connected components. Let $\mu\subseteq Z$ be the torsion subgroup of $Z$. By \cite{EmertonA}, Prop. 6.4.1 the inclusion of the unique maximal compact open subgroup $Z_0$ into $Z$ induces a (noncanonical) isomorphism $Z_0\times\mathbb{Z}^r\cong Z$ for some unique $r\geq 0$. Hence, there is a (noncanonical) isomorphism of rigid groups $$\hat{Z}_K\car \hat{Z_0}_K\times_K\Gm^r.$$ To proceed further we impose a mild condition on the group $Z_0$. First of all, being abelian profinite $Z_0$ contains a unique open pro-$p$-Sylow subgroup $Z_0(p)$. The torsion part $Z_0(p)^{\rm tor}$ of the latter group is finite and a direct factor so that $\mu$ is finite. Any complement $Z_0(p)^{\rm fl}$ (as $\mathbb{Z}_p$-module) in $Z_0(p)$ to $Z_0(p)^{\rm tor}$ has finite index in $Z_0(p)$ and hence is open according to \cite{DDMS}, Thm. 1.17. It is therefore naturally endowed with a structure of abelian locally $L$-analytic group. As such is has an open subgroup which is isomorphic, as locally $L$-analytic group, to the standard group $o^d$ (in the sense of \cite{B-L}, Thm. III. 7.3.4). We {\bf assume} in the following that the torsion part in $Z_0(p)$ admits a complement $Z_0(p)^{\rm fl}$ which is isomorphic, as locally $L$-analytic group to $o^d$. We fix such an isomorphism.
\vskip8pt Example: Let $\mathbb{T}$ be a linear algebraic torus over $L$ and $Z=\mathbb{T}(L)$ its group of $L$-rational points. Then $Z$ is topologically finitely generated. Indeed, this is immediate for the split part of $Z$ and follows for the anisotropic part of $Z$ by compactness, cf. \cite{BorelTits}, Cor. \S9.4. Now assume that $\mathbb{T}$ is split over $L$ so that we may identify $Z=(L^\times)^d$ and $Z_0=(o^\times)^d$. Suppose the ramification index $e$ of $L/\mathbb{Q}_p$ satisfies $1>e/(p-1)$. A possible choice for $Z_0(p)^{\rm fl}$ is given by $(1+\pio)^d.$ Moreover, the usual logarithm series followed by multiplication with $\pi^{-1}$ provides a locally $L$-analytic group isomorphism $Z_0(p)^{\rm fl}\car o^d$, e.g. \cite{NeukirchI}, Prop. 5.5.
\vskip8pt
By assumption the inclusion $ \mu\times Z_0(p)^{\rm fl}\subseteq Z$ induces a (noncanonical) isomorphism of locally $L$-analytic groups \[\mu\timeso^d\times \mathbb{Z}^r\stackrel{\cong}{\longrightarrow} Z\] with unique $r\geq 0$. Applying the functor $\hat{(\cdot)}_K$ gives a (noncanonical) isomorphism of rigid groups \[ \hat{Z}_K\car \hat{\mu}_K\times_K\hol^d_K\times_K\Gm^r.\]
The space $\hat{\mu}_K$ is a finite disjoint union of points $Sp~K_i,~i=1,...,s$ (with finite field extensions $K_i/K$). Since each $\hol_{K_i}^d$ is connected and $\mathbb{G}_{m,K_i}^r$ is geometrically connected, their fibre product over $K_i$ remains connected by \cite{DucrosExcellent}, Cor. 8.4. Hence, $$\pi_0(\hat{Z}_K)=s\leq\#\mu {\rm~~~and~~~}{\rm dim~}\hat{Z}_K=d+r.$$
We conclude with the remark that, in this situation, the natural projection morphism $\hat{\iota}: \hat{Z}_K\rightarrow\hat{\mu}_K\times_K\hol^d_K$ induces an isomorphism $$Pic(\hat{Z}_K)\car Pic(\hat{\mu}_K\times_K\hol_K^d)=\oplus_{i=1,...,s} Pic(\hol^d_{K_i}).$$ Indeed, using an argument with cohomology and inverse limits similar to subsect. \ref{cohomologylimits} one is reduced to show that the natural projection $\holnk^d\times_K\G_{m,n}^r\rightarrow \holnk^d$ induces an isomorphism on Picard groups for all $n$. Here, $\G^r_{m,n}\subset\G^r_m$ is the annulus defined by
$|p|^n\leq z_i\leq |p|^{-n}, i=1,...,r$. A finite induction reduces to the case $d=r=1$. This case follows then by properties of the one dimensional affinoids $\holnk$ and the fact that $Pic(\G_{m,n})=1$. We leave the remaining details to the interested reader.
\end{document} | arXiv |
\begin{document}
\title[Edgeworth expansion for coefficients] {Edgeworth expansion for the coefficients of random walks on the general linear group}
\author{Hui Xiao} \author{Ion Grama} \author{Quansheng Liu}
\curraddr[Xiao, H.]{ Universit\'{e} de Bretagne-Sud, LMBA UMR CNRS 6205, Vannes, France} \email{[email protected]} \curraddr[Grama, I.]{ Universit\'{e} de Bretagne-Sud, LMBA UMR CNRS 6205, Vannes, France} \email{[email protected]} \curraddr[Liu, Q.]{ Universit\'{e} de Bretagne-Sud, LMBA UMR CNRS 6205, Vannes, France} \email{[email protected]}
\begin{abstract} Let $(g_n)_{n\geqslant 1}$ be a sequence of independent and identically distributed random elements with law $\mu$ on the general linear group $\textup{GL}(V)$, where $V=\bb R^d$. Consider the random walk $G_n : = g_n \ldots g_1$, $n \geqslant 1$. Under suitable conditions on $\mu$, we establish the first-order Edgeworth expansion for the coefficients $\langle f, G_n v \rangle$ with $v \in V$ and $f \in V^*$,
in which a new additional term appears compared to the case of vector norm $\|G_n v\|$. \end{abstract}
\date{\today} \subjclass[2010]{Primary 60F05, 60F15, 60F10; Secondary 37A30, 60B20} \keywords{Random walks on groups; coefficients; central limit theorem; Edgeworth expansion; Berry-Esseen bound}
\maketitle
\section{Introduction}
Since the pioneering work of Furstenberg and Kesten \cite{FK60}, the study of random walks on linear groups has attracted a great deal of attention, see for instance the work of Le Page \cite{LeP82}, Guivarc'h and Raugi \cite{GR85}, Bougerol and Lacroix \cite{BL85}, Goldsheid and Margulis \cite{GM89}, Benoist and Quint \cite{BQ16b}, and the references therein.
Of particular interest is the study of asymptotic properties of the random walk $G_n : = g_n \ldots g_1$, $n \geqslant 1$,
where $(g_n)_{n \geqslant 1}$ is a sequence of independent and identically distributed random elements with law $\mu$ on the general linear group $\textup{GL}(V)$ with $V = \bb R^d$. One natural and important way to describe the random walk $(G_n)_{n\geqslant 1}$ is to investigate the growth rate of the coefficients $\langle f, G_n v \rangle$, where $v \in V$, $f \in V^*$ and $\langle \cdot, \cdot \rangle$ is the duality bracket: $\langle f, v \rangle = f(v)$. Bellman \cite{Bel54} conjectured that the classical central limit theorem should hold true for $\langle f, G_n v \rangle$ in the case when $g_n$ are positive matrices. This conjecture was proved by Furstenberg and Kesten \cite{FK60}, who established the strong law of large numbers and central limit theorem under the condition that the matrices $g_n$ are strictly positive and that all the coefficients of $g_n$ are comparable.
For further developments we refer to Kingman \cite{Kin73}, Cohn, Nerman and Peligrad \cite{CNP93}, Hennion \cite{Hen97}.
As noticed by Furstenberg \cite{Fur63}, the analysis developed in \cite{FK60} for positive matrices breaks down for invertible matrices. It turns out that the situation of invertible matrices is much more complicated and delicate. Guivarc'h and Raugi \cite{GR85} established the strong law of large numbers for the coefficients of products of invertible matrices under an exponential moment condition: for any $v \in V \setminus \{0\}$ and $f \in V^* \setminus \{0\}$, \begin{align}\label{Ch7_SLLN_Entry0a}
\lim_{n\to\infty} \frac{1}{n} \log | \langle f, G_n v \rangle | = \lambda \quad \mbox{a.s.}, \end{align} where $\lambda \in \bb R$ is a constant independent of $f$ and $v$, called the first Lyapunov exponent of $\mu$. It is worth mentioning that the result \eqref{Ch7_SLLN_Entry0a} does not follow from the classical subadditive ergodic theorem of Kingman \cite{Kin73}, nor from its recent version by Gou\"ezel and Karlsson \cite{GK20}. The central limit theorem for the coefficients has also been established in \cite{GR85} under the exponential moment condition:
if $\int_{ \textup{GL}(V) } N(g)^{\varepsilon} \mu(dg) < \infty$ with $N(g) = \max \{ \|g\|, \| g^{-1} \| \}$ for some $\varepsilon > 0$, then for any $t \in \bb R$, \begin{align}\label{Ch7_CLT_Entry0a}
\lim_{n \to \infty} \bb{P} \left( \frac{ \log | \langle f, G_n v \rangle |
- n \lambda}{ \sigma \sqrt{n} } \leqslant t \right) = \Phi(t), \end{align} where $\Phi$ is the standard normal distribution function on $\bb R$
and $\sigma^2 > 0$ is the asymptotic variance of $ \frac{1}{\sqrt n} \log | \langle f, G_n v \rangle |.$ Recently, using the martingale approximation method, Benoist and Quint \cite{BQ16} have improved \eqref{Ch7_CLT_Entry0a} by relaxing the exponential moment condition to the optimal second moment $\int_{ \textup{GL}(V) } (\log N(g))^{2} \mu(dg) < \infty$.
An important and interesting problem is the estimation of the rate of convergence in \eqref{Ch7_CLT_Entry0a}. Very recently, under the exponential moment condition, Cuny, Dedecker, Merlev\`ede and Peligrad \cite{CDMP21b} established a rate of convergence of order $\log n/\sqrt{n}$. Dinh, Kaufmann and Wu \cite{DKW21, DKW21b} improved this result by giving the optimal rate $1/\sqrt{n}$ under the same exponential moment assumption: there exists a constant $c > 0$ such that for all $n \geqslant 1$, $t \in \bb R$,
$v \in V$ and $f \in V^*$ with $\|v\| = \|f\| =1$, \begin{align} \label{BerryEsseen_Coeffaa-Intro}
\left| \bb{P} \left( \frac{\log |\langle f, G_n v \rangle| - n \lambda }{ \sigma \sqrt{n} } \leqslant t \right)
- \Phi(t) \right| \leqslant \frac{c }{\sqrt{n}}. \end{align}
The objective of this paper is to further elaborate on the central limit theorem \eqref{Ch7_CLT_Entry0a} by establishing the first-order Edgeworth expansion for the coefficients
under the exponential moment condition. We prove that
as $n\to \infty$,
uniformly in $t \in \bb R$, $x=\bb R v \in \bb P(V)$ and $ y = \bb R f \in \bb P(V^*)$ with $\|v\| = \|f\| =1$, \begin{align}\label{Edgeworth-Coeff-Intro}
& \bb{P} \left( \frac{\log |\langle f, G_n v \rangle| - n \lambda }{ \sigma \sqrt{n} } \leqslant t \right) \notag\\
& = \Phi(t) + \frac{\Lambda'''(0)}{ 6 \sigma^3 \sqrt{n}} (1-t^2) \phi(t)
- \frac{ b_{1}(x) + d_{1}(y) }{ \sigma \sqrt{n} } \phi(t) + o \Big( \frac{ 1 }{\sqrt{n}} \Big), \end{align} where $\phi$ denotes the standard normal density, $\Lambda'''(0)$, $b_{1}(x)$, $d_{1}(y)$ are defined in Section \ref{Sec-main-result}. Notice that the asymptotic bias terms $b_{1}(x)$ and $d_{1}(y)$ are new compared with the classical Edgeworth expansion for sums of independent real random variables \cite{Pet75};
$d_{1}(y)$ is also new compared with the Edgeworth expansion for the vector norm $\|G_n v \|$ \cite{XGL19b}.
In fact, we will establish a stronger result, that is, the first-order Edgeworth expansion
for the couple $(\varphi(G_n \!\cdot\! x), \log |\langle f, G_n v \rangle|)$ with a target function $\varphi$ on $\bb P(V)$, cf.\ Theorem \ref{Thm-Edge-Expan-Coeff001}.
Moreover, we prove a similar result under the changed measure, which can be useful for studying moderate deviations with explicit rates of convergence. Clearly, the expansion \eqref{Edgeworth-Coeff-Intro} implies the Berry-Esseen bound \eqref{BerryEsseen_Coeffaa-Intro}.
The proof of the Edgeworth expansion for the coefficient $\langle f, G_n v \rangle$
turns out to be much more complicated than that for the norm cocycle $\sigma (G_n, x)= \log \frac{\| G_n v \|}{\|v\|}$, $x = \bb R v \in \bb P(V)$ recently established in \cite{XGL19b}. One of the difficulties is that
$\log | \langle f, G_n v \rangle |$ is not a cocycle and cannot be studied with the same approach as $\sigma (G_n, x)$. Our starting point is the following decomposition which relates the coefficient to the norm cocycle:
for any $x = \bb R v \in \bb P(V)$ and $y = \bb R f \in \bb P(V^*)$ with $\|f\|=1$, \begin{align}\label{Ch7_Intro_Decom0a}
\log |\langle f, G_n v \rangle| = \sigma (G_n, x) + \log \delta(G_n \!\cdot\! x, y), \end{align}
where $\delta(x, y) = \frac{|\langle f, v \rangle|}{\|f\| \|v\|}$.
For the proof of the Edgeworth expansion \eqref{Edgeworth-Coeff-Intro}, we first use a partition $(\chi_{n,k}^y)_{k \geqslant 1}$ of the unity to discretize the component $\log \delta(G_n \!\cdot\! x, y)$ in \eqref{Ch7_Intro_Decom0a}. This allows us to reduce the study of the coefficient to that of the couple formed by norm cocycle $\sigma(G_n, x)$ and the target function $\chi_{n,k}^y(G_n \!\cdot\! x)$. It turns out that the Edgeworth expansion for the couple $ ( \chi_{n,k}^y (G_n \!\cdot\! x), \sigma(G_n, x))$
established recently in \cite{XGL19b} is not appropriate for our proof because the reminder terms therein are not precise enough. We need to track the dependence of the remainder term on the H\"older norm of the function $\varphi = \chi_{n,k}^y$, see Theorem \ref{Thm-Edge-Expan}.
In contrast to the previous work \cite{DKW21}, the partition of the unity that we use should become finer and finer as $n \to \infty$, in order to recover the term $d_{1}(y)$, see Lemma \ref{new bound for delta020}. Finally, another delicate point is to
patch up the expansions for couples $(\chi_{n,k}^y (G_n \!\cdot\! x), \sigma(G_n, x))$ by means of the H\"older regularity of the invariant measure $\nu$ and the linearity in $\varphi$ of the asymptotic bias term $b_{\varphi}(x)$.
\section{Main results}\label{Sec-main-result}
For any integer $d \geqslant 1$, denote by $V = \bb R^d$ the $d$-dimensional Euclidean space. We fix a basis $e_1, \ldots, e_d$ of $V$ and the associated norm on $V$ is defined by $\|v\|^2 = \sum_{i=1}^d |v_i|^2$ for $v = \sum_{i=1}^d v_i e_i \in V$. Let $V^*$ be the dual vector space of $V$ and its dual basis is denoted by $e_1^*, \ldots, e_d^*$ so that $e_i^*(e_j)= 1$ if $i=j$ and $e_i^*(e_j)= 0$ if $i\neq j$. Let $\wedge^2 V$ be the exterior product of $V$
and we use the same symbol $\| \cdot \|$ for the norms induced on $\wedge^2 V$ and $V^*$. We equip $\bb P (V)$ with the angular distance \begin{align}\label{Angular-distance}
d(x, x') = \frac{\| v \wedge v' \|}{ \|v\| \|v'\| } \quad \mbox{for} \ x= \bb R v \in \bb P(V), \ x' = \bb R v' \in \bb P(V). \end{align} We use the symbol $\langle \cdot, \cdot \rangle$ to denote the dual bracket defined by $\langle f, v \rangle = f(v)$ for any $v \in V$ and $f \in V^*$. Set \begin{align*}
\delta(x,y) = \frac{| \langle f, v \rangle |}{\|f\| \|v\| } \quad \mbox{for} \ x= \bb R v \in \bb P(V), \ y = \bb R f \in \bb P(V^*). \end{align*} Denote by $\mathscr{C}(\bb{P}(V) )$ the space of complex-valued continuous functions on $\bb{P}(V)$,
equipped with the norm $\|\varphi\|_{\infty}: = \sup_{x\in \bb{P}(V) } |\varphi(x)|$ for $\varphi \in \mathscr{C}(\bb{P}(V) )$. Let $\gamma>0$ be a constant and set \begin{align*}
\|\varphi\|_{\gamma}: = \|\varphi\|_{\infty} + [\varphi]_{\gamma}, \quad \mbox{where} \
[\varphi]_{\gamma} = \sup_{x, x' \in \bb{P}(V): x \neq x'} \frac{|\varphi(x)-\varphi(x')|}{ d(x, x')^{\gamma} }. \end{align*} Consider the Banach space \begin{align*}
\mathscr{B}_{\gamma}:= \left\{ \varphi\in \mathscr{C}(\bb P(V)): \|\varphi\|_{\gamma}< \infty \right\}, \end{align*} which consists of complex-valued $\gamma$-H\"older continuous functions on $\bb P(V)$. Denote by $\mathscr{L(B_{\gamma},B_{\gamma})}$ the set of all bounded linear operators from $\mathscr{B}_{\gamma}$ to $\mathscr{B}_{\gamma}$ equipped with the operator norm
$\left\| \cdot \right\|_{\mathscr{B}_{\gamma} \to \mathscr{B}_{\gamma}}$. The topological dual of $\mathscr B_\gamma$ endowed with the induced norm is denoted by $\mathscr B'_\gamma$. Let $\mathscr B_\gamma^*$ be the Banach space of $\gamma$-H\"older continuous functions on $\mathbb P(V^*)$ endowed with the norm \begin{align*}
\| \varphi \|_{\mathscr B_\gamma^*} =
\sup_{y\in \mathbb P(V^*) } |\varphi(y)|
+ \sup_{ y,y'\in \bb P(V^*): \, y \neq y' } \frac{ |\varphi(y)-\varphi(y')| }{ d(y,y')^{\gamma} }, \end{align*}
where $d(y,y') = \frac{\| f \wedge f' \|}{ \|f\| \|f'\| }$ for $y= \bb R f \in \bb P(V^*)$ and $y' = \bb R f' \in \bb P(V^*)$.
Let $\textup{GL}(V)$ be the general linear group of the vector space $V$. The action of $g \in \textup{GL}(V)$ on a vector $v \in V$ is denoted by $gv$, and the action of $g \in \textup{GL}(V)$ on a projective line $x = \bb R v \in \bb P(V)$ is denoted by $g \cdot x = \bb R gv$.
For any $g \in \textup{GL}(V)$, let $\| g \| = \sup_{v \in V \setminus \{0\} } \frac{\| g v \|}{\|v\|}$
and denote $N(g) = \max \{ \|g\|, \| g^{-1} \| \}$. Let $\mu$ be a Borel probability measure on $\textup{GL}(V)$.
We shall use the following exponential moment condition.
\begin{conditionA} \label{Ch7Condi-Moment} There exists a constant $\varepsilon >0$ such that $\int_{ \textup{GL}(V) } N(g)^{\varepsilon} \mu(dg) < \infty$. \end{conditionA}
Let $\Gamma_{\mu}$ be the smallest closed subsemigroup generated by the support of the measure $\mu$. An endomorphism $g$ of $V$ is said to be proximal
if it has an eigenvalue $\lambda$ with multiplicity one and all other eigenvalues of $g$ have modulus strcitly less than $|\lambda|$. We shall need the following strong irreducibility and proximality condition.
\begin{conditionA}\label{Ch7Condi-IP} {\rm (i)(Strong irreducibility)} No finite union of proper subspaces of $V$ is $\Gamma_{\mu}$-invariant.
{\rm (ii)(Proximality)} $\Gamma_{\mu}$ contains a proximal endomorphism. \end{conditionA}
Define the norm cocycle $\sigma: \textup{GL}(V) \times \bb P(V) \to \bb R$ as follows: \begin{align*}
\sigma (g, x) = \log \frac{\|gv\|}{\|v\|} \quad \mbox{for any} \ g \in \textup{GL}(V) \ \mbox{and} \ x = \bb R v \in \bb P(V). \end{align*} Recall that the first Lyapunov exponent $\lambda$ is defined by \eqref{Ch7_SLLN_Entry0a}. By \cite[Proposition 3.15]{XGL19b}, under \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP},
the following limit exists and is independent of $x \in \bb P(V)$: \begin{align}\label{Def-sigma} \sigma^2: = \lim_{n \to \infty} \frac{1}{n} \bb E \left[ (\sigma (G_n, x) - n \lambda)^2 \right] \in (0, \infty). \end{align} For any $s \in (-s_0, s_0)$ with $s_0 >0$ small enough, we define the transfer operator $P_s$ as follows: for any bounded measurable function $\varphi$ on $\bb P(V)$, \begin{align}\label{Def_Ps001} P_s \varphi(x) = \int_{ \textup{GL}(V) } e^{s \sigma(g, x)} \varphi(g \!\cdot\! x) \mu(dg), \quad x \in \bb P(V). \end{align} It will be shown in Lemma \ref{Ch7transfer operator} that there exists a constant $s_0 >0$ such that for any $s \in (-s_0, s_0)$, the operator $P_s \in \mathscr{L(B_{\gamma},B_{\gamma})}$ has a unique dominant eigenvalue $\kappa(s)$ with $\kappa(0) = 1$ and the mapping $s \mapsto \kappa(s)$ being analytic. We denote $\Lambda = \log \kappa$.
Under \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}, the Markov chain $(G_n \!\cdot\! x)_{n \geqslant 0}$ has a unique invariant probability measure $\nu$ on $\bb P(V)$ such that for any bounded measurable function $\varphi$ on $\bb P(V)$, \begin{align} \label{Ch7mu station meas} \int_{\bb P(V)} \int_{\textup{GL}(V)} \varphi(g \!\cdot\! x) \mu(dg) \nu(dx)
= \int_{ \bb P(V) } \varphi(x) \nu(dx) = : \nu(\varphi).
\end{align}
For any $\varphi \in \mathscr{B}_{\gamma}$, define the functions \begin{align} \label{drift-b001} b_{\varphi}(x): = \lim_{n \to \infty}
\mathbb{E} \big[ ( \sigma(G_n, x) - n \lambda ) \varphi(G_n \!\cdot\! x) \big], \quad x \in \bb P(V) \end{align} and \begin{align} \label{drift-d001} d_{\varphi}(y) := \int_{\bb P(V)} \varphi(x) \log \delta(x, y)\nu(dx), \quad y \in \bb P(V^*). \end{align} It will be shown in Lemmas \ref{Lem-Bs} and \ref{Lem-ds} that both functions $b_{\varphi}$ and $d_{\varphi}$ are well-defined and $\gamma$-H\"older continuous.
Denote $\phi(u) = \frac{1}{\sqrt{2 \pi}} e^{- u^2/2}$, $u \in \bb R$. Let
$\Phi(t) = \int_{- \infty}^t \phi(u) du$, $t \in \bb R$ be the standard normal distribution function.
In many applications it is of primary interest to give an estimation of the rate of convergence in the Gaussian approximation \eqref{Ch7_CLT_Entry0a}. In this direction we establish the following first-order Edgeworth expansion for the coefficients $\langle f, G_n v \rangle$.
\begin{theorem}\label{Thm-Edge-Expan-Coeff001} Assume \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}. Then, there exists a constant $\gamma >0$ such that for any $\varepsilon > 0$, uniformly in
$t \in \bb R$, $x=\bb R v \in \bb P(V)$, $ y = \bb R f \in \bb P(V^*)$ with $\|v\| = \|f\| =1$, and
$\varphi \in \mathscr{B}_{\gamma}$, as $n\to \infty$, \begin{align}\label{EdgeworthExpan} & \mathbb{E}
\Big[ \varphi(G_n \!\cdot\! x) \mathds{1}_{ \big\{ \frac{\log |\langle f, G_n v \rangle| - n \lambda }{\sigma \sqrt{n}} \leqslant t \big\} } \Big]
= \nu(\varphi) \Big[ \Phi(t) + \frac{\Lambda'''(0)}{ 6 \sigma^3 \sqrt{n}} (1-t^2) \phi(t) \Big]
\notag\\ & \qquad\qquad\qquad
- \frac{ b_{\varphi}(x) + d_{\varphi}(y) }{ \sigma \sqrt{n} } \phi(t)
+ \nu(\varphi) o \Big( \frac{ 1 }{\sqrt{n}} \Big) + \lVert \varphi \rVert_{\gamma} O \Big( \frac{ 1 }{n^{1 - \varepsilon} } \Big). \end{align} \end{theorem}
When compared with the standard Edgeworth expansion for sums of independent random variables (cf.\ \cite{Pet75}), we see that two new terms $b_{\varphi}(x)$ and $d_{\varphi}(y)$ show up, which are explained by the presence of an asymptotic bias for this model. We should also note that the Edgeworth expansion \eqref{EdgeworthExpan} for the coefficients is different from
that for the norm cocycle $\sigma (G_n, x)$ obtained in \cite{XGL19b},
namely, by the presence of the term $d_{\varphi}(y)$.
The difficulty in proving this precise expansion for coefficient $\langle f, G_n v \rangle$
consists in obtaining the exact expression of this new asymptotic bias term $d_{\varphi}(y)$.
As a consequence of Theorem \ref{Thm-Edge-Expan-Coeff001} one can get the Berry-Esseen bound \eqref{BerryEsseen_Coeffaa-Intro} with the optimal convergence rate, under the exponential moment condition.
It is an open problem how to relax the exponential moment condition \ref{Ch7Condi-Moment} for the Edgeworth expansion and for the Berry-Esseen bound. Solving it seems very challenging. Even for the easier case of the norm cocycle, the Berry-Esseen bound $O(n^{-1/2})$ is not known under the optimal third moment condition; it is only known under the fourth moment condition, see \cite{CDMP21}.
For positive matrices, the Edgeworth expansion \eqref{EdgeworthExpan} and the Berry-Esseen bound \eqref{BerryEsseen_Coeffaa-Intro} have been recently obtained using a different approach in a forthcoming paper \cite{XGL21b} under optimal moment conditions.
Finally we would like to mention that all the results of the paper remain valid when $V$ is $\bb C^d$ or $\bb K^d$, where $\bb K$ is any local field.
\section{Proof of the Edgeworth expansion}
\subsection{Preliminary results}\label{subsec-Pz}
For any $z \in \bb{C}$, we define the complex transfer operator $P_z$ as follows: for any bounded measurable function $\varphi$ on $\bb P(V)$, \begin{align}\label{Def_Pz_Ch7} P_z \varphi(x) = \int_{\textup{GL}(V)} e^{z \sigma(g, x)} \varphi(g \!\cdot\! x) \mu(dg), \quad x \in \bb P(V). \end{align}
Throughout this paper let $B_{s_0}(0): = \{ z \in \bb{C}: |z| < s_0 \}$ be the open disc with center $0$ and radius $s_0 >0$ in the complex plane $\bb C$. The following result shows that the operator $P_z$ has spectral gap properties when $z \in B_{s_0}(0)$; we refer to \cite{LeP82, HH01, GL16, BQ16b, XGL19b} for the proof based on the perturbation theory of linear operators. Recall that $\mathscr{B}_{\gamma}'$ is the topological dual space of the Banach space $\mathscr{B}_{\gamma}$, and that $\mathscr{L(B_{\gamma},B_{\gamma})}$ is the set of all bounded linear operators from $\mathscr{B}_{\gamma}$ to $\mathscr{B}_{\gamma}$ equipped with the operator norm
$\left\| \cdot \right\|_{\mathscr{B}_{\gamma} \to \mathscr{B}_{\gamma}}$.
\begin{lemma}[\cite{BQ16b, XGL19b}] \label{Ch7transfer operator} Assume \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}. Then, there exists a constant $s_0 >0$ such that for any $z \in B_{s_0}(0)$ and $n \geqslant 1$, \begin{align}\label{Ch7Pzn-decom} P_z^n = \kappa^n(z) \nu_z \otimes r_z + L_z^n, \end{align} where \begin{align*} z \mapsto \kappa(z) \in \bb{C}, \quad z \mapsto r_z \in \mathscr{B}_{\gamma} , \quad z \mapsto \nu_z \in \mathscr{B}_{\gamma}' , \quad z \mapsto L_z \in \mathscr{L(B_{\gamma},B_{\gamma})} \end{align*} are analytic mappings which satisfy, for any $z \in B_{s_0}(0)$,
\begin{itemize} \item[{\rm(a)}]
the operator $M_z: = \nu_z \otimes r_z$ is a rank one projection on $\mathscr{B}_{\gamma}$,
i.e. $M_z \varphi = \nu_z(\varphi) r_z$ for any $\varphi \in \mathscr{B}_{\gamma}$;
\item[{\rm(b)}]
$M_z L_z = L_z M_z =0$, $P_z r_z = \kappa(z) r_z$ with $\nu(r_z) = 1$, and $\nu_z P_z = \kappa(z) \nu_z$;
\item[{\rm(c)}]
$\kappa(0) = 1$, $r_0 = 1$, $\nu_0 = \nu$ with $\nu$ defined by \eqref{Ch7mu station meas}, and
$\kappa(z)$ and $r_z$ are strictly positive for real-valued $z \in (-s_0, s_0)$. \end{itemize} \end{lemma}
Using Lemma \ref{Ch7transfer operator}, a change of measure can be performed below. For any $s \in (-s_0, s_0)$ with $s_0>0$ sufficiently small,
any $x \in \bb P(V)$ and $g \in \textup{GL}(V)$, denote \begin{align*} q_n^s(x, g) = \frac{ e^{s \sigma(g, x) } }{ \kappa^{n}(s) } \frac{ r_s(g \!\cdot\! x) }{ r_s(x) }, \quad n \geqslant 1. \end{align*} Since the eigenvalue $\kappa(s)$ and the eigenfunction $r_s$ are strictly positive for $s \in (-s_0, s_0)$, using $P_s r_s = \kappa(s) r_s$ we get that \begin{align*} \bb Q_{s,n}^x (dg_1, \ldots, dg_n) = q_n^s(x, G_n) \mu(dg_1) \ldots \mu(dg_n), \quad n \geqslant 1, \end{align*} are probability measures and form a projective system on $\textup{GL}(V)^{\bb{N}}$. By the Kolmogorov extension theorem, there exists a unique probability measure $\bb Q_s^x$ on $\textup{GL}(V)^{\bb{N}}$ with marginals $\bb Q_{s,n}^x$. We write $\bb{E}_{\bb Q_s^x}$ for the corresponding expectation and the change of measure formula holds: for any $s \in (-s_0, s_0)$, $x \in \bb P(V)$, $n\geqslant 1$ and bounded measurable function $h$ on $(\bb P(V) \times \bb R)^{n}$, \begin{align}\label{Ch7basic equ1} & \frac{1}{ \kappa^{n}(s) r_{s}(x) } \bb{E} \Big[ r_{s}(G_n \!\cdot\! x) e^{s \sigma(G_n, x) } h \Big( G_1 \!\cdot\! x, \sigma(G_1, x), \dots, G_n \!\cdot\! x, \sigma(G_n, x) \Big)
\Big] \nonumber\\ & = \bb{E}_{\bb{Q}_{s}^{x}} \Big[ h \Big( G_1 \!\cdot\! x, \sigma(G_1, x), \dots, G_n \!\cdot\! x, \sigma(G_n, x) \Big) \Big]. \end{align} Under the changed measure $\bb Q_s^x$, the process $(G_n \!\cdot\! x)_{n \geqslant 0}$ is a Markov chain with the transition operator $Q_s$ given as follows: for any $\varphi \in \mathscr{C}(\bb P(V))$, \begin{align*} Q_{s}\varphi(x) = \frac{1}{\kappa(s)r_{s}(x)}P_s(\varphi r_{s})(x), \quad x \in \bb P(V). \end{align*} Under \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}, it was shown in \cite{XGL19b} that the Markov operator $Q_s$ has a unique invariant probability measure $\pi_s$ given by \begin{align}\label{ExpCon-Qs} \pi_s(\varphi) = \frac{ \nu_s( \varphi r_s) }{ \nu_s(r_s) } \quad \mbox{for any } \varphi \in \mathscr{C}(\bb P(V)). \end{align} By \cite[Proposition 3.13]{XGL19b}, the following strong law of large numbers for the norm cocycle under the changed measure $\bb Q_s^x$ holds: under \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}, for any $s\in (-s_0, s_0)$ and $x \in \bb P(V)$, \begin{align*} \lim_{n \to \infty} \frac{ \sigma(G_n, x) }{n} = \Lambda'(s), \quad \bb Q_s^x\mbox{-a.s.} \end{align*} where $\Lambda(s) = \log \kappa(s)$.
We need the following H\"older regularity of the invariant measure $\pi_s$. \begin{lemma}[\cite{GQX20}] \label{Lem_Regu_pi_s00} Assume \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}. Then there exist constants $s_0 >0$ and $\eta > 0$ such that \begin{align} \label{Regu_pi_s_00} \sup_{ s\in (-s_0, s_0) } \sup_{y \in \bb P(V^*) }
\int_{\bb P(V) } \frac{1}{ \delta(x, y)^{\eta} } \pi_s(dx) < + \infty. \end{align} \end{lemma}
We also need the following property:
\begin{lemma}[\cite{GQX20}] \label{Lem_Regu_pi_s} Assume \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}. Then, for any $\varepsilon >0$, there exist constants $s_0 >0$ and $c, C >0$ such that for all $s \in (-s_0, s_0)$, $n \geqslant k \geqslant 1$, $x \in \bb P(V)$ and $y \in \bb P(V^*)$, \begin{align}\label{Regu_pi_s} \bb Q_s^x \Big( \log \delta(G_n \!\cdot\! x, y) \leqslant -\varepsilon k \Big) \leqslant C e^{- ck}. \end{align} \end{lemma}
Note that \eqref{Regu_pi_s} is stronger than the exponential H\"{o}lder regularity of the invariant measure $\pi_s$ stated in Lemma \ref{Lem_Regu_pi_s00}.
\subsection{Proof of Theorem \ref{Thm-Edge-Expan-Coeff001}} \label{sec-proof of Edgeworth exp} In fact we shall prove a more general version of Theorem \ref{Thm-Edge-Expan-Coeff001} under the changed measure $\mathbb{Q}_{s}^{x}$. The proof for the case $s =0$ requires the same effort, so we decide to consider the more general setting. For any $s \in (-s_0, s_0)$ and $\varphi \in \mathscr{B}_{\gamma}$, define \begin{align}\label{Def-bsvarphi} b_{s, \varphi}(x): = \lim_{n \to \infty}
\mathbb{E}_{\mathbb{Q}_{s}^{x}} \big[ ( \sigma(G_n, x) - n \Lambda'(s) ) \varphi(G_n \!\cdot\! x) \big], \quad x \in \bb P(V) \end{align} and \begin{align} \label{drift-d001bis} d_{s,\varphi}(y)= \int_{\bb P(V)} \varphi(x) \log \delta(x, y) \pi_s(dx), \quad y \in \bb P(V^*). \end{align} These functions are well-defined and $\gamma$-H\"older continuous, as shown in Lemmas \ref{Lem-Bs} and \ref{Lem-ds} below. In particular, we have $b_{0,\varphi} = b_{\varphi}$ and $d_{0,\varphi} = d_{\varphi}$,
where $b_{\varphi}$ and $d_{\varphi}$ are defined in \eqref{drift-b001} and \eqref{drift-d001}, respectively.
Our goal of this subsection is to establish the following first-order Edgeworth expansion for the coefficients $\langle f, G_n v \rangle$
under the changed measure $\mathbb{Q}_{s}^x$.
Note that $\sigma_s = \sqrt{\Lambda''(s)}$, which is strictly positive under \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}.
\begin{theorem}\label{Thm-Edge-Expan-Coeff001extended} Assume \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}. Then, for any $\varepsilon > 0$,
there exist $\gamma >0$ and $s_0 > 0$ such that uniformly in $s \in (-s_0, s_0)$,
$t \in \bb R$, $x=\bb R v \in \bb P(V)$, $ y = \bb R f \in \bb P(V^*)$ with $\|v\| = \|f\| =1$, and
$\varphi \in \mathscr{B}_{\gamma}$, as $n\to \infty$, \begin{align*} & \mathbb{E}_{\mathbb{Q}_{s}^x}
\Big[ \varphi(G_n \!\cdot\! x) \mathds{1}_{ \big\{ \frac{\log |\langle f, G_n v \rangle| - n \Lambda'(s) }{\sigma_s \sqrt{n}} \leqslant t \big\} } \Big]
= \pi_s(\varphi) \Big[ \Phi(t) + \frac{\Lambda'''(s)}{ 6 \sigma_s^3 \sqrt{n}} (1-t^2) \phi(t) \Big]
\notag\\ & \qquad\qquad\qquad\qquad\qquad - \frac{ b_{s,\varphi}(x) + d_{s,\varphi}(y) }{ \sigma_s \sqrt{n} } \phi(t)
+ \pi_s(\varphi) o \Big( \frac{ 1 }{\sqrt{n}} \Big) + \lVert \varphi \rVert_{\gamma} O \Big( \frac{ 1 }{n^{1-\varepsilon} }\Big). \end{align*} \end{theorem} Theorem \ref{Thm-Edge-Expan-Coeff001} follows from Theorem \ref{Thm-Edge-Expan-Coeff001extended} by taking $s=0$.
The remaining part of the paper is devoted to establishing Theorem \ref{Thm-Edge-Expan-Coeff001extended}. We begin with some properties of the function $b_{s, \varphi}$ (cf.\ \eqref{Def-bsvarphi}). \begin{lemma}[\cite{XGL19b}] \label{Lem-Bs} Assume \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}. Then,
there exist constants $s_0 >0$, $\gamma>0$ and $c>0$ such that $b_{s,\varphi} \in \mathscr{B}_{\gamma}$
and $\| b_{s,\varphi}\|_{\gamma} \leqslant c \| \varphi \| _{\gamma}$ for any $s\in (-s_0, s_0)$. \end{lemma}
In addition to Lemma \ref{Lem-Bs}, we shall need the following result on the function $d_{s,\varphi}$ defined in \eqref{drift-d001bis}. \begin{lemma} \label{Lem-ds} Assume \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}. Then, there exists $s_0 >0$ such that for any $s\in (-s_0, s_0)$, the function $d_{s,\varphi}$ is well-defined. Moreover, there exist constants $\gamma>0$ and $c>0$ such that $d_{s,\varphi} \in \mathscr{B}^*_{\gamma}$
and $\| d_{s,\varphi}\|_{\gamma} \leqslant c \| \varphi \|_{\infty}$ for any $s\in (-s_0, s_0)$. \end{lemma}
\begin{proof} Without loss of generality, we assume that $\varphi$ is non-negative. Since $\log a\leqslant a$ for any $a\geqslant 0$ (with the convention that $\log 0 = -\infty$), we have that for any $\eta\in (0,1)$, \begin{align}\label{Inequlity-log-eta} - \eta \log \delta(x, y) \leqslant \delta(x, y)^{-\eta}, \end{align} so that \begin{align*} -d_{s,\varphi}(y)
\leqslant \frac{\| \varphi \| _{\infty}}{\eta} \int_{\bb P(V)} \frac{1}{ \delta(x, y)^{\eta} } \pi_s(dx). \end{align*} Choosing $\eta$ small enough, by Lemma \ref{Lem_Regu_pi_s00}, the latter integral is bounded by some constant uniformly in $y\in \bb P(V^*)$
and $s \in (-s_0, s_0)$, which proves that $d_{s,\varphi}$ is well-defined and $\|d_{s,\varphi}\|_{\infty} \leqslant c \| \varphi \|_{\infty}$ for some constant $c>0$.
To estimate $[d_{s,\varphi}]_{\gamma}$, we first note that for any $y'=\bb R f'\in \bb P(V^*)$, $y''=\bb R f''\in \bb P(V^*)$ and any $\gamma>0$, \begin{align*}
\left| \log \delta(x, y')-\log \delta(x, y'') \right|
&= \left| \log \delta(x, y')-\log \delta(x, y'') \right|
\mathds 1_{ \left\{ \left| \frac{\delta(x, y')-\delta(x, y'')}{\delta(x, y'')} \right|^{\gamma} > \frac{1}{2^{\gamma}} \right\} } \notag\\
& \quad + \left| \log \delta(x, y')-\log \delta(x, y'') \right|
\mathds 1_{ \big\{ \big|\frac{\delta(x, y')-\delta(x, y'')}{\delta(x, y'')} \big| \leqslant \frac{1}{2} \big\} } \notag\\
& =: I_1 + I_2. \end{align*} For $I_1$, we easily get \begin{align*}
I_1 \leqslant 2^{\gamma}\left( \left| \log \delta(x, y') \right| + \left| \log \delta(x, y'') \right| \right)
\left|\frac{\delta(x, y')-\delta(x, y'')}{\delta(x, y'')} \right|^{\gamma}. \end{align*}
For $I_2$, since $|\log(1+a)| \leqslant 2 |a|$ for any $|a| \leqslant \frac{1}{2}$, we deduce that
\begin{align*}
I_2 & = \left| \log \delta(x, y')-\log \delta(x, y'') \right|^{1 - \gamma}
\left| \log \left[ 1 + \frac{ \delta(x, y') - \delta(x, y'') }{ \delta(x, y'') } \right] \right|^{\gamma}
\mathds 1_{ \big\{ \big|\frac{\delta(x, y')-\delta(x, y'')}{\delta(x, y'')} \big| \leqslant \frac{1}{2} \big\} } \notag\\
& \leqslant 2^{\gamma} \left| \log \delta(x, y')-\log \delta(x, y'') \right|^{1-\gamma}
\left|\frac{\delta(x, y')-\delta(x, y'')}{\delta(x, y'')} \right|^{\gamma}. \end{align*} Therefore, \begin{align*}
\left| \log \delta(x, y')-\log \delta(x, y'') \right|
&\leqslant 2^{\gamma}\left( \left| \log \delta(x, y') \right| + \left| \log \delta(x, y'') \right| \right)
\left|\frac{\delta(x, y')-\delta(x, y'')}{\delta(x, y'')} \right|^{\gamma} \\
& \quad + 2^{\gamma} \left| \log \delta(x, y')-\log \delta(x, y'') \right|^{1-\gamma}
\left|\frac{\delta(x, y')-\delta(x, y'')}{\delta(x, y'')} \right|^{\gamma}. \end{align*} By \eqref{Inequlity-log-eta}, it holds that $- \gamma \log \delta(x, y) \leqslant \delta(x, y)^{-\gamma}$ for any $\gamma \in (0,1)$. Hence there exists a constant $c_{\gamma}>0$ such that \begin{align*}
&|\log \delta(x, y')-\log \delta(x, y'')| \\ &\leqslant c_{\gamma} \left(\delta(x, y')^{-\gamma} \delta(x, y'')^{-\gamma} + \delta(x, y'')^{ -2\gamma} \right)
\left|\delta(x, y')-\delta(x, y'')\right|^ {\gamma}\\ & \quad + c_{\gamma} \left(\delta(x, y')^{-\gamma (1-\gamma) } \delta(x, y'')^{-\gamma} + \delta(x, y'')^{-\gamma (1-\gamma)-\gamma} \right)
\left|\delta(x, y')-\delta(x, y'')\right|^ {\gamma}\\ &\leqslant c_{\gamma} \left(\delta(x, y')^{-\gamma} \delta(x, y'')^{-\gamma} + \delta(x, y'')^{-2\gamma} \right)
\left|\delta(x, y')-\delta(x, y'')\right|^ {\gamma}\\ &\leqslant c_{\gamma} \left(\delta(x, y')^{-2\gamma} + \delta(x, y'')^{-2\gamma} \right)
\left|\delta(x, y')-\delta(x, y'')\right|^ {\gamma}. \end{align*}
Since $\| \frac{f'}{\| f' \|} - \frac{f''}{\| f \|} \| \leqslant \sqrt{2} d(y',y'')$ where $d(y',y'')$ is the angular distance on $\bb P(V^*)$, we have $$
\left|\delta(x, y')-\delta(x, y'')\right|= \left|\frac{ \langle f', v \rangle }{\|v\| \|f'\|} - \frac{ \langle f'', v \rangle }{\|v\| \|f''\|}\right|
\leqslant \Big\| \frac{f'}{\| f' \|} - \frac{f''}{\| f \|} \Big\| \leqslant \sqrt{2} d(y',y''). $$ By the definition of the function $d_{s,\varphi}$, using the above bounds, we obtain \begin{align*}
\frac{| d_{s,\varphi}(y')-d_{s,\varphi}(y'')|}{d(y',y'')^{\gamma}}
\leqslant c_{\gamma} \| \varphi \|_{\infty} \int_{\bb P(V)} \left(\delta(x, y')^{-2\gamma} + \delta(x, y'')^{-2\gamma} \right) \pi_s(dx). \end{align*} By Lemma \ref{Lem_Regu_pi_s00}, the last integral is bounded by some constant uniformly in $y',y''\in \bb P(V^*)$ and $s \in (-s_0, s_0)$, by choosing $\gamma >0$ sufficiently small.
This, together with the fact that $\|d_{s,\varphi}\|_{\infty} \leqslant c \| \varphi \|_{\infty}$, proves that
$d_{s,\varphi} \in \scr B_{\gamma}^*$ and $\| d_{s,\varphi}\|_{\gamma} \leqslant c \| \varphi \|_{\infty}$. \end{proof}
In the proof of Theorem \ref{Thm-Edge-Expan-Coeff001extended} we shall make use of the following Edgeworth expansion for the couple $(G_n \cdot x, \sigma(G_n, x))$ with a target function $\varphi$ on $G_n \cdot x$, which slightly improves \cite[Theorem 5.3]{XGL19b} by giving more accurate reminder terms. This improvement will be important for establishing Theorem \ref{Thm-Edge-Expan-Coeff001extended}.
\begin{theorem}\label{Thm-Edge-Expan} Assume \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}. Then, there exist constants $s_0 >0$ and $\gamma >0$ such that, as $n \to \infty$, uniformly in $s \in (-s_0, s_0)$, $x \in \bb P(V)$, $t \in \bb R$ and $\varphi \in \mathscr{B}_{\gamma}$, \begin{align*}
\mathbb{E}_{\mathbb{Q}_s^x}
\Big[ \varphi(G_n \cdot x) \mathds{1}_{ \big\{ \frac{\sigma(G_n, x) - n \Lambda'(s) }{\sigma_s \sqrt{n}} \leqslant t \big\} } \Big]
& = \pi_s(\varphi) \Big[ \Phi(t) + \frac{\Lambda'''(s)}{ 6 \sigma_s^3 \sqrt{n}} (1-t^2) \phi(t) \Big]
- \frac{ b_{s,\varphi}(x) }{ \sigma_s \sqrt{n} } \phi(t) \notag\\ & \quad + \pi_s(\varphi) o \Big( \frac{ 1 }{\sqrt{n}} \Big) + \lVert \varphi \rVert_{\gamma} O \Big( \frac{ 1 }{n} \Big). \end{align*} \end{theorem}
\begin{proof} For any $x \in \bb P(V)$, define \begin{align*} F(t) & = \mathbb{E}_{\mathbb{Q}_s^x} \Big[ \varphi(G_n \!\cdot\! x) \mathds{1}_{ \big\{ \frac{\sigma(G_n, x) - n \Lambda'(s) }{\sigma_s \sqrt{n}} \leqslant t \big\} } \Big]
+ \frac{ b_{s,\varphi}(x) }{ \sigma_s \sqrt{n} } \phi(t),
\quad t \in \mathbb{R}, \notag\\ H(t) & = \mathbb{E}_{\mathbb{Q}_s^x} [ \varphi(G_n \!\cdot\! x) ]
\Big[ \Phi(t) + \frac{\Lambda'''(s)}{ 6 \sigma_s^3 \sqrt{n}} (1-t^2) \phi(t) \Big], \quad t\in \mathbb{R}. \end{align*} Since $F(-\infty) = H(-\infty) = 0$ and $F(\infty) = H(\infty)$, applying Proposition 4.1 of \cite{XGL19b} we get that \begin{align}\label{BerryEsseen001}
\sup_{t \in \mathbb{R}} \big| F(t) - H(t) \big| \leqslant \frac{1}{\pi } ( I_1 + I_2 + I_3 + I_4), \end{align} where \begin{align*}
I_1 & = o \Big( \frac{ 1 }{\sqrt{n}} \Big) \sup_{t \in \mathbb{R}} |H'(t)|,
\quad I_2 \leqslant C e^{-cn} \|\varphi \|_{\gamma},
\quad I_3 \leqslant \frac{c}{n} \|\varphi \|_{\gamma}, \quad
I_4 \leqslant \frac{c}{n} \|\varphi \|_{\gamma}. \end{align*} Here the bounds for $I_2$, $I_3$ and $I_4$ are obtained in \cite{XGL19b}. It is easy to see that \begin{align*} I_1 = o \Big( \frac{ 1 }{\sqrt{n}} \Big) \mathbb{E}_{\mathbb{Q}_s^x} \Big[ \varphi(G_n \!\cdot\! x) \Big]. \end{align*} This, together with the fact that \begin{align*}
\mathbb{E}_{\mathbb{Q}_s^x} \Big[ \varphi(G_n \!\cdot\! x) \Big] \leqslant \pi_s(\varphi) + C e^{-cn} \|\varphi \|_{\gamma} \end{align*} (cf.\ \cite{XGL19b}), proves the theorem. \end{proof}
In the following we shall construct a partition $(\chi_{n,k}^y)_{k \geqslant 0}$ of the unity on the projective space $\bb P(V)$, which is similar to the partitions in \cite{XGL19d, GQX20, DKW21}. In contrast to \cite{XGL19d, GQX20}, there is no escape of mass in our partition, which simplifies the proofs. Our partition becomes finer when $n \to \infty$, which allows us to obtain precise expressions for remainder terms in the central limit theorem and thereby to establish the Edgeworth expansion for the coefficients.
Let $U$ be the uniform distribution function on the interval $[0,1]$: $U(t)=t$ for $t\in [0,1]$, $U(t)=0$ for $t <0$ and $U(t)=1$ for $t > 1$.
Let $a_n=\frac{1}{\log n}$.
Here and below we assume that $n \geqslant 18$ so that $a_n e^{a_n} \leqslant \frac{1}{2}$. For any integer $k\geqslant 0$, define \begin{align*} U_{n,k}(t)= U\left(\frac{t-(k-1) a_n}{a_n}\right), \qquad h_{n,k}(t)=U_{n,k}(t) - U_{n,k+1}(t), \quad t \in \bb R. \end{align*} It is easy to see that $U_{n,m} = \sum_{k=m}^\infty h_{n,k}$ for any $m\geqslant 0$. Therefore, for any $t\geqslant 0$ and $m\geq0$, we have \begin{align} \label{unity decomposition h-001} \sum_{k=0}^{\infty} h_{n,k} (t) =1, \quad \sum_{k=0}^{m} h_{n,k} (t) + U_{n,m+1} (t) =1. \end{align} Note that for any $k\geqslant 0$, \begin{align} \label{h_kLip001}
\sup_{s,t\geqslant 0: s \neq t} \frac{ | h_{n,k}(s) - h_{n,k}(t) |}{|s-t|} \leqslant \frac{1}{a_{n}}. \end{align} For any $x \in \bb P(V)$ and $y \in \bb P(V^*)$, set \begin{align}\label{Def-chi-nk} \chi_{n,k}^y(x)=h_{n,k}(-\log \delta(x, y)) \quad \mbox{and} \quad \overline \chi_{n,k}^y(x)= U_{n,k} ( -\log \delta(x, y) ), \end{align} where we recall that $-\log\delta(x, y) \geqslant 0$ for any $x\in \bb P (V)$ and $y \in \bb P(V^*)$. From \eqref{unity decomposition h-001}
we have the following partition of the unity on $\bb P(V)$: for any $x\in \bb P (V)$, $y \in \bb P(V^*)$ and $m\geqslant 0$, \begin{align} \label{Unit-partition001} \sum_{k=0}^{\infty} \chi_{n,k}^y (x) =1, \quad \sum_{k=0}^{m} \chi_{n,k}^y (x) + \overline \chi_{n,m+1}^y (x) =1. \end{align} Denote by $\supp (\chi_{n,k}^y)$ the support of the function $\chi_{n,k}^y$. It is easy to see that for any $k\geqslant 0$ and $y\in \bb P(V^*)$, \begin{align} \label{on the support on chi_k-001}
-\log \delta(x, y) \in [a_n (k-1), a_n(k+1)] \quad \mbox{for any}\ x\in \supp (\chi_{n,k}^y). \end{align}
\begin{lemma} \label{lemmaHolder property001} There exists a constant $c>0$ such that for any $\gamma\in(0,1]$,
$k\geqslant 0$ and $y\in \bb P(V^*)$, it holds
$\chi_{n,k}^y\in \scr B_{\gamma}$ and, moreover, \begin{align} \label{Holder prop ohCHI_k-001}
\| \chi_{n,k}^y \|_{\gamma} \leqslant \frac{c e^{\gamma k a_n}}{a_{n}^\gamma}. \end{align} \end{lemma}
\begin{proof}
Since $\|\chi_{n,k}^y\|_{\infty} \leqslant 1$, it is enough to give a bound for the modulus of continuity: \begin{align*}
[\chi_{n,k}^y]_{\gamma} = \sup_{x',x''\in \bb P(V): x' \neq x''}\frac{|\chi_{n,k}^y(x') - \chi_{n,k}^y(x'')|}{d(x',x'')^{\gamma}}, \end{align*} where $d$ is the angular distance on $\bb P(V)$ defined by \eqref{Angular-distance}.
Assume that $x'=\bb R v'\in \bb P(V)$ and $x''=\bb R v''\in \bb P(V)$ are such that $\|v'\|=\|v''\|=1$. We note that \begin{align} \label{angular dist-bound001}
\| v'-v''\| \leqslant \sqrt{2}d(x',x''). \end{align} For short, denote $B_k=((k-1)a_n,ka_n]$. Note that the function $h_{n,k}$ is increasing on $B_k$ and decreasing on $B_{k+1}$. Set for brevity $t'=-\log \delta(x', y)$ and $t''=-\log \delta(x'', y)$. First we consider the case when $t'$ and $t''$ are such that $t',t''\in B_{k}$. Using \eqref{Def-chi-nk}, \eqref{h_kLip001}
and the fact that $|h_{n,k}| \leqslant 1$, we have that for any $\gamma \in (0,1]$, \begin{align} \label{bounddCHI-001}
&|\chi_{n,k}^y(x')- \chi_{n,k}^y(x'')| = |h_{n,k}(t')- h_{n,k}(t'')|^{1-\gamma} |h_{n,k}(t')- h_{n,k}(t'')|^{\gamma} \notag\\
&\leqslant 2 |h_{n,k}(t')- h_{n,k}(t'')|^{\gamma} \leqslant 2 \frac{|t'-t''|^{\gamma}}{a_{n}^{\gamma}}
= \frac{2}{a_{n}^{\gamma}} |\log u'-\log u''|^{\gamma}, \end{align} where we set for brevity $u'=\delta(x', y)$ and $u''= \delta(x'', y)$. Since $u'=e^{-t'}$, $u''=e^{-t''}$ and $t,t'\in B_{k}$, we have
$u''\geqslant e^{-k a_n}$ and $| u' -u''| \leqslant e^{-(k-1) a_n }-e^{-ka_n}$. Therefore, for $n \geqslant 18$, \begin{align*}
\Big| \frac{u'}{u''}-1 \Big|= \Big| \frac{u'-u''}{u''} \Big| \leqslant \frac{ e^{-(k-1)a_n} -e^{-ka_n} }{e^{-ka_n}} =e^{a_n}-1\leqslant a_n e^{a_n} \leqslant \frac{1}{2}, \end{align*}
which, together with the inequality $|\log(1+a)| \leqslant 2 |a|$ for any $|a| \leqslant \frac{1}{2}$, implies \begin{align} \label{h_kLip002}
|\log u' - \log u''|= \left| \log\left(1+ \frac{u' - u''}{u''} \right) \right| \leqslant 2 \frac{|u'-u''|}{u''}. \end{align}
Since $u''\geqslant e^{-k a_n}$, using the fact that $\|v'\|=\|v''\|=1$ and \eqref{angular dist-bound001}, we get \begin{align} \label{Lipschitz-u}
\frac{|u'-u''|}{u''}
&\leqslant e^{k a_n} |\delta(x', y)- \delta(x'', y)| = e^{k a_n} \frac{|f(v') - f(v'') |}{\|f\|} \notag\\
& \leqslant e^{k a_n} \| v'-v'' \| \leqslant \sqrt{2} e^{ ka_n} d(x',x''). \end{align} Therefore, from \eqref{bounddCHI-001}, \eqref{h_kLip002} and \eqref{Lipschitz-u}, it follows that for $\gamma \in (0,1]$, \begin{align} \label{bounddCHI-010}
|\chi_{n,k}^y(x')- \chi_{n,k}^y(x'')| \leqslant 6 \frac{e^{\gamma k a_n}}{a_{n}^\gamma}d(x',x'')^\gamma. \end{align} The case $t',t''\in B_k$ is treated in the same way.
To conclude the proof we shall consider the case when $t'=-\log\delta(x', y)\in B_{k-1}$ and $t''=-\log\delta(x'', y)\in B_{k}$; the other cases can be handled in the same way. We shall reduce this case to the previous ones. Let $x^*\in \bb P(V)$ be the point on the geodesic line $[x',x'']$ on $\bb P(V)$ such that $d(x',x'')=d(x',x^*)+d(x^*,x'')$ and $t^*=-\log\delta(y,x^*)=ka_n.$ Then \begin{align} \label{bounddCHI-011}
|\chi_{n,k}^y(x')- \chi_{n,k}^y(x'')|
&\leqslant |\chi_{n,k}^y(x')- \chi_{n,k}^y(x^*)| + |\chi_{n,k}^y(x'')- \chi_{n,k}^y(x^*)| \notag\\ &\leqslant 6 \frac{e^{\gamma k a_n}}{ a_{n}^\gamma }d(x',x^*)^{\gamma} + 6 \frac{e^{\gamma k a_n}}{ a_{n}^\gamma }d(x'',x^*)^{\gamma} \notag\\ &\leqslant 12 \frac{e^{\gamma k a_n}}{ a_{n}^\gamma }d(x',x'')^{\gamma}. \end{align} From \eqref{bounddCHI-010} and \eqref{bounddCHI-011} we conclude that $[\chi_{n,k}^y]_{\gamma}\leqslant 12 \frac{e^{ \gamma k a_n }}{a_{n}^{\gamma}}$, which shows \eqref{Holder prop ohCHI_k-001}. \end{proof}
We need the following bounds. Let $M_n=\floor{A\log^2 n}$, where $A>0$ is a constant and $n$ is large enough. For any measurable function $\varphi$ on $\bb P(V)$, it is convenient to denote \begin{align} \label{varphi-nk-001} \varphi_{n,k}^y=\varphi \chi_{n,k}^y\quad\mbox{for}\quad 0 \leqslant k \leqslant M_n-1,\quad \varphi_{n,M_n}^y=\varphi \overline \chi_{n,M_n}^y. \end{align}
\begin{lemma} \label{new bound for delta020} Assume \ref{Ch7Condi-Moment} and \ref{Ch7Condi-IP}. Then there exist constants $s_0 >0$ and $c >0$ such that for any $s \in (-s_0, s_0)$, $y\in \bb P(V^*)$ and any non-negative bounded measurable function $\varphi$ on $\bb P(V)$, \begin{align*}
\sum_{k=0}^{M_n} (k+1) a_n \pi_s(\varphi_{n,k}^y) \leqslant -d_{s,\varphi}(y) + 2 a_n \pi_s(\varphi) \end{align*} and \begin{align*}
\sum_{k=0}^{M_n} (k-1) a_n \pi_s(\varphi_{n,k}^y) \geqslant -d_{s,\varphi}(y) - 2 a_n \pi_s(\varphi) - c \frac{ \|\varphi \|_{\infty}}{n^2}. \end{align*} \end{lemma} \begin{proof} Recall that $d_{s,\varphi}(y)$ is defined in \eqref{drift-d001bis}. Using \eqref{varphi-nk-001} and \eqref{Unit-partition001} we deduce that \begin{align*} -d_{s,\varphi}(y) & = - \sum_{k=0}^{M_n} \int_{\bb P (V)} \varphi_{n,k}^y(x) \log\delta(x, y) \pi_s(dx) \notag \\ &\geqslant \sum_{k=0}^{M_n} (k-1) a_n \pi_s(\varphi_{n,k}^y)
= \sum_{k=0}^{M_n} (k+1) a_n \pi_s(\varphi_{n,k}^y) - 2 a_n \pi_s(\varphi), \end{align*} which proves the first assertion of the lemma.
Using the Markov inequality and the H\"older regularity of the invariant measure $\pi_s$ (Lemma \ref{Lem_Regu_pi_s00}),
we get that there exists a small constant $\eta >0$ such that \begin{align*} &- \int_{\bb P(V)} \varphi_{n,M_n}^y(x)\log \delta(x, y) \pi_s(dx) \notag\\
& \leqslant c \|\varphi \|_{\infty} \int_{\bb P(V)} \frac{e^{-\eta A\log n}}{\delta(x, y)^{\eta}} \delta(x, y)^{-\eta} \pi_s(dx) \notag \\
& = c \|\varphi \|_{\infty} e^{-\eta A \log n} \int_{\bb P(V)} \delta(x, y)^{-2\eta} \pi_s(dx)
\leqslant c \frac{ \|\varphi \|_{\infty}}{ n^2 }, \end{align*} where in the last inequality we choose $A >0$ to be sufficiently large so that $\eta A \geqslant 2$. Therefore, \begin{align*} -d_{s,\varphi}(y) &= - \sum_{k=0}^{M_n} \int_{\bb P (V)} \varphi_{n,k}^y(x) \log\delta(x, y) \pi_s(dx) \notag \\
&\leqslant \sum_{k=0}^{M_n-1} (k+1) a_n \pi_s(\varphi_{n,k}^y) + c \frac{ \|\varphi \|_{\infty}}{ n^2 } \notag \\
& \leqslant \sum_{k=0}^{M_n-1} (k-1) a_n \pi_s(\varphi_{n,k}^y) + 2 a_n \pi_s(\varphi) + c \frac{ \|\varphi \|_{\infty}}{ n^2 } \notag\\
& \leqslant \sum_{k=0}^{M_n} (k-1) a_n \pi_s(\varphi_{n,k}^y) + 2 a_n \pi_s(\varphi) + c \frac{ \|\varphi \|_{\infty}}{ n^2 }. \end{align*} This proves the second assertion of the lemma. \end{proof}
\begin{proof}[Proof of Theorem \ref{Thm-Edge-Expan-Coeff001extended}] Without loss of generality, we assume that the target function $\varphi$ is non-negative. With the notation in \eqref{varphi-nk-001}, we have that for $t \in \bb R$, \begin{align}\label{Initial decompos-001aa} I_n(t) &: =\bb{E}_{\mathbb{Q}_s^x} \left[ \varphi(G_n \!\cdot\! x)
\mathds{1}_{ \big\{ \frac{\log |\langle f, G_n v \rangle| - n\Lambda'(s) }{ \sigma_s \sqrt{n} } \leqslant t \big\} } \right] \notag \\ & = \sum_{k=0}^{M_n} \bb{E}_{\mathbb{Q}_s^x} \left[ \varphi_{n,k}^y (G_n \!\cdot\! x)
\mathds{1}_{ \big\{ \frac{\log |\langle f, G_n v \rangle| - n\Lambda'(s) }{ \sigma_s \sqrt{n} } \leqslant t \big\} }\right] =: \sum_{k=0}^{M_n} F_{n,k}(t). \end{align} For $0\leqslant k\leqslant M_n - 1$, using \eqref{Ch7_Intro_Decom0a} and the fact that $-\log \delta(x, y) \leqslant (k+1)a_n$ when $x \in \supp \varphi_{n,k}^y$, we get \begin{align} \label{boundFfbar-001} F_{n,k}(t) \leqslant \bb{E}_{\mathbb{Q}_s^x} \left[ \varphi_{n,k}^y (G_n \!\cdot\! x) \mathds{1}_{ \big\{ \frac{\sigma(G_n, x) - n\Lambda'(s) }{ \sigma_s \sqrt{n} } \leqslant t + \frac{(k+1) a_n}{\sigma_s \sqrt{n}} \big\} } \right] =: H_{n,k}(t). \end{align} For $k=M_n$, we have \begin{align} \label{boundFfbar-002} F_{n,M_n}(t)
&\leqslant
\bb{E}_{\mathbb{Q}_s^x} \left[ \varphi_{n,M_n}^y (G_n \!\cdot\! x) \mathds{1}_{ \big\{ \frac{ \sigma(G_n,x) - n\Lambda'(s) }{ \sigma_s \sqrt{n} } \leqslant t +\frac{(M_n+1) a_n}{\sigma_s \sqrt{n}} \big\} }\right] \notag \\ &\quad +\bb{E}_{\mathbb{Q}_s^x} \left[ \varphi_{n,M_n}^y (G_n \!\cdot\! x) \mathds{1}_{ \big\{ -\log \delta(G_n \cdot x, y) \geqslant (M_n +1)a_n \big\} }\right] \notag \\ &=: H_{n,M_n}(t) + W_{n}. \end{align} By Lemma \ref{Lem_Regu_pi_s} and choosing $A >0$ large enough, we get \begin{align} \label{W_n001}
W_{n} &\leqslant \|\varphi \|_{\infty} \mathbb{Q}_s^x ( -\log \delta(G_n \cdot x, y) \geqslant A \log n ) \notag\\
&\leqslant \frac{c_0}{n^{c_1 A}} \|\varphi \|_{\infty}
\leqslant \frac{c_0}{n^{2}} \|\varphi \|_{\infty}. \end{align} Now we deal with $H_{n,k}(t)$ for $0 \leqslant k \leqslant M_n$. Denote for short $t_{n,k} = t +\frac{(k+1) a_n}{\sigma_s \sqrt{n}}$. Applying the Edgeworth expansion (Theorem \ref{Thm-Edge-Expan}) we obtain that, uniformly in $s \in (-s_0, s_0)$, $x \in \bb P(V)$, $t \in \bb R$, $0 \leqslant k \leqslant M_n$ and $\varphi \in \mathscr{B}_{\gamma}$, as $n \to \infty$, \begin{align*}
H_{n,k}(t)
& = \pi_s(\varphi_{n,k}^y) \Big[ \Phi(t_{n,k}) + \frac{\Lambda'''(s)}{ 6 \sigma_s^3 \sqrt{n}} (1-t_{n,k}^2) \phi(t_{n,k}) \Big]
\notag\\
& \quad - \frac{ b_{s, \varphi_{n,k}^y}(x) }{ \sigma_s \sqrt{n} } \phi(t_{n,k})
+ \pi_s(\varphi_{n,k}^y) o \Big( \frac{ 1 }{\sqrt{n}} \Big)
+ \lVert \varphi_{n,k}^y \rVert_{\gamma} O \Big( \frac{ 1 }{n} \Big). \end{align*} Recall that $a_n=\frac{1}{\log n}$ and $M_n=\floor{A\log^2 n}$. By the Taylor expansion we have, uniformly in $s \in (-s_0, s_0)$, $x \in \bb P(V)$, $t \in \bb R$ and $0 \leqslant k \leqslant M_n$, \begin{align*} \Phi(t_{n,k}) & =\Phi(t) + \phi(t) \frac{(k+1) a_n}{\sigma_s \sqrt{n}} + O\Big(\frac{\log^2 n}{n} \Big) \end{align*} and \begin{align*} (1-t_{n,k}^2) \phi(t_{n,k}) &= (1-t^2) \phi(t) + O\left( \frac{\log n }{\sqrt{n}} \right). \end{align*} Moreover, using Lemma \ref{Lem-Bs}, we see that \begin{align*} \frac{ b_{s, \varphi_{n,k}^y}(x) }{ \sigma_s \sqrt{n} } \phi(t_{n,k}) = \frac{ b_{s, \varphi_{n,k}^y}(x) }{ \sigma_s \sqrt{n} } \phi(t)
+ \| \varphi_{n,k}^y \|_{\gamma} O \Big(\frac{\log n}{n}\Big). \end{align*} Using these expansions and \eqref{boundFfbar-001}, \eqref{boundFfbar-002} and \eqref{W_n001}, we get that there exists a sequence $(\beta_n)_{n \geqslant 1}$ of positive numbers satisfying $\beta_n \to 0$ as $n \to \infty$, such that for any $0 \leqslant k \leqslant M_n$, \begin{align}\label{F_nkbound001}
F_{n,k}(t)
& \leqslant \pi_s(\varphi_{n,k}^y) \Big[ \Phi(t) + \frac{\Lambda'''(s)}{ 6 \sigma_s^3 \sqrt{n}} (1-t^2) \phi(t) \Big]\notag\\
&\quad - \frac{ b_{s, \varphi_{n,k}^y}(x) }{ \sigma_s \sqrt{n} } \phi(t)
+ \frac{\phi(t)}{\sigma_s \sqrt{n}} \pi_s(\varphi_{n,k}^y)(k+1) a_n \notag\\
& \quad + \pi_s(\varphi_{n,k}^y) \frac{ \beta_n }{\sqrt{n}}
+ \lVert \varphi_{n,k}^y \rVert_{\gamma} \frac{ c \log n }{n}. \end{align} By Lemma \ref{lemmaHolder property001}, it holds that for any $\gamma \in (0, 1]$ and $0 \leqslant k \leqslant M_n$, \begin{align}\label{HolderNorm-varphik} \lVert \varphi_{n,k}^y \rVert_{\gamma} \leqslant c \lVert \varphi \rVert_{\infty} n^{\gamma A} \log^{\gamma} n + \lVert \varphi \rVert_{\gamma}. \end{align} From \eqref{Def-bsvarphi}, it follows that $b_{s,\varphi}(x) = \sum_{k= 0}^{M_n} b_{s, \varphi_{n,k}^y}(x)$. Therefore, summing up over $k$ in \eqref{F_nkbound001}, using \eqref{HolderNorm-varphik} and taking $\gamma >0$ to be sufficiently small such that $\gamma A < \varepsilon/2$, we obtain \begin{align*}
I_n(t) = \sum_{k= 0}^{M_n} F_{n,k}(t) & \leqslant \pi_s(\varphi)
\Big[ \Phi(t) + \frac{\Lambda'''(s)}{ 6 \sigma_s^3 \sqrt{n}} (1-t^2) \phi(t) \Big] \notag\\ & \quad - \frac{ b_{s,\varphi}(x) }{ \sigma_s \sqrt{n} } \phi(t) + \frac{\phi(t)}{\sigma_s \sqrt{n}} \sum_{k = 0}^{M_n} \pi_s(\varphi_{n,k}^y)(k+1) a_n \notag\\ & \quad + \pi_s(\varphi) \frac{ \beta_n }{\sqrt{n}} +
\lVert \varphi \rVert_{\gamma} \frac{ c }{n^{1 - \varepsilon}}. \end{align*} Using Lemma \ref{new bound for delta020} and the fact that $a_n \to 0$ as $n \to \infty$, we obtain the desired upper bound.
The lower bound is established in the same way. Instead of \eqref{boundFfbar-001} we use the following lower bound, which is obtained using \eqref{Ch7_Intro_Decom0a} and the fact that $-\log \delta(x, y) \geqslant (k-1)a_n$ for $x \in \supp \varphi_{n,k}^y$ and $0\leqslant k\leqslant M_n$, \begin{align} \label{boundFfbar-003} F_{n,k}(t) \geqslant \bb{E}_{\mathbb{Q}_s^x} \left[ \varphi_{n,k}^y (G_n \!\cdot\! x) \mathds{1}_{ \big\{ \frac{\sigma(G_n, x) - n\Lambda'(s) }{ \sigma_s \sqrt{n} } \leqslant t +\frac{(k-1) a_n}{\sigma_s\sqrt{n}} \big\} } \right]. \end{align} Proceeding in the same way as in the proof of the upper bound,
using \eqref{boundFfbar-003} instead of \eqref{boundFfbar-001} and \eqref{boundFfbar-002}, we get \begin{align*}
I_n(t) = \sum_{k= 0}^{M_n} F_{n,k}(t) & \geqslant \pi_s(\varphi)
\Big[ \Phi(t) + \frac{\Lambda'''(s)}{ 6 \sigma_s^3 \sqrt{n}} (1-t^2) \phi(t) \Big] \notag\\ &\quad - \frac{ b_{s, \varphi}(x) }{ \sigma_s \sqrt{n} } \phi(t) + \frac{\phi(t)}{\sigma_s \sqrt{n}} \sum_{k = 0}^{M_n} \pi_s(\varphi_{n,k}^y)(k-1) a_n \notag\\ & \quad + \pi_s(\varphi) o \Big( \frac{ 1 }{\sqrt{n}} \Big) +
\lVert \varphi \rVert_{\gamma} O \Big( \frac{1 }{n^{1 - \varepsilon}} \Big). \end{align*} The lower bound is obtained using again Lemma \ref{new bound for delta020} and the fact that $a_n \to 0$ as $n \to \infty$. \end{proof}
\end{document} | arXiv |
\begin{document}
\title{Optimal quantum estimation of the coupling constant of Jaynes-Cummings interaction} \author{Marco G. Genoni\inst{1}\fnmsep\thanks{\email{[email protected]}} \and Carmen Invernizzi\inst{2} }
\institute{QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, UK \and Dipartimento di Fisica, Universit\`a degli Studi di Milano, I-20133 Milano, Italia}
\abstract{ We address the estimation of the coupling constant of the Jaynes-Cummings Hamiltonian for a coupled qubit-oscillator system. We evaluate the quantum Fisher Information (QFI) for the system undergone the Jaynes-Cummings evolution, considering that the probe initial state is prepared in a Fock state for the oscillator and in a generic pure state for the qubit; we obtain that the QFI is exactly equal to the number of excitations present in the probe state. We then focus on the two subsystems, namely the qubit and the oscillator alone, deriving the two QFIs of the two reduced states, and comparing them with the previous result. Next we focus on feasible measurements on the system, and we
find out that if population measurement on the qubit and Fock number measurement on the oscillator are performed together, the Cramer-Rao bound is saturated, that is the corresponding Fisher Information (FI) is always equal to the QFI. We compare also the performances of these measurements performed alone, that is when one of the two subsystem is ignored. We show that, when the qubit is prepared in either the ground or the excited state, the local measurements are still optimal. } \maketitle
\section{Introduction} \label{intro}
The Jaynes-Cummings (JC) model \cite{JCmodel} is one of the paradigmatic examples of {\em hybrid systems}, where a two-level system, modelled by a spin-$1/2$, is coupled to a
quantized mode of a harmonic oscillator. This is the typical situation in quantum
electrodynamics (QED) \cite{Haroche} where a single atom can be coupled to a cavity mode,
for example in the microwave \cite{microwave,microwave2} and in the optical
\cite{optical} regime. The same model describes accurately other interesting physical systems, such as a single ion or a neutral atom in a trap \cite{Leibfried}, and the interaction of artificial atoms with resonators in circuit QED systems \cite{circuit00,circuit01}. All these {\em hybrid systems} are considered essential for the development of quantum information processing and in general of future quantum technologies \cite{wiring,qinternet}. The aim of quantum information is to characterize the peculiar properties of quantum systems and exploit them to perform tasks that would be not achievable in a classical context \cite{NC}. For these purposes it is necessary to characterize the value of quantities that are not directly accessible either in principle or due to experimental impediments. This is the case of relevant quantities like phase \cite{metrology,qphase1,qphase2}, entanglement \cite{estent,estentEXP} or temperature \cite{temperature}, that cannot correspond to any quantum observable or also the coupling constant of different kinds of interactions \cite{JIsing,loss,Kerr}. In this paper we apply the techniques of local quantum estimation theory (QET) \cite{Helstrom,Braunstein,Brody,lqe} to the problem of estimating the coupling constant of the JC Hamiltonian. In this framework, the characterization of the estimation of the parameter is provided by the Fisher information (FI) which represents an infinitesimal distance between probability distributions and gives the ultimate precision attainable by an estimator via the Cramer-Rao theorem. Its quantum counterpart, the quantum Fisher information (QFI), is related to the degree of distinguishability of a quantum state from its neighbours \cite{Bures,Uhlmann,Wootters} and gives the ultimate bound to the precision on the estimate allowed by quantum mechanics. \\ In our estimation problem we consider the initial probe state prepared in a Fock state, as regards the oscillator and in a generic pure state, as regards the qubit. We address the overall estimation properties, by evaluating the QFI for the whole system undergone the JC evolution. We also focus on the two subsystems alone, i.e. we consider the qubit subsystem obtained as the partial trace over the harmonic oscillator and {\em viceversa}, and we evaluate the QFI of the corresponding reduced states, in order to identify how much information about the parameter of interest is contained in each subsystem. Moreover we consider as possible feasible measurements on the coupled system, the measurement of the population of the excited state performed on the qubit and a measurement of the Fock number performed on the harmonic oscillator. We evaluate the FI for the collective measurement and observe that it allows one to achieve the ultimate bound on the precision given by the quantum Cramer-Rao bound. Finally we consider the FI for the same measurements performed on the two corresponding subsystems alone, that is when one of the two subsystems is ignored. We show that, when the qubit is prepared in either the ground or the excited state, both single local measurements are still optimal, that is the corresponding FIs are equal to the QFI of the whole systems. \\ The paper is structured as follows: in Sec. \ref{sec:1} we describe the JC model and the unitary dynamics for the coupled system. In Sec. \ref{sec:2} we review some concepts of QET and illustrate the quantum Cramer-Rao bound focusing to the case of a pure-state unitary family. In Sec. \ref{sec:3} we show in details our results and finally, Sec. \ref{conclusions} gives some concluding remarks.
\section{The model}\label{sec:1} The JC model describes the interaction between a single-mode bosonic field and a two-level system (qubit). Its dynamics is exactly solvable and the model has been widely investigated experimentally \cite{Haroche}.
The bosonic field is described upon introducing an annihilation and a creation operator, satisfying $[a,a^\dagger] = \mathbbm{1}$, and the corresponding Fock states $\{ |n\rangle \}_{n=0}^\infty\:$, {\em i.e.} the eigenstates of the number operator $N=a^\dag a$, which provide a basis of the infinite-dimensional Hilbert space. In the rest of the manuscript we will adopt the quantum optics terminology calling {\em photons} the bosonic excitations, however the results presented are also valid for the other physical settings described by the JC model. \\ The two-level system (qubit) is characterized by a
ground state $|g\rangle$ and an excited state $|e\rangle$ which are eigenstates of the Pauli operator $\sigma_z$ and in turn form a basis for the two-dimensional Hilbert space of the qubit. Any pure state of the whole system composed by the bosonic field and the qubit can be written as
$|\Psi\rangle = \sum_{j=g,e} \sum_n c_{j,n} |j,n\rangle$, where we denote with $|j,n\rangle=|j\rangle \otimes |n\rangle$ the tensor product between the state of the qubit and the state of the field.
The JC Hamiltonian reads \begin{align} \mathcal H = \frac{\hbar \omega_q}{2} \sigma_z + \hbar\omega_f \: a^\dag a + \frac{\hbar g}{2} (\sigma_+ a + \sigma_- a^\dagger ) \end{align} where the operators $\sigma_\pm$ are the qubit ladder operators $\sigma_+ =\ketbra{e}{g}$ and $\sigma_-=\ketbra{g}{e}$. The two operators $\sigma_+ a$ and $\sigma_- a^\dagger$ correspond
respectively to a transition from the lower level $|g\rangle$ to the
upper level $|e\rangle$ together with the emission of a photon,
and the transition from $|g\rangle$ to $|e\rangle$ together with the annihilation of a photon. It is clear that the interaction couples, for a given integer $n$, the states $\ket{e,n}$ and $\ket{g,n+1}$.
Upon choosing a suitable rotating frame and by considering the resonance condition $\omega_q = \omega_f$, one rewrites the Hamiltonian $\mathcal H$ in the so called interaction picture as \begin{align} \mathcal H_{JC}= \frac{\hbar g}{2} (\sigma_+ a+ \sigma_-a^\dagger ) . \end{align} The corresponding evolution unitary operator for an interaction time $\tau$ reads \begin{align} U_{JC}(\Omega) = \exp \left( -\frac{i}{\hbar}\mathcal H_{JC} \tau \right) = \exp (- i \Omega G ) \end{align} where we defined \begin{align} G_{JC}=\frac{\sigma_+a + \sigma_-a^\dagger }{2} \:\:\:\: \textrm{and} \:\:\:\: \Omega = g \tau. \end{align} The parameter $\Omega$ is the quantity of interest when we want to control the JC dynamics and in the following sections we will focus on its estimation properties.
In our treatment we assume that at time $t=0$ the probe state is prepared in a pure state and no initial correlations between the qubit and the field are present, in formula \begin{align}
\varrho(0)=|\Psi_0\rangle\langle\Psi_0| \end{align}
with $|\Psi_0\rangle = |\psi_q\rangle \otimes |\psi_f\rangle$. In particular the qubit at time $t=0$ is prepared in a pure superposition of ground and excited states, \begin{align}
|\psi_q\rangle = \cos\frac{\theta}{2} |e\rangle+ \sin\frac{\theta}{2} |g\rangle \label{eq:superpos} \end{align}
while the bosonic field is prepared in a Fock state $|\psi_f\rangle=|n\rangle$. Notice that preparation of Fock states have been proposed theoretically and proved experimentally both in cavity QED, ion trapping and circuit-QED systems \cite{Haroche,Cirac,Wineland,Walther,Bertet,HaroceFB,circuit,circuit2}. Given this probe preparation, the evolution of the system reads \begin{align} \varrho(\Omega)=U_{JC}(\Omega)\varrho(0)U_{JC}^\dagger(\Omega). \end{align} and upon tracing the evolved state over the bosonic field or the qubit degrees of freedom, we obtain the states \begin{align} \varrho_q(\Omega)=\Tr_f[U_{JC}(\Omega)\varrho(0)U_{JC}^\dagger(\Omega)] \\ \varrho_f(\Omega) =\Tr_q[U_{JC}(\Omega)\varrho(0)U_{JC}^\dagger(\Omega)] \label{eq:subsystems} \end{align} which describe respectively the qubit state and the harmonic oscillator subsystem state at time $t$.
In details, the reduced qubit density operator reads in the basis $\{ |e\rangle, |g\rangle\}$ as \begin{align} \varrho_q(\Omega)=\left (\begin{array}{cc} \varrho_{ee} &\varrho_{eg} \\ \varrho_{ge} &\varrho_{gg}
\end{array}\right ), \label{eq:qubitreduced} \end{align} where \begin{align} \varrho_{gg}=&\cos^2\frac \theta 2\sin^2 \left(\frac{\Omega}{2} \sqrt{n+1}\right) +\sin^2\frac\theta 2 \cos^2\left (\frac{\Omega}{2}\sqrt n\right) \nonumber \\ \varrho_{ee} =& 1-\varrho_{gg} \nonumber \\ \varrho_{eg}=& \frac 1 2\sin\theta\cos\left(\frac \Omega 2\sqrt n\right)\cos\left(\frac\Omega 2\sqrt{n+1}\right) =\varrho_{ge} \label{eq:qubitreduced3}. \end{align} On the other hand, the reduced density operator for the bosonic field is diagonal in the Fock basis: \begin{align}
\varrho_f(\Omega) = p_{n-1} |n-1\rangle\langle n-1| + p_n |n\rangle\langle n| + p_{n+1}
|n+1\rangle\langle n+1|,\label{eq:bosonreduced} \end{align} where \begin{align} p_{n-1} &= \sin^2 \frac\theta 2\left( \frac{1 - \cos \left(\Omega \sqrt{n} \right)}{2}\right) \nonumber \\ p_n &= \frac12 \left( 1 + \cos^2 \frac\theta 2 \cos\left (\Omega \sqrt{n+1}\right) + \sin^2 \frac\theta 2 \cos\left (\Omega \sqrt{n}\right) \right) \nonumber \\ p_{n+1} &= \cos^2 \frac\theta 2 \left(\frac{1 - \cos \left(\Omega \sqrt{n+1} \right)}{2}\right).
\label{eq:bosonreduced2} \end{align}
\section{Local quantum estimation theory}\label{sec:2} In this section we review the basic concepts of local quantum estimation theory that will be applied later to the qubit-oscillator system.
An estimation problem consists into choosing a measurement somehow related with the parameter of interest and then defining an estimator, i.e. a function from the set of the measurement outcomes to the parameter space, in order to infer the value of the quantity that we want to estimate. Classically, given the conditional probability
$p(j|\Omega)$ of measuring the outcome $j$ when the value to be estimated is $\Omega$, optimal estimators are those saturating the Cramer-Rao inequality, which establishes that the variance $\hbox{Var}(\Omega)$ of any unbiased estimator is lower bounded by \begin{align} \hbox{Var}(\Omega)\geq \frac{1}{M F(\Omega)} , \end{align} where $M$ is the number of measurements of the sample and $F(\Omega)$ the Fisher information (FI) \begin{align}
F(\Omega)=\sum_{j} p(j|\Omega)[\partial_\Omega \ln p(j|\Omega)]^2. \label{eq:FI} \end{align} In quantum mechanics, according to the Born rule one has
$p(j|\Omega)=\Tr[\varrho_\Omega \Pi_j ]$, where $\varrho_\Omega$ is a family of quantum states which depend on the parameter $\Omega$ and the operators $\{\Pi_j\}$ are the elements of the probability operator-valued measure (POVM) describing the quantum measurement. By defining the symmetric logarithmic derivative (SLD) operator $L_\Omega$ by means of the following equation \begin{align} \frac{\partial\varrho_\Omega}{\partial\Omega} = \frac{ L_{\Omega} \varrho_\Omega + \varrho_\Omega L_\Omega}2, \end{align} the classical Fisher Information in Eq. (\ref{eq:FI}) can be rewritten as \begin{align} F(\Omega) = \sum_j \frac{{\rm Re}(\Tr[\varrho_\Omega \Pi_j L_\Omega ])^2}{\Tr[\varrho_\Omega \Pi_j]} \end{align} which establishes the maximum precision for the estimation of the parameter $\Omega$ for a fixed measurement $\{\Pi_j\}$. Moreover, maximizing the FI over all the possible measurements, one can show that \begin{align} F(\Omega) \leq H(\Omega) = \Tr [\varrho_\Omega L_\Omega^2 ] . \end{align} The quantity $H(\Omega)$ is called quantum Fisher information (QFI) and define the corresponding quantum Cramer-Rao bound \begin{align} \hbox{Var}(\Omega)\geq\frac{1}{M H(\Omega)}, \end{align} which gives the ultimate limit to the precision allowed from quantum mechanics for the estimation of the parameter $\Omega$ labelling a given quantum statistical model $\varrho_\Omega$.
When the set of quantum states $\varrho_\Omega$ is given in a diagonal form $\varrho_\Omega= \sum_k \lambda_k |\phi_k\rangle \langle \phi_k |$, the QFI can be evaluated as \begin{align} H(\Omega) = \sum_k \frac{(\partial_\Omega \lambda_k)^2}{\lambda_k} + 2 \sum_{k,l} \frac{(\lambda_k - \lambda_l)^2}{\lambda_k + \lambda_l}
|\langle\phi_l | \partial_\Omega \phi_k\rangle|^2.
\label{eq:QFIr} \end{align} A particular case is given when the family of $\varrho_\Omega$ is a unitary pure-state family, \begin{align}
\varrho_{\Omega} = U_\Omega |\Psi_0\rangle\langle \Psi_0|U_\Omega^\dagger \qquad U_\Omega = \exp ( - i G \Omega) \end{align}
where $G$ is the generator of the transformation and $|\Psi_0\rangle$ the initial pure probe state. In this case one can prove that the QFI is independent on the actual value of the parameter $\Omega$ and it is proportional to the fluctuations of the generator on the probe state, i.e. \begin{align} H(\Omega) &= 4\: \langle \Delta^2 G \rangle_0 \nonumber \\
&= 4\left( \langle \Psi_0 | G^2 |\Psi_0\rangle - \langle \Psi_0 | G |\Psi_0\rangle ^2 \right) \label{eq:unitaryfam} \end{align} One can thus rewrite the quantum Cramer-Rao bound and read it as a Heisenberg-like uncertainty relation \begin{align} \hbox{Var}(\Omega )\: \langle \Delta^2 G \rangle_0 \geq \frac{1}{4 M}. \end{align}
\section{Results}\label{sec:3} In this section we report the results for the estimation of the parameter $\Omega$ in the JC model. We first derive the ultimate limit to the estimation by evaluating the QFI for the whole coupled system composed by the qubit and the harmonic oscillator. We then consider the two subsystems alone, by performing a partial trace either on the bosonic field or the qubit, and evaluate the QFI of the reduced states $\varrho_q(\Omega)$ and $\varrho_f(\Omega)$ given respectively in Eqs. (\ref{eq:qubitreduced}) and (\ref{eq:bosonreduced}) . We also consider the performances of two feasible measurements: the photon number measurement on the bosonic field and the population measurement on the qubit. We first evaluate the FI for the collective measurement on the total system and finally, we consider the FI for the same measurements performed on the two subsystems alone, namely the FI for population measurements on the qubit subsystem and the FI for photon number measurements on the oscillator subsystem alone.
\subsection{Quantum Fisher Information}
In the following we evaluate the QFI for the pure state $\varrho(\Omega)$ obtained upon choosing as a probe state $|\Psi_0\rangle = |\psi_q\rangle \otimes |n\rangle$, where the state of the qubit is a general superposition $|\psi_q\rangle$ as in Eq. (\ref{eq:superpos}) and
the field is prepared into a Fock state $|n\rangle$. \\ We consider the evolution of the whole system and assume that both the degrees of freedom of the qubit and of the field are accessible and measurable, i.e. our statistical model is a pure state unitary family and the corresponding QFI for the parameter $\Omega$ can be evaluated as in Eq. (\ref{eq:unitaryfam}) with the generator $G_{JC}=(a \sigma_+ + a^\dagger \sigma_-)/2$. The calculation leads to \begin{align} H(\Omega) = n+ \cos^2\frac\theta{2}. \end{align}
Note that the QFI $H(\Omega)$ has the minimum value for $\theta=\pi$, where $H(\Omega)=n$, and the maximum is $H(\Omega)=n+1$ for $\theta=0$. Thus the optimal preparation for the qubit state which gives the maximum value of the QFI, corresponds to the excited state $|\psi_q\rangle=|e\rangle$. In particular one can observe that the QFI $H(\Omega)$ of the total system is equal to the total number of excitations of the probe state, \begin{align}
H(\Omega) = \langle \Psi_0 | E |\Psi_0\rangle \:\:\: {\rm where} \:\:\:\: E=a^{\dag}a + \sigma_+\sigma_-, \end{align} showing that the more excitations we have, the more precise will be the estimation of the JC coupling constant.\\ We now consider the case where the degrees of freedom of one of the two subsystems are not accessible. The reduced states of the qubit and the bosonic field are given respectively in Eqs. (\ref{eq:qubitreduced}) and (\ref{eq:bosonreduced}) and the corresponding QFIs are denoted as $H_q(\Omega)$ and $H_f(\Omega)$. We evaluate them by means of Eq. (\ref{eq:QFIr})
\begin{figure}
\caption{ QFIs of the two subsystems as a function of $\theta$ with $n=3$ and for different values of the parameter $\Omega$ (blue lines, $\Omega=1.0$; red lines, $\Omega=1.5$). The dashed lines denote the QFI $H_f(\Omega)$ of the bosonic field, while the solid lines denote the QFI $H_q(\Omega)$ of the qubit subsystem. The black solid line on top, represents the QFI $H(\Omega)$ of the qubit-oscillator total system. }
\label{f:QFI}
\end{figure}
and plot their behaviour in Fig. \ref{f:QFI} as a function of $\theta$ and for different values of the parameter $\Omega$. We observe that the bosonic field alone contains more information about the parameter $\Omega$ than the qubit subsystem, that is \begin{align}
H_f(\Omega) \geq H_q(\Omega). \end{align} Moreover, we also observe that if the qubit is prepared in either the ground or excited state, the values of the QFI of the total system $H(\Omega)$ coincides to the QFIs of the two reduced subsystems. In formula we have that, for all the possible values of $\Omega$, \begin{align} H_q(\Omega)[\theta=0]=H_f(\Omega)[\theta=0] &=n+1 \\ H_q(\Omega)[\theta=\pi]=H_f(\Omega)[\theta=\pi] &=n. \end{align} This result shows that, for particular choices of the input state, a measure performed on one of the subsystems, either the qubit or the oscillator, can attain the ultimate precision on the estimate of the parameter $\Omega$ given by the QFI $H(\Omega)$.
\subsection{Fisher information for Fock and population measurements} In order to address the performances of feasible measurements for the estimation of the parameter $\Omega$, we consider the FI for the photon number measurement on the bosonic field and the population measurement on the qubit system. If we consider a collective measurement performed on both subsystems, the non-zero conditional probabilities that we have to take into account are \begin{align}
p(e, n+1 | \Omega) &= \cos^2 \frac\theta 2 \left( \frac{1-\cos\left(\Omega \sqrt{n+1} \right) }{2}\right) \\
p(e, n | \Omega) &= \sin^2 \frac\theta 2 \left( \frac{1+\cos\left(\Omega \sqrt{n} \right) }{2}\right) \\
p(g, n | \Omega) &= \cos^2 \frac\theta 2 \left( \frac{1+\cos\left(\Omega \sqrt{n+1} \right) }{2}\right) \\
p(g, n-1 | \Omega) &= \cos^2 \frac\theta 2 \left( \frac{1-\cos\left(\Omega \sqrt{n} \right) }{2}\right), \end{align}
where $p(j,n|\Omega)$ denotes the conditional probability of obtaining the qubit in the state $j$ and the bosonic field with $n$ excitations, when the parameter has the value $\Omega$. The corresponding Fisher information is evaluated by means of Eq. (\ref{eq:FI}) giving \begin{align} F(\Omega) = H(\Omega) = n + \cos^2 \frac\theta 2. \end{align} Therefore the Fisher information $F(\Omega)$ for the Fock and population measurements saturates the quantum Cramer-Rao bound i.e. the measurement considered is always optimal.
We now study the two measurements separately, i.e. we evaluate the Fisher information for population measurements on the qubit $F_q(\Omega)$ and the Fisher information for the photon number measurements on the harmonic oscillator $F_f(\Omega)$. In particular we want to compare them with the corresponding QFIs for the two subsystems $H_q(\Omega)$ and $H_f(\Omega)$. The population measurement on the qubit corresponds to measure the Pauli operator $\sigma_z$,
whose eigenstates are indeed $\{ |e\rangle, |g\rangle \}$. The conditional probabilities for the qubit to be found in the ground or the excited state are given by the diagonal elements of the reduced state $\varrho_q(\Omega)$, that is \begin{align}
p(g |\Omega)&= \varrho_{gg} = \cos^2\frac \theta 2\sin^2 \left(\frac{\Omega}{2} \sqrt{n+1}\right) +\sin^2\frac\theta 2 \cos^2\left (\frac{\Omega}{2}\sqrt n\right) \\
p(e|\Omega) &= \varrho_{ee} =1-\varrho_{gg}. \end{align} On the other hand, the non-zero conditional probabilites for the number measurement on the bosonic fields are \begin{align}
p(n-1|\Omega) = p_{n-1} &= \sin^2 \frac\theta 2\left( \frac{1 - \cos \left(\Omega \sqrt{n} \right)}{2}\right) \nonumber \\
p(n|\Omega) = p_n &= \frac12 \left( 1 + \cos^2 \frac\theta 2 \cos\left (\Omega \sqrt{n+1}\right) + \sin^2 \frac\theta 2 \cos\left (\Omega \sqrt{n}\right) \right) \nonumber \\
p(n+1|\Omega) = p_{n+1} &= \cos^2 \frac\theta 2 \left(\frac{1 - \cos \left(\Omega \sqrt{n+1} \right)}{2}\right) \end{align}
\begin{figure}
\caption{ (From top to bottom) Black solid line: QFI $H(\Omega)$ and FI $F(\Omega)$ for collective Fock and population measurement. Dashed blue line: QFI $H_f(\Omega)$ and FI $F_f(\Omega)$ for Fock measurement on the harmonic oscillator subsystem alone. Dashed red line: QFI $H_q(\Omega)$. Dotted red line: FI $F_q(\Omega)$ for population measurement on the qubit subsystem alone. The parameters considered are $\Omega=1.25$ and $n=3$. }
\label{f:FI}
\end{figure}
The corresponding FIs, $F_q(\Omega)$ and $F_f(\Omega)$, are evaluated by means of Eq. (\ref{eq:FI}) and are plotted in Fig. \ref{f:FI}. Since the field subsystem state $\varrho_f(\Omega)$ is diagonal in the Fock number basis, we have that the Fock number measurement is always optimal, i.e. the Fisher information is equal to the quantum Fisher information of the state $\varrho_f(\Omega)$ \begin{align} F_f(\Omega) = H_f(\Omega) \qquad \forall \: \: \Omega,\theta, n. \end{align} On the other hand, the population measurement is in general not optimal but it saturates the subsystem quantum Cramer-Rao bound when $\theta=0$ or $\theta=\pi$, that is when the qubit is prepared
respectively in the excited state $|e\rangle$ or in the ground state $|g\rangle$. In these special cases all the quantities we have evaluated so far are equal, that is the FIs for the measurements on the two subsystems are equal not only to the corresponding QFIs of the reduced states $H_f(\Omega)$ and $H_q(\Omega)$, but also to the QFI of the total system $H(\Omega)$. In formula we have that \begin{align} F_q(\Omega) = F_f(\Omega) = H(\Omega) = \begin{cases} n+1 \:\:\: \textrm{if}\:\: \theta=0 \\ n \qquad\:\: \textrm{if}\:\: \theta=\pi \end{cases} \end{align} This interesting feature suggests that for these particular choices of the qubit preparation, in order to attain the ultimate limit posed by the quantum Cramer-Rao bound, we can decide to measure only one of the two subsystems neglecting the other degrees of freedom.
\section{Conclusions}\label{conclusions}
In this paper we addressed the estimation of the coupling constant of the Jaynes-Cummings Hamiltonian considering as an input probe state the field prepared into a Fock state $|n\rangle$ and the qubit prepared in a generic superposition of excited and ground state.\\ We derived the QFI for the whole system, i.e. the qubit-oscillator system evolved under the JC Hamiltonian and observed that it is equal to the number of excitations of the input probe state and independent on the value of the parameter to be estimated. We then derived the QFI for the two subsystems, i.e. the QFI of the qubit and the oscillator state. In this case we have found that in principle the parameter is better estimated when the qubit subsystem is ignored, rather than the bosonic field.\\ We then considered a feasible detection scheme where a measurement of the population of the excited state is performed on the qubit and a measurement of the Fock number is performed on the harmonic oscillator. The FI for the corresponding collective measurement turned out to be equal to the corresponding QFI, saturating the quantum Cramer-Rao bound and providing the optimal estimate of the parameter.\\ Finally we considered the FI for the same measurements performed separately on the two subsystems alone, namely by ignoring the degrees of freedom of one subsystem. The surprising result is that, if the qubit is prepared either in the ground or in the excited state, both the measurements performed on the single subsystems, and ignoring the remaining one, are optimal. This result is relevant in many practical situations where one of the two subsystems is not experimentally accessible. Moreover since the bound obtained does not depend on the value of the parameter, it is not necessary to tune the measurement by means of two-step or adaptive estimation strategies in order to attain the optimality.\\ As we stressed above, our study shows the existence of a link between the estimation of the JC coupling constant and the amount of excitations present in the probe state. Since the preparation of a high-number Fock state is still experimentally challenging, as a future outlook, it would be interesting to study different preparations of the probe states, for example by considering the bosonic field in a coherent state with a high number of photons. It will be relevant in this case, also to understand if the measurements that in this case has been proved to be optimal, remain optimal with such a
different preparation of the probe.
\section{Acknowledgments} The authors acknowledge useful discussions with Stefano Olivares and Matteo Paris. MGG acknowledges support from UK EPSRC (grant EP/I026436/1).
\end{document} | arXiv |
New CAST Limit on the Axion-Photon Interaction (1705.02290)
CAST collaboration: V. Anastassopoulos, S. Aune, K. Barth, A. Belov, H. Brauninger, G. Cantatore, J. M. Carmona, J. F. Castel, S. A. Cetin, F. Christensen, J. I. Collar, T. Dafni, M. Davenport, T. A. Decker, A. Dermenev, K. Desch, C. Eleftheriadis, G. Fanourakis, E. Ferrer-Ribas, H. Fischer, J. A. Garcia, A. Gardikiotis, J. G. Garza, E. N. Gazis, T. Geralis, I. Giomataris, S. Gninenko, C. J. Hailey, M. D. Hasinoff, D. H. H. Hoffmann, F. J. Iguaz, I. G. Irastorza, A. Jakobsen, J. Jacoby, K. Jakovcic, J. Kaminski, M. Karuza, N. Kralj, M. Krcmar, S. Kostoglou, Ch. Krieger, B. Lakic, J. M. Laurent, A. Liolios, A. Ljubicic, G. Luzon, M. Maroudas, L. Miceli, S. Neff, I. Ortega, T. Papaevangelou, K. Paraschou, M. J. Pivovaroff, G. Raffelt, M. Rosu, J. Ruz, E. Ruiz Choliz, I. Savvidis, S. Schmidt, Y. K. Semertzidis, S. K. Solanki, L. Stewart, T. Vafeiadis, J. K. Vogel, S. C. Yildiz, K. Zioutas
May 5, 2017 hep-ex, physics.ins-det
During 2003--2015, the CERN Axion Solar Telescope (CAST) has searched for $a\to\gamma$ conversion in the 9 T magnetic field of a refurbished LHC test magnet that can be directed toward the Sun. In its final phase of solar axion searches (2013--2015), CAST has returned to evacuated magnet pipes, which is optimal for small axion masses. The absence of a significant signal above background provides a world leading limit of $g_{a\gamma} < 0.66 \times 10^{-10} {\rm GeV}^{-1}$ (95% C.L.) on the axion-photon coupling strength for $m_a \lesssim 0.02$ eV. Compared with the first vacuum phase (2003--2004), the sensitivity was vastly increased with low-background x-ray detectors and a new x-ray telescope. These innovations also serve as pathfinders for a possible next-generation axion helioscope.
Search for chameleons with CAST (1503.04561)
V. Anastassopoulos, M. Arik, S. Aune, K. Barth, A. Belov, H. Bräuninger, G. Cantatore, J. M. Carmona, S. A. Cetin, F. Christensen, J. I. Collar, T. Dafni, M. Davenport, K. Desch, A. Dermenev, C. Eleftheriadis, G. Fanourakis, E. Ferrer-Ribas, P. Friedrich, J.Galán, J. A. García, A. Gardikiotis, J. G. Garza, E. N. Gazis, T. Geralis, I. Giomataris, C. Hailey, F. Haug, M. D. Hasinoff, D. H. H. Hoffmann, F. J. Iguaz, I. G. Irastorza, J. Jacoby, A. Jakobsen, K. Jakovčić, J. Kaminski, M. Karuza, M. Kavuk, M. Krčmar, C. Krieger, A. Krüger, B. Lakić, J. M. Laurent, A. Liolios, A. Ljubičić, G. Luzón, S. Neff, I. Ortega, T. Papaevangelou, M. J. Pivovarov, G. Raffelt, H. Riege, M. Rosu, J. Ruz, I. Savvidis, S. K. Solanki, T. Vafeiadis, J. A. Villar, J. K. Vogel, S. C. Yildiz, K. Zioutas (CAST Collaboration), P. Brax, I. Lavrentyev, A. Upadhye
March 18, 2016 hep-ph, physics.ins-det, astro-ph.SR
In this work we present a search for (solar) chameleons with the CERN Axion Solar Telescope (CAST). This novel experimental technique, in the field of dark energy research, exploits both the chameleon coupling to matter ($\beta_{\rm m}$) and to photons ($\beta_{\gamma}$) via the Primakoff effect. By reducing the X-ray detection energy threshold used for axions from 1$\,$keV to 400$\,$eV CAST became sensitive to the converted solar chameleon spectrum which peaks around 600$\,$eV. Even though we have not observed any excess above background, we can provide a 95% C.L. limit for the coupling strength of chameleons to photons of $\beta_{\gamma}\!\lesssim\!10^{11}$ for $1<\beta_{\rm m}<10^6$.
New solar axion search in CAST with $^4$He filling (1503.00610)
M. Arik, S. Aune, K. Barth, A. Belov, H. Bräuninger, J. Bremer, V. Burwitz, G. Cantatore, J. M. Carmona, S. A. Cetin, J. I. Collar, E. Da Riva, T. Dafni, M. Davenport, A. Dermenev, C. Eleftheriadis, N. Elias, G. Fanourakis, E. Ferrer-Ribas, J. Galán, J. A. García, A. Gardikiotis, J. G. Garza, E. N. Gazis, T. Geralis, E. Georgiopoulou, I. Giomataris, S. Gninenko, M. Gómez Marzoa, M. D. Hasinoff, D. H. H. Hoffmann, F. J. Iguaz, I. G. Irastorza, J. Jacoby, K. Jakovčić, M. Karuza, M. Kavuk, M. Krčmar, M. Kuster, B. Lakić, J. M. Laurent, A. Liolios, A. Ljubičić, G. Luzón, S. Neff, T. Niinikoski, A. Nordt, I. Ortega, T. Papaevangelou, M. J. Pivovaroff, G. Raffelt A. Rodríguez, M. Rosu, J. Ruz, I. Savvidis, I. Shilon, S. K. Solanki, L. Stewart, A. Tomás, T. Vafeiadis, J. Villar, J. K. Vogel, S. C. Yildiz, K. Zioutas
June 11, 2015 hep-ex, physics.ins-det
The CERN Axion Solar Telescope (CAST) searches for $a\to\gamma$ conversion in the 9 T magnetic field of a refurbished LHC test magnet that can be directed toward the Sun. Two parallel magnet bores can be filled with helium of adjustable pressure to match the X-ray refractive mass $m_\gamma$ to the axion search mass $m_a$. After the vacuum phase (2003--2004), which is optimal for $m_a\lesssim0.02$ eV, we used $^4$He in 2005--2007 to cover the mass range of 0.02--0.39 eV and $^3$He in 2009--2011 to scan from 0.39--1.17 eV. After improving the detectors and shielding, we returned to $^4$He in 2012 to investigate a narrow $m_a$ range around 0.2 eV ("candidate setting" of our earlier search) and 0.39--0.42 eV, the upper axion mass range reachable with $^4$He, to "cross the axion line" for the KSVZ model. We have improved the limit on the axion-photon coupling to $g_{a\gamma}< 1.47\times10^{-10} {\rm GeV}^{-1}$ (95% C.L.), depending on the pressure settings. Since 2013, we have returned to vacuum and aim for a significant increase in sensitivity.
CAST solar axion search with 3^He buffer gas: Closing the hot dark matter gap (1307.1985)
M. Arik, S. Aune, K. Barth, A. Belov, S. Borghi, H. Brauninger, G. Cantatore, J. M. Carmona, S. A. Cetin, J. I. Collar, E. Da Riva, T. Dafni, M. Davenport, C. Eleftheriadis, N. Elias, G. Fanourakis, E. Ferrer-Ribas, P. Friedrich, J. Galan, J. A. Garcia, A. Gardikiotis, J. G. Garza, E. N. Gazis, T. Geralis, E. Georgiopoulou, I. Giomataris, S. Gninenko, H. Gomez, M. Gomez Marzoa, E. Gruber, T. Guthorl, R. Hartmann, S. Hauf, F. Haug, M. D. Hasinoff, D. H. H. Hoffmann, F. J. Iguaz, I. G. Irastorza, J. Jacoby, K. Jakovcic, M. Karuza, K. Konigsmann, R. Kotthaus, M. Krcmar, M. Kuster, B. Lakic, P. M. Lang, J. M. Laurent, A. Liolios, A. Ljubicic, V. Lozza, G. Luzon, S. Neff, T. Niinikoski, A. Nordt, T. Papaevangelou, M. J. Pivovaroff, G. Raffelt, H. Riege, A. Rodriguez, M. Rosu, J. Ruz, I. Savvidis, I. Shilon, P. S. Silva, S. K. Solanki, L. Stewart, A. Tomas, M. Tsagri, K. van Bibber, T. Vafeiadis, J. Villar, J. K. Vogel, S. C. Yildiz, K. Zioutas
Sept. 15, 2014 hep-ex, physics.ins-det, astro-ph.IM
The CERN Axion Solar Telescope (CAST) has finished its search for solar axions with 3^He buffer gas, covering the search range 0.64 eV < m_a <1.17 eV. This closes the gap to the cosmological hot dark matter limit and actually overlaps with it. From the absence of excess X-rays when the magnet was pointing to the Sun we set a typical upper limit on the axion-photon coupling of g_ag < 3.3 x 10^{-10} GeV^{-1} at 95% CL, with the exact value depending on the pressure setting. Future direct solar axion searches will focus on increasing the sensitivity to smaller values of g_a, for example by the currently discussed next generation helioscope IAXO.
Conceptual Design of the International Axion Observatory (IAXO) (1401.3233)
E. Armengaud, F. T. Avignone, M. Betz, P. Brax, P. Brun, G. Cantatore, J. M. Carmona, G. P. Carosi, F. Caspers, S. Caspi, S. A. Cetin, D. Chelouche, F. E. Christensen, A. Dael, T. Dafni, M. Davenport, A.V. Derbin, K. Desch, A. Diago, B. Döbrich, I. Dratchnev, A. Dudarev, C. Eleftheriadis, G. Fanourakis, E. Ferrer-Ribas, J. Galán, J. A. García, J. G. Garza, T. Geralis, B. Gimeno, I. Giomataris, S. Gninenko, H. Gómez, D. González-Díaz, E. Guendelman, C. J. Hailey, T. Hiramatsu, D. H. H. Hoffmann, D. Horns, F. J. Iguaz, I. G. Irastorza, J. Isern, K. Imai, A. C. Jakobsen, J. Jaeckel, K. Jakovčić, J. Kaminski, M. Kawasaki, M. Karuza, M. Krčmar, K. Kousouris, C. Krieger, B. Lakić, O. Limousin, A. Lindner, A. Liolios, G. Luzón, S. Matsuki, V. N. Muratova, C. Nones, I. Ortega, T. Papaevangelou, M. J. Pivovaroff, G. Raffelt, J. Redondo, A. Ringwald, S. Russenschuck, J. Ruz, K. Saikawa, I. Savvidis, T. Sekiguchi, Y. K. Semertzidis, I. Shilon, P. Sikivie, H. Silva, H. ten Kate, A. Tomas, S. Troitsky, T. Vafeiadis, K. van Bibber, P. Vedrine, J. A. Villar, J. K. Vogel, L. Walckiers, A. Weltman, W. Wester, S. C. Yildiz, K. Zioutas
Jan. 14, 2014 hep-ex, physics.ins-det
The International Axion Observatory (IAXO) will be a forth generation axion helioscope. As its primary physics goal, IAXO will look for axions or axion-like particles (ALPs) originating in the Sun via the Primakoff conversion of the solar plasma photons. In terms of signal-to-noise ratio, IAXO will be about 4-5 orders of magnitude more sensitive than CAST, currently the most powerful axion helioscope, reaching sensitivity to axion-photon couplings down to a few $\times 10^{-12}$ GeV$^{-1}$ and thus probing a large fraction of the currently unexplored axion and ALP parameter space. IAXO will also be sensitive to solar axions produced by mechanisms mediated by the axion-electron coupling $g_{ae}$ with sensitivity $-$for the first time$-$ to values of $g_{ae}$ not previously excluded by astrophysics. With several other possible physics cases, IAXO has the potential to serve as a multi-purpose facility for generic axion and ALP research in the next decade. In this paper we present the conceptual design of IAXO, which follows the layout of an enhanced axion helioscope, based on a purpose-built 20m-long 8-coils toroidal superconducting magnet. All the eight 60cm-diameter magnet bores are equipped with focusing x-ray optics, able to focus the signal photons into $\sim 0.2$ cm$^2$ spots that are imaged by ultra-low-background Micromegas x-ray detectors. The magnet is built into a structure with elevation and azimuth drives that will allow for solar tracking for $\sim$12 h each day.
CAST constraints on the axion-electron coupling (1302.6283)
K. Barth, A. Belov, B. Beltran, H. Brauninger, J. M. Carmona, J.I. Collar, T. Dafni, M. Davenport, L. Di Lella, C. Eleftheriadis, J. Englhauser, G. Fanourakis, E. Ferrer Ribas, H. Fischer, J. Franz, P. Friedrich, J. Galan, J. A. Garcia, T. Geralis, I. Giomataris, S. Gninenko, H. Gomez, M. D. Hassinoff, F. H. Heinsius, D. H. H. Hoffmann, I.G. Irastorza, J. Jacoby, K. Jakovcic, D. Kang, K. Konigsmann, R. Kotthaus, K. Kousouris, M. Krcmar, M. Kuster, B. Lakic, A. Liolios, A. Ljubicic, G. Lutz, G. Luzon, D.W. Miller, T. Papaevangelou, M.J. Pivovaroff, G. Raffelt, J. Redondo, H. Riege, A. Rodriguez, J. Ruz, I. Savvidis, Y. Semertzidis, L. Stewart, K. Van Bibber, J.D. Vieira, J.A. Villar, J.K. Vogel, L. Walckiers, K. Zioutas
April 19, 2013 hep-ph, astro-ph.SR
In non-hadronic axion models, which have a tree-level axion-electron interaction, the Sun produces a strong axion flux by bremsstrahlung, Compton scattering, and axio-recombination, the "BCA processes." Based on a new calculation of this flux, including for the first time axio-recombination, we derive limits on the axion-electron Yukawa coupling g_ae and axion-photon interaction strength g_ag using the CAST phase-I data (vacuum phase). For m_a < 10 meV/c2 we find g_ag x g_ae< 8.1 x 10^-23 GeV^-1 at 95% CL. We stress that a next-generation axion helioscope such as the proposed IAXO could push this sensitivity into a range beyond stellar energy-loss limits and test the hypothesis that white-dwarf cooling is dominated by axion emission.
IAXO - The International Axion Observatory (1302.3273)
J. K. Vogel, F. T. Avignone, G. Cantatore, J. M. Carmona, S. Caspi, S. A. Cetin, F. E. Christensen, A. Dael, T. Dafni, M. Davenport, A. V. Derbin, K. Desch, A. Diago, A. Dudarev, C. Eleftheriadis, G. Fanourakis, E. Ferrer-Ribas, J. Galan, J. A. Garcia, J. G. Garza, T. Geralis, B. Gimeno, I. Giomataris, S. Gninenko, H. Gomez, C. J. Hailey, T. Hiramatsu, D. H. H. Hoffmann, F. J. Iguaz, I. G. Irastorza, J. Isern, J. Jaeckel, K. Jakovcic, J. Kaminski, M. Kawasaki, M. Krcmar, C. Krieger, B. Lakic, A. Lindner, A. Liolios, G. Luzon, I. Ortega, T. Papaevangelou, M. J. Pivovaroff, G. Raffelt, J. Redondo, A. Ringwald, S. Russenschuck, J. Ruz, K. Saikawa, I. Savvidis, T. Sekiguchi, I. Shilon, H. Silva, H. H. J. ten Kate, A. Tomas, S. Troitsky, K. van Bibber, P. Vedrine, J. A. Villar, L. Walckiers, W. Wester, S. C. Yildiz, K. Zioutas
Feb. 13, 2013 hep-ex, physics.ins-det
The International Axion Observatory (IAXO) is a next generation axion helioscope aiming at a sensitivity to the axion-photon coupling of a few 10^{-12} GeV^{-1}, i.e. 1-1.5 orders of magnitude beyond sensitivities achieved by the currently most sensitive axion helioscope, the CERN Axion Solar Telescope (CAST). Crucial factors in improving the sensitivity for IAXO are the increase of the magnetic field volume together with the extensive use of x-ray focusing optics and low background detectors, innovations already successfully tested at CAST. Electron-coupled axions invoked to explain the white dwarf cooling, relic axions, and a large variety of more generic axion-like particles (ALPs) along with other novel excitations at the low-energy frontier of elementary particle physics could provide additional physics motivation for IAXO.
Results and perspectives of the solar axion search with the CAST experiment (1209.6347)
E. Ferrer-Ribas, M. Arik, S. Aune, K. Barth, A. Belov, S. Borghi, H. Bräuninger, G. Cantatore, J. M. Carmona, S. A. Cetin, J. I. Collar, T. Dafni, M. Davenport, C. Eleftheriadis, N. Elias, C. Ezer, G. Fanourakis, P. Friedrich, J. Galán, J. A. García, A. Gardikiotis, J. G. Garza, E. N. Gazis, T. Geralis, I. Giomataris, S. Gninenko, H. Gómez, E. Gruber, T. Guthörl, R. Hartmann, F. Haug, M. D. Hasinoff, D. H. H. Hoffmann, F. J. Iguaz, I. G. Irastorza, J. Jacoby, K. Jakovčić, M. Karuza, K. Königsmann, R. Kotthaus, M. Krčmar, M. Kuster, B. Lakić, J. M. Laurent, A. Liolios, A. Ljubičić, V. Lozza, G. Lutz, G. Luzón, J. Morales, T. Niinikoski, A. Nordt, T. Papaevangelou, M. J. Pivovaroff, G. Raffelt, T. Rashba, H. Riege, A. Rodríguez, M. Rosu, J. Ruz, I. Savvidis, P. S. Silva, S. K. Solanki, L. Stewart, A. Tomás, M. Tsagri, K. van Bibber, T. Vafeiadis, J. Villar, J. K. Vogel, S. C. Yildiz, K. Zioutas (CAST Collaboration)
Oct. 30, 2012 hep-ex, astro-ph.CO
The status of the solar axion search with the CERN Axion Solar Telescope (CAST) will be presented. Recent results obtained by the use of $^3$He as a buffer gas has allowed us to extend our sensitivity to higher axion masses than our previous measurements with $^4$He. With about 1 h of data taking at each of 252 different pressure settings we have scanned the axion mass range 0.39 eV$ \le m_{a} \le $ 0.64 eV. From the absence of an excess of x rays when the magnet was pointing to the Sun we set a typical upper limit on the axion-photon coupling of g$_{a\gamma} \le 2.3\times 10^{-10}$ GeV$^{-1}$ at 95% C.L., the exact value depending on the pressure setting. CAST published results represent the best experimental limit on the photon couplings to axions and other similar exotic particles dubbed WISPs (Weakly Interacting Slim Particles) in the considered mass range and for the first time the limit enters the region favored by QCD axion models. Preliminary sensitivities for axion masses up to 1.16 eV will also be shown reaching mean upper limits on the axion-photon coupling of g$_{a\gamma} \le 3.5\times 10^{-10}$ GeV$^{-1}$ at 95% C.L. Expected sensibilities for the extension of the CAST program up to 2014 will be presented. Moreover long term options for a new helioscope experiment will be evoked.
CAST search for sub-eV mass solar axions with 3He buffer gas (1106.3919)
M. Arik, S. Aune, K. Barth, A. Belov, S. Borghi, H. Bräuninger, G. Cantatore, J. M. Carmona, S. A. Cetin, J. I. Collar, T. Dafni, M. Davenport, C. Eleftheriadis, N. Elias, C. Ezer, G. Fanourakis, E. Ferrer-Ribas, P. Friedrich, J. Galán, J. A. García, A. Gardikiotis, E. N. Gazis, T. Geralis, I. Giomataris, S. Gninenko, H. Gómez, E. Gruber, T. Guthörl, R. Hartmann, F. Haug, M. D. Hasinoff, D. H. H. Hoffmann, F. J. Iguaz, I. G. Irastorza, J. Jacoby, K. Jakovčić, M. Karuza, K. Königsmann, R. Kotthaus, M. Krčmar, M. Kuster, B. Lakić, J. M. Laurent, A. Liolios, A. Ljubičić, V. Lozza, G. Lutz, G. Luzón, J. Morales, T. Niinikoski, A. Nordt, T. Papaevangelou, M. J. Pivovaroff, G. Raffelt, T. Rashba, H. Riege, A. Rodríguez, M. Rosu, J. Ruz, I. Savvidis, P. S. Silva, S. K. Solanki, L. Stewart, A. Tomás, M. Tsagri, K. van Bibber, T. Vafeiadis, J. Villar, J. K. Vogel, S. C. Yildiz, K. Zioutas
Oct. 4, 2012 hep-ex
The CERN Axion Solar Telescope (CAST) has extended its search for solar axions by using 3He as a buffer gas. At T=1.8 K this allows for larger pressure settings and hence sensitivity to higher axion masses than our previous measurements with 4He. With about 1 h of data taking at each of 252 different pressure settings we have scanned the axion mass range 0.39 eV < m_a < 0.64 eV. From the absence of excess X-rays when the magnet was pointing to the Sun we set a typical upper limit on the axion-photon coupling of g_ag < 2.3 x 10^{-10} GeV^{-1} at 95% CL, the exact value depending on the pressure setting. KSVZ axions are excluded at the upper end of our mass range, the first time ever for any solar axion search. In future we will extend our search to m_a < 1.15 eV, comfortably overlapping with cosmological hot dark matter bounds.
Relativistic kinematics beyond Special Relativity (1206.5961)
J. M. Carmona, J. L. Cortes, F. Mercati
Sept. 28, 2012 hep-th, gr-qc
In the context of departures from Special Relativity written as a momentum power expansion in the inverse of an ultraviolet energy scale M, we derive the constraints that the relativity principle imposes between coefficients of a deformed composition law, dispersion relation, and transformation laws, at first order in the power expansion. In particular, we find that, at that order, the consistency of a modification of the energy-momentum composition law fixes the modification in the dispersion relation. We therefore obtain the most generic modification of Special Relativity that preserves the relativity principle at leading order in 1/M.
Uncertainties in Constraints from Pair Production on Superluminal Neutrinos (1203.2585)
J. M. Carmona, J. L. Cortes, D. Mazon
Sept. 28, 2012 hep-ph
The use of the vacuum lepton pair production process ($\nu \to \nu e^- e^+ $), a viable reaction for superluminal neutrinos, to put constraints on Lorentz violations requires a dynamical framework. Different choices of dynamical matrix elements and modified dispersion relations for neutrinos, leading to numerical factors differing by one order of magnitude in the results for the pair production decay width, are used to show the uncertainties on these constraints.
NEXT-100 Technical Design Report (TDR). Executive Summary (1202.0721)
NEXT Collaboration: V. Álvarez, F. I. G. M. Borges, S. Cárcel, J. M. Carmona, J. Castel, J. M. Catalá, S. Cebrián, A. Cervera, D. Chan, C. A. N. Conde, T. Dafni, T. H. V. T. Dias, J. Díaz, M. Egorov, R. Esteve, P. Evtoukhovitch, L. M. P. Fernandes, P. Ferrario, A. L. Ferreira, E. Ferrer-Ribas, E. D. C. Freitas, V. M. Gehman, A. Gil, I. Giomataris, A. Goldschmidt, H. Gómez, J. J. Gómez-Cadenas, K. González, D. González-Díaz, R. M. Gutiérrez, J. Hauptman, J. A. Hernando Morata, D. C. Herrera, V. Herrero, F. J. Iguaz, I. G. Irastorza, V. Kalinnikov, D. Kiang, L. Labarga, I. Liubarsky, J. A. M. Lopes, D. Lorca, M. Losada, G. Luzón, A. Marí, J. Martín-Albo, A. Martínez, T. Miller, A. Moiseenko, F. Monrabal, C. M. B. Monteiro, J. M. Monzó, F. J. Mora, L. M. Moutinho, J. Muñoz Vidal, H. Natal da Luz, G. Navarro, M. Nebot, D. Nygren, C. A. B. Oliveira, R. Palma, J. Pérez, J. L. Pérez Aparicio, J. Renner, L. Ripoll, A. Rodríguez, J. Rodríguez, F. P. Santos, J. M. F. dos Santos, L. Segui, L. Serra, D. Shuman, C. Sofka, M. Sorel, J. F. Toledo, A. Tomás, J. Torrent, Z. Tsamalaidze, D. Vázquez, E. Velicheva, J. F. C. A. Veloso, J. A. Villar, R. C. Webb, T. Weber, J. White, N. Yahlali
April 16, 2012 hep-ex, physics.ins-det
In this Technical Design Report (TDR) we describe the NEXT-100 detector that will search for neutrinoless double beta decay (bbonu) in Xe-136 at the Laboratorio Subterraneo de Canfranc (LSC), in Spain. The document formalizes the design presented in our Conceptual Design Report (CDR): an electroluminescence time projection chamber, with separate readout planes for calorimetry and tracking, located, respectively, behind cathode and anode. The detector is designed to hold a maximum of about 150 kg of xenon at 15 bar, or 100 kg at 10 bar. This option builds in the capability to increase the total isotope mass by 50% while keeping the operating pressure at a manageable level. The readout plane performing the energy measurement is composed of Hamamatsu R11410-10 photomultipliers, specially designed for operation in low-background, xenon-based detectors. Each individual PMT will be isolated from the gas by an individual, pressure resistant enclosure and will be coupled to the sensitive volume through a sapphire window. The tracking plane consists in an array of Hamamatsu S10362-11-050P MPPCs used as tracking pixels. They will be arranged in square boards holding 64 sensors (8 times8) with a 1-cm pitch. The inner walls of the TPC, the sapphire windows and the boards holding the MPPCs will be coated with tetraphenyl butadiene (TPB), a wavelength shifter, to improve the light collection.
The International Axion Observatory (IAXO) (1201.3849)
I. G. Irastorza, F. T. Avignone, G. Cantatore, S. Caspi, J. M. Carmona, T. Dafni, M. Davenport, A. Dudarev, G. Fanourakis, E. Ferrer-Ribas, J. Galan, J. A. Garcia, T. Geralis, I. Giomataris, S. Gninenko, H. Gomez, D. H. H. Hoffmann, F. J. Iguaz, K. Jakovcic, M. Krcmar, B. Lakic, G. Luzon, A. Lindner, M. Pivovaroff, T. Papaevangelou, G. Raffelt, J. Redondo, A. Rodrıguez, S. Russenschuck, J. Ruz, I. Shilon, H. Ten Kate, A. Tomas, S. Troitsky, K. van Bibber, J. A. Villar, J. Vogel, L. Walckiers, K. Zioutas
Jan. 18, 2012 hep-ex
The International Axion Observatory (IAXO) is a new generation axion helioscope aiming at a sensitivity to the axion-photon coupling of a few 10$^{12}$ GeV$^{-1}$, i.e. 1 - 1.5 orders of magnitude beyond the one currently achieved by CAST. The project relies on improvements in magnetic field volume together with extensive use of x-ray focusing optics and low background detectors, innovations already successfully tested in CAST. Additional physics cases of IAXO could include the detection of electron-coupled axions invoked to solve the white dwarfs anomaly, relic axions, and a large variety of more generic axion-like particles (ALPs) and other novel excitations at the low-energy frontier of elementary particle physics. This contribution is a summary of our paper [1] to which we refer for further details.
SiPMs coated with TPB : coating protocol and characterization for NEXT (1201.2018)
V. Álvarez, J. Agramunt, M. Ball, M. Batallé, J. Bayarri, F. I. G. Borges, H. Bolink, H. Brine, S. Cárcel, J. M. Carmona, J. Castel, J. M. Catalá, S. Cebrián, A. Cervera, D. Chan, C. A. N. Conde, T. Dafni, T. H. V. T. Dias, J. Díaz, R. Esteve, P. Evtoukhovitch, J. Ferrando, L. M. P. Fernandes, P. Ferrario, A. L. Ferreira, E. Ferrer-Ribas, E. D. C. Freitas, S. A. García, A. Gil, I. Giomataris, A. Goldschmidt, E. Gómez, H. Gómez, J. J. Gómez-Cadenas, K. González, R. M. Gutiérrez, J. Hauptman, J. A. Hernando-Morata, D. C. Herrera, V. Herrero, F. J. Iguaz, I. G. Irastorza, V. Kalinnikov, L. Labarga, I. Liubarsky, J. A. M. Lopes, D. Lorca, M. Losada, G. Luzón, A. Marí, J. Martin-Albo, A. M. Méndez, T. Miller, A. Moisenko, F. Monrabal, C. M. B. Monteiro, J. M. Monzó, F.J. Mora, J. Muñoz Vidal, H. Natal da Luz, G. Navarro, M. Nebot, D. Nygren, C. A. B. Oliveira, R. Palma, J. L. Pérez Aparicio, J. Pérez, E. Radicioni, M. Quinto, J. Renner, L. Ripoll, A. Rodriguez, J. Rodriguez, F. P. Santos, J. M. F. dos Santos, L. Seguí, L. Serra, D. Shuman, C. Sofka, M. Sorel, A. Soriano, H. Spieler, J. F. Toledo, J. Torrent Collell, A. Tomás, Z. Tsamalaidze, D. Vázquez, E. Velicheva, J. F. C. A. Veloso, J. A. Villar, R. Webb, T. Weber, J. T. White, N. Yahlali
Silicon photomultipliers (SiPM) are the photon detectors chosen for the tracking readout in NEXT, a neutrinoless {\beta}{\beta} decay experiment which uses a high pressure gaseous xenon time projection chamber (TPC). The reconstruction of event track and topology in this gaseous detector is a key handle for background rejection. Among the commercially available sensors that can be used for tracking, SiPMs offer important advantages, mainly high gain, ruggedness, cost-effectiveness and radio-purity. Their main drawback, however, is their non sensitivity in the emission spectrum of the xenon scintillation (peak at 175 nm). This is overcome by coating these sensors with the organic wavelength shifter tetraphenyl butadienne (TPB). In this paper we describe the protocol developed for coating the SiPMs with TPB and the measurements performed for characterizing the coatings as well as the performance of the coated sensors in the UV-VUV range.
Towards a new generation axion helioscope (1103.5334)
I. G. Irastorza, F. T. Avignone, S. Caspi, J. M. Carmona, T. Dafni, M. Davenport, A. Dudarev, G. Fanourakis, E. Ferrer-Ribas, J. Galan, J. A. Garcia, T. Geralis, I. Giomataris, H. Gomez, D. H. H. Hoffmann, F. J. Iguaz, K. Jakovcic, M. Krcmar, B. Lakic, G. Luzon, M. Pivovaroff, T. Papaevangelou, G. Raffelt, J. Redondo, A. Rodriguez, S. Russenschuck, J. Ruz, I. Shilon, H. Ten Kate, A. Tomas, S. Troitsky, K. van Bibber, J. A. Villar, J. Vogel, L. Walckiers, K. Zioutas
May 23, 2011 hep-ex
We study the feasibility of a new generation axion helioscope, the most ambitious and promising detector of solar axions to date. We show that large improvements in magnetic field volume, x-ray focusing optics and detector backgrounds are possible beyond those achieved in the CERN Axion Solar Telescope (CAST). For hadronic models, a sensitivity to the axion-photon coupling of $\gagamma\gtrsim {\rm few} \times 10^{-12}$ GeV$^{-1}$ is conceivable, 1--1.5 orders of magnitude beyond the CAST sensitivity. If axions also couple to electrons, the Sun produces a larger flux for the same value of the Peccei-Quinn scale, allowing one to probe a broader class of models. Except for the axion dark matter searches, this experiment will be the most sensitive axion search ever, reaching or surpassing the stringent bounds from SN1987A and possibly testing the axion interpretation of anomalous white-dwarf cooling that predicts $m_a$ of a few meV. Beyond axions, this new instrument will probe entirely unexplored ranges of parameters for a large variety of axion-like particles (ALPs) and other novel excitations at the low-energy frontier of elementary particle physics.
Search for solar axion emission from 7Li and D(p,gamma)3He nuclear decays with the CAST gamma-ray calorimeter (0904.2103)
CAST Collaboration: S. Andriamonje, S. Aune, D. Autiero, K. Barth, A. Belov, B. Beltran, H. Brauninger, J. M. Carmona, S. Cebrian, J. I. Collar, T. Dafni, M. Davenport, L. Di. Lella, C. Eleftheriadis, J. Englhauser, G. Fanourakis, E. Ferrer. Ribas, H. Fischer, J. Franz, P. Friedrich, T. Geralis, I. Giomataris, S. Gninenko, H. Gomez, M. Hasinoff, F. H. Heinsius, D. H. H. Hoffmann, I. G. Irastorza, J. Jacoby, K. Jakovcic, D. Kang, K. Konigsmann, R. Kotthaus, M. Krcmar, K. Kousouris, M. Kuster, B. Lakic, C. Lasseur, A. Liolios, A. Ljubicic, G. Lutz, G. Luzon, D. W. Miller, J. Morales, A. Ortiz, T. Papaevangelou, A. Placci, G. Raffelt, H. Riege, A. Rodriguez, J. Ruz, I. Savvidis, Y. Semertzidis, P. Serpico, L. Stewart, J. D. Vieira, J. Villar, J. Vogel, L. Walckiers, K. Zioutas
March 6, 2010 hep-ex
We present the results of a search for a high-energy axion emission signal from 7Li (0.478 MeV) and D(p,gamma)3He (5.5 MeV) nuclear transitions using a low-background gamma-ray calorimeter during Phase I of the CAST experiment. These so-called "hadronic axions" could provide a solution to the long-standing strong-CP problem and can be emitted from the solar core from nuclear M1 transitions. This is the first such search for high-energy pseudoscalar bosons with couplings to nucleons conducted using a helioscope approach. No excess signal above background was found.
Search for 14.4 keV solar axions emitted in the M1-transition of 57Fe nuclei with CAST (0906.4488)
CAST Collaboration: S. Andriamonje, S. Aune, D. Autiero, K. Barth, A. Belov, B. Beltrán, H. Bräuninger, J. M. Carmona, S. Cebrián, J. I. Collar, T. Dafni, M. Davenport, L. Di Lella, C. Eleftheriadis, J. Englhauser, G. Fanourakis, E. Ferrer-Ribas, H. Fischer, J. Franz, P. Friedrich, T. Geralis, I. Giomataris, S. Gninenko, H. Gómez, M. Hasinoff, F. H. Heinsius, D. H. H. Hoffmann, I. G. Irastorza, J. Jacoby, K. Jakovčić, D. Kang, K. Königsmann, R. Kotthaus, M. Krčmar, K. Kousouris, M. Kuster, B. Lakić, C. Lasseur, A. Liolios, A. Ljubičić, G. Lutz, G. Luzón, D. Miller, J. Morales, A. Ortiz, T. Papaevangelou, A. Placci, G. Raffelt, H. Riege, A. Rodríguez, J. Ruz, I. Savvidis, Y. Semertzidis, P. Serpico, L. Stewart, J. Vieira, J. Villar, J. Vogel, L. Walckiers, K. Zioutas
Dec. 4, 2009 hep-ex
We have searched for 14.4 keV solar axions or more general axion-like particles (ALPs), that may be emitted in the M1 nuclear transition of 57Fe, by using the axion-to-photon conversion in the CERN Axion Solar Telescope (CAST) with evacuated magnet bores (Phase I). From the absence of excess of the monoenergetic X-rays when the magnet was pointing to the Sun, we set model-independent constraints on the coupling constants of pseudoscalar particles that couple to two photons and to a nucleon g_{a\gamma} |-1.19 g_{aN}^{0}+g_{aN}^{3}|<1.36\times 10^{-16} GeV^{-1} for m_{a}<0.03 eV at the 95% confidence level.
The case for a directional dark matter detector and the status of current experimental efforts (0911.0323)
S. Ahlen, N. Afshordi, J. B. R. Battat, J. Billard, N. Bozorgnia, S. Burgos, T. Caldwell, J. M. Carmona, S. Cebrian, P. Colas, T. Dafni, E. Daw, D. Dujmic, A. Dushkin, W. Fedus, E. Ferrer, D. Finkbeiner, P. H. Fisher, J. Forbes, T. Fusayasu, J. Galan, T. Gamble, C. Ghag, I. Giomataris, M. Gold, H. Gomez, M. E. Gomez, P. Gondolo, A. Green, C. Grignon, O. Guillaudin, C. Hagemann, K. Hattori, S. Henderson, N. Higashi, C. Ida, F. J. Iguaz, A. Inglis, I. G. Irastorza, S. Iwaki, A. Kaboth, S. Kabuki, J. Kadyk, N. Kallivayalil, H. Kubo, S. Kurosawa, V. A. Kudryavtsev, T. Lamy, R. Lanza, T. B. Lawson, A. Lee, E. R. Lee, T. Lin, D. Loomba, J. Lopez, G. Luzon, T. Manobu, J. Martoff, F. Mayet, B. Mccluskey, E. Miller, K. Miuchi, J. Monroe, B. Morgan, D. Muna, A. St. J. Murphy, T. Naka, K. Nakamura, M. Nakamura, T. Nakano, G. G. Nicklin, H. Nishimura, K. Niwa, S. M. Paling, J. Parker, A. Petkov, M. Pipe, K. Pushkin, M. Robinson, A. Rodriguez, J. Rodriguez-Quintero, T. Sahin, R. Sanderson, N. Sanghi, D. Santos, O. Sato, T. Sawano, G. Sciolla, H. Sekiya, T. R. Slatyer, D. P. Snowden-Ifft, N. J. C. Spooner, A. Sugiyama, A. Takada, M. Takahashi, A. Takeda, T. Tanimori, K. Taniue, A. Tomas, H. Tomita, K. Tsuchiya, J. Turk, E. Tziaferi, K. Ueno, S. Vahsen, R. Vanderspek, J. Vergados, J. A. Villar, H. Wellenstein, I. Wolfe, R. K. Yamamoto, H. Yegoryan
Nov. 1, 2009 astro-ph.CO
We present the case for a dark matter detector with directional sensitivity. This document was developed at the 2009 CYGNUS workshop on directional dark matter detection, and contains contributions from theorists and experimental groups in the field. We describe the need for a dark matter detector with directional sensitivity; each directional dark matter experiment presents their project's status; and we close with a feasibility study for scaling up to a one ton directional detector, which would cost around $150M.
Probing eV-scale axions with CAST (0810.4482)
CAST Collaboration: E. Arik, S. Aune, D. Autiero, K. Barth, A. Belov, B. Beltrán, S. Borghi, G. Bourlis, F. S. Boydag, H. Bräuninger, J. M. Carmona, S. Cebrián, S. A. Cetin, J. I. Collar, T. Dafni, M. Davenport, L. Di Lella, O. B. Dogan, C. Eleftheriadis, N. Elias, G. Fanourakis, E. Ferrer-Ribas, H. Fischer, P. Friedrich, J. Franz, J. Galán, T. Geralis, I. Giomataris, S. Gninenko, H. Gómez, R. Hartmann, M. Hasinoff, F. H. Heinsius, I. Hikmet, D. H. H. Hoffmann, I. G. Irastorza, J. Jacoby, K. Jakovčić, D. Kang, K. Königsmann, R. Kotthaus, M. Krčmar, K. Kousouris, M. Kuster, B. Lakić, C. Lasseur, A. Liolios, A. Ljubičić, G. Lutz, G. Luzón, D. Miller, J. Morales, T. Niinikoski, A. Nordt, A. Ortiz, T. Papaevangelou, M. J. Pivovaroff, A. Placci, G. Raffelt, H. Riege, A. Rodríguez, J. Ruz, I. Savvidis, Y. Semertzidis, P. Serpico, R. Soufli, L. Stewart, K. van Bibber, J. Villar, J. Vogel, L. Walckiers, K. Zioutas
Jan. 9, 2009 hep-ex
We have searched for solar axions or other pseudoscalar particles that couple to two photons by using the CERN Axion Solar Telescope (CAST) setup. Whereas we previously have reported results from CAST with evacuated magnet bores (Phase I), setting limits on lower mass axions, here we report results from CAST where the magnet bores were filled with \hefour gas (Phase II) of variable pressure. The introduction of gas generated a refractive photon mass $m_\gamma$, thereby achieving the maximum possible conversion rate for those axion masses \ma that match $m_\gamma$. With 160 different pressure settings we have scanned \ma up to about 0.4 eV, taking approximately 2 h of data for each setting. From the absence of excess X-rays when the magnet was pointing to the Sun, we set a typical upper limit on the axion-photon coupling of $\gag\lesssim 2.17\times 10^{-10} {\rm GeV}^{-1}$ at 95% CL for $\ma \lesssim 0.4$ eV, the exact result depending on the pressure setting. The excluded parameter range covers realistic axion models with a Peccei-Quinn scale in the neighborhood of $f_{\rm a}\sim10^{7}$ GeV. Currently in the second part of CAST Phase II, we are searching for axions with masses up to about 1.2 eV using \hethree as a buffer gas.
Solar axion search with the CAST experiment (0810.1874)
CAST Collaboration:E. Arik, S. Aune, D. Autiero, K. Barth, A. Belov, B. Beltrán, S. Borghi, F. S. Boydag, H. Bräuninger, G. Cantatore, J. M. Carmona, S. A. Cetin, J. I. Collar, T. Dafni, M. Davenport, L. Di Lella, O.B. Dogan, C. Eleftheriadis, N. Elias, G. Fanourakis, E. Ferrer-Ribas, H. Fischer, J. Franz, J. Galán, E. Gazis, T. Geralis, I. Giomataris, S. Gninenko, H. Gómez, M. Hasinoff, F. H. Heinsius, I. Hikmet, D. H. H. Hoffmann, I. G. Irastorza, J. Jacoby, K.Jakovčić, D. Kang, T. Karageorgopoulou, M. Karuza, K. Königsmann, R. Kotthaus, M. Krčmar, K. Kousouris, M. Kuster, B. Lakić, C. Lasseur, A. Liolios, A. Ljubičić, V. Lozza, G. Lutz, G. Luzón, D. Miller, J. Morales, T. Niinikoski, A. Nordt, A. Ortiz, T. Papaevangelou, M. J. Pivovaroff, A. Placci, G. Raiteri, G. Raffelt, H. Riege, A. Rodríguez, J. Ruz, I. Savvidis, Y. Semertzidis, P. Serpico, S. K. Solanki, R. Soufli, L. Stewart, M. Tsagri, K. van Bibber, J5D. Villar, J. Vogel, L. Walckiers, K. Zioutas
Oct. 10, 2008 hep-ex
The CAST (CERN Axion Solar Telescope) experiment is searching for solar axions by their conversion into photons inside the magnet pipe of an LHC dipole. The analysis of the data recorded during the first phase of the experiment with vacuum in the magnet pipes has resulted in the most restrictive experimental limit on the coupling constant of axions to photons. In the second phase, CAST is operating with a buffer gas inside the magnet pipes in order to extent the sensitivity of the experiment to higher axion masses. We will present the first results on the $^{4}{\rm He}$ data taking as well as the system upgrades that have been operated in the last year in order to adapt the experiment for the $^{3}{\rm He}$ data taking. Expected sensitivities on the coupling constant of axions to photons will be given for the recent $^{3}{\rm He}$ run just started in March 2008.
Interpretation of neutrino oscillations based on new physics in the infrared (0709.2267)
J. M. Carmona, J. L. Cortes, J. Indurain
June 4, 2008 hep-th, hep-ph
An interpretation of neutrino oscillations based on a modification of relativistic quantum field theory at low energies, without the need to introduce a neutrino mass, is seen to be compatible with all observations.
Background studies and shielding effects for the TPC detector of the CAST experiment (0706.1636)
G. Luzón, B. Beltrán, J. M. Carmona, S. Cebrián, H. Gómez, I. G. Irastorza, J. Morales, A. Ortíz, A. Rodríguez, J. Ruz, J. A. Villar
June 12, 2007 astro-ph
Sunset solar axions traversing the intense magnetic field of the CERN Axion Solar Telescope (CAST) experiment may be detected in a Time Projection Chamber (TPC) detector, as X-rays signals. These signals could be masked, however, by the inhomogeneous background of materials in the experimental site. A detailed analysis, based on the detector characteristics, the background radiation at the CAST site, simulations and experimental results, has allowed us to design a shielding which reduces the background level by a factor of ~4 compared to the detector without shielding, depending on its position, in the energy range between 1 and 10 keV. Moreover, this shielding has improved the homogeneity of background measured by the TPC.
Status of IGEX dark matter search at Canfranc Underground Laboratory (astro-ph/0211535)
I. G. Irastorza, A. Morales, C. E. Aalseth, F. T. Avignone III, R. L. Brodzinski, J. M. Carmona, S. Cebrian, E. Garcia, I. V. Kirpichnikov, A. A. Klimenko, G. Luzon, H. S. Miley, J. Morales, A. Ortiz de Solorzano, S. B. Osetrov, V. S. Pogosov, J. Puimedon, J. H. Reeves, M. L. Sarsa, A. A. Smolnikov, A. G. Tamanyan, A. A. Vasenko, S. I. Vasiliev, J. A. Villar
Nov. 25, 2002 astro-ph, hep-ex
One IGEX 76Ge double-beta decay detector is currently operating in the Canfranc Underground Laboratory in a search for dark matter WIMPs, through the Ge nuclear recoil produced by the WIMP elastic scattering. In this talk we report on the on-going efforts to understand and eventually reject the background at low energy. These efforts have led to the improvement of the neutron shielding and to partial reduction of the background, but still the remaining events are not totally identified. A tritium contamination or muon-induced neutrons are considered as possible sources, simulations and experimental test being still under progress. According to the success of this study we comment the prospects of the experiment as well as those of its future extension, the GEDEON dark matter experiment.
Status of the ANAIS experiment at Canfranc (hep-ex/0211050)
S. Cebrian, J. Amare, J. M. Carmona, E. Garcia, I. G. Irastorza, G. Luzon, A. Morales, J. Morales, A. Ortiz de Solorzano, J. Puimedon, M.L. Sarsa, J. A. Villar
Nov. 20, 2002 hep-ex
The present status of the ANAIS experiment (Annual Modulation with NaI's) is shown. ANAIS is intended to use more than 100 kg of NaI(Tl) in the Canfranc Underground Laboratory (Spain) searching for seasonal modulation effects in the WIMP signal; in a first stage, a prototype (one single 10.7 kg crystal) has been developed in order to obtain the best conditions regarding the energy threshold and the radioactive background in the low energy region as well as to check the stability of the environmental conditions. The first results corresponding to an exposure of 2069.85 kg day show an average background level of 1.2 counts/(keV kg day) from threshold ($E_{thr} \sim 4$ keV, even using one single photomultiplier) up to 10 keV.
Order parameter fluctuations and thermodynamic phase transitions in finite spin systems and fragmenting nuclei (nucl-th/0105030)
J. M. Carmona, J. Richert, P. Wagner
Feb. 18, 2002 nucl-th, cond-mat
We show that in small and low density systems described by a lattice gas model with fixed number of particles the location of a thermodynamic phase transition can be detected by means of the distribution of the fluctuations related to an order parameter which is chosen to be the size of the largest fragment. We show the correlation between the size of the system and the observed order of the transition. We discuss the implications of this correlation on the analysis of experimental fragmentation data. | CommonCrawl |
\begin{document}
\title{Violation of Mermin's version of a Bell inequality in a classical statistical model} \begin{abstract} We investigate a classical statistical model and show that Mermin's version of a Bell inequality is violated. We get this violation, if the measurement modifies the ensemble, a feature, which is also characteristic for measurement processes for quantum systems. \end{abstract}
\section{Introduction} Important distinctions between classical and Quantum systems are the nonlocal correlations between parts of a quantum system. John Bell formulated this difference in a powerful inequality~\cite{Bell:111654} which rules out local hidden variables as explanations for the measurement outcomes.
Merwin~\cite{Mermin:1981gb} gave a nice example of a Bell-type inequality. It was nicely described by Preskill~\cite{Preskill:1998aa} and Maccone~\cite{Maccone:2013aa}. A non-technical explanation of Bell's inequality was published by Alford~\cite{Alford:2015xpa}. In this article we investigate a classical statistical model and show that Mermin's version of a Bell inequality is violated. We get this violation, if the measurement modifies the ensemble, a feature, which is also characteristic for the measurement process for quantum systems.
Recently, Jaroslaw Duda suggested to investigate the violation of Bell inequalities in a model of maximal entropy random walk~\cite{Duda2009aa}. In the following we follow Duda's idea and present a slightly modified version. We describe an example of a random walk where we follow as close as possible to the presentation of Maccone~\cite{Maccone:2013aa} concerning Mermin's version of a Bell inequality.
\section{Mermin's version of a Bell inequality}
We observe objects $A$ which have three properties. These objects appear always in pairs with equal properties \begin{equation}\label{pairs} x_A = x_B,\ y_A = y_B,\ z_A=z_B\quad\textrm{with}\quad x, y, z\in\{0,1\} \end{equation} Since we know, that $A$ and $B$ have the same properties we need to check the combination of different properties only. In order not to disturb the measurements we test the first property at $A$ and the second property at $B$. Due to condition~(\ref{pairs}) $(x_A,y_B)$ gives the same result as $(x_B,y_A)$. Therefore, we can omit the indices $A$ and $B$. There are three possibilities for the test of pairs \begin{equation}\label{3paare} (x,y), (y,z), (z,x). \end{equation} Whatever the probabilities of triples $x, y, z$ are, we get according to Mermin \begin{equation}\label{Ungleichung} P_{x=y}+P_{y=z}+P_{z=x}\ge 1, \end{equation} as we can easily understand from the diagram in Fig.~\ref{mermin}, where the size of the area is proportional to the probability of the indicated relations. \begin{figure}
\caption{The area indicates the probability for the indicated relation between the values $x, y, z$.}
\label{mermin}
\end{figure}
Assuming that the objects can make certain flips of their properties $x, y, z$ we get random walk models. With the aim to finally violate the inequality~(\ref{Ungleichung}) we allow only certain flips of the properties in one time step. In Fig.~\ref{cube} these flips are indicated by blue edges. Only one of the properties can be flipped in one time step, from $0$ to $1$ or from $1$ to $0$. It is also allowed than none of the properties flips. Therefore, there are 20 flips possible.\begin{figure}
\caption{Three properties $x$, $y$ and $z$ of objects $A$ and $B$ can take the values $0$ and $1$ only. We allow for 20 transitions between adjacent sites of the eight $xyz$-triples which are indicated by thick lines. Transitions between adjacent sites can be way and back. Also none of the properties may flip in a time step.}
\label{cube}
\end{figure}
Using the order of sites \begin{equation}\label{Ordnung} (111),(110),(100),(101),(001),(011),(010),(000) \end{equation} we can specify the possible flips by the adjacency matrix \begin{equation}\label{MatM} M=\begin{pmatrix} 1&0&0&0&0&0&0&0\\ 0&1&1&0&0&0&1&0\\ 0&1&1&1&0&0&0&0\\ 0&0&1&1&1&0&0&0\\ 0&0&0&1&1&1&0&0\\ 0&0&0&0&1&1&1&0\\ 0&1&0&0&0&1&1&0\\ 0&0&0&0&0&0&0&1 \end{pmatrix}. \end{equation} For the random walk we assumes that all paths are equally probable. The number of paths passing at time $t$ at the eight points we store in a vector with eight numbers \begin{equation}\label{Folge} n_t=M^tn_0. \end{equation} Since the new distribution depends on the previous only, the process~(\ref{Folge}) is a Markov process.
Starting with $n_0(i)=\delta_{4i}$ we get the sequence \begin{equation}\begin{aligned}\label{ersteWerte} &n_0=(0, 0, 0, 1, 0, 0, 0, 0)\\ &n_1=(0, 0, 1, 1, 1, 0, 0, 0)\\ &n_2=(0, 1, 2, 3, 2, 1, 0, 0)\\ &n_3=(0, 3, 6, 7, 6, 3, 2, 0)\\ &n_4=(0,11,16,19,16,11, 8, 0)\\ &n_5=(0,35,46,51,46,35,30, 0) \end{aligned}\end{equation} which is soon approaching a constant distribution in the region $R=[2,7]$. If the process started at $t=-\infty$ we get at any finite time the same constant distribution, with the probability $p(i)=1/6$ that a path is arriving in a point $i\in R$. Since we have at every point in the region $R$ either $x=y$ or $y=z$ or $z=x$ and never $x=y=z$ we get for the probabilities $P$ of these paths \begin{equation}\label{Wxyz} P_{x=y}+P_{y=z}+P_{z=x}=1, \end{equation} in agreement with Mermin's version of a Bell inequality~(\ref{Ungleichung}).
To get a violation of this inequality, we define a measurement process which modifies the ensemble. This measurement process needs one time step. We request, that during this step the measured coordinates can not be changed. Further we never measure all three properties at once since the mearsurement of the third property would change the outcome of the two other properties.
With these assumptions we fulfil Maccones suppositions in~\cite{Maccone:2013aa}, ``that the values of these properties are predetermined (counterfactual definiteness) and not generated by their measurement, and that the determination of the property of one object will not influence any property of the other object (locality).''
Due to the Markov property we need to investigate only the measurement step since we know that an infinite number of steps before the measurement lead to a constant distribution in $R$. We show in Fig.~\ref{messung} the possible paths for $x$ and $y$ measurements. It is important that during an $xy$-measurement $z$ is not observed and can be modified. \begin{figure}
\caption{An infinite long Markov process according to Eq.~(\ref{Folge}) leads to a uniform distribution of possible paths. Here the measurement steps are shown. In the measurement steps the two measured values are not allowed to change. The third value is not measured and can vary.}
\label{messung}
\end{figure} Only two of the 10 trajectories in the $xy$-measurement step belong to the results $x=y$ leading to the probality $P_{x=y}=\frac{2}{10}$. In contradiction to the inequality~(\ref{Ungleichung}) we conclude \begin{equation}\label{Wxyz} P_{x=y}+P_{y=z}+P_{z=x}=\frac{6}{10}. \end{equation}
We observe that there are 18 possibile trajectories during one time step. In an $xy$-measurement only 10 of these trajectories are realised and only two of them contribute to the probability $P_{xy}$. We conclude that the modification of the ensemble of paths in the measurement process is the reason for the violation of the Bell inequality~(\ref{Ungleichung}).
Also in quantum mechanics a measurement modifies the state, if it is not done in an eigenstate of the corresponding operator. E.g. if a spin state is in the superposition $\frac{1}{2}|0\rangle+\frac{\sqrt 3}{2}|1\rangle$ of eigenstates of $s_3$, then by a measurement of $s_3$ the state makes with probability $\frac{1}{4}$ a transition to the state $|0\rangle$ or with probability $\frac{3}{4}$ to $|1\rangle$.
\end{document} | arXiv |
James Gray (mathematician)
James Gordon Gray FRSE MInstEE (1876 – 6 November 1934) was a Scottish mathematician and physicist.
James Gray
Born1876
Glasgow, Scotland
Died6 November 1934
Dowanhill, Glasgow
NationalityScottish
CitizenshipUnited Kingdom
Alma materGlasgow University (BSc Eng, DSc 1908)
Scientific career
FieldsMathematics
Physics
InstitutionsAssistant lecturer of physics, Glasgow University (1904)
Senior lecturer of physics, Glasgow University (1908)
Professor of applied physics, Glasgow University (1920–34) Glasgow University.
Life
He was born in Glasgow in 1876, the third of eight children to Andrew Gray and his wife, Annie Gordon. He was educated at Friars Grammar School, in Bangor, Caernarvonshire, Wales, where his father was employed by the university.[1] He attended the University College of North Wales until 1899, when his father and family moved back to Glasgow.[2]
He studied engineering at Glasgow University and graduated BScEng. He then was employed by the university as a lecturer in physics from 1904. The university gave him a doctorate (DSc) in 1908. During the First World War he assisted with naval and aerial defence.[2]
From 1920 to 1934 he was professor of applied physics at Glasgow University.
In 1909 he was elected a fellow of the Royal Society of Edinburgh. His proposers were his father, Andrew Gray, William Jack, Cargill Gilston Knott and George Chrystal.
He died in Dowanhill in Glasgow on 6 November 1934.[3]
Publications
• Dynamics (1911) co-written with his father
References
1. Waterston, Charles D; Macmillan Shearer, A (July 2006). Former Fellows of the Royal Society of Edinburgh 1783–2002: Biographical Index (PDF). Vol. I. Edinburgh: The Royal Society of Edinburgh. ISBN 978-0-902198-84-5. Archived from the original (PDF) on 4 October 2006. Retrieved 27 January 2011.
2. O'Connor, John J.; Robertson, Edmund F., "James Gordon Gray", MacTutor History of Mathematics Archive, University of St Andrews
3. BIOGRAPHICAL Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X. Archived from the original (PDF) on 24 January 2013. Retrieved 31 July 2016.
Authority control
International
• VIAF
National
• Germany
• Israel
Academics
• zbMATH
Other
• SNAC
| Wikipedia |
\begin{document}
\title[Measure and Dimension of Sums and Products]{Measure and Dimension of Sums and Products}
\author{Kyle Hambrook} \address{San Jose State University, One Washington Square, San Jose, CA 95192}
\author{Krystal Taylor} \address{The Ohio State University, Columbus, OH 43210}
\thanks{We thank Alex Iosevich and Izabella {\L}aba for their helpful comments.} \keywords{Hausdorff dimension, Fourier dimension, Minkowski sum, Minkowski product, fractals}
\subjclass[2010]{Primary 28A78, 28A80, 42A38, 42B10}
\date{}
\begin{abstract} We investigate the Lebesgue measure, Hausdorff dimension, and Fourier dimension of sets of the form
$RY + Z, $ where $R \subseteq (0,\infty)$ and $Y, Z \subseteq \mathbb{R}^d$. We prove a theorem on the Lebesgue measure and Hausdorff dimension of $RY+Z$; The theorem is a generalized variant of some theorems of Wolff and Oberlin in which $Y$ is the unit sphere, but its proof is much simpler. We also prove a deeper existence theorem: For each $\alpha \in [0,1]$ and for each non-empty compact set $R \subseteq (0,\infty)$, there exists a compact set $Y \subseteq [1,2]$ such that $\dim_F(Y) = \dim_H(Y) = \overline{\dim_M}(Y) = \alpha$
and $\dim_F(RY) \geq \min\{ 1, \dim_F(R) + \dim_F(Y)\}$.
This theorem verifies a weak form of a more general conjecture, and it can be used to produce new Salem sets from old ones.
\end{abstract}
\maketitle
\section{Introduction}
In this paper, we study the Lebesgue measure, Hausdorff dimension, and Fourier dimension of sets in $\mathbb{R}^d$ of the form \begin{align*}
RY+Z = \bigcup_{(r,z) \in R \times Z} (rY+z) = \cbr{ry+z : r \in R, y \in Y, z \in Z}. \end{align*} where $R \subseteq (0,\infty)$ and $Y,Z \subseteq \mathbb{R}^d$ are non-empty sets.
This problem has been considered before when $Y$ is a smooth surface with non-vanishing curvature; see \cite{bourgain-averages, marstrand-packing, mitsis-thesis, oberlin-2006, oberlin-2007, STmeasure, STinterior, stein-maximal-spherical-means, wolff-1997, wolff-smoothing}, In each of these references, the non-vanishing curvature assumption
is an essential ingredient, as it implies $Y$ supports a Borel probability measure whose Fourier transform decays at $\infty$.
Our aim is to understand what happens when $Y$ is an arbitrary set.
Fourier decay of measures on $Y$ will turn out to play an important role.
We assume throughout that the sets $R$, $Y$, and $Z$ are compact in order to guarantee that the set $RY+Z$ is Borel measurable. We use $\mathcal{L}_d(A)$, $\dim_H(A)$, $\dim_F(A)$,
and $\overline{\dim}_M(A)$, respectively, to denote the $d$-dimensional Lebesgue measure, Hausdorff dimension, Fourier dimension,
and upper Minkowski (or box-counting) dimension of a set $A \subseteq \mathbb{R}^d$. Definitions and basic properties of these dimensions are given in Section \ref{dimensions}.
The expression $a \lesssim b$ means $a \leq C b$ for some positive constant $C$ whose precise value is irrelevant in the context. The expression $a \approx b$ means $a \lesssim b$ and $b \lesssim a$.
\subsection{Main Results}\label{main-results}
This paper has two main results: Theorem \ref{main-thm-1} and Theorem \ref{non-uniform thm sets}. The first motivates the second.
\begin{thm}\label{main-thm-1} Let $R \subseteq (0,\infty)$ and $Y, Z \subseteq \mathbb{R}^d$ be non-empty compact sets.
Let $\delta$ be the maximum of $\dim_F(RY)+\dim_H(Z)$ and $\dim_H(RY)+\dim_F(Z)$. \begin{enumerate}[(a)] \item If $\delta > d$, then $\mathcal{L}_d(RY+Z) > 0$ \item If $\delta \leq d$, then $\dim_{H}(RY+Z) \geq \delta$. \end{enumerate} \end{thm}
This theorem makes precise the intuition that the Hausdorff dimension of $RY + Z$ will be at least the ``total'' dimension of $R$, $Y$, and $Z$, and that $RY +Z$ will have positive Lebesgue measure whenever the ``total'' dimension exceeds the ambient dimension $d$. It is a generalized variant of Theorems of Wolff \cite{wolff-1997, wolff-smoothing} and Oberlin \cite{oberlin-2006,oberlin-2007}, which are restricted to the special case where $Y$ is the unit sphere in $\mathbb{R}^d$.
A thorough discussion of Theorem \ref{main-thm-1} is delayed until Section \ref{lebesgue results}.
We turn now to motivate our second main result: Theorem \ref{non-uniform thm sets}.
In light of Theorem \ref{main-thm-1}, lower bounds on $\dim_F(RY)$ are important.
The following proposition gives two general bounds. Part (a) is immediate from the definition of Fourier dimension. Part (b) is essentially Theorem 7 of Bourgain \cite{bourgain-2010}. \begin{prop}\label{basic estimates} \hspace{1pt} \begin{enumerate}[(a)] \item If $R \subseteq \mathbb{R}$ contains a non-zero point and $Y \subseteq \mathbb{R}^d$, then $RY$ contains a dilate of $Y$, hence $\dim_F(RY) \geq \dim_F(Y)$. \item If $R$ and $Y$ are non-empty compact subsets of $(0,\infty)$,
then $\dim_F(RY) \geq \dim_H(R) + \dim_H(Y) - 1$. \end{enumerate} \end{prop}
One might hope for the following estimate, at least for some specific sets $Y$: \begin{align}\label{hope} \dim_F(RY) \geq \min\cbr{d,\dim_F(R) + \dim_F(Y)}. \end{align}
As the following example shows, \eqref{hope} does not hold in general.
\begin{example}
Any subset of a $(d-1)$-dimensional linear subspace of $\mathbb{R}^d$ has Fourier dimension $0$ (see, e.g., \cite[p.41]{Mat15}, \cite[p.64]{wolff-book-2003}). Let $Y$ be such a subset. Assume also that $Y$ is compact. Let $R \subseteq \mathbb{R}$ be a compact set with $\dim_F(R)>0$. Then $RY$ is of the same form as $Y$, and so $\dim_F(RY) = \dim_F(Y) = 0 < \dim_F(R)$. \end{example}
However, if $Y$ equals $S_{d-1}$ (the $(d-1)$-dimensional unit sphere in $\mathbb{R}^d$), then \eqref{hope} does hold:
\begin{prop}\label{sphere thm sets} For every non-empty compact set $R \subseteq (0,\infty)$, $$\dim_F(RS_{d-1}) \geq \dim_F(R) + \dim_F(S_{d-1}).$$ \end{prop}
Note that $\dim_F(S_{d-1}) = \dim_H(S_{d-1}) = \overline{\dim_M}(S_{d-1}) = d-1$ (see, e.g., \cite{Mat95, wolff-book-2003}).
We wonder if there are other sets in $\mathbb{R}^d$ like the sphere. More precisely, we wonder if the following conjecture is true.
\begin{conjecture}\label{fractal conjecture sets} For every $\alpha \in [0,d]$, there exists a compact set $Y \subseteq \mathbb{R}^d$ such that $\dim_F (Y) = \dim_H (Y) = \overline{\dim_M}(Y) = \alpha$ and such that,
for every non-empty compact set $R \subseteq (0,\infty)$,
\eqref{hope} holds. \end{conjecture}
We are not able to prove this conjecture. However, if we restrict ourselves to dimension $d=1$ and allow $Y$ to depend on $R$, we are able to prove the following weaker version of the conjecture, which is the second main result of this paper.
\begin{thm}\label{non-uniform thm sets} For every $\alpha \in [0,1]$ and every non-empty compact set $R \subseteq (0,\infty)$, there exists a compact set $Y \subseteq [1,2]$ such that $\dim_F (Y) = \dim_H (Y) = \overline{\dim_M}(Y) = \alpha$ and \eqref{hope} holds.
\end{thm} The proof of Theorem \ref{non-uniform thm sets} is inspired by a construction of Salem sets due to {\L}aba and Pramanik \cite{LP}. See also \cite{chen, hambrook-thesis, HL2013, HL2016}, where generalizations of the {\L}aba-Pramanik construction were used to show the sharpness of fractal Fourier restriction theorems.
As an immediate consequence of Theorems \ref{main-thm-1} and \ref{non-uniform thm sets}, we have: \begin{corollary} Suppose $Y, Z \subseteq \mathbb{R}$ are compact sets such that $Y$ contains a non-zero point and $\dim_F(Y)+\dim_H(Z) > 0$. Let $\alpha \in [0,1]$ such that $\dim_F(Y)+\dim_H(Z) > 1-\alpha$. Then there exists a compact set $R \subseteq (0,\infty)$ such that $\dim_H (R) = \dim_F (R) = \alpha$ and $RY+Z$ has positive Lebesgue measure. \end{corollary}
The proofs of Theorem \ref{main-thm-1}, Proposition \ref{sphere thm sets}, and Theorem \ref{non-uniform thm sets} are given in Sections \ref{main-thm-1 proof}, \ref{sphere thm sets proof}, and \ref{non-uniform thm sets proof}, respectively. The following construction will be used in all the proofs.
\begin{definition}\label{measure defn} Given a
Borel measure $\mu$ on $\mathbb{R}$ and a
Borel measure $\nu$ on $\mathbb{R}^d$, we define the Borel measure $\mu \cdot \nu$ on $\mathbb{R}^d$ by \begin{align*} \int_{\mathbb{R}^d} f(z) d (\mu \cdot \nu)(z) = \int_{\mathbb{R}^d} \int_{\mathbb{R}} f(ry) d\mu(r) d\nu(y). \end{align*}
It is readily verified that $(\mu \cdot \nu)(\mathbb{R}^d) = \mu(\mathbb{R})\nu(\mathbb{R}^d)$ and $\mathrm{supp}(\mu \cdot \nu) = \mathrm{supp}(\mu)\mathrm{supp}(\nu)$. \end{definition}
\subsection{Dimensions}\label{dimensions}
In this section, we state the definitions and necessary properties of Hausdorff, Fourier, and upper Minkowski dimension. For more details, see \cite{falconer-book-fractal-geometry, Mat95, wolff-book-2003}. The support of a Borel measure $\mu$ on $\mathbb{R}^d$, denoted $\mathrm{supp}(\mu)$, is
the smallest closed set $F$ with $\mu(\mathbb{R}^d \setminus F) = 0$.
For each $A \subseteq \mathbb{R}^d$, let $\mathcal{M}(A)$ be the set of all non-zero finite Borel measures on $\mathbb{R}^d$ with compact support contained in $A$.
For every $\mu \in \mathcal{M}(\mathbb{R}^d)$ and $0 < s < d$, the $s$-energy of $\mu$ is \begin{align*}
I_s(\mu) = \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} |x-y|^{-s} d\mu(x) d\mu(y) = c(d,s) \int_{\mathbb{R}^d} |\widehat{\mu}(\xi)|^2 |\xi|^{s-d} d\xi, \end{align*}
where $c(d,s) = \pi^{s-d/2} \Gamma((d-s)/2) / \Gamma(s/2)$.
There are numerous equivalent definitions of Hausdorff dimension. For our purposes, the following is most convenient: The Hausdorff dimension of a Borel set $A \subseteq \mathbb{R}^d$ is
\begin{align}\label{capacity dimension} \dim_H(A) = \sup\cbr{ 0 < s < d : I_s(\mu) < \infty \text{ for some } \mu \in \mathcal{M}(A) }. \end{align} The Fourier dimension of a Borel set $A \subseteq \mathbb{R}^d$ is
\begin{align}\label{fourier dimension} \dim_F(A) = \sup\cbr{0 < s < d :
\sup_{\xi \in \mathbb{R}^d} |\widehat{\mu}(\xi)|^2 |\xi|^s < \infty \text{ for some } \mu \in \mathcal{M}(A)}. \end{align}
The Fourier dimension of a measure $\mu \in \mathcal{M}(\mathbb{R}^d)$ is \begin{align}\label{fourier dimension measure}
\dim_F(\mu) = \sup\cbr{ 0 < s < d: \sup_{\xi \in \mathbb{R}^d} |\widehat{\mu}(\xi)|^2 |\xi|^s}. \end{align}
The upper Minkowski dimension of a non-empty bounded set $A \subseteq \mathbb{R}^d$ is \begin{align}\label{upper Minkowski dimension} \overline{\dim_M}(A) = \limsup_{\epsilon \to 0^+} \frac{\log N(A,\epsilon)}{\log(1 / \epsilon)}. \end{align} where
$N(A,\epsilon)$ is the smallest number of $\epsilon$-balls needed to cover $A$.
\begin{lemma}\label{dimension properties} Let $A$ and $B$ be non-empty bounded Borel subsets of $\mathbb{R}^d$. \begin{enumerate}[(a)] \item If $f:A \to \mathbb{R}^n$ is a Lipschitz map, then $\dim_H(f(A)) \leq \dim_H(A)$. \item $\dim_F(A) \leq \dim_H(A) \leq \overline{\dim}_{M}(A)$. \item $\dim_H(A) + \dim_H(B) \leq \dim_H(A \times B) \leq \dim_H(A) + \overline{\dim_M}(B)$. \item If there is a $\mu \in \mathcal{M}(A)$ and positive numbers
$r_0$ and $s$ such that \begin{align*}
\mu(B(x,r)) \approx r^s \quad \text{ for all } x \in A, \,\, 0 < r \leq r_0, \end{align*} then $\dim_H(A) = \overline{\dim_M}(A) = s$. \end{enumerate} \end{lemma}
\subsection{ \texorpdfstring{Reverse of \eqref{hope} and Salem Sets}{Reverse of () and Salem Sets} } \label{equality}
In this section, we find sufficient conditions for the reverse of inequality \eqref{hope} and for $RY$ to be a Salem set. A Borel set $A \subseteq \mathbb{R}^d$ is called Salem if $\dim_F(A) = \dim_H(A)$. \begin{comment} Salem sets
are useful examples in exploring
classic theorems and problems in harmonic analysis (such as Marstrand's projection theorem, Falconer's distance problem \cite{mattila-1987}, Stein's Fourier restriction problem \cite{mock}, and the containment of arithmetic progressions \cite{LP}). \end{comment} See \cite{LP}, \cite{mattila-1987}, \cite{Mat15}, \cite{mock} for some illustrations of the construction of Salem sets and their usefulness in harmonic analysis.
Let $R \subseteq \mathbb{R}$ and $Y \subseteq \mathbb{R}^d$ be non-empty compact sets. By the definition of Fourier dimension, $\dim_F(RY) \leq d$. By parts (a),(b),(c) of Lemma \ref{dimension properties} and because the map $f(x,y) = xy$ is Lipschitz on bounded sets,
\begin{align*} \dim_F(RY) \leq \dim_H(RY)
\leq \dim_H(R \times Y) \leq \dim_H(R) + \overline{\dim_M}(Y). \end{align*}
From this inequality, the following observations are immediate: \begin{enumerate}[(i)] \item If $R$ is a Salem set and $Y$ satisfies $\dim_F(Y) = \dim_H(Y) = \overline{\dim_M}(Y)$, then the reverse of \eqref{hope} holds, i.e., $$ \dim_F(RY) \leq \min\cbr{d,\dim_F(R) + \dim_F(Y)}. $$ \item If, in addition to the hypotheses of (i), \eqref{hope} holds, then $RY$ is a Salem set. \end{enumerate}
The hypotheses of (i) and (ii) are satisfied if, for example, $Y$ is the unit sphere $S_{d-1}$ or a set furnished by Theorem \ref{non-uniform thm sets}, The construction of Salem sets is generally non-trivial; (ii) gives us a way to produce new Salem sets from old ones.
\subsection{Discussion of Theorem \ref{main-thm-1}}\label{lebesgue results}
Let $S_{d-1}$ denote the $(d-1)$-dimensional unit sphere in $\mathbb{R}^d$. Theorem \ref{main-thm-1} is a generalized variant of the following theorems of Wolff and Oberlin.
\begin{thm}[Wolff, Oberlin]\label{real wolff thm} Let $K \subseteq (0,\infty) \times \mathbb{R}^d$ be a non-empty compact set. If $\dim_H(K) > 1$, then $$ \mathcal{L}_d\rbr{\bigcup_{(r,z) \in K} (rS_{d-1}+z)} > 0. $$
\end{thm}
\begin{thm}[Wolff]\label{haus wolff thm} Let $K \subseteq (0,\infty) \times \mathbb{R}^d$ be a non-empty compact set. Let $C \subseteq \mathbb{R}^d$ be the set of centers of the spheres $\cbr{rS_{d-1}+z : (r,z) \in K}$, i.e., $C = \cbr{z: (r,z) \in K \text{ for some } r }$. If $\dim_H(K) \leq 1$ and $\dim_H( C ) \leq 1$, then \begin{align*} \dim_H \rbr{ \bigcup_{(r,z) \in K} (rS_{d-1}+z) } \geq \dim_H(C) + d-1. \end{align*} \end{thm}
\begin{thm}[Oberlin]\label{haus oberlin thm} Let $K \subseteq (0,\infty) \times \mathbb{R}^d$ be a non-empty compact set. If $\dim_H(K) \leq 1$ and $\dim_H(K) < (d-1)/2$, then \begin{align*} \dim_H \rbr{ \bigcup_{(r,z) \in K} (rS_{d-1}+z) } \geq \dim_H(K) + d-1. \end{align*} \end{thm}
Theorem \ref{real wolff thm} is due to Wolff \cite[Corollary 3]{wolff-smoothing} when $d=2$ and Oberlin \cite[Corollary 1]{oberlin-2006} when $d \geq 3$. Theorem \ref{haus oberlin thm} is due to Oberlin \cite[Theorem $3_S$]{oberlin-2007}.
Theorem \ref{haus wolff thm} is due to Wolff \cite[Corollary 5.4]{wolff-1997} (see also the remark following the proof). In fact, Wolff and Oberlin proved slightly more general results; we have stated special cases to simplify comparison to Theorem \ref{main-thm-1}.
Wolff obtained Theorem \ref{real wolff thm} for $d = 2$ as a corollary of localized $L^p$ estimates on functions with Fourier support near the light cone; the proof
involves intricate bounds on circle tangencies. For $d=2$, the proof of Theorem \ref{haus wolff thm} is based on an $L^3$-$L^3$ circular maximal inequality, whose proof in turn is also based on bounds on circle tangencies. For $d \geq 3$, Theorem \ref{haus wolff thm} follows from an an easier $L^2$-$L^2$ spherical maximal inequality. Oberlin obtained Theorem \ref{real wolff thm} for $d \geq 3$ and Theorem \ref{haus oberlin thm} by way of estimates for spherical averaging operators. Mitsis \cite{mitsis-thesis} previously proved a special case of Theorem \ref{real wolff thm} with an additional hypothesis on the Hausdorff dimension of the set of centers $C = \cbr{ z: (r,z) \in K \text{ for some } r }$; the methods are similar to those used by Wolff for Theorem \ref{haus wolff thm}. An alternative proof of Mitsis' result using the technology of spherical maximal operators in a fractal setting follows as an immediate consequence of the work of Krause, Iosevich, Sawyer, Taylor, and Uriarte-Tuero \cite{Maximal}.
In contrast to the proofs of Theorems \ref{real wolff thm}, \ref{haus wolff thm}, \ref{haus oberlin thm}, our proof of Theorem \ref{main-thm-1} is extremely short and uses only elementary Fourier analysis and geometric measure theory. As described in Section \ref{main-results}, Theorem \ref{main-thm-1} also leads to the problem of of obtaining lower bounds on the Fourier dimension of the Minkowski product of two sets, which motivates Theorem \ref{non-uniform thm sets}.
We call Theorem \ref{main-thm-1} a generalized variant of Theorems \ref{real wolff thm}, \ref{haus wolff thm}, \ref{haus oberlin thm}, because Theorem \ref{main-thm-1} allows an arbitrary set $Y$ in place of the sphere $S_{d-1}$. However, Theorem \ref{main-thm-1} is not a true generalization. For one thing, the union
$RY+Z = \bigcup_{(r,z) \in R \times Z } (rY+Z)$ in Theorem \ref{main-thm-1} is over the Cartesian product $R \times Z$, while the union in Theorems \ref{real wolff thm}, \ref{haus wolff thm}, \ref{haus oberlin thm}, is over a more general set $K$.
To compare Theorem \ref{main-thm-1} to Theorems \ref{real wolff thm}, \ref{haus wolff thm}, \ref{haus oberlin thm} on the same footing, let $Y=S_{d-1}$, let $K = R \times Z$ (where $R \subseteq (0,\infty)$ and $Z \subseteq \mathbb{R}^d$ are compact sets), and let $\delta$ be as in Theorem \ref{main-thm-1}. Arguing as in Section \ref{equality}, we find
\begin{align}\label{inequality} \dim_H(K) + d-1 = \dim_H(R \times S_{d-1} \times Z)
\geq \dim_H(R S_{d-1}) + \dim_H(Z) \geq \delta. \end{align}
\begin{comment} Since the map $f(x,y)=xy$ is Lipschitz on bounded sets and since $\overline{\dim_M}(S_{d-1}) = \dim_{H}(S_{d-1}) = d-1$, (a), (b), and (c) of Lemma \ref{dimension properties} \begin{align*} \dim_H(K) + d-1 = \dim_H(K \times S_{d-1}) = \dim_H(R \times S_{d-1} \times Z) \geq \dim_H(R \times S_{d-1}) + \dim_H(Z) \geq \dim_H(R S_{d-1}) + \dim_H(Z) \geq \delta \end{align*} \end{comment} From \eqref{inequality}, we see that the hypothesis of Theorem \ref{main-thm-1}(a) ($\delta > d$) is stronger than the hypothesis of Theorem \ref{real wolff thm} ($\dim_H(K) > 1$), while the conclusions are the same. Thus
Theorem \ref{real wolff thm} is stronger than Theorem \ref{main-thm-1}(a). The situation is more complicated when comparing Theorem \ref{main-thm-1}(b) to Theorems \ref{haus wolff thm} and \ref{haus oberlin thm}. By \eqref{inequality}, the hypothesis of Theorem \ref{main-thm-1}(b) is weaker than the hypotheses of Theorems \ref{haus wolff thm} and \ref{haus oberlin thm}, but the conclusion of Theorem \ref{main-thm-1}(b) is also weaker.
Theorem \ref{main-thm-1}(b) fills some gaps in Theorems \ref{haus wolff thm} and \ref{haus oberlin thm}. In particular, Theorem \ref{haus oberlin thm} misses the endpoint case $\dim_H(K)=1$ when $d=3$ and misses the interval $1/2 \leq \dim_H(K) \leq 1$ when $d=2$. Meanwhile, Theorem \ref{haus wolff thm} can only establish that $\dim_H(\bigcup_{(r,z) \in K} rS_{d}+z)$ is at least $\dim_H(C) + d-1$, rather than at least $\dim_H(K) + d-1$. As the next example shows, Theorem \ref{main-thm-1}(b) can give superior information in these cases.
\begin{example}\label{gaps filled 1} Let $d=3$. Let $A$ be a compact subset of $(0,\infty)$ such that $\dim_F(A)=\dim_H(A)=\overline{\dim_M}(A) = 1/4$ (Theorem \ref{non-uniform thm sets} furnishes such a set). Let $R=A$, $Z = A^3$, and $K = R \times Z$. By Lemma \ref{dimension properties}(c), $\dim_H(Z) = 3/4$ and $\dim_H(K) = 1$. Since $\dim_H(K) \geq (d-1)/2$, Theorem \ref{haus oberlin thm} gives no information about the Hausdorff dimension of $\bigcup_{(r,z) \in K} (rS_{2}+z)$. Since $Z = C = \cbr{z : (r,z) \in K \text{ for some } r }$, Theorem \ref{haus wolff thm} implies only that the Hausdorff dimension of $\bigcup_{(r,z) \in K} (rS_{2}+z)$ is at least $2 + 3/4$. However, by Proposition \ref{sphere thm sets}, $ \dim_F(RS_2) + \dim_H(Z) \geq \dim_{F}(R)+\dim_F(S_{2})+\dim_H(Z) = 3. $ So Theorem \ref{main-thm-1}(b) implies the Hausdorff dimension of $\bigcup_{(r,z) \in K} (rS_{2}+z) = R S_{2} + Z$ equals $3$. (Taking $R=A$ and $Z=A^2$ gives a similar example when $d=2$.) \end{example}
\begin{comment} \begin{example}\label{gaps filled 2} When $d=2$, Theorems \ref{haus oberlin thm} and \ref{haus wolff thm} both fail to address the endpoint case $\dim_H(K)=\dim_H(\cbr{z : (r,z) \in K \text{ for some } r }) = 1$. But Theorem \ref{main-thm-1} partially covers it.
For example, let $Z$ be any compact subset of $\mathbb{R}^2$ with $\dim_H(Z)=1$. Let $K = \cbr{1} \times Z$. Then $Z = \cbr{z : (r,z) \in K \text{ for some } r }$. By Lemma \ref{dimension properties}(c), $\dim_H(K) = \dim_H(\cbr{z : (r,z) \in K \text{ for some } r }) = \dim_H(Z) = 1$. So Theorems \ref{haus oberlin thm} and \ref{haus wolff thm} give no information about the Hausdorff dimension of $S_{1}+Z = \bigcup_{(r,z) \in K} (rS_{1}+z)$. On the other hand,
$ \delta \geq \dim_{F}(S_{1})+\dim_H(Z) = 2. $ So Theorem \ref{main-thm-1}(b) implies the Hausdorff dimension of $S_{d-1}+Z = \bigcup_{(r,z) \in K} (rS_{d-1}+z)$ equals $2$. \end{example}
\begin{remark} In connection with the last example, more is known about sets of the sets of the form $S_{d-1}+Z$ with $Z \subseteq \mathbb{R}^d$. Taylor and Simon \cite{STmeasure} showed: (i) If $\dim_H(Z) \leq 1$, then $\dim_H(S_{d-1} + Z) = d-1 + \dim_H(Z)$. (ii) If $d=2$ and $Z$ has finite non-zero 1-dimensional Hausdorff measure,
then $\mathcal{L}_2(S_1+Z)=0$ if and only if $Z$ is purely unrectifiable. In \cite{STinterior}, Simon and Taylor established conditions under which sets of the form $S_{1} + Z$ with $Z \subseteq \mathbb{R}^d$ have non-empty interior. \end{remark} \end{comment}
Theorem \ref{real wolff thm} can be generalized to any
$(d-1)$-dimensional surface with non-vanishing Gaussian curvature \cite{alex}. However, there is no hope of replacing the sphere in Theorem \ref{real wolff thm} by an arbitrary $(d-1)$-dimensional set, as the following example shows.
\begin{example}\label{cone example} Let $Y$ be the intersection of a $(d-1)$-dimensional cone through the origin
with an open spherical shell centered at the origin (such as the set $\cbr{x \in \mathbb{R}^d: 1/2 < |x| < 2}$). Then $RY$ is of the same form whenever $R \subseteq (0,\infty)$ contains a non-zero point. Let $R = [a,b] \subseteq (0,\infty)$ be a compact interval, let $Z \subseteq \mathbb{R}^d$ be a compact set with $\dim_H(Z) \in (0,1)$, and let $K = R \times Z$. By Lemma \ref{dimension properties}(c), $\dim_H(K) \geq \dim_H(R) + \dim_H(Z) > 1$. But $\bigcup_{(r,z) \in K} (rY+z) = RY+Z$ has Hausdorff dimension $\leq d-1 + \dim_H(Z) < d$, hence has $d$-dimensional Lebesgue measure $0$. To see that $\dim_H(RY+Z) \leq d-1 + \dim_H(Z)$, we first observe that, as $RY$ is a smooth $(d-1)$-dimensional manifold, its Hausdorff and upper Minkowski dimensions are both equal to $d-1$. (To see this, one can appeal to Lemma \ref{dimension properties}(d) and take $\mu$ to be the restriction of the $(d-1)$-dimensional Hausdorff measure to $RY$). Thus, by Lemma \ref{dimension properties}(c), $\dim_H(RY \times Z) = d-1 +\dim_H(Z)$. Then, since $f(x,y) = x+y$ is Lipschitz, Lemma \ref{dimension properties}(a) implies $\dim_H(RY + Z) \leq d-1 +\dim_H(Z)$. \end{example}
\section{Proof of Theorem \ref{main-thm-1}}\label{main-thm-1 proof}
\subsection{Proof of Part (a)}
Assume $\dim_F(RY)+\dim_H(Z) \geq \dim_H(RY)+\dim_F(Z)$. The proof in the opposite case is similar. Assume $\dim_F(RY)+\dim_H(Z) > d$. Choose $0 < \alpha < \dim_F(RY)$ and $0 < \beta < \dim_H(Z)$ such that $\alpha+\beta = d$.
Choose $\mu \in \mathcal{M}(RY)$ and $\nu \in \mathcal{M}(Z)$ such that
$\sup_{\xi \in \mathbb{R}^d} |\widehat{\mu}(\xi)|^2 |\xi|^{\alpha} < \infty$ and $I_{\beta}(\nu) < \infty$.
Then $\mu \ast \nu \in \mathcal{M}(RY+Z)$ and \begin{align*}
\int |\widehat{\mu \ast \nu}(\xi)|^2 d\xi
= \int |\widehat{\mu}(\xi)|^2 |\widehat{\nu}(\xi)|^2 d\xi
\lesssim \int |\xi|^{(d-\alpha) - d} |\widehat{\nu}(\xi)|^2 d\xi
= I_{\beta}(\nu) < \infty. \end{align*} Since $\widehat{\mu \ast \nu}$ is in $L^2$,
a standard argument (e.g. see \cite[Theorem 3.3]{Mat15}) shows $\mu \ast \nu$ is absolutely continuous with $L^2$ density, hence $\mathcal{L}_d(RY+Z) > 0$.
\subsection{Proof of Part (b)}
Assume $\dim_F(RY)+\dim_H(Z) \geq \dim_H(RY)+\dim_F(Z)$. The proof in the opposite case is similar. Let $s \in (0,d)$ with $\dim_F(RY)+\dim_H(Z) > s$. If either $\dim_F(RY) = 0$ or $\dim_H(Z)=0$, then the result is immediate; assume otherwise. Choose $0 < \alpha < \dim_F(RY)$ and $0 < \beta < \dim_H(Z)$ such that $\alpha+\beta = s$.
Choose $\mu \in \mathcal{M}(RY)$ and $\nu \in \mathcal{M}(Z)$ such that
$\sup_{\xi \in \mathbb{R}^d} |\widehat{\mu}(\xi)|^2 |\xi|^{\alpha} < \infty$ and $I_{\beta}(\nu) < \infty$.
Then $\mu \ast \nu \in \mathcal{M}(RY+Z)$ and \begin{align*} I_{s}(\mu \ast \nu)
&= \int |\widehat{\mu \ast \nu}(\xi)|^2 |\xi|^{s-d} d\xi
= \int |\widehat{\mu}(\xi)|^2 |\widehat{\nu}(\xi)|^2 |\xi|^{s-d} d\xi \\
&\lesssim \int |\xi|^{(s-\alpha)-d} |\widehat{\nu}(\xi)|^2 d\xi
= I_{\beta}(\nu) < \infty. \end{align*}
By \eqref{capacity dimension} and our choice of $s$, $\dim_H(RY+Z) \geq \dim_F(RY)+\dim_H(Z)$.
\section{Proof of Proposition \ref{sphere thm sets}}\label{sphere thm sets proof}
Let $\sigma$ be the surface measure on the $(d-1)$-dimensional unit sphere $S_{d-1} \subseteq \mathbb{R}^d$.
The following asymptotic is well-known (e.g., see \cite{Mat15, wolff-book-2003}):
For
$\xi \in \mathbb{R}^d$, \begin{align*} \notag \widehat{\sigma}(\xi) &=
2|\xi|^{-(d-1)/2}\cos\rbr{ 2\pi \rbr{|\xi| - \frac{d-1}{8}} }
+ O\rbr{|\xi|^{-(d+1)/2}} \\
&=
|\xi|^{-(d-1)/2} \rbr{ c_d e^{2\pi i |\xi| }
+ \overline{c_d} e^{-2 \pi i |\xi|} } + O\rbr{|\xi|^{-(d+1)/2}}, \end{align*} where $c_d = e^{-\pi i (d-1)/4}$. Let $R \subseteq (0,\infty)$.
Let $\mu \in \mathcal{M}(R)$ be arbitrary. Choose $a,b>0$ such that $\mathrm{supp}(\mu) \subseteq [a,b] \subseteq (0,\infty)$. Define $\mu_0$ by $d\mu_0(r) = r^{\frac{(d-1)}{2}} d \mu(r)$.
Then $\mathrm{supp}(\mu_0) = \mathrm{supp}(\mu)$, $\mu_0 \in \mathcal{M}(R)$, and $\mu_0 \cdot \sigma \in \mathcal{M}(RS_{d-1})$. Furthermore,
\begin{align*} \widehat{\mu_0 \cdot \sigma}(\xi)
= \int_a^b \widehat{\sigma}(r\xi) d\mu_0(r)
=
|\xi|^{-(d-1)/2} \rbr{
c_d \widehat{\mu}(-|\xi|)
+ \overline{c_d} \widehat{\mu}(|\xi|) }
+ O\rbr{|\xi|^{-(d+1)/2}} \end{align*} for all sufficiently large $\xi \in \mathbb{R}^d$. Therefore $\dim_F(RS_{d-1}) \geq \dim_F(\mu_0 \cdot \sigma) \geq \dim_F(\mu) + d-1$ and (consequently) $\dim_F(RS_{d-1}) \geq \dim_F(R) + d-1$.
\section{Proof of Theorem \ref{non-uniform thm sets}} \label{non-uniform thm sets proof}
\begin{comment} Theorem \ref{non-uniform thm sets} follows immediately from the following theorem in terms of measures. We will use Definition \ref{measure defn} in the following Theorem.
\begin{thm}\label{non-uniform thm measures} Let $\alpha \in [0,1]$ and let $\nu$ be a non-zero finite Borel measure on $\mathbb{R}$ with compact support not containing $0$. Then there is a Borel probability measure $\mu$ on $\mathbb{R}$ such that $\mathrm{supp}(\mu) \subseteq [1,2]$, $\dim_H(\mu) = \dim_F(\mu) = \alpha$, and $\dim_F(\mu \cdot \nu) \geq \min\cbr{1,\dim_F(\mu) + \dim_F(\nu)}$. \end{thm} \end{comment}
In light of parts (b) and (d) of Lemma \ref{dimension properties} and the definition of Fourier dimension, Theorem \ref{non-uniform thm sets} is implied by:
\begin{thm}\label{non-uniform thm measures} Let $\alpha \in [0,1]$. Let $R$ be a non-empty compact subset of $(0,\infty)$. Let $\nu \in \mathcal{M}(R)$. Then there is a Borel probability measure $\mu$ on $\mathbb{R}$ such that \begin{enumerate}[(a)] \item $\mathrm{supp}(\mu) \subseteq [1,2]$, \item $\mu(B(x,r)) \approx r^{\alpha}$ for all $x \in \mathrm{supp}(\mu)$ and $0 < r \leq 1$, \item $\dim_F(\mu) \geq \alpha$, \item $\dim_F(\mu \cdot \nu) \geq \min\cbr{1,\dim_F(\mu) + \dim_F(\nu)}$. \end{enumerate} \end{thm}
The following two sections are devoted to proving Theorem \ref{non-uniform thm measures}. In the first section, we give a general construction of a measure $\mu$ that satisfies parts (a) and (b) of Theorem \ref{non-uniform thm measures}. Nothing in the first section will depend on $\nu$. In the second section, we specialize the construction of $\mu$ to prove parts (c) and (d). The specialization will depend on $\nu$.
Before we begin, we dispense with a trivial case: If $\alpha = 0$, then taking $\mu$ to be a point mass gives the desired result. Hereafter, we assume $\alpha \in (0,1]$.
\subsection{Proof of Theorem \ref{non-uniform thm measures}: General Construction}
For every $n \in \mathbb{Z}_{> 0}$, we use the notation $[n] = \cbr{0,1,\ldots,n-1}$ For sequences $(t_j)_{j=1}^{\infty}$ and $(n_j)_{j=1}^{\infty}$ of positive integers, we use the notation $T_j = t_1 \cdots t_j$ and $N_j = n_1 \cdots n_j$. We also use the empty product convention, so that $T_0 = N_0 = 1$.
Fix an integer $n_{\ast} \geq 2$. Fix sequences $(t_j)_{j=1}^{\infty}$ and $(n_j)_{j=1}^{\infty}$ of positive integers such that, for all $j \in \mathbb{Z}_{> 0}$, we have $2 \leq n_j \leq n_{\ast}$, $1 \leq t_j \leq n_j$, and $T_j \approx N_j^{\alpha}$.
We recursively define two families of sets: $$ \cbr{A_j : j \in \mathbb{Z}_{\geq 0}} \text{ and } \cbr{B_{j+1,a} : j \in \mathbb{Z}_{\geq 0}, a \in A_j}. $$ Define $A_{0} = \cbr{1}$.
Assuming that $A_j$ has been defined for a fixed $j \in \mathbb{Z}_{\geq 0}$, for each $a \in A_{j}$ choose a set $B_{j+1,a} \subseteq N_{j+1}^{-1}[n_{j+1}]$ such that $|B_{j+1,a}| = t_{j+1}$.
Later we make a specific choice for the sets $B_{j+1,a}$, but for now the sets $B_{j+1,a}$ are arbitrary. Define $A_{j+1} = \bigcup_{a \in A_j} \rbr{a+B_{j+1,a}}$.
Note that this recursive definition implies that $A_j \subseteq [1,2)$ and $|A_j| = T_j$ for all $j \in \mathbb{Z}_{\geq 0}$.
Question: What are the sets $A_j$ and $B_{j+1,a}$? Answer: They are sets of endpoints in the following Cantor set construction.
Start with the interval $[1,2]$. Divide it into $n_1$ intervals of length $1/N_1$, keep $t_1$ of these intervals, and discard the rest. For each of the kept intervals, we do the following: Divide it into $n_2$ intervals of length $1/N_2$, keep $t_2$ of these intervals, and discard the rest. This gives, in total, $T_2$ intervals of length $1/N_2$. Continuing in this way, at the $j$-th stage we have $T_j$ intervals of length $1/N_j$. The set of left endpoints of these intervals is $A_j$. For each of these intervals, we do the following: Divide it into $n_{j+1}$ intervals of length $1/N_{j+1}$, keep $t_{j+1}$ of these intervals, and discard the rest. If
$a$ is the left endpoint of the interval we started with, the set of left endpoints of the intervals kept is $a + B_{j+1,a}$.
The union of all the sets $a + B_{j+1,a}$ (as $a$ ranges over $A_{j}$) is $A_{j+1}$. The Cantor set constructed is \begin{align}\label{cantor set} \bigcap_{j=0}^{\infty} \bigcup_{a \in A_j} [a,a+N_j^{-1}]. \end{align}
For each $j \in \mathbb{Z}_{\geq 0}$, define $\mu_j$ to be the probability measure whose density with respect to Lebesgue measure on $\mathbb{R}$ is $$ \mu_j = \frac{N_j}{T_j} \sum_{a \in A_j} \mathbf{1}_{[a,a+N_{j}^{-1}]}. $$
Note that we have abused notation by using the same symbol for a measure and its density; we will continue to do this. For each $j \in \mathbb{Z}_{\geq 0}$, \begin{align}\label{supp mu j formula} \mathrm{supp}(\mu_j) = \bigcup_{a \in A_j} [a,a+N_{j}^{-1}]. \end{align}
Moreover, for every $j,k \in \mathbb{Z}_{\geq 0}$ with $j \leq k$ and for every $a \in A_j$, \begin{align}\label{mu k formula} \mu_k([a,a+N_{j}^{-1}]) = T_j^{-1}. \end{align}
\begin{lemma}\label{weak convergence} The sequence $(\mu_j)_{j=0}^{\infty}$ converges weakly (i.e., in distribution) to a probability measure $\mu$. \end{lemma} \begin{proof} For each $j \in \mathbb{Z}_{\geq 0}$, let $F_j$ be the cumulative distribution function of $\mu_j$, i.e., $F_j(t)=\mu_j((-\infty,t])$ for all $t \in \mathbb{R}$. Let $t \in \mathbb{R}$. If $t \leq \min A_j$, then $F_{j}(t) = F_{j+1}(t) = 0$. Now assume $t \geq \min A_j$. Let $a(t)$ be the unique element of $A_j$ such that $a(t) \leq t < a(t)+N^{-j}$. Since $F_{j}(a) = F_{j+1}(a)$ for each $a \in A_j$ and because of \eqref{mu k formula}, we have \begin{align*}
|F_{j+1}(t) - F_{j}(t)|
&\leq |\mu_{j+1}((a(t),t]) - \mu_{j}((a(t),t])| \\ &\leq \mu_{j+1}([a(t),a(t)+N_j^{-1}]) + \mu_{j}([a(t),a(t)+N_j^{-1}]) \\ &= 2T_{j}^{-1} \approx N_{j}^{-\alpha} \leq 2^{-j\alpha}. \end{align*}
It follows that $(F_j)_{j=0}^{\infty}$ is a uniformly Cauchy and (consequently) uniformly convergent sequence of continuous cumulative distribution functions. Therefore the limit $F$ is a continuous cumulative distribution function of a Borel probability measure $\mu$ on $\mathbb{R}$. Hence $(\mu_j)_{j=0}^{\infty}$ converges weakly to $\mu$.
\end{proof}
Combining \eqref{mu k formula} with Lemma \ref{weak convergence} gives, for every $j \in \mathbb{Z}_{\geq 0}$ and $a \in A_j$, \begin{align}\label{mu formula} \mu([a,a+N_{j}^{-1}]) = T_j^{-1}. \end{align}
\begin{lemma}\label{support lemma} The support of $\mu$ is \begin{align*} \mathrm{supp}(\mu) = \bigcap_{j=0}^{\infty} \mathrm{supp}(\mu_j) = \bigcap_{j=0}^{\infty} \bigcup_{a \in A_j} [a,a+N_j^{-1}]. \end{align*} \end{lemma} \begin{proof} The second equality is immediate from \eqref{supp mu j formula}. For the first equality, we consider $\subseteq$ and $\supseteq$ separately.
$\subseteq:$ We prove the contrapositive. Suppose $x \in \mathbb{R} \setminus \mathrm{supp}(\mu_{j_0})$ for some $j_0 \in \mathbb{Z}_{\geq 0}$. Since $\mathbb{R} \setminus \mathrm{supp}(\mu_{j_0})$ is open, there is an open ball $B(x,r_0)$ contained in $\mathbb{R} \setminus \mathrm{supp}(\mu_{j_0})$. Since $(\mathrm{supp}(\mu_j))_{j=0}^{\infty}$ is a decreasing sequence of sets, $B(x,r_0)$ is contained in $\mathbb{R} \setminus \mathrm{supp}(\mu_{j})$ for every $j \geq j_0$. Thus $\mu_j(B(x,r_0))=0$ for every $j \geq j_0$. Choose a continuous function $\phi:\mathbb{R} \to \mathbb{R}$ such that $\mathbf{1}_{B(x,r_0/2)} \leq \phi \leq \mathbf{1}_{B(x,r_0)}$. Then, since $\mu_{j} \to \mu$ weakly, we have $\mu(B(x,r_0/2)) \leq \int \phi d\mu = \lim_{j \to \infty} \int \phi d\mu_j \leq \lim_{j \to \infty} \mu_j(B(x,r_0)) = 0$. So $\mu(B(x,r_0/2))=0$. It follows that $x \in \mathbb{R} \setminus \mathrm{supp}(\mu)$.
$\supseteq:$ Let $x \in \bigcap_{j=0}^{\infty} \mathrm{supp}(\mu_j)$. Let $B(x,r)$ be any open ball centered at $x$. Choose $j$ large enough that $N_{j}^{-1} < r/2$. Since $x \in \mathrm{supp}(\mu_j)$, we have $\mu_j(B(x,r/2)) > 0$, hence $B(x,r/2)$ intersects $[a,a+N_{j}^{-1}]$ for some $a \in A_j$. Therefore $B(x,r)$ contains $[a,a+N_{j}^{-1}]$. Choose a continuous function $\phi:\mathbb{R} \to \mathbb{R}$ such that $\mathbf{1}_{[a,a+N_{j}^{-1}]} \leq \phi \leq \mathbf{1}_{B(x,r)}$. By \eqref{mu k formula}, $T_j^{-1} = \mu_k([a,a+N_{j}^{-1}]) \leq \int \phi d\mu_k$ for all $k \geq j$. Then, since $\mu_{k} \to \mu$ weakly, we have $T_j^{-1} \leq \lim_{k \to \infty} \int \phi d\mu_k = \int \phi d\mu \leq \mu(B(x,r))$. So $\mu(B(x,r)) > 0$. Since $B(x,r)$ was arbitrary, it follows that $x \in \mathrm{supp}(\mu)$. \end{proof}
The measure $\mu$ is the so-called natural measure on the Cantor set \eqref{cantor set}.
\begin{lemma}\label{Hausdorff mu}
For every interval $I$ with diameter $|I|$, $
\mu(I) \lesssim |I|^{\alpha}. $ \end{lemma} \begin{proof} \begin{comment} For every $\epsilon > 0$ and $j \in \mathbb{Z}_{\geq 0}$, \begin{align*} \mathcal{H}^{\alpha+\epsilon}(\mathrm{supp}(\mu)) \leq \sum_{a \in A_j} \mathcal{H}^{\alpha+\epsilon}([a,a+N_j^{-1}]) = T_j N_j^{-\alpha-\epsilon} \approx N_j^{-\epsilon}. \end{align*} Letting $j \to \infty$, we see that $\mathcal{H}^{\alpha+\epsilon}(\mathrm{supp}(\mu))=0$ for all $\epsilon>0$, hence $\dim_H ( \mathrm{supp}(\mu) ) \leq \alpha$. This implies $\dim_H(\mu) \leq \alpha$.
To show that $\dim_H(\mu) \geq \alpha$, it suffices to show that It suffices to show that $$
\mu(I) \lesssim |I|^{\alpha} $$
for every interval $I$ in $\mathbb{R}$, where $|I|$ denotes the diameter of $I$. \end{comment}
If $|I| > 1$, then $$
\mu(I) \leq \mu(\mathbb{R}) = 1 \leq |I|^{\alpha} $$ since $\mu$ is a probability measure.
Now suppose $|I| \leq 1$.
Choose $j_0 \in \mathbb{Z}_{\geq 0}$ such that $N_{j_+1}^{-1} \leq |I| \leq N_{j_0}^{-1}$. Assume $I$ intersects $\text{supp}(\mu)$ (otherwise $\mu(I) = 0$).
Then $I$ intersects an interval $[a,a+N_{j_0}^{-1}]$ for some $a \in A_{j_0}$.
Since $|I| \leq N_{j_0}^{-1}$ and $A_j \subseteq N_{j_0}^{-1}[N_{j_0}]$, there are at most two such intervals; call them $J_1$ and $J_2$. By \eqref{mu formula}, the choice of the sequences $(n_j)$ and $(t_j)$, and the choice of $j_0$, we have \begin{align*} \mu(I) = \mu(I \cap J_1) + \mu(I \cap J_2) \leq \mu(J_1) + \mu(J_2) = \dfrac{2}{T_{j_0}} \approx \frac{1}{N_{j_0 +1}^{\alpha}}
\leq |I|^{\alpha} \end{align*} \end{proof}
\begin{lemma}\label{lower regularity}
For every interval $I$ with center in $E$ and diameter $|I| \leq 1$, $
\mu(I) \gtrsim |I|^{\alpha}. $ \end{lemma} \begin{proof} Choose ${j_0} \in \mathbb{Z}_{> 0}$ such that
$2N_{{j_0}+1}^{-1} \leq |I| \leq 2N_{{j_0}}^{-1}$. Since the center of $I$ belongs to $E = \bigcap_{j=0}^{\infty} \bigcup_{a \in A_j} [a,a+N_j^{-1}]$, it belongs to the interval $[a,a+N_{{j_0}+1}^{-1}]$ for some $a \in A_{{j_0}+1}$.
Then, since $2N_{{j_0}+1}^{-1} \leq |I|$, $[a,a+N_{{j_0}+1}^{-1}] \subseteq I$. By \eqref{mu formula}, the choice of the sequences $(n_j)$ and $(t_j)$, and the choice of $j_0$, we have \begin{align*} \mu(I) \geq \mu([a,a+N_{{j_0}+1}^{-1}]) = \frac{1}{T_{{j_0}+1}} \approx \frac{1}{N_{{j_0}+1}^{\alpha}} = \frac{1}{2^{\alpha} n_{{j_0}+1}^{\alpha}} \frac{2^{\alpha}}{N_{{j_0}}^{\alpha}}
\geq \frac{1}{2^{\alpha} n_{*}^{\alpha}} |I|^{\alpha}. \end{align*} \end{proof}
\subsection{Proof of Theorem \ref{non-uniform thm measures}: Fourier Decay}
We will now prove that the sets $B_{j+1,a}$ can be chosen so that
(c) and (d) of Theorem \ref{non-uniform thm measures} hold. Before beginning, we outline the proof and make two remarks.
The idea of the proof is to choose the sets $B_{j+1,a}$
using the probabilistic method. More specifically, for a given $j$ and a given set $A_j$ of numbers $a$, we choose each $B_{j+1,a}$ uniformly at random from a finite collection of possible sets. Thus the differences (i) $\widehat{\mu_{j+1}}(\xi) - \widehat{\mu_j}(\xi)$ and (ii) $\widehat{\mu_{j+1} \cdot \nu}(\xi) - \widehat{\mu_j \cdot \nu}(\xi)$ can be written as sums of finitely many independent random variables. We use Hoeffding's large deviation inequality to show that, with positive probability, the differences (i) and (ii) are small. We deduce that there exists a choice of the sets $B_{j+1,a}$ which makes (i) and (ii) small. This probabilistic argument is the heart of the proof. It appears as Lemma \ref{choice tele lemma} below. After Lemma \ref{choice tele lemma}, we use telescoping sum and geometric series arguments to deduce the desired decay estimates for $\widehat{\mu}$ and $\widehat{\mu \cdot \nu}$. Before getting to Lemma \ref{choice tele lemma}, we give several preparatory results.
\begin{remark} The sets $B_{j+1,a}$ depend on $\nu$ because they are chosen (probabilistically) to make $\widehat{\mu_{j+1} \cdot \nu}(\xi) - \widehat{\mu_j \cdot \nu}(\xi)$ small.
It would be possible to work with a finite or countably infinite collection of measures $\nu_1$, $\nu_2$, $\ldots$ and choose the sets $B_{j+1,a}$ so that $\widehat{\mu_{j+1} \cdot \nu_i}(\xi) - \widehat{\mu_j \cdot \nu_i}(\xi)$ is small for each $i$. (As an exercise, we invite the reader to verify this by modifying the proof of Lemma \ref{choice tele lemma}). Countable subadditivity of the probability measure is what makes this possible. However, we do not know how to choose the sets $B_{j+1,a}$ so that $\widehat{\mu_{j+1} \cdot \nu}(\xi) - \widehat{\mu_j \cdot \nu}(\xi)$ is small for all measures $\nu$. If we could do that, then we could make the measure $\mu$ independent of $\nu$ in Theorem \ref{non-uniform thm measures}, and hence make the set $Y$ independent of $R$ in Theorem \ref{non-uniform thm sets}. \end{remark}
Now we begin the proof.
Let $j \in \mathbb{Z}_{\geq 0}$. We write the densities of $\mu_j$ and $\mu_{j+1}$ in more convenient forms.
By partitioning $[0,N_{j}^{-1}]$ into intervals of length $N_{j+1}^{-1}$, we see that
\begin{align*} \mu_j = \left(\frac{N_{j+1}}{T_{j}}\right) \left( \frac{1}{n_{j+1}}\right) \sum_{a \in A_j} \sum_{b \in N_{j+1}^{-1}[n_{j+1}]} \mathbf{1}_{a+b+[0,N_{j+1}^{-1}]}. \end{align*}
Since $A_{j+1} = \bigcup_{a \in A_j} \rbr{a + B_{j+1,a}}$, we also have \begin{align*} \mu_{j+1} = \left(\frac{N_{j+1}}{T_{j}}\right) \left( \frac{1}{t_{j+1}}\right)
\sum_{a \in A_j} \sum_{b \in B_{j+1,a}} \mathbf{1}_{a+b+[0,N_{j+1}^{-1}]}. \end{align*} For each $a \in A_j$, $b \in B_{j+1,a}$, and $\xi \in \mathbb{R}$, define \begin{align} \label{I defn} I(a,b,j,\xi) &= \int_{[0,1]} e^{-2 \pi i (\xi / N_{j+1})(aN_{j+1}+bN_{j+1}+x)} dx, \\ \label{J defn} J(a,b,j,\xi) &= \int_{\mathbb{R}} \int_{[0,1]} e^{-2 \pi i (\xi / N_{j+1})(aN_{j+1}+bN_{j+1}+x)y} dx d\nu(y), \\ \label{X defn} X_a(j,\xi) &= \frac{1}{t_{j+1}} \sum_{b \in B_{j+1,a}} I(a,b,j,\xi) - \frac{1}{n_{j+1}} \sum_{b \in N_{j+1}^{-1}[n_{j+1}]} I(a,b,j,\xi), \\ \label{Y defn} Y_a(j,\xi) &= \frac{1}{t_{j+1}} \sum_{b \in B_{j+1,a}} J(a,b,j,\xi) - \frac{1}{n_{j+1}} \sum_{b \in N_{j+1}^{-1}[n_{j+1}]} J(a,b,j,\xi). \end{align}
It follows that, for all $\xi \in \mathbb{R}$, \begin{align} \label{mu equation} \widehat{\mu_{j+1}}(\xi) - \widehat{\mu_{j}}(\xi) = \frac{1}{T_j} \sum_{a \in A_j} X_a(j,\xi) \\ \label{m equation} \widehat{\mu_{j+1} \nu}(\xi) - \widehat{\mu_{j} \nu}(\xi) = \frac{1}{T_j} \sum_{a \in A_j} Y_a(j,\xi). \end{align}
By direct calculation, we have \begin{lemma}\label{I X lemma} For each $j \in \mathbb{Z}_{j \geq 0}$, $a \in A_{j}$, $b \in B_{j+1,a}$, and $\xi \in \mathbb{R}$, we have \begin{align}\label{I bound}
|I(a,b,j,\xi)| &\leq \min\cbr{1,N_{j+1}/|\xi|)}, \\ \label{X bound}
|X_a(j,\xi)| &\leq 2\min\cbr{1,N_{j+1}/|\xi|)}. \end{align} \end{lemma}
Define $g:[0,\infty) \to [0,\infty)$ by $$
g(x) = (1+x)^{-1/2} + \sup\cbr{ |\widehat{\nu}(tx)| : t \in \mathbb{R}, |t| \geq 1 } $$ The following properties of $g$ are straightforward to verify. \begin{lemma} \label{g lemma}\hspace{1pt} \begin{enumerate}[(a)]
\item $g$ is non-increasing
\item For all $\xi \in \mathbb{R}$, $|\widehat{\nu}(\xi)| \leq g(|\xi|)$.
\item For all $0 \leq \beta \leq 1$,
$\sup_{\xi \in \mathbb{R}} |\widehat{\nu}(\xi)| (1+|\xi|)^{\beta/2} < \infty$
if and only if $\sup_{\xi \in \mathbb{R}} g(|\xi|) (1+|\xi|)^{\beta/2} < \infty$. \end{enumerate} \end{lemma}
Recall that $\mathrm{supp}(\nu)$ is compact and does not contain $0$. Choose a
Schwartz function $\phi:\mathbb{R} \to \mathbb{C}$ such that $\phi(y) = 1/y$ for all $y \in \mathrm{supp}(\nu)$.
\begin{lemma}\label{phi nu decay lemma} For all $x \in \mathbb{R}$,
$|(\widehat{\phi \nu})(x)| \lesssim g(\tfrac{1}{2}|x|).$
\end{lemma} \begin{proof} Write \begin{align*}
|(\widehat{\phi \nu})(x)| = |(\widehat{\phi} \ast \widehat{\nu})(x)| \leq \int_{\mathbb{R}} |\widehat{\phi}(y)| \widehat{\nu}(|x-y|) dy. \end{align*}
Bound the integral above by the sum of the integrals over $R_1 = \cbr{y: \frac{1}{2}|x| \leq |x-y|}$ and $R_2 = \cbr{y: \frac{1}{2}|x| \leq |y|}$.
For the integral over $R_1$, use (a) and (b) of Lemma \ref{g lemma}. For the integral over $R_2$, note $|\widehat{\phi}(y)|^{1/2} \lesssim g(|y|)$ for all $y \in \mathbb{R}$ (because $\widehat{\phi}$ is Schwartz), then use (a) of Lemma \ref{g lemma}.
\end{proof}
\begin{lemma}\label{J Y lemma} There is a constant $C_0 > 0$ such that for each $j \in \mathbb{Z}_{\geq 0}$, $a \in A_{j}$, $b \in B_{j+1,a}$, and $\xi \in \mathbb{R}$, we have \begin{align} \label{J bound}
|J(a,b,j,\xi)| &\leq C_0 g(\tfrac{1}{2}|\xi|) \min\cbr{1,N_{j+1}/|\xi|)}, \\ \label{Y bound}
|Y_a(j,\xi)| &\leq 2 C_0 g(\tfrac{1}{2}|\xi|) \min\cbr{1,N_{j+1}/|\xi|)}. \end{align} \end{lemma} \begin{proof} Note \eqref{Y bound} is immediate from \eqref{J bound}. Integrating $y$ in \eqref{J defn} shows that \begin{align*} J(a,b,j,\xi) = \int_{[0,1]} \widehat{\nu}((\xi / N_{j+1})(aN_{j+1}+bN_{j+1}+x)) dx \end{align*}
Since $|(\xi / N_{j+1})(aN_{j+1}+bN_{j+1}+x)| \geq |\xi|$ for each $x \in [0,1]$, (a) and (b) of Lemma \ref{g lemma} give \begin{align*}
|J(a,b,j,\xi)| \leq g(|\xi|). \end{align*}
On the other hand, integrating $x$ in \eqref{J defn} shows that \begin{align*} J(a,b,j,\xi) = \int_{\mathbb{R}} e^{-2 \pi i (\xi / N_{j+1})(aN_{j+1}+bN_{j+1})y} \frac{e^{-2 \pi i (\xi / N_{j+1}) y} - 1}{-2 \pi i (\xi / N_{j+1}) y} d\nu(y). \end{align*}
After multiplying by $-2 \pi i (\xi / N_{j+1})$, using that $\phi(y) = 1/y$ for all $y \in \mathrm{supp}(\nu)$, and integrating $y$, we see that $-2 \pi i (\xi / N_{j+1}) J(a,b,j,\xi)$ is $$ = \widehat{\phi \nu}((\xi / N_{j+1})(aN_{j+1}+bN_{j+1}+1)) - \widehat{\phi \nu}((\xi / N_{j+1})(aN_{j+1}+bN_{j+1})). $$
Since $|(\xi / N_{j+1})(aN_{j+1}+bN_{j+1}+x)| \geq |\xi|$ for each $x \in [0,1]$, Lemma \ref{phi nu decay lemma} and (a) of Lemma \ref{g lemma} give \begin{align*}
|J(a,b,j,\xi)| \lesssim \frac{1}{\pi} \left(\frac{N_{j+1}}{|\xi|}\right) g\left(\frac{1}{2}|\xi|\right). \end{align*}
\end{proof}
We need the following fact about averages over random subsets.
\begin{lemma}\label{mean zero} Let $t \leq n$ be positive integers. Let $A$ be a finite set of size $n$, and let $F: A \to \mathbb{C}$. Let $\mathcal{B}_t$ be the collection of all size $t$ subsets of $A$, and let $B$ be a set chosen uniformly at random from $\mathcal{B}_t$.
Then \begin{align*} \mathbb{E}\rbr{ \frac{1}{t} \sum_{x \in B} F(x) } = \frac{1}{n} \sum_{x \in A} F(x). \end{align*} \end{lemma} \begin{proof} There are $\binom{n}{t}$ sets in $\mathcal{B}_t$. For each $x \in A$, there are $\binom{n-1}{t-1}$ sets in $\mathcal{B}_t$ that contain $x$. Therefore \begin{align*} \mathbb{E}\rbr{ \frac{1}{t} \sum_{x \in B} F(x) } = \frac{1}{{\textstyle \binom{n}{t}} \cdot t } \sum_{B \in \mathcal{B}_t} \sum_{x \in B} F(x) = \frac{1}{{\textstyle \binom{n}{t}} \cdot t } \sum_{x \in A} \binom{n-1}{t-1} F(x) = \frac{1}{n} \sum_{x \in A} F(x). \end{align*} \end{proof}
We need the following version of Hoeffding's inequality for complex-valued random variables.
\begin{lemma}\label{bern}
Suppose $Z_1,\ldots,Z_t$ are independent complex-valued random variables satisfying $\mathbb{E}(Z_i)=0$ and $|Z_i| \leq c$ for $i=1,\ldots,t$, where $c$ is a positive constant. For all $u > 0$, \begin{align*} \mathbb{P}\rbr{ \abs{\frac{1}{t} \sum_{i=1}^{t} Z_i} \geq c u } \leq 4 \exp\rbr{ -\frac{1}{4} t u^2 }. \end{align*} \end{lemma} \begin{proof}
Apply the standard Hoeffding inequality to the real and imaginary parts of $Z_1,\ldots,Z_t$. \end{proof}
\begin{lemma}\label{choice tele lemma}
Define $d_0$ by
$ 1/d_0 = \max\cbr{ \mathrm{diam}\rbr{\mathrm{supp}(\mu)}, \mathrm{diam}\rbr{\mathrm{supp}(\mu \cdot \nu)} }. $
Fix a real number $\zeta_0$ such that \begin{align*}
\zeta_0 > \sum_{k \in \mathbb{Z}} \frac{2}{1+d_0^2|k|^2}. \end{align*}
It is possible to choose the sets $B_{j+1,a}$ such that, for every $j \in \mathbb{Z}_{\geq 0}$ and every $\xi \in d_0 \mathbb{Z} = \cbr{d_0 k : k \in \mathbb{Z}}$,
\begin{align}\label{tele ineq mu}
|\widehat{\mu_{j+1}}(\xi) - \widehat{\mu_j}(\xi)|
&< 2 T_j^{-1/2} \ln^{1/2}(4\zeta_0(1 + |\xi|^2)) \min\cbr{1,N_{j+1}/|\xi|}, \\ \label{tele ineq m}
|\widehat{\mu_{j+1} \nu}(\xi) - \widehat{\mu_j \nu}(\xi)|
&< 2 C_0 T_j^{-1/2} \ln^{1/2}(4\zeta_0(1 + |\xi|^2)) g(\tfrac{1}{2}|\xi|) \min\cbr{1,N_{j+1}/|\xi|}, \end{align} where $C_0$ is the constant from Lemma \ref{J Y lemma}. \end{lemma}
\begin{proof}
Fix $j \in \mathbb{Z}_{\geq 0}$ and assume a set $A_j \subseteq [1,2)$ satisfying $|A_j|=T_j$ is given.
To simplify notation in what follows, we write $N=N_{j+1}$, $n=n_{j+1}$, and $t = t_{j+1}$. For each $a \in A_j$, suppose we choose $B_{j+1,a}$ independently and uniformly at random from the collection of all size $t$ subsets of $N^{-1}[n]$.
Now fix $\xi \in \mathbb{R}$.
Then $\cbr{X_a(j,\xi): a \in A_j }$ is a set of independent complex-valued random variables, and the same is true of $\cbr{Y_a(j,\xi): a \in A_j }$.
Moreover, for each $a \in A_j$, we find that $\mathbb{E}(X_a(j,\xi))=0$ by applying Lemma \ref{mean zero} with $F(b) = I(a,b,j,\xi)$ for $b \in N^{-1}[n]$. Likewise, we find that $\mathbb{E}(Y_a(j,\xi))=0$ by applying Lemma \ref{mean zero} with $F(b) = J(a,b,j,\xi)$.
We also have the bounds on $|X_a(j,\xi)|$ and $|Y_a(j,\xi)|$ from Lemma \ref{I X lemma} and Lemma \ref{J Y lemma}, respectively.
Define \begin{align*}
u_{j,\xi} &= \sqrt{ T_j^{-1} \ln\rbr{4\zeta_0(1+|\xi|^2)} } \end{align*}
Let $E^1(\xi)$ be the event that $\abs{\frac{1}{T_j} \sum_{a \in A_j} X_a(j,\xi)} < 2 u_{j,\xi} \min\cbr{1,N/|\xi|}$.
Let $E^2(\xi)$ be the event that $\abs{\frac{1}{T_j} \sum_{a \in A_j} Y_a(j,\xi)} < 2 C_0 u_{j,\xi} g(\frac{1}{2}|\xi|) \min\cbr{1,N/|\xi|}$.
For each $\xi \in \mathbb{R}$, Lemma \ref{bern} implies \begin{align*}
\mathbb{P}\rbr{(E^i(\xi))^c} &\leq 4 \exp\rbr{-\frac{1}{4} T_j u_{j,\xi}^2} = \frac{1}{\zeta_0(1+|\xi|^2)}.
\end{align*} Therefore, by De Morgan's laws and countable subaddivity,
the probability that both $E^1(d_0 k)$ and $E^2(d_0 k)$ hold for all $k \in \mathbb{Z}$ is
$$
\geq 1 - \sum_{k \in \mathbb{Z}} \rbr{\mathbb{P}\rbr{(E^1(d_0 k))^c} + \mathbb{P}\rbr{(E^2(d_0 k))^c}}
\geq 1 - \sum_{k \in \mathbb{Z}} \frac{2}{\zeta_0(1+d_0^2|k|^2)} > 0.
$$ \begin{comment} Therefore \begin{align*} &\mathbb{P}\rbr{ E'(d_0 k) \text{ and } E''(d_0 k) \text{ for all } k \in \mathbb{Z} } = \mathbb{P}\rbr{\bigcap_{k \in \mathbb{Z}} \rbr{ E'(d_0 k) \cap E'(d_0 k) } } \\ &\geq 1 - \sum_{k \in \mathbb{Z}} \rbr{\mathbb{P}\rbr{(E'(d_0 k))^c} + \mathbb{P}\rbr{(E''(d_0 k))^c}}
\geq 1 - \sum_{k \in \mathbb{Z}} \frac{2}{\zeta_0(1+d_0^2|k|^2)} > 0. \end{align*} \end{comment}
In light of \eqref{mu equation} and \eqref{m equation}, this implies there is some choice of the sets $B_{j+1,a}$ $(a \in A_j)$ such that \eqref{tele ineq mu} and \eqref{tele ineq m} hold for every $\xi \in d_0 \mathbb{Z}$. \end{proof}
\begin{lemma}\label{mu mu nu decay lemma} With the sets $B_{j+1,a}$ chosen as in Lemma \ref{choice tele lemma}, \begin{align} \label{mu decay}
|\widehat{\mu}(\xi)| &\lesssim (1+|\xi|)^{-\alpha/2} \ln^{1/2}(4\zeta_0(1+|\xi|^2)) \quad \forall \xi \in \mathbb{R}, \\ \label{m decay}
|\widehat{\mu \cdot \nu}(\xi)| &\lesssim (1+|\xi|)^{-\alpha/2} g\left(\frac{1}{2}|\xi|\right) \ln^{1/2}(4\zeta_0(1+|\xi|^2)) \quad \forall \xi \in \mathbb{R}. \end{align}. \end{lemma} \begin{proof} We prove only \eqref{m decay}, as the proof of \eqref{mu decay} is similar and simpler. We begin by making two reductions.
First, by a standard argument (see Kahane \cite[pp.252-253]{kahane-book}), we only need to prove \eqref{m decay} for $\xi = d_0 k \in d_0 \mathbb{Z}$.
For the second reduction, note that for every $0 \neq \xi \in \mathbb{R}$, we have \begin{align*} \widehat{{\mu_{0} \cdot \nu}}(\xi) &= \int_{\mathbb{R}} \int_{1}^{2} e^{-2\pi i x y \xi} dx d\nu(y) = \int_{\mathbb{R}} \frac{e^{-2\pi i y \xi} - e^{-2\pi i y (2\xi)}}{-2 \pi i y \xi} d\nu(y) \\ &= \int_{\mathbb{R}} \frac{e^{-2\pi i y \xi} - e^{-2\pi i y (2\xi)}}{-2 \pi i \xi} \phi(y) d\nu(y) = \frac{1}{-2 \pi i \xi} \rbr{ \widehat{\phi \nu}(\xi) - \widehat{\phi \nu}(2\xi) }. \end{align*}
By Lemma \ref{phi nu decay lemma} and (a) of Lemma \ref{g lemma}, $|\widehat{{\mu_{0} \cdot \nu}}(\xi)| \lesssim (1+|\xi|)^{-1} g\left(\frac{1}{2}|\xi|\right)$ for all $0 \neq \xi \in \mathbb{R}$. The same inequality holds when $\xi=0$ by direct calculation. Therefore, by the triangle inequality, we just need to prove \eqref{m decay} with the left-hand side replaced by $|\widehat{\mu \cdot \nu}(\xi) - \widehat{{\mu_{0} \cdot \nu}}(\xi)|$.
Lemma \ref{weak convergence} says $\mu_j \to \mu$ weakly. Then the dominated convergence theorem shows that ${\mu_{j} \cdot \nu} \to {\mu \cdot \nu}$ weakly. Therefore, for every $\xi \in \mathbb{R}$, $\widehat{\mu \cdot \nu}(\xi) = \lim_{j \to \infty} \widehat{\mu_{j} \cdot \nu}(\xi)$, hence \begin{align}\label{sum above}
|\widehat{\mu \cdot \nu}(\xi) - \widehat{\mu_{0} \cdot \nu}(\xi)| \leq \sum_{j=0}^{\infty} |\widehat{\mu_{j+1} \cdot \nu}(\xi) - \widehat{\mu_{j} \cdot \nu}(\xi)|. \end{align} If $\xi = 0$, each term of the sum
in \eqref{sum above} is zero, by direct calculation. Now assume $\xi = d_0 k \in d_0 \mathbb{Z}$, $\xi \neq 0$. By Lemma \ref{choice tele lemma}, the sum
in \eqref{sum above} is \begin{align*}
&\leq 2 C_0 g(\tfrac{1}{2}|\xi|) \ln^{1/2}(4\zeta_0(1+|\xi|^2)) \rbr{
\sum_{j: N_{j+1} > |\xi|} T_j^{-1/2}
+ \sum_{j: N_{j+1} \leq |\xi|} T_j^{-1/2} \frac{N_{j+1}}{|\xi|} }. \end{align*}
To estimate the last two sums, recall that $2 \leq n_{j} \leq n_{\ast}$, $N_j = n_1 \cdots n_j$, $N_0=T_0=1$, and $T_{j} \approx N_{j}^{\alpha}$ for all $j \in \mathbb{Z}_{> 0}$. Thus the first sum is \begin{align*}
\approx |\xi|^{-\alpha/2} \sum_{j: N_{j+1} > |\xi|} (N_{j+1}/|\xi|)^{-\alpha/2}
\leq |\xi|^{-\alpha/2} \sum_{k=0}^{\infty} 2^{-k \alpha/2}
\lesssim |\xi|^{-\alpha/2}. \end{align*} And the second sum is \begin{align*}
\approx |\xi|^{-\alpha/2} \sum_{j: N_{j+1} \leq |\xi|} (N_{j+1}/|\xi|)^{1-\alpha/2}
\leq |\xi|^{-\alpha/2} \sum_{k=0}^{\infty} 2^{-k(1-\alpha/2)}
\lesssim |\xi|^{-\alpha/2}. \end{align*} \end{proof}
\begin{lemma} With the sets $B_{j+1,a}$ chosen as in Lemma \ref{choice tele lemma}, $\dim_F(\mu) \geq \alpha$ and $\dim_F(\mu \cdot \nu) \geq \min\cbr{1,\dim_F(\mu) + \dim_F(\nu)}$. \end{lemma} \begin{proof} Lemma \ref{mu mu nu decay lemma}
implies $\dim_{F}(\mu) \geq \alpha$. Lemma \ref{mu mu nu decay lemma} and (c) of Lemma \ref{g lemma} imply $\dim_F(\mu \cdot \nu) \geq \dim_F(\mu) + \dim_F(\nu)$. \end{proof}
\end{document} | arXiv |
ICIS Technical Reports:
[All BiBTeX entries for this year]
J.W.G.M. Hubbers, and A.H.M. ter Hofstede. An Investigation of the Formal Foundations of Object-Oriented Conceptual Data Modeling. Technical report, Computing Science Institute, University of Nijmegen, Nijmegen, The Netherlands, 1996, To appear.
[ Missing PDF ] [ Bibtex ]
E. Hoenkamp, L. Schomaker, P. van Bommel, C.H.A. Koster, and Th.P. van der Weide. Profile: A Proactive Information Filter. Technical report: CSI-N9602, February, Radboud University Nijmegen, 1996.
Information Systems. Publication List 1991-1996. Technical report: CSI-N9603, March, Radboud University Nijmegen, 1996.
J.W.G.M. Hubbers, and P. van Bommel. Using Graph Rewrite Rules in MISS. Technical report: CSI-N9604, May, Computing Science Institute, University of Nijmegen, Nijmegen, The Netherlands, 1996.
A.I. Bleeker. Informatie zoeken op inhoud: de volgende stap. Technical report: CSI-N9605, May, Radboud University Nijmegen, 1996.
Maristella Agosti, and Araminte Bleeker. Indexing and Evaluation: Two Main Problems in Content-based Multimedia Retrieval. Technical report: CSI-N9606, July, Radboud University Nijmegen, 1996.
Gy�rgy Kov�cs, and P. van Bommel. Overview of F-logicfrom Database Transformation Perspective. Technical report: CSI-N9607, September, Radboud University Nijmegen, 1996.
M.C.A. Devillers. The Binary Decision Machine. Technical report: CSI-N9609, October, Radboud University Nijmegen, 1996.
P.J.M. Frederiks, and Th. P. A Note on Valid Instance Categoriesfor Conceptual Data Modeling. Technical report: CSI-N9610, December, Radboud University Nijmegen, 1996.
P.J.M. Frederiks, and Th.P. van der Weide. Verification and Design for Information Architectures. Technical report: CSI-N9611, December, Radboud University Nijmegen, 1996.
P.J.M. Frederiks, and Th.P. van der Weide. From a File-Oriented View to an Object-Oriented View. Technical report: CSI-R9601, January, Computing Science Institute, University of Nijmegen, Nijmegen, The Netherlands, 1996.
The last three decades the architecture of information systems has evolved from file-oriented, via data-oriented and communication-oriented towards an object-oriented view. Hand in hand with this architectural evolution the way a user communicates with the information system is changed. In this paper we discuss the relation between the different architectures and their associated man-machine communication. We introduce the concept of information grammar, and show that this grammar can be seen as a common point of convergence. Together with the evolution, we can identify an evolution of methods for information system development. The construction of an information grammar is also addressed in this paper.
P.J.M. Frederiks, C.H.A. Koster, and Th.P. van der Weide. Validation of Object-Oriented Analysis Models using Informal Language. Technical report: CSI-R9609, May, Computing Science Institute, University of Nijmegen, Nijmegen, The Netherlands, 1996.
In this paper a conceptual model for object-oriented analysis is introduced. Three submodels are described which can be seen as milestones during the analysis phase. Each (sub)model has a corresponding paraphrasing mechanism which may be used (1) to provide a description of the structure of the model and (2) to generate sample instantiations. This paraphrasing mechanism is intended to enable the domain expert to validate the model by sentences in (semi-)natural language.
P.J.M. Frederiks, and Th.P. van der Weide. Cognitive Requirements for Natural Language Based Conceptual Modeling. Technical report: CSI-R9610, June, Computing Science Institute, University of Nijmegen, Nijmegen, The Netherlands, 1996.
In this paper we discuss the consequences of a natural language based modeling process for those who are involved in this process, i.e. domain experts and system analysts. For both domain experts and system analysts the cognitive requirements in a natural language based conceptual modeling process are presented as axiom-like requirements.
J.J. Sarbo. Representing subsumption by concept lattices. Technical report: CSI-R9611, June, Computing Science Institute, University of Nijmegen, Nijmegen, The Netherlands, 1996.
B.P.F. Jacobs. Hybrid Systems of Coalgebras plus Monoid Actions. Technical report: CSI-R9614, September, Radboud University Nijmegen, 1996.
Hybrid systems combine discrete and continuous dynamics. Weintroduce a semantics for such systems consisting of a coalgebratogether with a monoid action. The coalgebra captures the (discrete)operations on a state space that can be used by a client (like inthe semantics of ordinary (non-temporal) object-oriented systems).The monoid action captures the influence of time on the state space,where the monoids that we consider are the natural numbers monoid$(\NNO,0,+)$ of discrete time, and the positive reals monoid $({\BbbR}_{\geq 0},0,+)$ of real time. Based on this semantics we develop ahybrid specification formalism with timed method applications: itinvolves expressions like $\sans{s}.\sans{meth}@\alpha$, with the following meaning: in state$\sans{s}$ let the state evolve for $\alpha$ units of time(according to the monoid action), and then apply the (coalgebraic)method $\sans{meth}$. In this formalism we specify various(elementary) hybrid systems, investigate their correctness, anddisplay their behaviour in simulations. We further define a suitable notion of homomorphism between ourhybrid models (of coalgebras plus monoid actions), in such a waythat minimal realizations (of the specified behaviour) appear asterminal models. We identify the terminal models of our examplespecifications, and give general constructions. This leads to aninvestigation of various topics related to terminality:bisimilarity, behaviour-realization adjunctions, refinement (with acoinductive proof method for correctness) and inheritance. In afinal section we briefly discuss non-homogeneous hybrid systems(with continuous inputs).
J.W.G.M. Hubbers, and A.H.M. ter Hofstede. On the Concepts Underlying Object-Oriented Conceptual Data Modeling. Technical report: CSI-R9615, September, Computing Science Institute, University of Nijmegen, Nijmegen, The Netherlands, 1996.
G. Kov�cs, and P. van Bommel. From Conceptual Model to OO Database via Intermediate Specification. Technical report: CSI-R9617, October, Radboud University Nijmegen, 1996.
When designing underlying databases of information systems, data arefirst modelled on conceptual level and then the obtained conceptual data modelis transformed to a database.The focus of this paper is the transformation ofconceptual models into object-oriented database systems.Fora conceptual schema, consisting of an information structure and a set ofintegrity constraints,both the structure and the constraints have to be translated into thetarget environment.In our approach this mapping is captured within the frameworkof a two level architecture. Conceptual models are first mapped to abstract intermediate specifications,which are then transformed to database schemas in a given object-oriented target database environment (e.g. SQL3, ODMG).To express intermediate representations of conceptual modelswe use F-logic, a logic-based abstract specification language forobject-oriented systems.In the present paper we focus on the first step of the overall transformation,i.e. the mapping of conceptual models into F-logic.Several transformation alternatives are discussed,and a corresponding graphical notation for specifying transformationalternatives is provided.
B.P.F. Jacobs. Behaviour-Refinement of Object-Oriented Specifications with Coinductive Correctness Proofs. Technical report: CSI-R9618, October, Radboud University Nijmegen, 1996.
A notion of refinement is defined in the context of coalgebraic specification of classes in object-oriented languages. It tells us whenobjects in a ``concrete'' class behave exactly like (or: simulate) objectsin an ``abstract'' class. The definition of refinement involves certain selection functions between procedure-inputs and attribute-outputs,which gives this notion considerable flexibility. Thecoalgebraic approach allows us to use coinductive proof methods inestablishing refinements (via (bi)simulations). This is illustrated inseveral examples.
M.S. klein Gebbinck, and Th. E. Accurate Area Estimation by Data-Driven Decomposition of Mixed Pixels. Technical report: CSI-R9622, December, Radboud University Nijmegen, 1996.
There are many image processing applications where the area of anobject has to be estimated as accurately as possible. A well-knownexample is the area estimation of agricultural fields, which is ofgreat importance for the management of the agricultural subsidy systemof the European Union. The area of an object can be estimated usingboth classification, which allocates a pixel to a single class, anddecomposition, which divides a pixel between several classes. Sincedecomposition is better at handling mixed pixels---pixels comprisingmultiple classes---which are often found at object boundaries, areaestimation by decomposition is expected to be more accurate. To testthis hypothesis, a data-driven decomposition method was developed andapplied to a series of artificial satellite images of increasingcomplexity. Data-driven decomposition was able to estimate thepercentage of each component of a pixel with an average error of~5\%.Narrow structures were processed correctly, and isolated pixels weredetected using a simple threshold. A quantitative comparison with theresults of three other methods found in the literature showed that thearea estimates of data-driven decomposition were significantly moreaccurate. This study suggests that data-driven decomposition is anaccurate area estimation method which is worth further research usingreal satellite images.
G. Kov�cs, and P. van Bommel. Designing and Implementing OO Database Using Conceptual Data Modelling. Technical report: CSI-R9624, December, Radboud University Nijmegen, 1996.
The focus of this paper is the transformation of conceptual data models (suchas ER, NIAM, PSM) to object-oriented databases.This transformation is capturedwithin the framework of a two level architecture. Conceptual modelsare first mapped to abstract intermediate specifications, which arethen transformed to database schemas in a given object-orienteddatabase environment. This enables us to treat different target systemsin a uniform way. As final implementation environtments we consider object-orientedas well as object-relational DBMSs, including the SQL3 andODMG-93 standards. Intermediate representations are expressed in F-logic, a logic-basedabstract specification language for object-oriented systems. Several transformation alternatives are discussed in a formalcontext, resulting in a collection of design options.%A corresponding graphical notation for design options%is provided.
P.J.M. Frederiks, and Th.P. van der Weide. Formalization, Integration, and Validation of Object-Oriented Analysis Models leading to an Information Grammar. Technical report: CSI-R9625, December, Radboud University Nijmegen, 1996.
In this paper the focus is on object-oriented analysis of information systems.We assume that the communication within an application domain can be describedby a logbook of events.In our view, the purpose of the analysis phase is to model the structure of this logbook.The resulting conceptual model is referred to as theinformation architecture, and is an integration of threeformal object-oriented analysis models with each a specificview on the application domain. Furthermore, the information architectureforms an abstraction of an underlying grammar, called the informationgrammar, for the communication within the application domain.This grammar can be used to validate the information architecturein a textual format by informed users. Furthermore, the information grammarcan be used to obtain the relevant data and processes of theapplication domain, and serves as a basis for the query language of userswith the information system.
M.S. klein Gebbinck, and Th.E. Schouten. Application of Data-Driven Decomposition to Landsat-TM Images for Crop Area Estimation. Technical report: CSI-R9626, December, Radboud University Nijmegen, 1996.
An accurate crop area estimation method based on satellite remotesensing imagery is needed to manage the agricultural subsidy system ofthe European Union. The area estimator can use either classification,which allocates a pixel to a single class, or decomposition, whichdivides a pixel between several classes, to determine the ground covertype(s) a pixel is composed of. While in early days classification wasmuch used, recently the decomposition approach has gained moreinterest, however, only on a per pixel basis. In a previous study, wedeveloped the data-driven decomposition method, which used spatialinformation to guide the decomposition process; on artificialLandsat-TM images this method proved to be far more accurate thantechniques based on classification or pixel-based decomposition. Toinvestigate whether data-driven decomposition also results in animproved area estimation when using real satellite images, the area of17 agricultural lots was determined from a large scale topographicalmap. After co-registration with a corresponding Landsat-TM image,application of data-driven decomposition gave an estimation that wasequally or more accurate than the estimates of a similar method basedon classification in 14 of the 17 cases. Furthermore, data-drivenclassification also showed to be better suited for handling the smallboundary structures that separated the agricultural fields. Theseresults suggest that the accuracy of data-driven decomposition ishigher than that of an area estimator based on classification whendealing with agricultural fields.
J.F. Groote, and J. Springintveld. Algebraic Verification of a Distributed Summation Algorithm. Technical report: CSI-R9627, December, Radboud University Nijmegen, 1996.
In this note we present an algebraic verification of Segall's Propagation of Information with Feedback (PIF) algorithm.This algorithm serves as a nice benchmark forverification exercises (see \cite{Chou,Va95,Hes96}).The verification is based on the methodology presented in \cite{GS95}and demonstrates its applicability to distributed algorithms. | CommonCrawl |
Evaluation of anti-inflammatory activity of Justicia secunda Vahl leaf extract using in vitro and in vivo inflammation models
Godswill Nduka Anyasor ORCID: orcid.org/0000-0003-2644-63961,
Azeezat Adenike Okanlawon1 &
Babafemi Ogunbiyi1
Justicia secunda Vahl. is a medicinal plant used in ethnomedical practice as therapy to manage inflammation. Therefore, this study was designed to evaluate the anti-inflammatory activity of methanol extract of J. secunda leaves (MEJSL) using in vitro and in vivo inflammation models.
Seventy-percent MEJSL was prepared following standard procedure. In vitro anti-inflammatory assays were performed using heat-induced bovine serum albumin (BSA) denaturation and erythrocyte membrane stabilization assays. Carrageenan and formaldehyde induced inflammation in rat models were used to evaluate the anti-inflammatory activity of MEJSL in vivo. Diclofenac sodium was used as a reference drug. In addition, liver and kidney function assays and hematological analysis were carried out.
Data revealed that varying concentrations of MEJSL significantly (P < 0.05) inhibited heat-induced BSA denaturation and stabilized erythrocyte membrane against hypotonicity-induced hemolysis when compared with diclofenac sodium in a concentration-dependent manner. In vivo study showed that 10 mg/kg body weight (b.w.) diclofenac sodium, 100 and 300 mg/kg b.w. MEJSL suppressed carrageenan-induced paw edema at the sixth hour by 71.14%, 83.08%, and 89.05%, respectively. Furthermore, 10 mg/kg b.w. diclofenac sodium, 100 and 300 mg/kg b.w. MEJSL inhibited formaldehyde-induced paw edema by 72.53%, 74.73%, and 76.48%, respectively. Animals treated with varying doses of MEJSL had reduced plasma aspartate aminotransferase and alanine aminotransferase activities; urea and creatinine concentrations; and modulated hematological parameters when compared with the untreated control group.
Findings from this study showed that MEJSL exhibited substantial anti-inflammatory actions in the in vitro and in vivo models. It also indicated that MEJSL anti-inflammatory mechanisms of action could be through interference with phase 2 inflammatory stressors, upregulation of cytoprotective genes, stabilization of inflammatory cell membranes and immunomodulatory activity.
Medicinal plants have been used since time immemorial to relieve symptoms and treat diseases. It has played a vital role in health care systems where a substantial population of the world depends on the use of herbs as medicine [1]. Nowadays, there is an increasing interest among researchers to investigate the pharmacological effects and potential mechanisms of action of sundry medicinal plants using in vitro and in vivo models [2, 3].
Justicia secunda Vahl. which belong to the family of Acanthaceae is commonly known as "bloodroot" and "sanguinaria" in Barbados and Venezuela respectively [4, 5]. In South-Eastern Nigeria, it is locally called "obara bundu". The Ogbia people of Otuoke-Otuaba, Bayelsa, Niger-Delta region of Nigeria calls it "asindiri" or "ohowaazara". Justicia secunda grows in humid soil around rivers or creeks and can be located in tropical and pantropical regions of the world [6].
In folklore medicine, J. secunda leaves are used for treatment of wound, anemia, and pain within the abdominal region [7]. The leaf decoction of J. secunda is consumed in some parts of Nigeria, Cote-d'Ivoire, and Congo for the purpose of improving hematocrit count [8]. Justicia secunda leaves have been demonstrated to possess anti-sickling, antimicrobial, antihypertensive and hematinic activities [4, 6, 8, 9]. The anti-inflammatory potential of J. secunda leaves in animal model for 24 h was reported by Onoja et al. [5]. Phytochemical evaluation of J. secunda leaves detected alkaloids, polyphenols, flavonoids, tannins, leucoanthocyanins, quinones and anthocyanins [8]. In addition, quindoline, luteolin, auranamide, secundarellone A, B and C, aurantamide acetate, and pyrrolidone derivatives have been documented for J. secunda leaves [10].
Inflammation is the body's protective mechanism elicited in response to mechanical injuries, microbial infections, burns, and other deleterious stimuli that may threaten the host health [11]. It can be classified as either acute or chronic inflammation. Acute inflammation occurs as an immediate response to trauma, usually between two hours while chronic inflammation occurs as an ongoing response to a longer-term medical condition [12, 13]. Chronic inflammation has been claimed to cause the most significant death in the World [14]. Clinically, inflammation is defined as a pathophysiological process characterized by pain, redness, edema, heat, and loss of tissue function [15]. This process involves changes in blood flow, increased permeability of vascular tissues, and tissue destruction via the activation and migration of leucocytes with the synthesis of reactive oxygen species (ROS), and local inflammatory mediators, including prostaglandins, leukotrienes, and platelet-activating factors induced by phospholipase A2, cyclooxygenases, and lipoxygenases [16, 17].
Conventional steroidal anti-inflammatory drugs and non-steroidal anti-inflammatory drugs (NSAID) used in the treatment of acute inflammatory disorders have been unsuccessful in the treatment of chronic inflammatory disorders including rheumatoid arthritis. These conventional anti-inflammatory drugs have also been associated with unwanted side effects [18, 19]. This has led to the search for an alternative remedy, especially from medicinal plants to treat these inflammatory disorders. Therefore, the aim of this study was to evaluate the anti-inflammatory effects of MEJSL, using in vitro and in vivo inflammation models with the rationale to provide an insight into the potential anti-inflammatory mechanisms of action.
Justicia secunda leaves were obtained in fresh condition from a farm at Usaka-Umuofor, Isiala Ngwa North, South-Eastern, Nigeria. The plant was identified and authenticated at Forestry Research Institute of Nigeria, Ibadan, Oyo State with voucher specimen number 112177.
Extraction procedure
Justicia secunda leaves were washed and dried in a hot air-oven at 40 °C for 48 h. They were pulverized using a mechanical blender and sieved to obtain the fine powder form. Eighty grams of pulverized J. secunda leaves was steeped in 640 mL 70% methanol (1:8), shaken intermittently for 48 h. The obtained suspension was filtered using Whatman No.1 filter paper and the filtrate concentrated in a rotary evaporator (Buchi Rotavapor RE-3; Buchi Labortecknic AG, Switzerland) at 40 °C. The obtained concentrate yield was 8.16 g, stored at 4 °C until further use.
Evaluation of in vitro anti-inflammation activity
Inhibition of heat-induced bovine serum albumin denaturation assay
Effect of methanol extract of J. secunda leaves (MEJSL) on heat-induced bovine serum albumin (BSA) denaturation assay was carried out using a method described by Chandra et al. [20] with minor modifications. The reaction mixtures consist of varying concentrations (100, 200, 300 and 500 μg/mL) of MEJSL or reference drug diclofenac sodium (2-[(2,6 dichlorophenyl)amino] benzene acetic acid sodium salt) (an NSAID), 1% w/v BSA and phosphate buffered saline (PBS, pH 6.4) separately while PBS was used as control. The reaction mixtures were incubated at 37 °C for 20 min and the temperature was increased to keep the samples at 70 °C for 5 min. After cooling, turbidity was measured at 660 nm using UV-visible spectrophotometer (Schimadzu Double Beam UV-2600, Japan). The control represents 100% protein denaturation. The percentage inhibition of BSA denaturation was calculated as stated below:
$$ \%\mathrm{inhibition}\ \mathrm{of}\ \mathrm{BSA}\ \mathrm{denaturation}=100\times \left[1-\left(\raisebox{1ex}{$\mathrm{A}2$}\!\left/ \!\raisebox{-1ex}{$\mathrm{A}1$}\right.\right)\right] $$
Where A1 = absorbance of the control, and A2 = absorbance of the test sample.
Erythrocyte membrane stabilization assay
The effect of MEJSL on hypotonicity-induced erythrocyte membrane hemolysis assay was performed following the method adopted by Shinde et al. [21] and modified by Oyedapo et al. [22]. Whole blood sample (5 mL) was obtained from human by venipuncture using a syringe and immediately transferred to an ethylenediaminetetraacetic acid (EDTA) bottle. The blood sample was centrifuged for 10 min at 3000 rpm (rpm) and the supernatant was carefully removed while the packed red blood cells were washed in freshly prepared isosaline solution (0.85% NaCl). Subsequently, the blood was washed and centrifuged repeatedly until the supernatant became clear. Stock red blood cell (10% v/v) was prepared in isosaline solution. The assay mixture contained 1 mL sodium phosphate buffer (pH 7.4, 0.15 mol/L), 2 mL hyposaline solution (0.36% w/v NaCl), 0.5 mL stock red blood cell suspension (10%, v/v) with 0.5 mL of MEJSL or diclofenac sodium (reference drug) of varying concentrations in test tubes. For the control, distilled water replaced hyposaline solution to induce 100% hemolysis. The different test tubes were incubated at 56 °C in a water bath for 30 min and then centrifuged at 5000 rpm. The hemoglobin content in each tube was estimated using UV-visible spectrophotometer (Schimadzu Double Beam UV-2600, Japan) at 560 nm.
$$ \%\mathrm{stabilization}=100-\left[\frac{\mathrm{optical}\ \mathrm{density}\ \mathrm{of}\ \mathrm{extract}}{\mathrm{optical}\ \mathrm{density}\ \mathrm{of}\ \mathrm{control}}\times 100\right] $$
Experimental animals
Fifty (50) male albino rats (Wistar strain) weighing between 100 and 180 g were purchased from the Animal Facility, Babcock University. The rats were allowed to acclimatized and fed with commercial pellet rat chow and water for 14 days. The rats were housed in plastic cages and maintained following the National Institute of Health (NIH) documentation on the guide for the care and use of laboratory animals [23].
Experimental animal design
The rats were randomly distributed into 10 groups by a feature of weight, for two separate investigations, consisting of 5 rats per group.
Experiment 1: carrageenan-induced inflammation model
Effect of MEJSL on carrageenan-induced inflammation was carried out as described by Winter et al. [24]. This experiment involved five groups of five rats each, the rats were fasted overnight and had free access to water prior to the day of the experiment. The experimental design was as follows: Group I: rats were orally administered with 1 mL 0.9% NaCl only (normal group); Group II: rats were orally administered with 1 mL 0.9% NaCl + induction of arthritis using carrageenan (control group); Group III: rats were orally administered with 10 mg/kg body weight (b.w.) diclofenac sodium + induction of arthritis using carrageenan (standard group); Group IV and V: rats were orally administered with 100 and 300 mg/kg b.w. MEJSL + induction of arthritis using carrageenan (test groups). Prior to the treatments, the initial paw edema of each rat was measured using a micrometer screw gauge. One hour after treatment, paw edema was induced by injecting 0.1 mL 1% solution of carrageenan into the left hind paw just beneath the plantar of aponeurosis. Subsequently, the increase in left paw edema was measured at an hour interval for 6 h post-treatment. The percentage inhibition of inflammation was calculated as stated below:
$$ \%\mathrm{inhibition}\ \mathrm{of}\ \mathrm{inflammation}=\left(1-\frac{Vt}{Vc}\right)\times 100 $$
Vt is the mean paw edema in the treated groups while Vt is the mean paw edema in the control group.
Experiment 2: formaldehyde-induced inflammation model
Effect of MEJSL on formaldehyde-induced inflammation assay was carried out by following a modified method described by Agnel and Shobana [25]. The study involved five groups of five rats each, the rats were fasted overnight and had free access to water prior to the day of the experiment which lasted for 7 days. The experimental design was as follows: Group I: rats were orally administered with 0.2 mL 0.9% NaCl only (normal group); Group II: rats were orally administered with 0.2 mL 0.9% NaCl + induction of arthritis using 2% v/v formaldehyde (control group); Group III: rats were orally administered with 10 mg/kg b.w. diclofenac sodium + induction of arthritis using 2% v/v formaldehyde (standard group); Group IV and V: rats were orally administered with 100 and 300 mg/kg b.w. MEJSL + induction of arthritis using 2% v/v formaldehyde (test groups). In the first and third days of treatment, 0.02 mL 2% v/v formaldehyde was injected into the left hind paw of rats just beneath the plantar of aponeurosis to induce arthritis. The increase in paw edema was measured using a micrometer screw gauge. This was done 30 min before the induction of arthritis, and every 24 h for 7 days. The percentage inhibition of inflammation was calculated as stated below:
On the eight-day of the formaldehyde-induced inflammation model study, blood samples were collected by using 5 mL hypodermal syringes through cardiac puncture and transferred into EDTA bottles and heparin bottles to avoid clotting. The whole blood samples in heparin bottles were spun in a centrifuge (UNICO C856 Power Spin™ Model LX, United States) at 3500 rpm for 5 min to obtain plasma which was used for liver and kidney function analysis while blood samples in EDTA bottles were used for hematological analysis.
Liver function analysis
Effect of MEJSL on aspartate aminotransferase (AST) and alanine aminotransferase (ALT) activities were assayed according to the method outlined in the Randox kit (Randox, United Kingdom).
Kidney function analysis
Effect of MEJSL on plasma creatinine and urea concentrations were measured following the quantitative colorimetric methods as outlined in the Randox kit (Randox, United Kingdom).
Hematological analysis
Hematological analysis was performed using whole blood samples in EDTA bottles to determine red blood cell (RBC), hematocrit (HCT), hemoglobin (HGB), platelet (PLT), white blood cell (WBC), lymphocyte (LYM), granulocyte (GRAN), basophil, eosinophil, and monocyte (EBM) counts using an autoanalyzer (Swelab Alfa 3-Part Hematology Analyzer, Boule Medicals, Spanga, Sweden) at Babcock University Teaching Hospital Medical Laboratory.
Data were expressed as mean ± standard error of mean (SEM) (n = 5). Sample T-test analytical method was used to evaluate the difference between means in the in vitro anti-inflammatory experiments. Linear regression was performed to determine 50 % inhibitory concentration (IC50). The difference between the experimental and control groups was determined using GraphPad Prism® version 7.0, the comparison carried out using one-way analysis of variance (ANOVA). The significant difference in the experimental groups was assessed using the least significant difference (LSD) post-hoc analysis to test the significance at P < 0.05.
In vitro anti-inflammation assays
Data in Table 1 showed that 100–500 μg/mL MEJSL and diclofenac sodium inhibited heat-induced BSA denaturation in a concentration-dependent manner. MEJSL (IC50 = 186.20 ± 2.25 μg/mL) significantly (P < 0.05) exhibited a higher inhibition of heat-induced BSA denaturation than diclofenac sodium (IC50 = 215.50 ± 4.11 μg/mL). Furthermore, Data in Table 2 showed that the 250–2000 μg/mL MEJSL and diclofenac sodium stabilized erythrocyte membrane against hypotonicity-induced hemolysis in a concentration-dependent manner. The MEJSL (480.40 ± 1.47 μg/mL) significantly (P < 0.05) stabilized erythrocyte membrane against hypotonicity-induced hemolysis than diclofenac sodium (637.40 ± 2.69 μg/mL).
Table 1 Effect of methanol extract of J. secunda leaves on heat-induced BSA denaturation
Table 2 Effect of methanol extract of J. secunda leaves on hypotonicity-induced hemolysis erythrocyte membrane
In vivo anti-inflammation assays
Carrageenan-induced inflammation model
Data in Fig. 1 showed that 100 and 300 mg/kg b.w. μg/mL MEJSL and 10 mg/kg b.w. diclofenac sodium treated animals induced with inflammation using carrageenan significantly (P < 0.05) inhibited paw edema in a dose-dependent manner when compared with the untreated control group for a period of 6 hours. Diclofenac sodium (10 mg/kg b.w.), 100 and 300 mg/kg b.w. MEJSL suppressed paw edema of rats at the sixth hour by 0.58 ± 0.03 mm (71.14%), 0.34 ± 0.09 mm (83.08%) and 0.22 ± 0.07 mm (89.05%), respectively when compared with those of untreated control group.
Percentage inhibition of carrageenan-induced paw edema in rats treated with 100 and 300 mg/kg methanol extract of Justicia secunda leaves. Data expressed as mean ± standard error of mean
Formaldehyde-induced inflammation model
Data in Fig. 2 indicated that orally administered 100 and 300 mg/kg b.w. MEJSL and 10 mg/kg b.w. diclofenac sodium to experimental animals induced with inflammation using 2% v/v formaldehyde significantly (P < 0.05) inhibited paw edema of rats when compared with the untreated control group in a dose-dependent manner. The 100 and 300 mg/kg b.w. MEJSL suppressed paw edema of rats by 74.73% (1.15 ± 0.30 mm) and 76.48% (1.07 ± 0.40 mm) which was found to be comparable with the animals treated with 10 mg/kg b.w. diclofenac sodium that inhibited paw edema by 75.53% (1.25 ± 0.33 mm), respectively.
Change in paw edema by different doses of methanol extract of Justicia secunda leaves in rat induced with inflammation using formaldehyde
Plasma alanine aminotransferase (ALT) activity
Data in Fig. 3 showed that untreated control animals (10.24 ± 0.55 U/l) had a significantly (P < 0.05) elevated plasma ALT activity when compared with the normal group (7.37 ± 0.48 U/l). Animals induced with inflammation using formaldehyde and treated with 100 mg/kg b.w. MEJSL (8.84 ± 0.37 U/l), 300 mg/kg b.w. MEJSL (7.94 ± 0.44 U/l), and diclofenac sodium (7.40 ± 0.28 U/l) had significantly (P < 0.05) reduced plasma ALT activity when compared with those of untreated group (10.24 ± 0.55 U/l). The MEJSL exhibited a dose-dependent reduction in plasma ALT activity. In addition, the plasma ALT activity in animals treated with 300 mg/kg b.w. MEJSL was comparable to those of diclofenac treated group.
Effects of methanol extract of J. secunda leaves on plasma alanine aminotransferase (ALT) activity in normal and formaldehyde-induced arthritic rats. MEJSL-indicates methanol extract of Justicia secunda leaves. Different letters indicate significantly different at P < 0.05
Plasma aspartate aminotransferase (AST) activity
Data in Fig. 4 showed that animals induced with inflammation using formaldehyde and treated with 100 mg/kg b.w. MEJSL (27.10 ± 2.31 U/l), 300 mg/kg b.w. MEJSL (22.40 ± 0.73 U/l) and 10 mg/kg diclofenac sodium (30.98 ± 2.68 U/l) had significantly (P < 0.05) reduced plasma AST activities when compared with untreated animals (52.55 ± 3.41 U/l). Furthermore, MEJSL exhibited a dose-dependent decrease in plasma AST activity. Animals treated with 300 mg/kg b.w. MEJSL (22.40 ± 0.73 U/l) had a significantly (P < 0.05) reduced plasma AST activity when compared with diclofenac sodium treated group (30.98 ± 2.68 U/l).
Effects of methanol extract of J. secunda leaves on the plasma aspartate aminotransferase (AST) activity in normal and formaldehyde-induced arthritic rats. MEJSL-indicates methanol extract of Justicia secunda leaves. Different letters indicate significantly different at P < 0.05
Plasma urea
Data in Fig. 5 showed that the untreated control group (40.40 ± 2.50 mg/dL) had a significantly (P < 0.05) higher plasma urea concentration when compared with the normal group (29.23 ± 2.54 mg/dL). Animals induced with inflammation using formaldehyde and treated with 100 mg/kg b.w. MEJSL (25.24 ± 2.14 mg/dL), 300 mg/kg b.w. MEJSL (26.12 ± 0.98 mg/dL) and 10 mg/kg b.w. diclofenac sodium (30.9 ± 1.68 mg/dL) had significantly (P < 0.05) decreased plasma urea concentrations when compared with the untreated control group. Furthermore, animals treated with MEJSL had significantly (P < 0.05) reduced plasma urea concentration when compared with diclofenac sodium treated group.
Effects of methanol extract of J. secunda leaves on the concentration of plasma urea in normal and formaldehyde-induced treated rats. MEJSL-indicates methanol extract of Justicia secunda leaves. Different letters indicate significantly different at P < 0.05
Plasma creatinine
Figure 6 showed that the untreated control group (1.62 ± 0.22 mg/dL) had a significantly (P < 0.05) elevated plasma creatinine concentration when compared with the normal group (0.95 ± 1.10 mg/dL). However, animals induced with inflammation using formaldehyde and treated with 100 mg/kg b.w. MEJSL (1.33 ± 0.13 mg/dL), 300 mg/kg b.w. MEJSL (1.08 ± 0.06 mg/dL) and 10 mg/kg b.w. diclofenac sodium (1.04 ± 0.13 mg/dL). The MEJSL treated animals exhibited a dose-dependent reduction in plasma creatinine concentrations. The 300 mg/kg b.w. MEJSL treated animals had comparable plasma creatinine concentration when compared with diclofenac sodium treated animals.
Effects of methanol extract of J. secunda leaves on plasma creatinine concentration in normal and formaldehyde-induced arthritic rats. MEJSL-indicates methanol extract of Justicia secunda leaves. Different letters indicate significantly different at P < 0.05
Results in Table 3 indicated that 100 and 300 mg/kg b.w. MEJSL treated animals induced with inflammation using formaldehyde had significantly (P < 0.05) higher red blood cell (RBC), hematocrit (HCT), hemoglobin (HGB) and platelet (PLT) counts when compared with the untreated control group. However, there was no significant difference (P > 0.05) in RBC, HCT, HGB and PLT counts between 10 mg/kg b.w. diclofenac sodium treated animals and untreated control group. Furthermore, 100 and 300 mg/kg b.w. MEJSL treated animals had significantly (P < 0.05) reduced white blood cell (WBC), lymphocyte (LYM), granulocyte (GRAN), eosinophil, basophil and monocyte (EBM) counts when compared with untreated control animal. However, WBC, LYM, GRAN and EBM counts in animals treated with diclofenac sodium were not significantly different (P > 0.05) when compared with untreated control animals.
Table 3 Effects of methanol extract of J. secunda leaves on hematological parameters of rats induced with inflammation using formaldehyde
Identification and isolation of medicinal products from plants have been an ever-increasing area of interest in the field of drug discovery [26]. In this present study, the anti-inflammatory potential of methanol extract of J. secunda leaves (MEJSL) using in vitro and in vivo inflammation models were evaluated.
The different concentrations of MEJSL exhibited high inhibition of heat-induced protein denaturation and stabilization of erythrocyte membrane against hypotonicity-induced hemolysis, when compared with those of diclofenac sodium. These data indicate that MEJSL could contain an anti-inflammatory property. Previous studies have demonstrated that plant extracts with anti-inflammatory property possess the capacity to inhibit protein denaturation and stabilize cell membrane against lysis [27, 28]. Protein denaturation is a biochemical reaction that occurs during chronic inflammatory response which could result into loss of tissue function [29, 30]. In addition, lysis of lysosomal membranes during chronic inflammation has been proven to release pro-inflammatory markers including activated neutrophils, proteases and histamines at the local site of tissue damage [31, 32]. Hence, medicinal plant extract that inhibit protein denaturation and stabilize cell membrane against lysis could serve as a potential source of lead anti-inflammatory drug candidates.
Furthermore, the in vivo anti-inflammatory potential of MEJSL was evaluated using two animal models: carrageenan and formaldehyde induced inflammations. In carrageenan and formaldehyde-induced inflammation models, the animals that were orally administered with 100 and 300 mg/kg b.w. MEJSL had substantially suppressed paw edema thickness when compared with those of untreated control animals. This indicates that MEJSL possesses anti-inflammatory activity in vivo. These findings supported the data from the in vitro anti-inflammatory study that purported MEJSL to possess anti-inflammatory property. Previously, it has been documented that carrageenan and formaldehyde are agents that have the capacity to elicit the release of inflammatory stressors including prostaglandin, serotonin, protease, lysosome, cytokine, and histamine in animal models [15]. Hence, it is suggested that the capacity of MEJSL to suppress rat paw edema in carrageenan and formaldehyde induced inflammation rat models might be through mitigation of the release and migration of inflammatory stressors to the site of tissue damage. Furthermore, bioactive compounds present in MEJSL could be responsible for the anti-inflammatory action. Previous studies reported that J. secunda leaves contain polyphenolic compounds, which have been shown to potentially exhibit anti-inflammatory effects [5, 8, 33].
In addition, the anti-inflammatory property of MEJSL in the animal models might be through activation of nuclear factor-erythroid 2 p45 related factor 2 (Nrf2) signaling pathway. Several studies have demonstrated that Nrf2 signaling pathway is a key target in the discovery of anti-inflammatory bioactive compounds. Nrf2 orchestrates the activation of anti-inflammatory and antioxidant gene expressions through the Keap (Kelch-like ECH-associated protein) 1/Nrf2/ARE (antioxidant response element) signaling pathway and inhibits the progression of inflammation, thereby protecting cells from injuries [34, 35]. A supplementation with hydroxytyrosol, a polyphenolic compound had been shown to reduce the inflammatory stressors in mice fed with high fat diet through activation of Nrf2 pathway and down-regulation of nuclear factor kappa B (NF-κB) [36, 37]. NF-κB has been shown to induce transcription of pro-inflammatory mediators which includes interleukins, cytokines and cyclooxygenase-2 (COX-2) [38]. Hence, plant bioactive compound that target and inhibit NF-κB could serve as anti-inflammatory agent. More so, a plant polyphenol, kaempferol has been demonstrated to attenuate interleukin (IL)-6-induced COX-2 expression in human monocytic THP-1 cells suggesting its beneficial role in chronic inflammation [39].
In this present study, experimental animals induced with inflammation using carrageenan and formaldehyde models were found to exhibit a biphasic inflammatory response to tissue damage. Previous studies stipulate that the first phase of inflammatory response in carrageenan-induced inflammation is usually due to the release of substance P, kinin-like substances, serotonin and histamine with enhanced production of prostaglandins at the localized site of tissue damage within 2 h, while the second inflammation phase is characterized by the release of bradykinin, leukotrienes, prostaglandins, and proteases [40]. Based on this premise, it can be deduced from the carrageenan-induced inflammation model that suppression of the first inflammatory phase at 2nd hour by MEJSL might be due to inhibition of early pro-inflammatory mediator's release, including histamine and serotonin. The suppressive effect noted during second inflammatory phase might be the resultant effect of cyclooxygenase inhibition by MEJSL [41]. It was also thought that the membrane-stabilizing property of MEJSL could prevent the release of proteases from lysosomes, thereby mitigating the release of pro-inflammatory factors.
Furthermore, the biphasic phenomenon in formaldehyde-induced inflammation model has been reported to be both neurogenic (first phase) and anti-inflammatory (second phase) [40]. MEJSL suppressed paw edema starting from the 2nd day of animal treatment. This further strengthens the claim that MEJSL possesses anti-inflammatory property. Previous studies have shown that the first phase of an inflammatory response results from the stimulation of pain receptors and it is usually insensitive to anti-inflammatory agents. This could account for the increase in rat paw thickness in all experimental groups from the time of treatment until "day 2" when suppression was noticed among the MEJSL and diclofenac sodium treated groups. More so, anti-inflammatory agents have been demonstrated to prominently block the second phase than the first phase in formaldehyde-induced inflammation model [42, 43]. The second phase of an inflammatory response is dependent on peripheral inflammation due to changes in the pain receptors by chemical mediators including prostaglandin-like substances [40]. Hence, evidence from this present study suggests that MEJSL may be suppressing the chemical mediators associated with inflammation. This data seems to be in agreement with the previous claim that J. secunda leaves could possess an anti-inflammatory activity [5].
Furthermore, MEJSL treated animals had reduced plasma AST and ALT activities when compared with untreated control animals. This revealed that MEJSL could possess hepatoprotective effect. It has been previously shown that elevated plasma AST and ALT activities are indicators of tissue damage which are measured clinically as a diagnostic evaluation of liver function [44]. Hepatocellular damage resulting from formaldehyde-induced oxidative and inflammatory stressors could cause leakage of cytoplasmic enzymes into the plasma with concomitant increase in plasma enzyme activities. This observed hepatoprotective effect of MEJSL might be due to the presence of bioactive compounds with antioxidant activity against ROS-induced tissue damage. Previous data had shown that J. secunda leaves possess antioxidant compounds [5]. It could also be due to the presence of electrophilic compounds in J. secunda leaves that activate Nrf2 signaling pathway. The activated Nrf2 is translocated to the nucleus where it transactivate ARE which in turn triggers the upregulation of cytoprotective antioxidant and drug detoxifying genes, thereby protecting tissues against oxidative and inflammatory stressors [34].
Furthermore, animals treated with MEJSL had reduced plasma urea and creatinine concentrations when compared with untreated control animals. The reduced plasma urea and creatinine concentrations could be indicative of MEJSL nephroprotective activity. Plasma urea and creatinine concentrations are measured primarily to assess kidney function where their elevation is indicative of under-excretion capacity of the kidney, suggesting kidney impairment [45]. Hence, findings from this present study connotes that MEJSL contains bioactive compounds with nephroprotective property. Previous report has also shown that plant polyphenols could elicit a chemoprotective effect against kidney damage [46].
Hematological analysis revealed that animals induced with inflammation using formaldehyde and treated with 100 and 300 mg/kg b.w. MEJSL had elevated RBC, HCT, and HGB counts when compared with the untreated control and normal groups. This suggests that MEJSL could possess erythropoietic property. Previous study had indicated the J. secunda leaves could contain blood-boosting principles and this perhaps account for its ethnobotanical use in the management of anemia [5, 8]. In addition, the elevated platelet count in the MEJSL treated animals could be indicative of its wound healing property. Platelet cells had been demonstrated to play a major role in the healing process of damaged tissues [47].
Further investigation showed that animals treated with MEJSL and diclofenac sodium had reduced WBC, LYM, GRAN, and EBM counts when compared with untreated control animals. This suggests that MEJSL could possess immunomodulatory activity. An elevation in the immunological/inflammatory cells has been reported to indicate a heightened inflammatory response to the local site of tissue damage [15, 48]. More so, cellular response played by infiltrating cells through the release lysosomal contents at the local site of tissue damage would aggravate the inflammatory stressors [49, 50]. During inflammation, immunological/inflammatory cells generate ROS that damage macromolecules and also produce large amount of pro-inflammatory mediators including cytokines, chemokines and prostaglandins. These pro-inflammatory markers further recruit macrophages and directly activate multiple signal transduction cascades and transcription factors such as NF-κB, MAPK (mitogen-activated protein kinase), and JAK (janus kinase)-STAT (signal transducers and activators of transcription) associated with inflammation signaling pathways [15]. The clinical use of non-steroidal anti-inflammatory drugs have also been reported to exhibit immunomodulatory activity on infiltrating cells at the local site of tissue damage [51].
This study suggested that the methanol extract of J. secunda leaves possess anti-inflammatory, hepatoprotective, nephroprotective and immunomodulatory activities. It also showed that the anti-inflammatory mechanisms of action might be through direct inhibition of phase 2 pro-inflammatory signals, activation of Nrf2 signaling pathways, upregulation of cytoprotective genes and stabilization of inflammatory cell membranes. Furthermore, it provides some scientific insight into the ethnobotanical use of J. secunda leaves in folklore medicine. It is recommended that MEJSL could be considered as a choice candidate in pharmaceutical anti-inflammatory drug development.
All data pertaining to this study have been included in the manuscript
antioxidant response elemen
Aspartate aminotransferase
COX:
Cyclooxygenase
EBM:
Eosinophil, Basophil and Monocyte
GRAN:
Granulocyte
JAK:
Janus kinase
Keap1:
Kelch-like ECH-associated protein 1
LYM:
MAPK:
mitogen-activated protein kinase
MEJSL:
Methanol extract of Justicia secunda leaves
NF-κB:
Nuclear factor kappa B
Nrf2:
Nuclear factor-erythroid 2 p45 related factor 2
NSAID:
Non-steroidal anti-inflammatory drugs
Red blood cell (RBC)
ROS:
Signal transducers and activators of transcription
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Authors express deep gratitude to the Department of Biochemistry, Babcock University for providing support and research facility that enabled the completion of this study. We thank Mrs. Anyasor, Chiamaka for editorial assistance.
This research work did not receive any specific funding
Department of Biochemistry, Benjamin S. Carson (Snr.) School of Medicine, Babcock University, Ilishan-Remo, Ogun State, Nigeria
Godswill Nduka Anyasor, Azeezat Adenike Okanlawon & Babafemi Ogunbiyi
Godswill Nduka Anyasor
Azeezat Adenike Okanlawon
Babafemi Ogunbiyi
GNA and AAO conceived and designed the experiment; GNA, AAO, and BO performed the experiment and acquired the data; GNA and BO analyzed and interpreted the data; GNA and AAO drafted the manuscript and all authors read and approved the manuscript.
Correspondence to Godswill Nduka Anyasor.
The animal experiment was performed in accordance with the animal care and use guideline set by the National Institute of Health and Babcock University Health Research Ethics Committee approval was obtained. Justicia secunda leaves were obtained in fresh condition from a farm at Usaka-Umuofor, Isiala Ngwa North, South-Eastern, Nigeria. The plant was identified and authenticated at Forestry Research Institute of Nigeria, Ibadan, Oyo State with voucher specimen number 112177.
The authors declare no financial and non-financial competing interest with regards to this work.
Anyasor, G.N., Okanlawon, A.A. & Ogunbiyi, B. Evaluation of anti-inflammatory activity of Justicia secunda Vahl leaf extract using in vitro and in vivo inflammation models. Clin Phytosci 5, 49 (2019). https://doi.org/10.1186/s40816-019-0137-8
Justicia secunda
Anti-inflammation
And immunology | CommonCrawl |
\begin{definition}[Definition:Finite Algebra over Ring]
Let $A$ be a commutative ring with unity.
Let $B$ be an $A$-algebra.
Then $B$ is '''finite''' over $A$ {{iff}} its underlying module is finitely generated.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Pseudocomplemented Lattice]
Let $\struct {L, \wedge, \vee, \preceq}$ be a lattice with smallest element $\bot$.
Then $\struct {L, \wedge, \vee, \preceq}$ is a '''pseudocomplemented lattice''' {{iff}} each element $x$ of $L$ has a pseudocomplement.
The pseudocomplement of $x$ is denoted $x^*$.
\end{definition} | ProofWiki |
\begin{document}
\maketitle
\begin{abstract}
In our earlier paper \cite{DLW}, it is proved that a homogeneous rigid, traction or impedance condition on one or two intersecting line segments together with a certain zero point-value condition implies that the solution to the Lam\'e system must be identically zero, which is referred to as the generalized Holmgren principle (GHP). The GHP enables us to solve a longstanding inverse scattering problem of determining a polygonal elastic obstacle of general impedance type by at most a few far-field measurements. In this paper, we include all the possible physical boundary conditions from linear elasticity into the GHP study with additionally the soft-clamped, simply-supported as well as the associated impedance-type conditions. We derive a comprehensive and complete characterisation of the GHP associated with all of the aforementioned physical conditions. As significant applications, we establish novel unique identifiability results by at most a few scattering measurements not only for the inverse elastic obstacle problem but also for the inverse elastic diffraction grating problem within polygonal geometry in the most general physical scenario. We follow the general strategy from \cite{DLW} in establishing the results. However, we develop technically new ingredients to tackle the more general and challenging physical and mathematical setups. It is particularly worth noting that in \cite{DLW}, the impedance parameters were assumed to be constant whereas in this work they can be variable functions.
\noindent{\bf Keywords:}~~Lam\'e system; unique continuation; generalized Holmgren's principle; inverse elastic problems; unique identifiability; polygonal obstacle; polygonal grating; single measurement
\noindent{\bf 2010 Mathematics Subject Classification:}~~35Q74, 35B34, 35J57, 35R30, 78J20, 74J25
\end{abstract}
\section{Introduction}
This paper is continuation of our earlier paper \cite{DLW} on a novel unique continuation principle for the Lam\'e operator and its applications to several challenging inverse elastic problems. We begin by briefly introducing the background and motivation of our study and referring to \cite{DLW} for more related discussions.
Let $\Omega\subset\mathbb{R}^2$ be a bounded open set. $\bmf u=(u_\ell)_{\ell=1}^2\in L^2(\Omega)^2$ is said to be a (generalised) Lam\'e eigenfunction if \begin{equation}\label{eq:lame} -\mathcal{L} (\bmf{u})=\kappa \bmf{u}\ \text { in }\ \Omega,\ \ \kappa \in\mathbb{R}_+, \end{equation} where \begin{equation}\label{eq:pdo1} \mathcal{L}(\bmf u) :=\mu \Delta \bmf u+(\lambda+\mu) \nabla(\nabla \cdot \bmf u ) \end{equation} and $\mathcal{L}$ is known as the Lam\'e operator that arises in the theory of linear elasticity. Here $\lambda,\mu$ are the Lam\'e constants satisfying the following strong convexity condition \begin{equation}\label{eq:convex} \mu>0 \text { and } \lambda+\mu>0. \end{equation} In \eqref{eq:lame}, it is noted that we do not assume any boundary condition on $\partial\Omega$ for $\bmf{u}$ since we shall be mainly concerned with a local property of $\mathbf{u}$. \eqref{eq:lame} arises in the study of the time-harmonic elastic wave scattering with $\mathbf{u}$ and $\omega:=\sqrt{\kappa}$ respectively signifying the elastic displacement field and the angular frequency. We next present six trace operators associated with $\bmf{u}$ in \eqref{eq:lame} that arise in different physical scenarios in the theory of linear elasticity. To that end, we let $\Gamma_h\Subset\Omega$ be a closed connected line segment, where $h\in\mathbb{R}_+$ signifies the length of the line segment. Let \begin{equation}\label{eq:nutau} {\nu }=(\nu_1,\nu_2)^\top \mbox{ and }
\boldsymbol{\tau}=(-\nu_2,\nu_1)^\top \end{equation} respectively, signify the unit normal and tangential vectors to $\Gamma_h$. The traction $T_\nu\bmf{u}$ on $\Gamma_h$ is defined by \begin{equation}\label{eq:Tu}
T_{\bmf{\nu }}\bmf{u}=2\mu\partial_{\bmf{\nu }}\bmf{u}+\lambda\bmf{\nu} \left(\nabla \cdot \bmf{u}\right)+\mu\boldsymbol{\tau}(\partial_{2}u_{1}-\partial_{1}u_{2}), \end{equation} where $$ \nabla\bmf u:=\begin{bmatrix}
\partial_1 u_1 & \partial_2 u_1 \cr
\partial_1 u_2 & \partial_2 u_2 \end{bmatrix},\quad \partial_\nu\bmf u:=\nabla\bmf u \cdot \nu, \quad \partial_j u_i:=\partial u_i/\partial x_j. $$ Set \begin{align}
&\mathcal{B}_1(\mathbf{u})=T_\nu\mathbf{u}\big |_{\Gamma_h}, \hspace*{1.9cm} \mathcal{B}_2(\mathbf{u})=\mathbf{u}\big |_{\Gamma_h}, \label{eq:b12}\\
& \mathcal{B}_3(\mathbf{u})\big |_{\Gamma_h }=\left( \begin{array}{c} {\nu} \cdot \mathbf{u} \\ \boldsymbol{\tau} \cdot T_{\mathbf{\nu}} \bmf{u} \\ \end{array} \right), \quad
\mathcal{B}_4(\mathbf{u}) \big |_{\Gamma_h }=\left( \begin{array}{c} \boldsymbol{\tau} \cdot \bmf{u} \\ {\nu} \cdot T_{\mathbf{\nu}} \mathbf{u} \\ \end{array} \right),\label{eq:tf}\\ &\mathcal{B}_5(\mathbf{u})=\mathcal{B}_1(\mathbf{u})+\boldsymbol{\eta}\mathcal{B}_2(\mathbf{u}),\quad \mathcal{B}_6(\mathbf{u})=\mathcal{B}_3(\mathbf{u})+\boldsymbol{\eta}\mathcal{B}_4(\mathbf{u})\label{eq:b56} \end{align} where $\boldsymbol{\eta}\in L^\infty(\Gamma_h)$ and $ \boldsymbol \eta\equiv\hspace*{-4mm}\backslash\, 0$. Physically, the condition $\mathcal{B}_j(\mathbf{u})=0$, $j=1,\ldots, 6$, corresponds, respectively, to the case that $\Gamma_h$ is a traction-free, rigid, soft-clamped, simply-supported, impedance or generalized-impedance line segment.
It is noted that $\mathbf{u}$ is real-analytic in $\Omega$ since $\mathcal{L}$ is an elliptic PDO with constant coefficients. The classical Holmgren's uniqueness principle (HUP) states that if any two different conditions from $\mathcal{B}_j(\mathbf{u})=\mathbf{0}$, $1\leq j\leq 4$, are satisfied on $\Gamma_h$, then $\mathbf{u}\equiv \mathbf{0}$ in $\Omega$; see \cite{DLW} for more relevant background discussion on this aspect. In \cite{DLW}, we show that under ultra weaker conditions in two scenarios, the Holmgren principle still holds, which is referred to as the generalized Holmgren principle (GHP). In the first scenario, it is proved that if only one homogeneous condition is fulfilled on $\Gamma_h$ plus a certain zero point-value condition, say $\boldsymbol{\tau}^\top \nabla \bmf{u} |_{\bmf{x}={\bmf{x}}_0 } {\nu}=\bmf{0}$ for any fixed $\bmf{x}_0\in\Gamma_h$ in the case that $\Gamma_h$ is rigid, then one must have $\mathbf{u}\equiv \mathbf{0}$ in $\Omega$. That is, one of the two homogeneous conditions on $\Gamma_h$ in the Holmgren principle can be replaced by a point-value condition. In the other scenario, two intersecting line segments $\Gamma_h^+$ and $\Gamma_h^-$ are involved. It is proved that if two homogeneous conditions are respectively fulfilled on $\Gamma_h^\pm$ (the two homogeneous conditions can be the same to or different from each other), then one generically has $\mathbf{u}\equiv \mathbf{0}$. It is noted that in the degenerate case where the two line segments are collinear and the two homogeneous conditions are different, the GHP is reduced to the standard Holmgren principle. However, in \cite{DLW}, only $\mathcal{B}_1, \mathcal{B}_2$ and $\mathcal{B}_5$ were considered. In this paper, we shall include all the boundary traces from \eqref{eq:b12}--\eqref{eq:b56} into the GHP study, and derive a comprehensive and complete characterisation of the GHP associated with all the possible physical situations. In addition, there are two more extensions that are worthy of noting: first, in \cite{DLW}, the impedance parameter $\boldsymbol{\eta}$ is always assumed to be constant, and in this paper, it can be a variable function; second, in \cite{DLW}, a certain geometric restriction was required on the intersecting angle in the case of two intersecting line segments, and in this paper, we completely relax such a restriction. In principle, we follow the general strategy from \cite{DLW} in establishing the new GHP results. However, we develop technically new ingredients to tackle the more general and challenging physical and mathematical setups.
It is remarked that the HUP holds in a more general geometric setup where the line segment $\Gamma_h$ can be relaxed to be an analytic curve. Indeed, it is a special case of the more general Unique Continuation Principle (UCP) for elliptic PDOs. But the HUP requires the analyticity of the solution, which is also the case in our study. This is the main reason that we require $\Gamma_h$ located inside the domain $\Omega$, and refer to the new uniqueness principle as the GHP. We believe the GHP generically does not hold if $\Gamma_h$ is curved. On the other hand, our study should be able to be extended to the higher dimensional case. But as shall be seen that even for the two-dimensional case associated with flat lines, the study involves highly technical and lengthy arguments. Hence, we defer the aforementioned extensions and generalisations to our future study.
The GHP is strongly motivated by our study of a longstanding geometrical problem in the inverse scattering theory. It is concerned with the unique determination of the geometrical shape of an obstacle by minimal/optimal or at least formally-determined scattering measurements, which constitutes an open problem in the literature. In recent years, global uniqueness results with formally-determined scattering data have been achieved in determining polygonal/polyhedral obstacles of the soft-clamped or simply-supported type \cite{ElschnerYama2010,LiuXiao}. The mathematical machinery therein is mainly based on certain reflection and path arguments that are of a global nature. The GHP enables us to develop a completely local argument in resolving the unique determination problem for polygonal obstacles of the traction-free, rigid or impedance type \cite{DLW}. In this paper, using the newly established results for the GHP, we can prove the global uniqueness in determining the shape of a general polygonal obstacle as well as its boundary impedance (if there is any) by formally-determined data (indeed at most a few scattering measurements) in the most general physical scenario. We are aware that in the non-polygonal/polyhedral setting with formally-determined scattering data, only local uniqueness/stability results were derived in the literature for rigid or traction-free obstacles, where one needs to a-priori know that the possible obstacles cannot deviate too much \cite{GM,RSS}. In addition to the inverse elastic obstacle problem, we also consider the inverse elastic diffraction grating problem of determining unbounded polygonal structures and establish several novel unique identifiability results by at most a few scattering measurements.
The rest of the paper is organized as follows. In Section \ref{sect:2}, we present an overview and summary of the GHP results for the convenience of the readers. Sections \ref{sect:3} and \ref{sec:4} are respectively devoted to the GHP in the presence of a single homogeneous line segment and two homogeneous line segments. In Section \ref{sect:5}, we present the unique identifiability results for the inverse elastic obstacle problem and the inverse diffraction grating problem.
\section{Overview and summary of the GHP results}\label{sect:2}
In order to have a complete and comprehensive study, the derivation of the GHP results in this paper is lengthy and technically involved. In order ease the reading of the audience, we provide a brief summary of the GHP results in this section. We first fix several notations as well as the geometric setup in our study.
\begin{defn}\label{def:class1p}
Suppose that $\psi(r)$ is a complex-valued function for $r \in \Sigma:=[0, r_0]$, where $r_0\in\mathbb{R}_+$. $\psi$ is said to belong to the class $\mathcal{A}(\Sigma)$ if it allows an absolutely convergent series representation as follows
\begin{equation}\label{eq:series1}
\psi(r)=a_0+\sum_{j=1}^\infty a_j r^j,
\end{equation}
where $a_0\in\mathbb{C}\backslash\{0\}$ and $a_j \in \mathbb C$.
\end{defn}
\begin{defn}\label{def:class1}
Let $\Gamma_h\Subset\Omega$ be a closed line segment and $\boldsymbol{\eta}(\mathbf{x})$, $\mathbf{x}\in\Gamma_h$, be a complex-valued function. For a given $\mathbf{x}_0\in\Gamma_h$, we treat $\boldsymbol{\eta}(\mathbf{x})$ as a function in terms of the distance away from $\mathbf{x}_0$ along the line, namely $r:=|\mathbf{x}-\mathbf{x}_0|$. $\boldsymbol{\eta}$ is said to belong to the class $\mathcal{A}(\Gamma_h)$ if for any $\mathbf{x}\in\Gamma_h$, there exists a small neighborhood of $\mathbf{x}_0$ on the line segment, say $\Sigma_{\epsilon(\mathbf{x}_0)}=[0, \epsilon(\mathbf{x}_0)]$ such that $\boldsymbol{\eta}\in\mathcal{A}(\Sigma_{\epsilon(\mathbf{x}_0)})$. \end{defn}
Throughout the rest of the paper, we always assume that the impedance parameters in \eqref{eq:b56} belong to the class $\boldsymbol{\eta}\in\mathcal{A}(\Gamma_h)$. In fact, one can easily verify that if $\boldsymbol{\eta}$ is analytic on $\Gamma_h$ and nonvanishing at any point, then it belongs to $\mathcal{A}(\Gamma_h)$. Nevertheless, we give the explicit definition for clarity and in particular, we shall frequently make use of the series form \eqref{eq:series1} in our subsequent arguments. The GHP that we shall establish is a local property, and we shall mainly prove that $\mathbf{u}$ is identically vanishing in a sufficiently small neighbourhood around a point on $\Gamma_h$. Hence, without loss of generality, we can simply assume that the series expansion \eqref{eq:series1} holds on the whole line segment if $\boldsymbol{\eta}\in \mathcal{A}(\Gamma_h)$. Moreover, in order to ease the exposition, we shall not distinguish between a line and a line segment in what follows, which should be clear from the context.
\begin{defn}\label{def:1} Recall the rigid, traction-free, soft-clamped, simply-supported, impedance and generalized-impedance lines introduced after \eqref{eq:b12}--\eqref{eq:b56}.
Set ${\mathcal R}_\Omega^{\kappa}$, ${\mathcal T}_\Omega^{\kappa} $, ${\mathcal I}_\Omega^{\kappa} $, ${\mathcal G}_\Omega^{\kappa}$, ${\mathcal F}_\Omega^{\kappa} $ and ${\mathcal H}_\Omega^{\kappa}$ to respectively denote the sets of rigid, traction-free, impedance, soft-clamped, simply-supported and generalized-impedance lines in $\Omega$ of $\bmf{u}$.
\end{defn}
\begin{defn}\label{def:2} Recall that the unit normal vector $\nu $ and the tangential vector $\boldsymbol{\tau}$ to $\Gamma_h$ are defined in \eqref{eq:nutau}. Define \begin{subequations}\notag \begin{align}
{\mathcal S}\left( {\mathcal R}_\Omega^{\kappa} \right):=& \{\Gamma_h \in {\mathcal R}_\Omega^{\kappa} ~|~ \exists \bmf{x}_0 \in \Gamma_h \mbox{ such that }
\boldsymbol{\tau}^\top \nabla \bmf{u} |_{\bmf{x}={\bmf{x}}_0 } {\nu}=\bmf{0} \},\label{eq:defa} \\
{\mathcal S}\left( {\mathcal T}_\Omega^{\kappa} \right):=& \{\Gamma_h \in {\mathcal T}_\Omega^{\kappa} ~|~ \exists \bmf{x}_0 \in \Gamma_h \mbox{ such that } \bmf{u}(\bmf{x}_0)=\bmf{0},\,
\boldsymbol{\tau}^\top \nabla \bmf{u} |_{\bmf{x}={\bmf{x}}_0 } {\nu}=\bmf{0}
\},\label{eq:defb}\\
{\mathcal S}\left( {\mathcal I}_\Omega^{\kappa} \right):=& \{\Gamma_h \in {\mathcal I}_\Omega^{\kappa} ~|~ \exists \bmf{x}_0 \in \Gamma_h \mbox{ such that } \bmf{u}(\bmf{x}_0)=\bmf{0},\,
\boldsymbol{\tau}^\top \nabla \bmf{u} |_{\bmf{x}={\bmf{x}}_0 } {\nu}=\bmf{0} \},\label{eq:defc}\\
{\mathcal S}\left( {\mathcal G}_\Omega^{\kappa} \right):=& \{\Gamma_h \in {\mathcal G}_\Omega^{\kappa} ~|~ \exists \bmf{x}_0 \in \Gamma_h \mbox{ such that } {\nu}^\top \nabla \bmf{u} |_{\bmf{x}={\bmf{x}}_0 } {\nu} =0 \},\label{eq:def1} \\
{\mathcal S}\left( {\mathcal F}_\Omega^{\kappa} \right):=& \{\Gamma_h \in {\mathcal F}_\Omega^{\kappa} ~|~ \exists \bmf{x}_0 \in \Gamma_h \mbox{ such that } \boldsymbol{\tau}^\top \nabla \bmf{u} |_{\bmf{x}={\bmf{x}}_0 } {\nu} =0 .\label{eq:def2} \end{align} \end{subequations} They are referred to as the singular sets of ${\mathcal R}_\Omega^{\kappa}$, ${\mathcal T}_\Omega^{\kappa} $, ${\mathcal I}_\Omega^{\kappa} $, ${\mathcal G}_\Omega^{\kappa}$, ${\mathcal F}_\Omega^{\kappa} $, respectively. \end{defn}
\begin{rem}
Taking $\mathcal{S}\left( {\mathcal R}_\Omega^{\kappa} \right)$ as an example, the presence of a singular rigid line $\Gamma_h\in \mathcal{S}\left( {\mathcal R}_\Omega^{\kappa} \right)$ means that the homogeneous condition $\mathbf{u}|_{\Gamma_h}=0$ and the point-value condition $\boldsymbol{\tau}^\top \nabla \bmf{u} |_{\bmf{x}={\bmf{x}}_0 } {\nu}=\bmf{0}$ are fulfilled for the line $\Gamma_h$ and a given point $\mathbf{x}_0\in\Gamma_h$. It is proved in \cite{DLW} that the presence of a singular rigid line implies that $\mathbf{u}\equiv\mathbf{0}$ in $\Omega$. In what follows, we shall show that the presence of any other singular lines also ensures $\mathbf{u}\equiv\mathbf{0}$ in $\Omega$. It is pointed out that we shall also introduce the singular set of the generalised-impedance lines in what follows, namely $\mathcal{S}(\mathcal{H}_\Omega^\kappa)$. But its definition involves the Fourier coefficients of $\mathbf{u}$, which we would need to introduce first. \end{rem}
\begin{figure}
\caption{Schematic of the geometry of two intersecting lines with an angle $\varphi_0$ with $0< \varphi_0\leq \pi $.}
\label{fig1}
\end{figure}
We next introduce the geometric setup of our study. Let two line segments respectively be defined by (see Fig.~\ref{fig1} for a schematic illustration): \begin{equation}\label{eq:gamma_pm}
\begin{split}
\Gamma_h^+&=\{\bmf{x} \in \mathbb R^2~|~\bmf{x}=r\cdot (\cos \varphi_0, \sin \varphi_0 )^\top ,\quad 0\leq r\leq h,\quad 0<\varphi_0\leq 2\pi \}, \\
\Gamma_h^-&=\{\bmf{x} \in \mathbb R^2~|~\bmf{x}=r\cdot (1, 0 )^\top,\quad 0\leq r\leq h \},\ \ h\in\mathbb{R}_+.
\end{split}
\end{equation} Clearly, the intersecting angle between $\Gamma_h^+$ and $\Gamma_h^-$ is \begin{equation}\label{eq:angle1} \angle(\Gamma_h^+,\Gamma_h^{-})=\varphi_0, \quad 0< \varphi_0 \leq 2\pi. \end{equation} We should emphasize that if the intersection of $\Gamma_h^+$ and $\Gamma_h^-$ degenerates, i.e., $\varphi_0=\pi$ or $\varphi_0=2\pi$, then $\Gamma_h^+$ and $\Gamma_h^-$ are actually lying on a same line. In this degenerate case, we let $\Gamma_h$ denote the line, which corresponds to the case with a single line in our subsequent study. Moreover, $\Gamma_h^\pm$ shall be the homogeneous lines in Definition~\ref{def:1} or the singular lines in Definition~\ref{def:2}. Since the PDO $\mathcal{L}$ defined in \eqref{eq:lame} is invariant under rigid motions, for any two of such lines that are intersecting in $\Omega$ (or one line in the degenerate case), we can always have two lines as introduced in \eqref{eq:gamma_pm} after a straightforward coordinate transformation such that the homogeneous conditions in Definitions~\ref{def:1} and \ref{def:2} are still satisfied on $\Gamma_h^\pm$. Furthermore, we assume that $h\in\mathbb{R}_+$ is sufficiently small such that $\Gamma_h^\pm$ are contained entirely in $\Omega$. If $\Gamma_h^\pm$ are impedance or generalized-impedance lines, we assume that the impedance parameters on $\Gamma_h^\pm$ are respectively $\boldsymbol{ \eta}_1$ and $\boldsymbol{\eta}_2$, where $\boldsymbol{ \eta}_1 \in \mathcal A(\Gamma_h^+ )$ and $\boldsymbol{\eta}_2\in \mathcal A(\Gamma_h^- )$. As also noted before that $\bmf{u}$ is analytic in $\Omega$, it is sufficient for us to consider the case that $0<\varphi_0\leq \pi$. In fact, if $\pi < \varphi_0 \leq 2\pi$, we see that $\Gamma_h^+$ belongs to the half-plane of $x_2<0$ (see Fig.~\ref{fig1}). Let $\widetilde{\Gamma}_h^+$ be the extended line segment of length $h$ in the half-plane of $x_2>0$. By the analytic continuation, we know that $\widetilde{\Gamma}_h^+$ is of the same type of $\Gamma_h^+$, namely $\bmf{u}$ satisfies the same homogeneous condition on $\widetilde{\Gamma}_h^+$ as that on $\Gamma_h^+$. Hence, instead of studying the intersection of $\Gamma_h^+$ and $\Gamma_h^-$, one can study the intersection of $\widetilde{\Gamma}_h^+$ and $\Gamma_h^-$. Clearly, $\angle(\widetilde{\Gamma}_h^+, \Gamma_h^-)\in (0,\pi]$.
With the above geometric setup, our argument shall be mainly confined in a neighborhood of the origin, namely $\mathbf{x}_0=\mathbf{0}$. We shall heavily rely on the Fourier expansion in terms of the radial wave functions of the solution $\bmf{u}$ to \eqref{eq:lame} around the origin, which we also fix in the following to facilitate our subsequent use. Most of the results can be conveniently found in \cite{DLW} that are summarized from \cite{DR95,SP}. To that end, throughout the rest of the paper, we set \begin{equation}\label{eq:kpks}
k_{{p}}=\sqrt{ \frac{\kappa }{\lambda+2 \mu} } \text { and } k_{{s}}=\sqrt{ \frac{\kappa }{\mu}}, \end{equation} which are known as the compressional and shear wave numbers, respectively.
\begin{lem}\cite[Lemma 2.3]{DLW} \label{lem:u2 exp} Let $J_m(t)$ be the first-kind Bessel function of the order $m\in \mathbb{N} \cup \{0\} $ and $\bmf{x}=r(\cos \varphi, \sin \varphi )^\top \in \mathbb R^2$. Let $\bmf{u}(\bmf{x})$ be a Lam\'e eigenfunction of \eqref{eq:lame}. The radial wave expression of $\mathbf{u}(\bmf{x})$ to \eqref{eq:lame} at the origin can be written as \begin{equation}\label{eq:u}
\begin{aligned}
\mathbf{u}(\mathbf{x})= \sum_{m=0}^{\infty} & \left\{ \frac{k_p}{2} a_{m} \mathrm{e}^{{\mathrm{i}} m \varphi} \left\{J_{m-1}\left(k_{p} r\right)\mathrm{e}^{-{\mathrm{i}} \varphi}\mathbf{e}_1
-J_{m+1}\left(k_{p}r\right)\mathrm{e}^{{\mathrm{i}} \varphi}\mathbf{e}_2 \right\}\right.\\
& + \frac{{\mathrm{i}} k_s}{2} b_{m} \mathrm{e}^{{\mathrm{i}} m \varphi} \left\{J_{m-1}\left(k_{s} r\right)\mathrm{e}^{-{\mathrm{i}} \varphi}\mathbf{e}_1
+J_{m+1}\left(k_{s} r\right)\mathrm{e}^{{\mathrm{i}} \varphi}
\mathbf{e}_2
\right\} \bigg\} .
\end{aligned} \end{equation} where and also throughout the rest of the paper, $\mathbf{e}_1:=(1,{\mathrm{i}})^\top \mbox{ and }\mathbf{e}_2:=(1,-{\mathrm{i}})^\top$. \end{lem}
\begin{rem}
In view of \eqref{eq:u}, we have
\begin{align} u_1 \left(\bmf{x}\right)= \sum_{m=0} ^\infty & \Big [ \frac{k_p}{2} a_m \left(\mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-1} \left(k_p r\right) - \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+1} \left(k_p r\right) \right)\nonumber \\ & + \frac{{\mathrm{i}} k_s}{2} b_m \left(\mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-1} \left(k_s r\right) + \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+1} \left(k_s r\right) \right) \Big ],\label{eq:u1}\\ u_2 \left(\bmf{x}\right)= \sum_{m=0} ^\infty & \Big [ \frac{{\mathrm{i}} k_p}{2} a_m \left( \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-1} \left(k_p r\right) + \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+1} \left(k_p r\right) \right)\nonumber \\ & + \frac{ k_s}{2} b_m \left(-\mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-1} \left(k_s r\right) + \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+1} \left(k_s r\right) \right) \Big ].\label{eq:u2} \end{align} \end{rem}
By the analyticity of $\bmf{u}$ in the interior domain of $\Omega$ and strong continuation principle, we have the following proposition. \begin{prop}\label{prop:1} Suppose $\mathbf{0}\in \Omega$ and $\bmf{u}$ has the expansion \eqref{eq:u} around the origin such that $a_m=b_m=0$ for $\forall m \in \mathbb{N}\cup \{0\}$. Then $$ \bmf{u} \equiv \bmf{0} \mbox{ in } \Omega . $$ \end{prop}
The corresponding Fourier representation of the boundary traction operator $T_{\bmf{\nu }}\mathbf u \Big |_{\Gamma^\pm_h }$ defined in \eqref{eq:Tu} can be found in the following lemma.
\begin{lem}\cite[Lemma 2.4]{DLW}\label{lem:Tu1 exp}
Let $\bmf{u}(\bmf{x})$ be a Lam\'e eigenfunction to \eqref{eq:lame} with the Fourier expansion \eqref{eq:u} and $\Gamma^\pm_h$ is defined in \eqref{eq:gamma_pm}. Then $ T_\nu \mathbf{u}\Big |_{\Gamma^+_h}$ possesses the following radial wave expansion at the origin \begin{equation}\label{eq:Tu1}
\begin{aligned}
T_\nu \mathbf{u}\Big |_{\Gamma^+_h} & = \sum_{m=0}^{\infty} \left\{\frac{{\mathrm{i}} k_p^2}{2} a_{m} \Big[ \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi_0} \mu J_{m-2}(k_p r) \mathbf{e}_1 + \mathrm{e}^{{\mathrm{i}} (m -1) \varphi_0} (\lambda+\mu) J_m(k_p r) \mathbf{e}_1\right.\\
& - \mathrm{e}^{{\mathrm{i}} (m+1) \varphi_0} \mu J_{m+2}\left(k_{p} r\right) \mathbf{e}_2 - \mathrm{e}^{{\mathrm{i}}( m +1)\varphi_0} \left(\lambda+\mu\right) J_m\left(k_{p} r\right) \mathbf{e}_2\Big] \\
&- \frac{k_s^2}{2} b_{m}\Big[ \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi_0} \mu J_{m-2}\left(k_{s} r\right) \mathbf{e}_1 - \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi_0} \mu J_{m+2}\left(k_{s} r\right) \mathbf{e}_2\Big]
\bigg\}.
\end{aligned} \end{equation}
Similarly, the radial wave expansion of $ T_\nu \mathbf{u}\Big |_{\Gamma^-_h}$ at the origin is given by \begin{equation}\label{eq:Tu2}
\begin{split}
T_\nu \mathbf{u}\Big |_{\Gamma^-_h}&= \sum_{m=0}^{\infty} \left\{-\frac{{\mathrm{i}} k_p^2}{2} a_{m} \mu J_{m-2}(k_p r) \mathbf{e}_1 - \frac{{\mathrm{i}} k_p^2}{2} a_{m} (\lambda+\mu) J_m(k_p r) \mathbf{e}_1\right.\\
& + \frac{k_s^2}{2} b_{m} \mu J_{m-2}\left(k_{s} r\right) \mathbf{e}_1 + \frac{{\mathrm{i}} k_p^2}{2} a_{m} \left(\lambda+\mu\right) J_m\left(k_{p} r\right) \mathbf{e}_2 \\
&+ \frac{{\mathrm{i}} k_p^2}{2} a_{m} \mu J_{m+2}\left(k_{p} r\right) \mathbf{e}_2 + \frac{k_s^2}{2} b_{m} \mu J_{m+2}\left(k_{s} r\right) \mathbf{e}_2
\bigg\}.
\end{split} \end{equation} \end{lem}
The explicit Fourier expansions of $\mathcal B_3(\mathbf u)$ and $\mathcal B_4(\mathbf u)$ defined in \eqref{eq:tf} are given in the following lemma. \begin{lem}
Let the unit normal vector $\nu $ and the tangential vector $\boldsymbol{\tau}$ to $\Gamma_h$ are defined in \eqref{eq:nutau}. Suppose that $\bmf{u}(\bmf{x})$ is a Lam\'e eigenfunction of \eqref{eq:lame}, $\mathcal B_3(\mathbf u)$ and $\mathcal B_4(\mathbf u)$
are defined in \eqref{eq:tf}. We have the Fourier expansions of $\mathcal B_3(\mathbf u)$ as
\begin{equation}\label{eq:third3}
\begin{aligned}
\mathcal B_3(\mathbf u)|_{\Gamma_h^-} &= \sum_{m=0}^{\infty} \bigg\{ \left[- \frac{{\mathrm{i}} k_p}{2} a_{m} \big (J_{m-1}(k_p r) + J_{m+1}(k_p r)\big )\right.\\
& + \frac{k_s}{2} b_m \big (J_{m-1}(k_s r) - J_{m+1}(k_s r) \big )\bigg]\hat{\mathbf{e}}_1 + \left[ \frac{{\mathrm{i}} k_p^2}{2} a_{m} \big (-J_{m-2}(k_p r)\right.\\
& + J_{m+2}(k_p r) \big )
+ \frac{k_s^2}{2} b_m \big (J_{m-2}(k_s r) + J_{m+2}(k_s r) \big )\bigg] \mu \hat{\mathbf{e}}_2 \bigg\},
\end{aligned} \end{equation} \vspace*{-2mm}
\begin{equation}\label{eq:third2}
\begin{aligned}
\mathcal B_3(\mathbf u)|_{\Gamma_h^+} &= \sum_{m=0}^{\infty} \mathrm{ e}^{{\mathrm{i}} m \varphi_0} \bigg\{ \left[ \frac{{\mathrm{i}} k_p}{2} a_{m} \big (J_{m-1}(k_p r) + J_{m+1}(k_p r)\big )\right.\\
& + \frac{k_s}{2} b_m \big (-J_{m-1}(k_s r) + J_{m+1}(k_s r) \big )\bigg]\hat{\mathbf{e}}_1 + \left[ \frac{{\mathrm{i}} k_p^2}{2} a_{m} \big (-J_{m-2}(k_p r)\right.\\
& + J_{m+2}(k_p r) \big )
+ \frac{k_s^2}{2} b_m \big (J_{m-2}(k_s r) + J_{m+2}(k_s r) \big )\bigg] \mu \hat{\mathbf{e}}_2 \bigg\},
\end{aligned} \end{equation} where and also throughout the rest of the paper, $\hat{\mathbf{e}}_1:=(0,1)^\top $ and $\hat{\mathbf{e}}_2:=(1,0)^\top $. Furthermore, it holds that \begin{equation}\label{eq:forth3}
\begin{aligned}
\mathcal B_4(\mathbf u)\big |_{\Gamma_h^-} = & \sum_{m=0}^{\infty} \bigg\{ \left[ \frac{ k_p}{2} a_{m} ( J_{m-1}(k_p r) - J_{m+1}(k_p r) )\right.\\
& + \frac{{\mathrm{i}} k_s}{2} b_m (J_{m-1}(k_s r) + J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 \\
& + \left[- \frac{ k_p^2}{2} a_{m} (J_{m-2}(k_p r)\mu + 2(\lambda+\mu) J_m(k_p r)+ J_{m+2}(k_p r) \mu)\right.\\
& - \frac{{\mathrm{i}} k_s^2}{2} b_m (J_{m-2}(k_s r) - J_{m+2}(k_s r) )\mu \bigg] \hat{\mathbf{e}}_2 \bigg\}.
\end{aligned}
\vspace*{-1mm} \end{equation} \begin{equation}\label{eq:forth2}
\begin{aligned}
\mathcal B_4(\mathbf u)\big |_{\Gamma_h^+} = & \sum_{m=0}^{\infty} \mathrm{ e}^{{\mathrm{i}} m \varphi_0} \bigg\{ \left[ \frac{ k_p}{2} a_{m} ( - J_{m-1}(k_p r) + J_{m+1}(k_p r) )\right.\\
& - \frac{{\mathrm{i}} k_s}{2} b_m (J_{m-1}(k_s r) + J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 \\
& + \left[- \frac{ k_p^2}{2} a_{m} (J_{m-2}(k_p r)\mu + 2(\lambda+\mu) J_m(k_p r)+ J_{m+2}(k_p r) \mu)\right.\\
& - \frac{{\mathrm{i}} k_s^2}{2} b_m (J_{m-2}(k_s r) - J_{m+2}(k_s r) )\mu \bigg] \hat{\mathbf{e}}_2 \bigg\}.
\end{aligned} \end{equation} \end{lem}
\begin{proof} On $\Gamma_h^+$, we have ${\nu}=(-\sin \varphi_0,\cos \varphi_0)^\top$ and $\boldsymbol{\tau}=(-\cos \varphi_0,-\sin \varphi_0)^\top$. By virtue of \eqref{eq:tf}, \eqref{eq:u} and \eqref{eq:Tu1}, as well as noting $$ \nu \cdot \mathbf e_1={\mathrm{i}} {\mathrm e}^{{\mathrm{i}} \varphi_0},\quad \nu \cdot \mathbf e_2=-{\mathrm{i}} {\mathrm e}^{-{\mathrm{i}} \varphi_0},\quad \boldsymbol{\tau} \cdot \mathbf e_1=- {\mathrm e}^{{\mathrm{i}} \varphi_0},\quad \boldsymbol{\tau} \cdot \mathbf e_2=- {\mathrm e}^{-{\mathrm{i}} \varphi_0}, $$ and by direct calculations, we can obtain \eqref{eq:third2} and \eqref{eq:forth2}. Similarly, one can derive \eqref{eq:third3} and \eqref{eq:forth3}. \end{proof}
\begin{lem}{\cite{CDLZ}}\label{lem:co exp} Suppose that for $0<h \ll 1$ and $t \in (0,h)$,\[\sum_{n=0}^{\infty}\alpha_{n}J_{n}(t)=0,\] where $J_{n}\left(t\right)$ is the n-th Bessel function of the first kind. Then \[\alpha_{n}=0,\quad n=0,1,2,\ldots \] \end{lem}
We are now in a position to present the summary of the main GHP results. Henceforth, if $\Gamma_h^+ $ is an impedance line or a generalized-impedance line associated with the parameter $\boldsymbol{\eta}_1$, according to Definition~\ref{def:class1}, we assume that $\boldsymbol{ \eta}_1$ is given by the following absolutely convergent series at $\mathbf 0\in \Gamma_h^+$: \begin{equation}\label{eq:eta1 ex}
\boldsymbol{\eta}_1=\eta_1+\sum_{j=1}^\infty \eta_{1,j} r^j,
\end{equation}
where $\eta_{1}\in\mathbb{C}\backslash\{0\}$, $\eta_{1,j}\in \mathbb C$ and $r\in [0,h]$. Similarly, when $\Gamma_h^-$ is an impedance line or a generalized-impedance line associated with the parameter $\boldsymbol{\eta}_2$, we suppose that $\boldsymbol{\eta}_2$ is given by the following absolutely convergent series at $\mathbf 0\in \Gamma_h^+$: \begin{equation}\label{eq:eta2 ex}
\boldsymbol{\eta}_2=\eta_2+\sum_{j=1}^\infty \eta_{2,j} r^j,
\end{equation}
where $\eta_{2}\in\mathbb{C}\backslash\{0\}$, $\eta_{2,j}\in \mathbb C$ and $r\in [0,h]$.
We first present four theorems on the GHP where only $\mathcal{B}_1, \mathcal{B}_2$ and $\mathcal{B}_5$ are involved. They are actually extended from our paper \cite{DLW}, where the boundary impedance parameters were always assumed to be constant, but in this paper, they can be variable functions as introduced in Definition~\ref{def:class1}.
\begin{thm}\label{thm:impedance line} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exits a singular impedance line $\Gamma_h\subset\Omega$ of $\bmf{u}$ with an impedance parameter $\boldsymbol{ \eta} \in \mathcal{A}(\Gamma_h ) $, then $\bmf{u}\equiv \bmf{ 0}$. \end{thm}
\begin{thm}\label{thm:54} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a rigid line and $\Gamma_h^+$ is an impedance line with the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, where the associated impedance parameter satisfies \eqref{eq:eta1 ex}, then $\bmf{u}\equiv \bmf{0}$.
\end{thm}
\begin{thm}\label{thm:55} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. Suppose there exit two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a traction-free line and $\Gamma_h^+$ is an impedance line associated with an impedance parameter $\boldsymbol{\eta}_2$ satisfying \eqref{eq:eta1 ex},
with the property that $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $ and $\bmf{u}$ vanishes at the intersecting point, namely $\bmf{u}(\bmf{0})=\bmf{0}$, then $\bmf{u}\equiv \bmf{0}$.
\end{thm}
\begin{thm}\label{eq:impedance line exp} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. Suppose there exist two impedance lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi$ satisfying that \begin{equation}\label{eq:lem53 cond}
\eta_1 \mathrm{e}^{-{\mathrm{i}} \varphi_0}+ \eta_2 \neq 0 \end{equation} is fulfilled and $\bmf{u}$ vanishes at the intersecting point, i.e. $\bmf{u}(\bmf{0})=\bmf{0}$, then $\bmf{u}\equiv \bmf{0}$. Here, $T_\nu \bmf{u}+\boldsymbol{ \eta}_1\bmf{u}=\mathbf{0}$ on $\Gamma_h^+$, $T_\nu \bmf{u}+\boldsymbol{ \eta}_2\bmf{u}=\mathbf{0}$ on $\Gamma_h^-$ with $\boldsymbol{ \eta}_1 \in \mathcal A(\Gamma_h^+ )$
and $\boldsymbol{ \eta}_2 \in \mathcal A(\Gamma_h^- )$.
\end{thm}
In addition to those from Theorems~\ref{thm:impedance line}--\ref{eq:impedance line exp}, there are two more results from \cite{DLW} stating that the existence of a singular rigid or traction-free line also ensures $\mathbf{u}\equiv\mathbf{0}$. Two remarks are in order. First, as mentioned earlier, those theorems are extended and generalised from our earlier paper \cite{DLW}. The proofs of these theorems in such as a generalised situation can be done by following the new strategies developed in this paper for the more general cases together with the arguments from \cite{DLW}. In order to save the space of the paper, we shall skip the corresponding proofs. Second, as can be seen from Theorems~\ref{thm:impedance line}--\ref{eq:impedance line exp}, if there are two line segments involved, we always assume that they are not collinear. Indeed, as discussed earlier, in the degenerate case where the two lines are collinear, the corresponding study is either reduced to the single line case or the standard HUP. This remark also applies to our subsequent study for the other cases.
By including the additional boundary trace conditions $\mathcal{B}_3, \mathcal{B}_4$ and $\mathcal{B}_6$ into the study, we can establish the GHP results in a similar flavour to Theorems~\ref{thm:impedance line}--\ref{eq:impedance line exp}. We summarize the corresponding findings in Table \ref{table:main results}, where the first column lists the presence of one or two homogeneous line segments, the second column lists the required additional conditions and the third column is the GHP conclusions.
\begin{table} [!htbp]
\centering
{
\begin{tabular}{|c|c|c|}
\hline
&\tabincell{c}{ Assumptions} & Conclusions\\
\hline
$\Gamma_h \in {\mathcal S} (\mathcal R^\kappa_\Omega )$ & / & $\mathbf u\equiv \mathbf 0$ \\
\hline
$\Gamma_h \in {\mathcal S} (\mathcal T^\kappa_\Omega )$ &/ & $\mathbf u\equiv \mathbf 0$ \\
\hline
$\Gamma_h \in {\mathcal S} (\mathcal I^\kappa_\Omega )$&/& $\mathbf u\equiv \mathbf 0$ \\
\hline
$\Gamma_h \in {\mathcal S} (\mathcal G^\kappa_\Omega )$&/ & $\mathbf u\equiv \mathbf 0$ \\
\hline
$\Gamma_h \in {\mathcal S} (\mathcal F^\kappa_\Omega )$&/ & $\mathbf u\equiv \mathbf 0$ \\
\hline
$ \Gamma_h^+ \in \mathcal H^\kappa_\Omega $& \tabincell{c}{
${\nu}^\top \nabla \bmf{u} {\nu} |_{\bmf{x}={\bmf{0}} }= \boldsymbol{\tau}^\top \nabla \bmf{u} {\nu} |_{\bmf{x}={\bmf{0}} }=0$; \\ $b_0=b_1=b_2=0$;\\ $\eta_1 \neq \pm {\mathrm{i}} \mbox{ and } \frac{\pm \sqrt{(\lambda + 3\mu)(\lambda + \mu)} - \mu {\mathrm{i}}}{\lambda + 2 \mu}$ }& $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^+,\Gamma_h^- \in \mathcal R^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $& $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^+,\Gamma_h^- \in \mathcal T^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ and $\mathbf u(\mathbf 0)=\mathbf 0$& $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal R^\kappa_\Omega,\Gamma_h^+ \in \mathcal T^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal R^\kappa_\Omega,\Gamma_h^+ \in \mathcal I^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal I^\kappa_\Omega,\Gamma_h^+ \in \mathcal I^\kappa_\Omega $& \tabincell{c}{$\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, $\mathbf u(\mathbf 0)=\mathbf 0$; \\ $\eta_1 \mathrm{e}^{-{\mathrm{i}} \varphi_0}+ \eta_2 \neq 0$ } & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
R^\kappa_\Omega,\Gamma_h^+ \in \mathcal G^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
T^\kappa_\Omega,\Gamma_h^+ \in \mathcal G^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
I^\kappa_\Omega,\Gamma_h^+ \in \mathcal G^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
R^\kappa_\Omega,\Gamma_h^+ \in \mathcal F^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
I^\kappa_\Omega,\Gamma_h^+ \in \mathcal F^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
I^\kappa_\Omega,\Gamma_h^+ \in \mathcal F^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
R^\kappa_\Omega,\Gamma_h^+ \in \mathcal H^\kappa_\Omega $&\tabincell{c}{$\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $; either $\eta_1\neq -\frac{{\mathrm{i}} \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}}{\lambda+\mu(1+\mathrm{e}^{2 {\mathrm{i}} \varphi_0})}$ \\ or $a_0= 0, \quad \eta_1\neq \mathrm i$} & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
T^\kappa_\Omega,\Gamma_h^+ \in \mathcal H^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ and $\eta_1 \neq \mathrm i$ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
I^\kappa_\Omega,\Gamma_h^+ \in \mathcal H^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ and $\eta_2 \neq \mathrm i$ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
G^\kappa_\Omega,\Gamma_h^+ \in \mathcal F^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
G^\kappa_\Omega,\Gamma_h^+ \in \mathcal H^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
H^\kappa_\Omega,\Gamma_h^+ \in \mathcal H^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $, $\eta_1=\eta_2=\eta\neq \pm {\mathrm{i}},-\frac{{\mathrm{i}} m}{m+2}$ for $\forall m\in \mathbb N$ & $\mathbf u\equiv \mathbf 0$\\
\hline
$\Gamma_h^-\in \mathcal
H^\kappa_\Omega,\Gamma_h^+ \in \mathcal F^\kappa_\Omega $& $\angle(\Gamma_h^+,\Gamma_h^-)\neq \pi $ & $\mathbf u\equiv \mathbf 0$\\
\hline
\end{tabular}
}
\caption{}
\label{table:main results} \end{table}
\section{GHP with the presence of a single singular line}\label{sect:3}
In this section, we prove that if $\Omega$ contains a singular soft-clamped, simply-supported or generalized-impedance line $\Gamma_h$ as introduced in Definition~\ref{def:2} of a generic Lam\'e eigenfunction $\bmf{u}$ to \eqref{eq:lame}, then $\bmf{u}$ is identically zero. According to our discussion made at the beginning of Section~\ref{sect:2}, we can assume that the point $\bmf{x}_0$ involved in Definition~\ref{def:2} is the origin, namely $\bmf{x}_0=\bmf{0}$. It is clear that the unit normal vectors $ \nu $ to $\Gamma_h^+$ as defined in \eqref{eq:gamma_pm} is $\pm (-\sin \varphi_0,\cos \varphi_0)^\top$. In this paper, we choose $\nu=(-\sin \varphi_0,\cos \varphi_0)^\top $. In such a case, the following conditions involved in Definition~\ref{def:2} \begin{equation}\notag
{\nu}^\top \nabla \bmf{u} {\nu} |_{\bmf{x}={\bmf{x}}_0 }=0 ,\, \boldsymbol{\tau}^\top \nabla \bmf{u} {\nu} |_{\bmf{x}={\bmf{x}}_0 }=0 \end{equation} turn out to be \begin{equation}\label{eq:32 cond}
{\nu}^\top \nabla \bmf{u} \boldsymbol{\nu}|_{ \mathbf x=\bmf{0} }=0 , \, \boldsymbol{\tau}^\top \nabla \bmf{u} {\nu} |_{ \mathbf x=\bmf{0} }=0 , \end{equation} where $\boldsymbol{\nu}=(-\sin {\varphi_0},\cos{\varphi_0})^\top$ and $\boldsymbol{\tau}=(-\cos {\varphi_0},-\sin{\varphi_0})^\top$.
Similar to Definition \ref{def:2}, we introduce the {\it singular generalized-impedance line} of $\mathbf u$ in the following definition. \begin{defn}\label{def:3} Let $\mathbf{0}\in \Gamma_h$ be a generalized-impedance line of $\mathbf u$ defined in Definition \ref{def:1}, where $\mathbf u$ has the Fourier expansion \eqref{eq:u} at the origin $\mathbf 0$ with the coefficients $a_\ell$ and $b_\ell$, $\ell \in \mathbb N \cup \{0\}$. If the Fourier coefficients satisfy \begin{equation}\label{eq:def3}
b_0=b_1=b_2=0, \end{equation} and \begin{equation}\label{eq:sg cond}
{\nu}^\top \nabla \bmf{u} {\nu}|_{ \mathbf x=\bmf{0} }=\boldsymbol{\tau}^\top \nabla \bmf{u} {\nu} |_{ \mathbf x=\bmf{0} }=0 , \end{equation}
we call $\Gamma_h$ a {\it singular generalized-impedance line} of $\mathbf u$. The set of the singular generalized-impedance line is denoted by $ {\mathcal S} \left({\mathcal H}_\Omega^{\kappa}\right).$ \end{defn}
In the following lemma, we characterize the algebraic relationship between the Fourier coefficients of \eqref{eq:u} from \eqref{eq:32 cond}, whose proof is postponed to the Appendix.
\begin{lem}\label{lem:condition} Let $\mathbf{u}$ be a Lam\'e eigenfunction of \eqref{eq:lame}, where $\mathbf{u}$ has the expansion \eqref{eq:u} around the origin. Consider the condition \eqref{eq:32 cond} on $\Gamma_h^+$ with ${\nu}=(-\sin {\varphi_0},\cos{\varphi_0})^\top$ and $\boldsymbol{\tau}=(-\cos {\varphi_0},-\sin{\varphi_0})^\top$. Then \begin{subequations}
\begin{align}
{\nu}^\top \nabla \bmf{u} {\nu} |_{ \mathbf x=\bmf{0} }=0\ \ &\mbox{ implies }\ \ 2 k_p^2 a_0 + \mathrm e^{ 2 {\mathrm{i}} \varphi_0}(k_p^2 a_2 + {\mathrm{i}} k_s^2 b_2)=0,\label{eq:gradient1} \\
\boldsymbol{\tau}^\top \nabla \bmf{u} {\nu}|_{ \mathbf x=\bmf{0} }=0\ \ &\mbox{ implies }\ \ 2 {\mathrm{i}} k_s^2 b_0 + \mathrm e^{ 2 {\mathrm{i}} \varphi_0}(k_p^2 a_2 + {\mathrm{i}} k_s^2 b_2)=0. \label{eq:gradient2}
\end{align} \end{subequations}
\end{lem}
\subsection{The case with a singular soft-clamped line} \begin{lem}\label{lem:third} Let $\mathbf{u}$ be a Lam\'e eigenfunction of \eqref{eq:lame}, where $\mathbf{u}$ has the expansion \eqref{eq:u} around the origin. Suppose that $\Gamma_h^+$ defined in \eqref{eq:gamma_pm} such that $\Gamma_h^+\in {\mathcal G}_\Omega^{\kappa}$. Then we have \begin{equation}\label{eq:1}
{\mathrm{i}} k_p a_1 - k_s b_1=0,\quad
{\mathrm{i}} k_p^2 a_2 - k_s^2 b_2 = 0, \end{equation} and \begin{equation}\label{eq:2}
2 k_s ^2 b_0+ \mathrm{e}^{2 {\mathrm{i}} \varphi_0} ({\mathrm{i}} k_p^2 a_2- k_s^2 b_2)=0,\quad
{\mathrm{i}} k_p ^3 a_1 - k_s^3 b_1 - \mathrm{e}^{2 {\mathrm{i}} \varphi_0}({\mathrm{i}} k_p ^3 a_3 - k_s^3 b_3)=0. \end{equation} Moreover, it holds that \begin{equation}\label{eq:a1b1}
b_0=a_1=b_1=0. \end{equation} Furthermore, suppose that \begin{equation}\label{eq:aLbL}
a_\ell=b_\ell=0 \end{equation} where $\ell=0,\ldots,m-1$ and $m\in {\mathbb N}$ with $m\geq 3 $, then \begin{align}
\label{eq:lem31}
a_\ell=b_\ell=0, \quad \forall \ell \in \mathbb N \cup \{0\}.
\end{align} \end{lem}
\begin{proof} Since $\Gamma_h^+$ is a soft-clamped line of $\bmf{u}$, by virtue of \eqref{eq:third2}, we have \begin{equation}\label{eq:third}
\begin{aligned} \bmf{0} = & \sum_{m=0}^{\infty} \mathrm{ e}^{{\mathrm{i}} m \varphi_0} \bigg\{ \left[ \frac{{\mathrm{i}} k_p}{2} a_{m} (J_{m-1}(k_p r) + J_{m+1}(k_p r) )\right.\\
& + \frac{k_s}{2} b_m (-J_{m-1}(k_s r) + J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 + \left[ \frac{{\mathrm{i}} k_p^2}{2} a_{m} (-J_{m-2}(k_p r) + J_{m+2}(k_p r) )\right.\\
& + \frac{k_s^2}{2} b_m (J_{m-2}(k_s r) + J_{m+2}(k_s r) )\bigg] \mu \hat{\mathbf{e}}_2 \bigg\},\quad 0 \leqslant r\leqslant h,
\end{aligned} \end{equation} where and also throughout the rest of the paper, $\hat{\mathbf{e}}_1=(0,1)^\top $, $\hat{\mathbf{e}}_2=(1,0)^\top $. Noting $J_{-m}(t)=(-1)^m J_{m}(t)$ (cf. \cite{Abr}), we obtain that \begin{equation}\label{eq:Jmfact}
J_{-m}(k_pr)=-J_{m}(k_p r), \quad J_{-m }(k_s r)=-J_{m}(k_s r),\quad m =1,2. \end{equation}
Using Lemma \ref{lem:co exp} and \eqref{eq:Jmfact}, comparing the coefficient of the term $r^0$ in both sides of \eqref{eq:third}, it holds that $$ ({\mathrm{i}} k_p a_1 - k_s b_1) \hat{\mathbf{e}}_1 + ( - {\mathrm{i}} k_p^2 a_2 + k_s^2 b_2) \mu \mathrm{e}^{ {\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_2= \bmf{0}. $$ Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linear independent, we obtain \eqref{eq:1}. Similarly, from Lemma \ref{lem:co exp}, we compare the coefficient of the term $r^1$ in both sides of \eqref{eq:third}, then it holds that $$ \big[2 k_s^2 b_0 + ({\mathrm{i}} k_p^2 a_2 - k_s^2 b_2) \mathrm{e}^{2 {\mathrm{i}} \varphi_0} \big] \hat{\mathbf{e}}_1 + \big[ ( {\mathrm{i}} k_p^3 a_1 - k_s^3 b_1)-( {\mathrm{i}} k_p^3 a_1 - k_s^3 b_1) \mu \mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}\big] \mu \mathrm{e}^{ {\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_2 = \bmf{0}. $$ Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linear independent, we readily obtain \eqref{eq:2}.
It can be seen that $b_0=0$ by utilizing the second equation of \eqref{eq:1} and the first equation of \eqref{eq:2}.
Next we compare the coefficient of the term $r^2$ in both sides of \eqref{eq:third} and can conclue that \begin{equation}\nonumber \begin{aligned} & \big[( {\mathrm{i}} k_p^3 a_1 + 3 k_s^3 b_1) + ( {\mathrm{i}} k_p^3 a_3 - k_s^3 b_3) \mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}\big] \hat{\mathbf{e}}_1+ \big[( 2 {\mathrm{i}} k_p^4 a_2 - 2 k_s^4 b_2)\\ &\quad - ( {\mathrm{i}} k_p^4 a_4 - k_s^4 b_4) \mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}\big] \mu \mathrm{e}^{ {\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_2= \bmf{0}. \end{aligned} \end{equation} Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linear independent, we obtain \begin{equation}\label{eq:3g} \bigg\{
\begin{array}{l}
{\mathrm{i}} k_p^3 a_1 + 3 k_s^3 b_1 + ( {\mathrm{i}} k_p^3 a_3 - k_s^3 b_3) \mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}=0,\\
2 {\mathrm{i}} k_p^4 a_2 - 2 k_s^4 b_2 - ( {\mathrm{i}} k_p^4 a_4 - k_s^4 b_4) \mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}=0.
\end{array} \end{equation}
Substituting the second equation of \eqref{eq:2} into the first equation of \eqref{eq:3g}, one has
\begin{equation}\label{eq:4g}
{\mathrm{i}} k_p^3 a_1 + k_s^3 b_1 = 0.
\end{equation} By the first equation of \eqref{eq:1} and \eqref{eq:4g}, it yields that \begin{equation}\label{eq:3}
{\mathrm{i}} k_p a_1 - k_s b_1=0,\quad
{\mathrm{i}} k_p^3 a_1- k_s^3 b_1=0. \end{equation} Since the determinant of the coefficient matrix of \eqref{eq:3} is $$
\left| \begin{matrix}
{\mathrm{i}} k_p & - k_s\\
{\mathrm{i}} k_p^3 & - k_s^3
\end{matrix} \right|={\mathrm{i}} k_p k_s (k_p^2 - k_s^2) \neq 0, $$ we can readily prove \eqref{eq:a1b1}.
Now we suppose that \eqref{eq:aLbL} holds. Substituting \eqref{eq:aLbL} into \eqref{eq:third}, it yields that \begin{equation}\label{eq:third1}
\begin{aligned}
\bmf{0} = & \sum_{m=\ell}^{\infty} \mathrm{ e}^{{\mathrm{i}} m \varphi_0} \bigg\{ \left[ \frac{{\mathrm{i}} k_p}{2} a_{m} (J_{m-1}(k_p r) + J_{m+1}(k_p r) )\right.\\
& + \frac{k_s}{2} b_m (-J_{m-1}(k_s r) + J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 + \left[ \frac{{\mathrm{i}} k_p^2}{2} a_{m} (-J_{m-2}(k_p r) + J_{m+2}(k_p r) )\right.\\
& + \frac{k_s^2}{2} b_m (J_{m-2}(k_s r) + J_{m+2}(k_s r) )\bigg] \mu \hat{\mathbf{e}}_2 \bigg\},\quad 0 \leqslant r\leqslant h.
\end{aligned} \end{equation} The lowest order of the power with respect to $r$ in the right hand of \eqref{eq:third1} is $m-1$. Comparing the coefficients of the terms $r^{m-1}$ in both sides of \eqref{eq:third1}, it holds that $$
( {\mathrm{i}} k_p^m a_m - k_s^m b_m ) \hat{\mathbf{e}}_1 + ( - {\mathrm{i}} k_p^{m+1} a_{m+1} + k_s^{m+1} b_{m+1}) \mathrm{e}^{ {\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_2= \bmf{0}. $$ Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent, we obtain that \begin{equation}\label{eq:4}
{\mathrm{i}} k_p^m a_m - k_s^m b_m=0,\quad
{\mathrm{i}} k_p^{m+1} a_{m+1} - k_s^{m+1} b_{m+1}=0. \end{equation} Comparing the coefficients of the terms $r^m$ in both sides of \eqref{eq:third1}, it holds that \begin{equation}\nonumber \begin{aligned} & ({\mathrm{i}} k_p^{m+1} a_{m+1} - k_s^{m+1} b_{m+1}) \mathrm{e}^{{\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_1\\ &\quad +\left({\mathrm{i}} m k_p^{m+2} a_m - m k_s^{m+2} b_m - ({\mathrm{i}} k_p^{m+2} a_{m+2} - k_s^{m+2} b_{m+2}) \mathrm{e}^{2 {\mathrm{i}} \varphi_0}\right)\mu \hat{\mathbf{e}}_2= \bmf{0}, \end{aligned} \end{equation} which further gives that
\begin{equation}\label{eq:5}
\bigg\{ \begin{array}{l} {\mathrm{i}} k_p^{m+1} a_{m+1} - k_s^{m+1} b_{m+1}=0,\\
{\mathrm{i}} m k_p^{m+2} a_m - m k_s^{m+2} b_m - ( {\mathrm{i}} k_p^{m+2} a_{m+2} - k_s^{m+2} b_{m+2})\mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0.
\end{array} \end{equation} Again comparing the coefficient of the term $r^{m+1}$ in both sides of \eqref{eq:third1}, then it holds that \begin{equation}\nonumber \begin{aligned} & \left( -{\mathrm{i}} m k_p^{m+2} a_m + (m+2) k_s^{m+2} b_m + ({\mathrm{i}} k_p^{m+2} a_{m+2} - k_s^{m+2} b_{m+2}) \mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}\right) \hat{\mathbf{e}}_1\\ & + \left( {\mathrm{i}} (m+1) k_p^{m+3} a_{m+1} -(m+1) m k_s^{m+3} b_{m+1} - ( {\mathrm{i}} k_p^{m+3} a_{m+3} - k_s^{m+3} b_{m+3}) \mathrm{e}^{2 {\mathrm{i}} \varphi_0} \right) \mu \mathrm{e}^{ {\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_2 \\ & = \bmf{0}. \end{aligned} \end{equation}
Therefore one has \begin{equation}\label{eq:6}
\bigg\{ \begin{array}{l} -{\mathrm{i}} m k_p^{m+2} a_m + (m+2) k_s^{m+2} b_m + ({\mathrm{i}} k_p^{m+2} a_{m+2} - k_s^{m+2} b_{m+2})\mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0,\\
{\mathrm{i}} (m+1) k_p^{m+3} a_{m+1} -(m+1) k_s^{m+3} b_{m+1} - ( {\mathrm{i}} k_p^{m+3} a_{m+3} - k_s^{m+3} b_{m+3})\mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0.
\end{array} \end{equation} Combining the second equation of \eqref{eq:5} with the first equation of \eqref{eq:6}, it can be derived that \begin{equation}\label{eq:7} b_m=0. \end{equation} Substituting \eqref{eq:7} into the first equation of \eqref{eq:4}, we can obtain \begin{equation}\label{eq:8} a_m=0. \end{equation} Repeating the above procedure for \eqref{eq:7} and \eqref{eq:8} step by step, we can prove that $a_\ell=b_\ell=0$ for $\ell=m+1,\ldots$. \end{proof}
\begin{thm}\label{thm:third} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exits a singular soft-clamped line $\Gamma_h \subset \Omega$ corresponding to $\mathbf u$, then $\bmf{u}\equiv \bmf{ 0}$. \end{thm}
\begin{proof}
Suppose that there exits a singular soft-clamped line $\Gamma_h$ of $\mathbf u$ as described at the beginning of this section. Therefore we have ${\nu}^\top \nabla \bmf{u} {\nu}|_{ \mathbf x=\bmf{0} } =0$, which implies that \eqref{eq:gradient1} holds.
Since $\Gamma_h$ is a soft-clamped line, from Lemma \ref{lem:third} we know that
\begin{equation}\label{eq:9}
a_1=b_1=0,
\end{equation} and \eqref{eq:1} and \eqref{eq:2} hold. In view of the second equation of \eqref{eq:1} and \eqref{eq:gradient1}, one has \begin{equation}\label{eq:9z}
{\mathrm{i}} k_p^2 a_2 - k_s^2 b_2=0,\quad
2 {\mathrm{i}} k_s^2 a_0 + k_p^2 a_2 + {\mathrm{i}} k_s^2 b_2=0, \end{equation} which implies \begin{equation}\label{eq:10}
a_0=0. \end{equation} Substituting \eqref{eq:10} into the first equation of \eqref{eq:2}, it yields that ${\mathrm{i}} k_p^2 a_2 - k_s^2 b_2=0$, which can be furthered substituted into the second equation of \eqref{eq:1} for deriving $b_0=0$.
Since we have proved that $a_\ell=b_\ell=0$ for $\ell=0,1$, the lowest order with respect to the power of $r$ in \eqref{eq:third} is 2. Using Lemma \ref{lem:co exp}, comparing the coefficients of the terms $r^2$ in both sides of \eqref{eq:third}, one can show that $$ ({\mathrm{i}} k_p ^3 a_3 - k_s^3 b_3) \mathrm{e}^{{\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_1+\left(2 {\mathrm{i}} k_p^4 a_2- 2 k_s^4 b_2+(-{\mathrm{i}} k_p^4 a_4+k_s^4 b_4)\mathrm{e}^{2 {\mathrm{i}} \varphi_0} \right) \mu \hat{\mathbf{e}}_2= \bmf{0}. $$ Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent, we obtain \begin{equation}\label{eq:12} {\mathrm{i}} k_p ^3 a_3 - k_s^3 b_3=0,\quad
2 {\mathrm{i}} k_p^4 a_2- 2 k_s^4 b_2 - ({\mathrm{i}} k_p^4 a_4 - k_s^4 b_4)\mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0. \end{equation} Similarly, from Lemma \ref{lem:co exp}, we compare the coefficients of the terms $r^3$ in both sides of \eqref{eq:third}, and can conclude that \begin{equation}\nonumber \begin{aligned}
& \left( -2{\mathrm{i}} k_p^4 a_2 + 4 k_s^4 b_2 + ({\mathrm{i}} k_p^4 a_4 - k_s^4 b_4) \mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}\right) \hat{\mathbf{e}}_1\\
& + \left( 3 {\mathrm{i}} k_p^5 a_3 -3 k_s^5 b_3 - ( {\mathrm{i}} k_p^5 a_5 - k_s^5 b_5) \mathrm{e}^{2 {\mathrm{i}} \varphi_0} \right) \mu \mathrm{e}^{ {\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_2
= \bmf{0}.
\end{aligned} \end{equation} Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent, we obtain that \begin{equation}\label{eq:13}
\bigg\{ \begin{array}{l} -2{\mathrm{i}} k_p^4 a_2 + 4 k_s^4 b_2 + ({\mathrm{i}} k_p^4 a_4 - k_s^4 b_4) \mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}=0,\\
3 {\mathrm{i}} k_p^5 a_3 -3 k_s^5 b_3 - ( {\mathrm{i}} k_p^5 a_5 - k_s^5 b_5) \mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0.
\end{array} \end{equation} Substituting the second equation of \eqref{eq:12} into the first equation of \eqref{eq:13}, one can show that \begin{equation}\label{eq:14} b_2=0. \end{equation} Substituting \eqref{eq:14} into the second equation of \eqref{eq:1}, we can obtain $
a_2=0.
$
By now we have proved $a_\ell=b_\ell=0$ for $\ell=0,1,2$. According to Lemma \ref{lem:third}, we know that \eqref{eq:lem31} holds. Therefore, from Proposition \ref{prop:1}, we readily have that $\bmf{u}\equiv \bmf{0}$ in $\Omega$.
The proof is complete. \end{proof}
\subsection{The case with a singular simply-supported line} \begin{lem}\label{lem:forth} Let $\mathbf{u}$ be a Lam\'e eigenfunction of \eqref{eq:lame}, where $\mathbf{u}$ has the expansion \eqref{eq:u} around the origin. Suppose that $\Gamma_h^+$ is defined in \eqref{eq:gamma_pm} such that $\Gamma_h^+\in {\mathcal F}_\Omega^{\kappa}$. Then we have \begin{equation}\label{eq:16}
\bigg\{ \begin{array}{l} 2 (\lambda+\mu) k_p^2 a_0 +(k_p^2 a_2+ {\mathrm{i}} k_s^2 b_2) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0,\\
k_p a_1 + {\mathrm{i}} k_s b_1 = 0,
\end{array} \end{equation} and \begin{equation}\label{eq:17} \bigg\{
\begin{array}{l}
2 k_p^2 a_0 - (k_p^2 a_2+ {\mathrm{i}} k_s^2 b_2) \mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0\\
-(2 \lambda +\mu) k_p ^3 a_1 + {\mathrm{i}} k_s^3 b_1 \mu -(k_p^3 a_3+ {\mathrm{i}} k_s^3 b_3) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0.
\end{array} \end{equation} Moreover, it holds that \begin{equation}\label{eq:a0 forth}
a_0=0. \end{equation} Furthermore, suppose that \begin{equation}\label{eq:aLbL4}
a_\ell=b_\ell=0, \end{equation} where $\ell=0,\ldots,m-1$ and $m\in {\mathbb N}$ with $m\geq 3 $, then $$ a_\ell=b_\ell=0, \quad \forall \ell \in \mathbb N \cup \{0\}. $$ \end{lem}
\begin{proof} Since $\Gamma_h^+$ is a simply-supported line of $\bmf{u}$, we have from \eqref{eq:forth2} that \begin{equation}\label{eq:forth}
\begin{aligned}
\bmf{0} = & \sum_{m=0}^{\infty} \mathrm{ e}^{{\mathrm{i}} m \varphi_0} \bigg\{ \left[ \frac{ k_p}{2} a_{m} ( - J_{m-1}(k_p r) + J_{m+1}(k_p r) )\right.\\
& - \frac{{\mathrm{i}} k_s}{2} b_m (J_{m-1}(k_s r) + J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 \\
& + \left[- \frac{ k_p^2}{2} a_{m} (J_{m-2}(k_p r)\mu + 2(\lambda+\mu) J_m(k_p r)+ J_{m+2}(k_p r) \mu)\right.\\
& - \frac{{\mathrm{i}} k_s^2}{2} b_m (J_{m-2}(k_s r) - J_{m+2}(k_s r) )\mu \bigg] \hat{\mathbf{e}}_2 \bigg\},\quad 0 \leqslant r\leqslant h,
\end{aligned} \end{equation} where $\hat{\mathbf{e}}_1=(0,1)^\top $, $\hat{\mathbf{e}}_2=(1,0)^\top $. Using Lemma \ref{lem:co exp} and \eqref{eq:Jmfact}, and comparing the coefficients of the terms $r^0$ in both sides of \eqref{eq:forth}, we have that $$ ( k_p a_1 + {\mathrm{i}} k_s b_1) \mathrm{e}^{{\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_1 + \left[2 (\lambda+\mu) k_p^2 a_0 + ( k_p^2 a_2 + {\mathrm{i}} k_s^2 b_2) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0} \right] \hat{\mathbf{e}}_2= \bmf{0}, $$ Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent, we obtain \eqref{eq:16}. Similarly, from Lemma \ref{lem:co exp}, we compare the coefficients of the terms $r^1$ in both sides of \eqref{eq:forth}, and derive that \begin{equation}\nonumber \begin{aligned}
&\left[ 2 k_p^2 a_0 - (k_p^2 a_2 + {\mathrm{i}} k_s^2 b_2) \mathrm{e}^{2 {\mathrm{i}} \varphi_0}\right] \hat{\mathbf{e}}_1 \\
& + \left[-(2\lambda+\mu) k_p ^3 a_1 + {\mathrm{i}} k_s^3 \mu b_1 - (k_p^3 a_3 + {\mathrm{i}} k_s^3 b_3)\mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0} \right] \mathrm{e}^{ {\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_2= \bmf{0}. \end{aligned} \end{equation} Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent, one obtains \eqref{eq:17}.
Multiplying $\mu$ on both sides of \eqref{eq:17}, then adding it to \eqref{eq:16}, by \eqref{eq:convex}, one can directly obtain \eqref{eq:a0 forth}.
Next we assume that \eqref{eq:aLbL4} is valid. Substituting \eqref{eq:aLbL4} into \eqref{eq:forth}, it yields that \begin{equation}\label{eq:forth1}
\begin{aligned}
\bmf{0} = & \sum_{m=\ell}^{\infty} \mathrm{ e}^{{\mathrm{i}} m \varphi_0} \bigg\{ \left[ \frac{ k_p}{2} a_{m} ( - J_{m-1}(k_p r) + J_{m+1}(k_p r) )\right.\\
& - \frac{{\mathrm{i}} k_s}{2} b_m (J_{m-1}(k_s r) + J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 \\
& + \left[- \frac{ k_p^2}{2} a_{m} (J_{m-2}(k_p r)\mu + 2(\lambda+\mu) J_m(k_p r)+ J_{m+2}(k_p r) \mu)\right.\\
& - \frac{{\mathrm{i}} k_s^2}{2} b_m (J_{m-2}(k_s r) - J_{m+2}(k_s r) )\mu \bigg] \hat{\mathbf{e}}_2 \bigg\},\quad 0 \leqslant r\leqslant h.
\end{aligned} \end{equation} The lowest order of the power with respect to $r$ in the right hand of \eqref{eq:forth1} is $m-1$. Comparing the coefficients of the terms $r^{m-1}$ in both sides of \eqref{eq:forth1}, it holds that $$
-( k_p^m a_m + {\mathrm{i}} k_s^m b_m ) \hat{\mathbf{e}}_1 - ( k_p^{m+1} a_{m+1} + {\mathrm{i}} k_s^{m+1} b_{m+1}) \mu \mathrm{e}^{ {\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_2= \bmf{0}, $$ which readily gives that \begin{equation}\label{eq:18}
k_p^m a_m + {\mathrm{i}} k_s^m b_m=0,\quad
k_p^{m+1} a_{m+1} + {\mathrm{i}} k_s^{m+1} b_{m+1}=0. \end{equation} Comparing the coefficients of the terms $r^m$ in both sides of \eqref{eq:forth1}, it holds that \begin{equation}\nonumber \begin{aligned} & - (k_p^{m+1} a_{m+1} + {\mathrm{i}} k_s^{m+1} b_{m+1}) \mathrm{e}^{{\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_1+ \big(((m-2)\mu-2\lambda) k_p^{m+2} a_m \\
& + {\mathrm{i}} m \mu k_s^{m+2} b_m - ( k_p^{m+2} a_{m+2} + {\mathrm{i}} k_s^{m+2} b_{m+2}) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}\big) \hat{\mathbf{e}}_2= \bmf{0}. \end{aligned} \end{equation} Therefore we have \begin{equation}\label{eq:19}
\bigg\{ \begin{array}{l} k_p^{m+1} a_{m+1} + {\mathrm{i}} k_s^{m+1} b_{m+1}=0,\\
((m-2)\mu-2\lambda) k_p^{m+2} a_m + {\mathrm{i}} m \mu k_s^{m+2} b_m - ( k_p^{m+2} a_{m+2} + {\mathrm{i}} k_s^{m+2} b_{m+2}) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0,
\end{array} \end{equation} by using the fact that $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent. Finally comparing the coefficients of the terms $r^{m+1}$ in both sides of \eqref{eq:forth1}, it holds that \begin{equation}\nonumber \begin{aligned} & \left( (m+2) k_p^{m+2} a_m + {\mathrm{i}} m k_s^{m+2} b_m - ( k_p^{m+2} a_{m+2} + {\mathrm{i}} k_s^{m+2} b_{m+2}) \mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}\right) \hat{\mathbf{e}}_1\\ & + \big( (((m-1)\mu-2\lambda) k_p^{m+3} a_{m+1} + {\mathrm{i}} (m+1)\mu k_s^{m+3} b_{m+1}) \mathrm{e}^{ {\mathrm{i}} \varphi_0}\\ & - (k_p^{m+3} a_{m+3} + {\mathrm{i}} k_s^{m+3} b_{m+3}) \mathrm{e}^{3 {\mathrm{i}} \varphi_0} \mu \big)\hat{\mathbf{e}}_2 = \bmf{0}, \end{aligned} \end{equation}
which further yields that \begin{equation}\label{eq:20}
\Bigg \{ \begin{array}{l}(m+2) k_p^{m+2} a_m + {\mathrm{i}} m k_s^{m+2} b_m - ( k_p^{m+2} a_{m+2} + {\mathrm{i}} k_s^{m+2} b_{m+2}) \mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}=0,\\
\begin{aligned}
&((m-1)\mu-2\lambda) k_p^{m+3} a_{m+1} + {\mathrm{i}} (m+1) \mu k_s^{m+3} b_{m+1}\\
& - (k_p^{m+3} a_{m+3} + {\mathrm{i}} k_s^{m+3} b_{m+3}) \mathrm{e}^{2 {\mathrm{i}} \varphi_0} \mu=0.
\end{aligned}
\end{array} \end{equation}
We multiply the first equation of \eqref{eq:20} by $\mu$ and subtract it from the second equation of \eqref{eq:19}, which readily gives that \begin{equation} \notag
2(\lambda+2\mu )a_m=0. \end{equation} Therefore one has \begin{equation}\label{eq:am21}
a_m=0 \end{equation}
by noting \eqref{eq:convex}. Substituting \eqref{eq:am21} into the first equation of \eqref{eq:18}, we can obtain \begin{equation}\label{eq:22} b_m=0. \end{equation} Repeating the above procedure for deriving \eqref{eq:am21} and \eqref{eq:22}, we can prove that $a_\ell=b_\ell=0$ for $\ell=m+1,\ldots$.
The proof is complete. \end{proof}
\begin{thm}\label{thm:fourth}
Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exits a singular simply-supported line $\Gamma_h \subset \Omega$ corresponding to $\mathbf u$, then $\bmf{u}\equiv \bmf{ 0}$. \end{thm}
\begin{proof}
Suppose that there exits a singular soft-clamped line $\Gamma_h$ of $\mathbf u$ as described at the beginning of this section. Therefore we have $\boldsymbol{\tau}^\top \nabla \bmf{u} {\nu}|_{\mathbf x=\bmf{0} }=0$, which implies that \eqref{eq:gradient2} holds.
According to Lemma \ref{lem:forth}, \eqref{eq:16} and \eqref{eq:17} hold since $\Gamma_h$ is a soft-clamped line of $\mathbf u$. Substituting the first equation of \eqref{eq:16} into the first equation of \eqref{eq:17}, we have \begin{equation}\label{eq:23}
a_0=0. \end{equation} Substituting \eqref{eq:23} into \eqref{eq:17} yields that \begin{equation}\label{eq:a_2337}
k_p^2 a_2+{\mathrm{i}} k_s^2 b_2=0. \end{equation} Substituting \eqref{eq:a_2337} into \eqref{eq:gradient2}, we have \begin{equation}\label{eq:24}
b_0=0. \end{equation} Using Lemma \ref{lem:co exp}, \eqref{eq:23} and \eqref{eq:24}, and comparing the coefficients of the terms $r^2$ in both sides of \eqref{eq:forth}, it holds that \begin{equation}\nonumber \begin{aligned} & \big(3 k_p ^3 a_1 + {\mathrm{i}} k_s^3 b_1 - (k_p^3 a_3 + {\mathrm{i}} k_s^3 b_3)\mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}\big)\hat{\mathbf{e}}_1\\ & +\big((-2 \lambda k_p^4 a_2+ 2 {\mathrm{i}} \mu k_s^4 b_2) - ( k_p^4 a_4+ {\mathrm{i}} k_s^4 b_4) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0} \big) \mathrm{e}^{ {\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_2= \bmf{0} . \end{aligned} \end{equation} Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent, we obtain \begin{equation}\label{eq:25}
\bigg\{ \begin{array}{l} 3 k_p ^3 a_1 + {\mathrm{i}} k_s^3 b_1 - (k_p^3 a_3 + {\mathrm{i}} k_s^3 b_3)\mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}=0,\\
-2 \lambda k_p^4 a_2+ 2 {\mathrm{i}} \mu k_s^4 b_2 - ( k_p^4 a_4+ {\mathrm{i}} k_s^4 b_4) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0.
\end{array} \end{equation} By multiplying the first equation of \eqref{eq:25} with $\mu$ and subtracting it from the second equation of \eqref{eq:17}, one can show that \begin{equation}\label{eq:27z} a_1=0. \end{equation} Substituting \eqref{eq:27z} into the second equation of \eqref{eq:16}, we can obtain \begin{equation}\label{eq:28z} b_1=0. \end{equation}
By now we have proved $a_\ell=b_\ell=0$ for $\ell=0,1$. Similarly, from Lemma \ref{lem:co exp}, we compare the coefficients of the terms $r^3$ in both sides of \eqref{eq:forth}, and can conclude that \begin{equation}\nonumber \begin{aligned} & \big(4 k_p ^4 a_2 + 2 {\mathrm{i}} k_s^4 b_2 - (k_p^4 a_4 + {\mathrm{i}} k_s^4 b_4)\mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}\big)\hat{\mathbf{e}}_1\\ & +\big(((\mu-2 \lambda) k_p^5 a_3 + 3 {\mathrm{i}} \mu k_s^5 b_3) - ( k_p^5 a_5 + {\mathrm{i}} k_s^5 b_5) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0} \big) \mathrm{e}^{ {\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_2= \bmf{0}. \end{aligned} \end{equation} Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent, we obtain that \begin{equation}\label{eq:26}
\bigg\{ \begin{array}{l} 4 k_p ^4 a_2 + 2 {\mathrm{i}} k_s^4 b_2 - (k_p^4 a_4 + {\mathrm{i}} k_s^4 b_4)\mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}=0,\\
(\mu-2 \lambda) k_p^5 a_3 + 3 {\mathrm{i}} \mu k_s^5 b_3 - ( k_p^5 a_5 + {\mathrm{i}} k_s^5 b_5) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0.
\end{array} \end{equation} Multiplying the first equation of \eqref{eq:26} by $\mu$ and subtracting it from the second equation of \eqref{eq:25}, it yields that \begin{equation}\label{eq:27} a_2=0. \end{equation} Substituting \eqref{eq:27}, \eqref{eq:23} into the first equation of \eqref{eq:17}, we can obtain \begin{equation}\label{eq:28} b_2=0. \end{equation}
Since $a_\ell=b_\ell=0$ for $\ell=0,1,2$, according to Lemma \ref{lem:third}, we have that $a_\ell=b_\ell=0, \forall \ell \in \mathbb N \cup \{0\}$. Therefore, from Proposition \ref{prop:1}, we know that $\bmf{u}\equiv \bmf{0}$ in $\Omega$.
The proof is complete. \end{proof}
\subsection{The case with a singular generalized-impedance line }
In this subsection, we shall establish the GHP for the presence of a singular generalize impedance line. Before that, we need several auxiliary lemmas. Using \eqref{eq:gradient1} and \eqref{eq:gradient2}, we can directly obtain that
\begin{lem}\label{lem:34} Suppose that \eqref{eq:sg cond} holds, then we have \begin{equation} {\mathrm{i}} k_p^2 a_0 + k_s^2 b_0=0. \label{eq:gradient3} \end{equation} \end{lem}
\begin{lem}\label{lem:g imp} Let $\mathbf{u}$ be a Lam\'e eigenfunction of \eqref{eq:lame}, where $\mathbf{u}$ has the expansion \eqref{eq:u} around the origin. Suppose that $\Gamma_h^+$ is defined in \eqref{eq:gamma_pm} such that $\Gamma_h^+\in {\mathcal H}_\Omega^{\kappa}$ with the generalized-impedance parameter $\boldsymbol{ \eta}_1 $ satisfying \eqref{eq:eta1 ex}.
Then we have \begin{equation}\label{eq:1b} \bigg\{
\begin{array}{l}
(1+ {\mathrm{i}} \eta_1)( {\mathrm{i}} k_p a_1 - k_s b_1) =0,\\
2 \eta_1 (\lambda+\mu) k_p^2 a_0 + (1 - {\mathrm{i}} \eta_1) ( {\mathrm{i}} k_p^2 a_2 - k_s^2 b_2) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0,
\end{array} \end{equation} and \begin{equation}\label{eq:g2} \bigg\{
\begin{array}{l}
2 k_s^2 b_0 + \mathrm{e}^{2 {\mathrm{i}} \varphi_0} (1+{\mathrm{i}} \eta_1) ({\mathrm{i}} k_p^2 a_2 - k_s^2 b_2) =0,\\
({\mathrm{i}} \mu - 2 \lambda \eta_1 - \mu \eta_1) k_p^3 a_1 + \mu ({\mathrm{i}} \eta_1 - 1) k_s^3 b_1 - \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0} ( 1 - {\mathrm{i}} \eta_1) ({\mathrm{i}} k_p^3 a^3 - k_s^3 b_3) =0.
\end{array} \end{equation}
\end{lem}
\begin{proof} Since $\Gamma_h^+$ is a generalized-impedance line of $\bmf{u}$, using \eqref{eq:third2} and \eqref{eq:forth2}, we have \begin{equation}\label{eq:GI}
\begin{aligned} \bmf{0} = & \sum_{m=0}^{\infty} \mathrm{ e}^{{\mathrm{i}} m \varphi_0} \bigg\{ \left[ \frac{{\mathrm{i}} k_p}{2} a_{m} (J_{m-1}(k_p r) + J_{m+1}(k_p r) )\right.\\
& + \frac{k_s}{2} b_m (-J_{m-1}(k_s r) + J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 + \left[ \frac{{\mathrm{i}} k_p^2}{2} a_{m} (-J_{m-2}(k_p r) + J_{m+2}(k_p r) )\right.\\
& + \frac{k_s^2}{2} b_m (J_{m-2}(k_s r) + J_{m+2}(k_s r) )\bigg] \mu \hat{\mathbf{e}}_2 \bigg\}\\
&+\boldsymbol{ \eta }_1 \bigg\{ \left[ \frac{ k_p}{2} a_{m} ( - J_{m-1}(k_p r) + J_{m+1}(k_p r) )\right.
- \frac{{\mathrm{i}} k_s}{2} b_m (J_{m-1}(k_s r) + J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 \\
& + \left[- \frac{ k_p^2}{2} a_{m} (J_{m-2}(k_p r)\mu + 2(\lambda+\mu) J_m(k_p r)+ J_{m+2}(k_p r) \mu)\right.\\
& - \frac{{\mathrm{i}} k_s^2}{2} b_m (J_{m-2}(k_s r) - J_{m+2}(k_s r) )\mu \bigg] \hat{\mathbf{e}}_2 \bigg\},\quad 0 \leqslant r\leqslant h,
\end{aligned} \end{equation} where $\hat{\mathbf{e}}_1=(0,1)^\top $, $\hat{\mathbf{e}}_2=(1,0)^\top $. Using Lemma \ref{lem:co exp}, \eqref{eq:Jmfact} and \eqref{eq:eta1 ex}, and comparing the coefficient of the term $r^0$ in both sides of \eqref{eq:GI}, we see that \begin{equation}\nonumber \begin{aligned} &\left[({\mathrm{i}} - \eta_1) k_p a_1 - (1+ {\mathrm{i}} \eta_1) k_s b_1\right] {\mathrm e}^{{\mathrm{i}} \varphi_0} \hat{\mathbf{e}}_1\\ &- \left[ 2 \eta (\lambda+\mu) k_p^2 a_0 +\left(({\mathrm{i}} + \eta_1) k_p^2 a_2 - (1 - {\mathrm{i}} \eta_1) k_s^2 b_2\right) \mu {\mathrm e}^{2 {\mathrm{i}} \varphi_0}\right] \hat{\mathbf{e}}_2=\bmf{0}, \end{aligned} \end{equation} which readily gives \eqref{eq:1b}.
Using Lemma \ref{lem:co exp} and \eqref{eq:Jmfact}, and comparing the coefficient of the term $r^1$ in both sides of \eqref{eq:GI}, together with \eqref{eq:eta1 ex}, we can show that \begin{equation}\nonumber \begin{aligned} & \left[2 k_s^2 b_0 + \mathrm{e}^{2 {\mathrm{i}} \varphi_0} (1+{\mathrm{i}} \eta) ({\mathrm{i}} k_p^2 a_2 - k_s^2 b_2)\right] \hat{\mathbf{e}}_1 + \\ & \left[\left(({\mathrm{i}} \mu - 2 \lambda \eta - \mu \eta) k_p^3 a_1 + \mu ({\mathrm{i}} \eta - 1) k_s^3 b_1\right) \mathrm{e}^{ {\mathrm{i}} \varphi_0} - \mu \mathrm{e}^{3 {\mathrm{i}} \varphi_0} ( 1 - {\mathrm{i}} \eta) ({\mathrm{i}} k_p^3 a^3 - k_s^3 b_3)\right] \hat{\mathbf{e}}_2 \\ & = \bmf{0}. \end{aligned} \end{equation} Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent, we readily obtain \eqref{eq:g2}.
The proof is complete. \end{proof}
\begin{lem}\label{lem:determinant} Let $\lambda,\mu$ be the Lam\'e constants satisfying the strong convexity condition \eqref{eq:convex}. Suppose that $k_{p}$ and $ k_{{s}} $ are the compressional and shear wave numbers defined in \eqref{eq:kpks} respectively. If $$ \eta_1 \neq \frac{\pm \sqrt{(\lambda + 3\mu)(\lambda + \mu)} - \mu {\mathrm{i}}}{\lambda + 2 \mu}, $$ then for $\forall m \in \mathbb{N} \cup \{0\}$ and $\forall \varphi_0 \in (0,\pi)$ \begin{equation} \notag
D_m=\left|\begin{array}{ccc} {\mathrm{i}} k_p^m & - k_s^m & 0\\ ({\mathrm{i}} m \mu + m \mu \eta_1 - 2 \lambda \eta_1 - 2 \mu \eta_1) k_p^{m+2} & m \mu ( {\mathrm{i}} \eta_1 - 1) k_s^{m+2} & \mu( {\mathrm{i}} \eta_1 - 1) \mathrm{ e}^{2 {\mathrm{i}} \varphi_0}\\ (m \eta_1 + 2 \eta_1 - {\mathrm{i}} m ) k_p^{m+2} & ( m + 2 + {\mathrm{i}} m \eta_1 ) k_s^{m+2} & (1 + {\mathrm{i}} \eta_1)\mathrm{ e}^{2 {\mathrm{i}}\varphi_0}\\
\end{array}\right| \neq {0}. \end{equation}
\end{lem} \begin{proof} By directly calculations, it can be verified that
\begin{equation}\label{eq:determinant1} D_m= -2 \mathrm{ e}^{2 {\mathrm{i}}\varphi_0} k_p^m k_s^m \left[ {\mathrm{i}} (\lambda + 2 \mu ) k_p^2 \eta_1 ^2 + (\lambda k_p^2 - \mu k_s^2) \eta_1 -{\mathrm{i}} \mu k_s^2\right]. \end{equation} Substituting \eqref{eq:kpks} into \eqref{eq:determinant1}, we obtain that \begin{equation}\label{eq:determinant2} D_m=-2 \mathrm{ e}^{2 {\mathrm{i}}\varphi_0} k_p^m k_s^m \kappa ^2 \Big [{\mathrm{i}} \eta_1 ^2 + \left(\frac{\lambda}{\lambda + 2 \mu} - 1 \right) \eta_1 - {\mathrm{i}} \Big ]. \end{equation}
It can be prove that $$ \eta_{\sf root}=\frac{\pm \sqrt{(\lambda + 3\mu)(\lambda + \mu)} - \mu {\mathrm{i}}}{\lambda + 2 \mu} $$ are roots of ${\mathrm{i}} \eta ^2 + \left(\frac{\lambda}{\lambda + 2 \mu} - 1 \right) \eta - {\mathrm{i}}$.
The proof is complete.
\end{proof}
\begin{thm}\label{thm:g im} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exits a singular generalized-impedance line $\Gamma_h\subset\Omega$ of $\bmf{u}$ associated with the parameter $\boldsymbol{ \eta}_1$ satisfying \begin{equation}\label{eq:thm33 cond}
\eta_1 \neq \pm {\mathrm{i}} \mbox{ and } \frac{\pm \sqrt{(\lambda + 3\mu)(\lambda + \mu)} - \mu {\mathrm{i}}}{\lambda + 2 \mu}, \end{equation}
where $\eta_1$ is the constant part of $\boldsymbol{\eta}_1$ defined in \eqref{eq:eta1 ex}, then $\bmf{u}\equiv \bmf{ 0}$. \end{thm} \begin{proof} Suppose that there exits a singular generalized-impedance line $\Gamma_h=\Gamma_h^+$ of $\mathbf u$ as described at the beginning of this section. From Lemma \ref{lem:g imp}, we know \eqref{eq:1b} holds.
Since $\Gamma_h^+ \in \mathcal S( \mathcal H_\Omega^\kappa) $, we know that \eqref{eq:def3} and \eqref{eq:sg cond} hold. Therefore \eqref{eq:gradient3} holds according to Lemma \ref{lem:34}. By virtue of \eqref{eq:def3}, substituting $b_0=0$ into \eqref{eq:gradient3}, we obtain
\begin{equation}\label{eq:2b}
a_0=0.
\end{equation} Substituting $b_1=b_2=0$ and \eqref{eq:2b} into \eqref{eq:1b}, under the assumption $\eta_1 \neq \pm {\mathrm{i}} $, we obtain
\begin{equation}\label{eq:3b}\notag
a_1=
a_2=0.
\end{equation}
By now we have proved that $a_\ell=0$ for $\ell=0,1,2$ under the assumptions $\Gamma_h^+ \in \mathcal S( \mathcal H_\Omega^\kappa) $ and $\eta_1 \neq \pm {\mathrm{i}}$. In the following we prove $a_\ell=b_\ell=0$ for $\forall m \in \mathbb N\backslash\{1,2\}$ by induction.
Suppose that
\begin{align}\label{eq:352 al}
a_{\ell}=b_{\ell}=0,\quad \ell=0,\ldots, m-1,
\end{align}
where $m\geq 3$ is a natural number. Substituting \eqref{eq:352 al} into \eqref{eq:GI}, one can show that
\begin{equation}\label{eq:GI1}
\begin{aligned} \bmf{0} = & \sum_{m=\ell}^{\infty} \mathrm{ e}^{{\mathrm{i}} m \varphi_0} \bigg\{ \left[ \frac{{\mathrm{i}} k_p}{2} a_{m} (J_{m-1}(k_p r) + J_{m+1}(k_p r) )\right.\\
& + \frac{k_s}{2} b_m (-J_{m-1}(k_s r) + J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 + \left[ \frac{{\mathrm{i}} k_p^2}{2} a_{m} (-J_{m-2}(k_p r) + J_{m+2}(k_p r) )\right.\\
& + \frac{k_s^2}{2} b_m (J_{m-2}(k_s r) + J_{m+2}(k_s r) )\bigg] \mu \hat{\mathbf{e}}_2 \bigg\}\\
&+\boldsymbol{ \eta}_1 \bigg\{ \left[ \frac{ k_p}{2} a_{m} ( - J_{m-1}(k_p r) + J_{m+1}(k_p r) )\right.
- \frac{{\mathrm{i}} k_s}{2} b_m (J_{m-1}(k_s r) + J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 \\
& + \left[- \frac{ k_p^2}{2} a_{m} (J_{m-2}(k_p r)\mu + 2(\lambda+\mu) J_m(k_p r)+ J_{m+2}(k_p r) \mu)\right.\\
& - \frac{{\mathrm{i}} k_s^2}{2} b_m (J_{m-2}(k_s r) - J_{m+2}(k_s r) )\mu \bigg] \hat{\mathbf{e}}_2 \bigg\},\quad 0 \leqslant r\leqslant h.
\end{aligned} \end{equation} The lowest order of the power with respect to $r$ in the right hand side of \eqref{eq:GI1} is $m-1$. Comparing the coefficients of the terms $r^{m-1}$ in both sides of \eqref{eq:GI1}, together with the use of \eqref{eq:eta1 ex}, one can directly verify that \begin{equation}\label{eq:5b}
( {\mathrm{i}} \eta_1 + 1)( {\mathrm{i}} k_p^m a_m - k_s^m b_m) = 0,\quad
({\mathrm{i}} \eta_1 - 1)({\mathrm{i}} k_p^{m+1} a_{m+1} - k_s^{m+1} b_{m+1})=0. \end{equation} Similarly, comparing the coefficients of the terms $r^m$ in both sides of \eqref{eq:GI1} and in view of \eqref{eq:eta1 ex}, we obtain that \begin{equation}\label{eq:6b}
\left\{ \begin{array}{l} ({\mathrm{i}} \eta_1 +1)({\mathrm{i}} k_p^{m+1} a_{m+1} - k_s^{m+1} b_{m+1})=0,\\
\begin{aligned}
&({\mathrm{i}} m \mu + m \mu \eta - 2 \lambda \eta_1 - 2 \mu \eta_1) k_p^{m+2} a_m + m \mu( {\mathrm{i}} \eta_1 - 1) k_s^{m+2} b_m\\
&+ \mu ( {\mathrm{i}} \eta_1 -1 ) ( {\mathrm{i}} k_p^{m+2} a_{m+2} - k_s^{m+2} b_{m+2}) \mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0.
\end{aligned}
\end{array}
\right. \end{equation} Finally comparing the coefficients of the terms $r^{m+1}$ in both sides of \eqref{eq:GI1}, together with the use of \eqref{eq:eta1 ex},
we obtain that \begin{equation}\label{eq:7b}
\left\{ \begin{array}{l}
\begin{aligned}
& (m \eta_1 + 2 \eta_1 - {\mathrm{i}} m ) k_p^{m+2} a_m + ( m + 2 + {\mathrm{i}} m \eta_1 ) k_s^{m+2} b_m \\
& + ( 1 + {\mathrm{i}} \eta_1 ) \left[{\mathrm{i}} k_p^{m+2} a_{m+2} - k_s^{m+2} b_{m+2}\right] \mathrm{e}^{ 2 {\mathrm{i}} \varphi_0}=0,\\
& [(m \mu -\mu -2 \lambda) \eta_1 + {\mathrm{i}} (m+1) \mu] k_p^{m+3} a_{m+1} + (m+1) \mu ( {\mathrm{i}} \eta_1 - 1 ) k_s^{m+3} b_{m+1}\\
& + \mu ( {\mathrm{i}} \eta_1-1 ) ( {\mathrm{i}} k_p^{m+3} a_{m+3} - k_s^{m+3} b_{m+3}) \mathrm{e}^{2 {\mathrm{i}} \varphi_0} =0.
\end{aligned}
\end{array}
\right. \end{equation} From \eqref{eq:5b}, \eqref{eq:6b}, \eqref{eq:7b}, we have \begin{align*}
&\left[\begin{array}{ccc} {\mathrm{i}} k_p^m & - k_s^m & 0\\ ({\mathrm{i}} m \mu + m \mu \eta_1 - 2 \lambda \eta_1 - 2 \mu \eta_1) k_p^{m+2} & m \mu ( {\mathrm{i}} \eta_1 - 1) k_s^{m+2} & \mu( {\mathrm{i}} \eta_1- 1) \mathrm{ e}^{2 {\mathrm{i}} \varphi_0}\\ (m \eta_1+ 2 \eta_1 - {\mathrm{i}} m ) k_p^{m+2} & ( m + 2 + {\mathrm{i}} m \eta_1 ) k_s^{m+2} & (1 + {\mathrm{i}} \eta_1)\mathrm{ e}^{2 {\mathrm{i}}\varphi_0} \end{array}\right] \\ &\quad \times
\left[\begin{array}{c}
a_m \\ b_m \\ c_{m+2}
\end{array}\right]=\bmf{0}, \end{align*} where $c_{m+2}= {\mathrm{i}} k_p^{m+2} a_{m+2} - k_s^{m+2} b_{m+2} $. By Lemma \ref{lem:determinant}, along with the use of \eqref{eq:thm33 cond}, one has $a_m=b_m=0$. Using the above induction procedure, we can prove that $a_m=b_m=0$ for $m \in \mathbb{N} \cup {0}$.
The proof is complete. \end{proof}
\section{GHP with the non-degenerate intersection of two homogeneous lines}\label{sec:4}
In this section, microlocal singularities of the Lam\' e eigenfunction $\mathbf u$ can be established when two homogeneous lines introduced in Definition~\ref{def:1} intersect with each other under the generic non-degenerate case. Microlocal singularities prevent the occurrence of such intersections unless the Lam\'e eigenfunction $\bmf{u}$ is identically vanishing. As discussed in the beginning of Section~\ref{sect:2}, we assume throughout this section that the aforementioned two homogeneous lines are given by $\Gamma_h^\pm$ in \eqref{eq:gamma_pm} with the intersecting angle $\varphi_0\in (0, \pi)$.
In the following three lemmas, we review some related results from \cite{DLW} for the rigid, traction-free and impedance lines of $\mathbf u$, which shall be needed in our subsequent analysis.
\begin{lem}\cite[Lemma 3.1]{DLW}\label{lem:rigid} Let $\mathbf{u}$ be a generalized Lam\'e eigenfunction to \eqref{eq:lame} with the radial wave expansion given in \eqref{eq:u} around the origin. Suppose there exists $\Gamma_h^-\in {\mathcal R}_\Omega^{\kappa} $. Then one has \begin{equation}\label{eq:1a}
k_p a_1 + {\mathrm{i}} k_s b_1=0,\quad
k_p^3a_1-{\mathrm{i}} k_s^3 b_1=0, \end{equation} and \begin{equation}\label{eq:2a}
k_p ^2 a_0+ {\mathrm{i}} k_s^2 b_0- k_p^2 a_2- {\mathrm{i}} k_s ^2 b_2=0,\quad
k_p ^2 a_0- {\mathrm{i}} k_s^2 b_0=0. \end{equation} Furthermore, suppose that \begin{equation}\label{eq:lem cond}
a_\ell=b_\ell=0, \end{equation} where $\ell=0,\ldots,m-1$ and $m\in {\mathbb N}$ with $m\geq 2 $, then $$ a_\ell=b_\ell=0, \quad \forall \ell \in \mathbb N \cup \{0\}. $$
\end{lem}
\begin{lem}\cite[Lemma 3.3]{DLW}\label{lem:tranction free} Let $\mathbf{u}$ be a Lam\'e eigenfunction to \eqref{eq:lame} with the radial wave expansion \eqref{eq:u} around the origin. Suppose that $\Gamma_h^- \in {\mathcal T}_\Omega^{\kappa} $. Then we have \begin{equation}\label{eq:3a} {\mathrm{i}} k_p^2 a_2- k_s^2 b_2=0,\quad a_0 =0, \end{equation} and \begin{equation}\label{eq:4a} k_s^3 b_1+{\mathrm{i}} k_p^3 a_3-k_s^3 b_3=0,\quad
a_1=0. \end{equation} Furthermore, suppose that $a_\ell=b_\ell=0$ for $\ell=0,1$, then \begin{equation}\label{eq:lem33}
a_\ell=b_\ell=0,\quad \forall \ell \in \mathbb N\cup \{0\}. \end{equation} \end{lem}
Assume that $\Gamma_h^-$ is an impedance line of $\mathbf u$ associated with the impedance parameter $\boldsymbol{\eta}_2$ belonging to the class $\mathcal A(\Gamma_h^- )$. By virtue of \eqref{eq:eta2 ex}, the following lemma can be directly obtained by modifying the corresponding proof of \cite[Lemma 3.4]{DLW}. The proof of this lemma is omitted.
\begin{lem} \label{lem:impedance line} Let $\mathbf{u}$ be a solution to \eqref{eq:lame} with the radial wave expansion \eqref{eq:u} around the origin. Suppose that there is an impedance line $\Gamma_h^-$ of $\bmf{u}$ with an impedance parameter $ \boldsymbol{ \eta}_2 $ satisfying \eqref{eq:eta2 ex}. Then we have \begin{equation}\label{eq:5a}
\eta_2 k_p a_1+{\mathrm{i}} \eta_2 k_s b_1 -{\mathrm{i}} k_p^2 \mu a_2+k_s^2 \mu b_2=0,\quad
a_0=0, \end{equation} and \begin{equation}\label{eq:6a}
- k_s^3 \mu b_1 + \eta_2 k_p^2 a_2+{\mathrm{i}} \eta_2 k_s^2 b_2 +{\mathrm{i}} k_p^3 \mu a_3-k_s^3 \mu b_3=0,\quad
a_1=0. \end{equation} Furthermore, if $a_\ell=b_\ell=0$ for $\ell=0,1$, then \begin{equation}\label{eq:a1bl=0}
a_\ell=b_\ell=0, \quad \forall \ell \in \mathbb N \cup \{0\}. \end{equation} \end{lem}
If there are two intersecting traction-free lines of $\mathbf u$ under certain generic conditions, it was proved that $\bmf{u}\equiv \bmf{0}$ in $\Omega$. In fact, we have
\begin{thm}\cite[Theorem 4.2]{DLW} \label{thm:two traction} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. Suppose there exit two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^\pm $ are two traction-free lines with the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, if $\bmf{u}(\bmf{0})=\bmf{0}$ and \begin{equation}\label{eq:Thm41 cond}
\varphi_0 \in (0, \varphi_{\sf root}), \end{equation}
where $\varphi_{\sf root}$ is the unique root of \begin{equation}\label{eq:varphi0}
g(\varphi):=\frac{4}{3}\cdot \frac{\varphi}{\cos^6(\varphi/2 ) }-1, \quad \varphi \in (0,\pi), \end{equation}
then $\bmf{u}\equiv \bmf{0}$.
\end{thm}
In Theorem \ref{thm:Tu thm} in what follows, we shall show that the assumption \eqref{eq:Thm41 cond} in Theorem \ref{thm:two traction} can be removed. In the proof of Theorem \ref{thm:Tu thm}, we need the complex geometrical optics solution $\mathbf v (\mathbf x)$ introduced in \cite{EBL} to establish the integral equality. In the following we first introduce the geometrical setup.
Let $B_h$ be the central disk of radius $h \in \mathbb{R}_+$. Let $\Gamma^\pm$ signify the infinite extension of $\Gamma_{h}^\pm$ in the half-space $x_2\geq 0$, where $\Gamma_h^\pm$ are defined in \eqref{eq:gamma_pm}. Consider the open sector \begin{equation}\label{eq:K}
\mathcal{K} =\left\{\bmf{x}=(x_1,x_2) \in \mathbb{R}^{2}~ |~ \bmf{x}\neq \bmf{0},\quad 0<\arg \left(x_{1}+\mathrm{i} x_{2}\right)<\varphi_{0}\right\},\quad \mathrm{i}:=\sqrt{-1}, \end{equation} which is formed by the two half-lines $\Gamma^-$ and $\Gamma^+$. In the sequel, we set \begin{equation}\label{eq:sh}
S_h=\mathcal{K}\cap B_h. \end{equation} Next we set \begin{equation}\label{eq:v} \bmf{v}(\bmf{x} )=\left(\begin{array}{c}{\exp (-s \sqrt{r} \exp({\mathrm{i}} \varphi/2))} \\ {\mathrm{i}} \cdot {\operatorname{exp}(-s \sqrt{r} \exp({\mathrm{i}} \varphi/2) )}\end{array}\right):= \left(\begin{array}{c}{v_1(\bmf{x})} \\ {v_2(\bmf{x})}\end{array}\right)=v_1(\bmf{x}) \bmf{e}_1, \end{equation} where $\bmf{x}=r\cdot (\cos \varphi, \sin \varphi ) $, $s \in \mathbb{R}_{+}$, $-\pi<\varphi\leqslant \pi$ and $\bmf{e}_1=(1,{\mathrm{i}})^\top$. $\bmf{v}$ is known as the Complex Geometrical Optics (CGO) solution for the Lam\'e operator and was first introduced in \cite{EBL}. In the following we set \begin{equation}\label{eq:deltaK}
\delta_{{\mathcal K}}=\min_{0<\varphi<\varphi_0} \cos(\varphi/2). \end{equation} It can seen that $\delta_{{\mathcal K}}$ is a positive constant depending ${\mathcal K}$.
We first derive the following three critical lemmas. \begin{lem}\label{lem:r1} Let $a \in \mathbb{C}$ and $\Re(a)>0$. For any given positive numbers $\alpha$ and $\varepsilon$ satisfying $\varepsilon < \mathrm e $, if $ \Re(a) \geq \frac{2 \alpha}{\mathrm e}$, one has $$
\left|\int_{\varepsilon}^{+ \infty} r^{\alpha} \mathrm{e}^{- a r} \mathrm{d} r \right| \leq \frac{2}{\Re(a)}\mathrm{e}^{- \varepsilon \Re(a)/2}. $$ \end{lem}
\begin{proof} Let $g(r)=\frac{\ln r}{r}$ where $r\in [\varepsilon, +\infty )$. $g(r)$ increases monotonically in the interval $[\varepsilon,e]$ and decreases monotonically in the interval $[e,+ \infty]$. One has $$ \max\limits_{r \in (\varepsilon,+\infty)}g(r)= \max\limits_{r \in (\varepsilon,+\infty)} \frac{\ln r}{r}= \frac{1}{e} . $$ Since $\frac{2 \alpha}{e} \leq \Re(a)$, we have $ 2 \alpha \frac{\ln r}{r} \leq \Re(a), \, \forall r \in [\varepsilon,+\infty]$. Therefore $$ r^\alpha \leq {\mathrm e}^{ \Re(a) /2 }, $$ which can then be used to obtain that \begin{equation}\nonumber \begin{aligned}
& \left|\int_{\varepsilon}^{+ \infty} r^{\alpha} \mathrm{e}^{- a r} \mathrm{d} r \right|
\leq \int_{\varepsilon}^{+ \infty} \mathrm{e}^{- \frac{1}{2}\Re(a) r }\mathrm d r= \frac{2}{\Re(a)}\mathrm{e}^{- \varepsilon \Re(\mu)/2}. \end{aligned} \end{equation}
The proof is complete. \end{proof}
\begin{lem}\label{lem:r2} For any given positive numbers $\alpha$ and $h$ satisfying $h < \mathrm e $, $\varphi \in (0, \varphi_0 )$ with $\varphi_0 \in (0,\pi)$, if \begin{equation}\label{eq:s cond}
s \geq \frac{2 \alpha}{\mathrm e \, \delta_{\mathcal K} }, \end{equation} where $\delta_{\mathcal K}$ is defined in \eqref{eq:deltaK}, we have
\begin{equation}\label{eq:1r1}
\int_{0}^{h} r^{\alpha} \mathrm{e}^{- s \mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}} r^{\frac{1}{2}}} \mathrm{d} r = \frac{ 2 \Gamma (2 \alpha + 2)}{(s \mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}}) ^ {2 \alpha + 2}} + {\mathcal O}(\mathrm{e}^{- \frac{s \sqrt{\varepsilon}\delta_{\mathcal K}}{2}})
\end{equation}
as $s\rightarrow +\infty. $
\end{lem}
\begin{proof} By using the change of variable $r=t^2$, we have \begin{equation}\label{eq:r2} \begin{aligned}
&\int_{0}^{h} r^{\alpha} \mathrm{e}^{- s \sqrt{r} \mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}} } \mathrm{d} r = 2\int_{0}^{\sqrt{h}} t^{ 2 \alpha+1} \mathrm{e}^{- s \mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}} t} \mathrm{d} t\\
&=2\left(\frac{ \Gamma (2 \alpha + 2)}{(s \mathbf{e}^{{\mathrm{i}} \frac{\varphi}{2}}) ^ {2 \alpha + 2}} - \int_{\sqrt{h}}^{+ \infty} t^{ 2 \alpha+1} \mathrm{e}^{- s \mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}} t} \mathrm{d} t\right),
\end{aligned}
\end{equation}
where in the last equality we use the Laplace transform.
Due to \eqref{eq:deltaK}, from \eqref{eq:s cond}, it can be verified that $s\Re({ \mathrm e}^{{\mathrm{i}} \varphi/2} ) > 2\alpha /{\mathrm e}$. From Lemma \ref{lem:r1}, we have
\begin{equation}\label{eq:r3}
\left|\int_{\sqrt{h}}^{+ \infty} t^{2 \alpha+1} \mathrm{e}^{- s \mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}} t} \mathrm{d} t \right| \leq \frac{2}{s \cos \frac{\varphi}{2}} \mathrm{e}^{-\frac{\sqrt{h}}{2} s \cos \frac{\varphi}{2}} \leq \frac{2}{s \delta_{\mathcal K}} \mathrm{e}^{- \frac{s \sqrt{h} \delta_{\mathcal K}}{2}}
\end{equation}
by using \eqref{eq:deltaK}. It is easy to see that \eqref{eq:r3} is exponentially decaying as $s\rightarrow +\infty$ . Therefore, we obtain \eqref{eq:1r1}.
The proof is complete. \end{proof}
\begin{lem}\cite[Lemma 2.12]{DLW}\label{lem:uv}
Let $\bmf{u}$ and $\bmf{v}$ be respectively given by \eqref{eq:u} and \eqref{eq:v}. Then the following expansion
\begin{equation}\label{eq:uv old}
\begin{aligned}
\mathbf{u} \cdot \mathbf{v} ={\mathrm e}^{-s \sqrt{ r} \mathrm e^{{\mathrm{i}} \varphi/2} } \sum_{m=0}^{\infty} & \mathrm{e}^{{\mathrm{i}} (m+1) \varphi} \left[- k_p a_{m}
J_{m+1}\left(k_{p}r\right)
+{\mathrm{i}} k_s b_{m}
J_{m+1}\left(k_{s} r\right)
\right ]
\end{aligned} \end{equation} convergences uniformly in $S_{2h}:=\mathcal{K}\cap B_{2h}$,
where $\mathcal K$ is defined in \eqref{eq:K}.
\end{lem}
\begin{thm}\label{thm:Tu thm} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. Suppose there exit two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^\pm $ are two traction-free lines with the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, if $\bmf{u}(\bmf{0})=\bmf{0}$, then $\bmf{u}\equiv \bmf{0}$. \end{thm}
\begin{proof} By virtue of \cite[Lemma 2.17]{DLW}, the following integral identity holds \begin{equation}\label{eq:CGO2}
I_3=I_1^+ + I_1^-+I_2, \end{equation} where \begin{subequations}\notag
\begin{align}
I_1^\pm&=\int_{\Gamma_h^\pm } \left[\left({T}_{\nu} \bmf{u}\right) \cdot \bmf{v}-\left({T}_{\nu} \bmf{v}\right) \cdot \bmf{u}\right] \mathrm{d} \sigma, \quad
I_2=\int_{\Lambda_h} \left[\left({T}_{\nu} \bmf{u}\right) \cdot \bmf{v}-\left({T}_{\nu} \bmf{v}\right) \cdot \bmf{u}\right] \mathrm{d} \sigma, \label{eq:CGO6a}\\
I_3&
=-\kappa \int_{S_h} \bmf{u} \cdot \bmf{v} \mathrm{d} \bmf{x}. \label{eq:I3 int}
\end{align}
\end{subequations}
Furthermore, it holds that
\begin{equation}\label{eq:I2I4}
\begin{split}
\left|I_2\right|& \leqslant \mathcal{C}_{{\mathcal K },B_h,\mu,\lambda}\|\bmf{u}\|_{H^2\left( {{\mathcal K}} \cap B_h \right)}\left(1+s\right){\mathrm e}^{-\delta_{{\mathcal K} } s \sqrt{h}},
\end{split}
\end{equation}
which is exponentially decays as $s\rightarrow +\infty$. Here $\delta_{\mathcal{K}} $ is a positive constant defined in \eqref{eq:deltaK}.
By virtue of \cite[Eq. (4.18)]{DLW}, it was derived that \begin{equation}\label{eq:EFI1}
I_1^++I_1^-= -90 \mu (1-\mathrm{e}^{-2{\mathrm{i}} \varphi_0}) k_s^4 b_0s^{-6} - r_{I_1^-,2}-r_{I_1^+,2}, \end{equation} where \begin{equation}\label{eq:r1+}
r_{I_1^+,2}= -{\mathrm{i}} s \mu\mathrm{e}^{{\mathrm{i}} \varphi_0/2 } \int_0^h {\mathrm e}^{s \sqrt{ r} \zeta(\varphi_0 ) } R_{2,\Gamma_h^+} \mathrm{d} r,\, r_{I_1^-,2} = {\mathrm{i}} s \mu\int_0^h {\mathrm e}^{- s \sqrt{ r} } R_{2,\Gamma_h^-} \mathrm{d} r.
\end{equation} Here \begin{equation} \notag \label{eq:RTvu+} \begin{aligned} &R_{2,\Gamma_h^+} = r^{\frac{7}{2}}\Bigg\{ a_0 \mathrm e^{ {\mathrm{i}} \varphi_0} \sum_{k=2}^{\infty} \frac{(-1)^{k+1} k_p^{2k+2}}{2^{2k+1}k! (k+1)!}r^{2k-3} \\ &+ \sum_{m=1}^{2} \sum_{k=1}^{\infty} a_m \mathrm e^{ {\mathrm{i}} (m+1)\varphi_0} \frac{(-1)^{k+1} k_p^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}r^{2k+m-3} \\ & + \sum_{m=3}^{\infty} \sum_{k=0}^{\infty} a_m \mathrm e^{ {\mathrm{i}} (m+1)\varphi_0} \frac{(-1)^{k+1} k_p^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}r^{2k+m-3} +{\mathrm{i}} b_0 \mathrm e^{ {\mathrm{i}} \varphi_0} \sum_{k=2}^{\infty} \frac{(-1)^k k_s^{2k+2}}{2^{2k+1}k! (k+1)!}r^{2k-3} \\ & + {\mathrm{i}} \sum_{m=1}^{2} \sum_{k=1}^{\infty} b_m \mathrm e^{ {\mathrm{i}} (m+1)\varphi_0} \frac{(-1)^k k_s^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}r^{2k+m-3}\\ & + {\mathrm{i}} \sum_{m=3}^{\infty} \sum_{k=0}^{\infty} b_m \mathrm e^{ {\mathrm{i}} (m+1)\varphi_0} \frac{(-1)^k k_s^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}r^{2k+m-3}\Bigg\}, \end{aligned} \end{equation}
and \begin{equation}\notag \label{eq:RTvu-} \begin{aligned} & R_{2,\Gamma_h^-} = r^{\frac{7}{2}}\Bigg\{ a_0 \sum_{k=2}^{\infty} \frac{(-1)^{k+1} k_p^{2k+2}}{2^{2k+1}k! (k+1)!}r^{2k-3} + \sum_{m=1}^{2} \sum_{k=1}^{\infty} a_m \frac{(-1)^{k+1} k_p^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}r^{2k+m-3} \\ & + \sum_{m=3}^{\infty} \sum_{k=0}^{\infty} a_m \frac{(-1)^{k+1} k_p^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}r^{2k+m-3} +{\mathrm{i}} b_0 \sum_{k=2}^{\infty} \frac{(-1)^k k_s^{2k+2}}{2^{2k+1}k! (k+1)!}r^{2k-3} \\ & + {\mathrm{i}} \sum_{m=1}^{2} \sum_{k=1}^{\infty} b_m \frac{(-1)^k k_s^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}r^{2k+m-3} + {\mathrm{i}} \sum_{m=3}^{\infty} \sum_{k=0}^{\infty} b_m \frac{(-1)^k k_s^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}r^{2k+m-3}\Bigg\}, \end{aligned} \end{equation}
By virtue of \cite[Eq. (4.19)]{DLW}, one can verify that \begin{equation}\label{eq:54 r1}
|r_{I_1^+,2}| \leq S_3 \cdot{\mathcal O}( s^{-8} ),|r_{I_1^-,2}| \leq S_3 \cdot{\mathcal O}( s^{-8} ), \end{equation} as $s\rightarrow +\infty$, where \begin{equation} \notag
\begin{aligned}
S_3 = & |a_0| \sum_{k=2}^{\infty} \frac{ k_p^{2k+2}}{2^{2k+1}k! (k+1)!}h^{2k-3} +| b_0 | \sum_{k=2}^{\infty} \frac{ k_s^{2k+2}}{2^{2k+1}k! (k+1)!}h^{2k-3} \\
& + \sum_{m=1}^{2}\sum_{k=1}^{\infty} |a_m| \frac{ k_p^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}h^{2k+m-3} \\
& + \sum_{m=1}^{2}\sum_{k=1}^{\infty} |b_m| \frac{ k_s^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}h^{2k+m-3} \\
& \quad + \sum_{m=3}^{\infty} \bigg| a_m \sum_{k=0}^{\infty} \frac{(-1)^{k} k_p^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}h^{2k+m-3} \\
&\quad + {\mathrm{i}} b_m \sum_{k=0}^{\infty} \frac{(-1)^k k_s^{2k+m+2}}{2^{2k+m+1}k! (k+m+1)!}h^{2k+m-3}\bigg |. \end{aligned} \end{equation}
Since $\Gamma_h^-$ is a traction-free line of $\mathbf u$, by \eqref{eq:3a}, we have $a_0=0$. Hence substituting \eqref{eq:uv old} into $I_3$, one can derive that \begin{equation}\label{eq:r6}
\begin{split}
I_3 & = -\kappa \int_0^h \int_0^{\varphi_0} \mathrm{e}^{-s\sqrt{r}\mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}}}\sum_{m=0}^{\infty} \mathrm{e}^{{\mathrm{i}} (m+1) \varphi} \left[- k_p a_{m}
J_{m+1}\left(k_{p}r\right)
+{\mathrm{i}} k_s b_{m}
J_{m+1}\left(k_{s} r\right)
\right ] r \mathrm{d} r \mathrm{d} {\varphi}\\
& =-\frac{{\mathrm{i}}}{2}\kappa b_0 k_s^2\int_0^{\varphi_0} \left(\mathrm{e}^{{\mathrm{i}} \varphi}\int_0^h \mathrm{e}^{-s\sqrt{r}\mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}}} r^2 \mathrm{d} r \right)\mathrm{d} {\varphi} - I_{3,1},
\end{split} \end{equation}
where \begin{equation}\label{eq:r4} \begin{aligned}
I_{3,1}=&{\mathrm{i}} b_0 \kappa \sum_{k=1}^{+\infty}\frac{(-1)^k k_s^{2k+2}}{2^{2k+1}k!(k+1)!}\int_0^h r^{2k+2} \mathrm{e}^{-s\sqrt{r}\mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}}} \mathrm{d} r \int_0^{\varphi_0} \mathrm{e}^{{\mathrm{i}} \varphi} \mathrm{d} {\varphi}\\
&+ \kappa \sum_{m=1}^{+\infty}\sum_{k=0}^{+\infty}\frac{(-1)^k ( - k_p^{2k+m+2} a_m + {\mathrm{i}} k_s^{2k+m+2} b_m)}{2^{2k+m+1}k!(k+m+1)!}\\
&\times
\int_0^h r^{2k+m+2} \mathrm{e}^{-s\sqrt{r}\mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}}} \mathrm{d} r \int_0^{\varphi_0} \mathrm{e}^{{\mathrm{i}} (m+1) \varphi} \mathrm{d} {\varphi}. \end{aligned} \end{equation}
Assume that \begin{equation}\label{eq:s h cond}
s \geq \frac{8}{\mathrm e \, \delta_{\mathcal K} },\quad 0<h <\mathrm e, \end{equation} from \eqref{eq:1r1}, we can deduce that \begin{equation}\label{eq:r5} \begin{aligned}
|I_{3,1}| & \leq |b_0| \kappa \sum_{k=1}^{+\infty}\frac{k_s^{2k+2} h^{2k-2}}{2^{2k}k!(k+1)!} \int_0^{\varphi_0} \frac{ \Gamma (10)}{(s \mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}}) ^ {10}} \mathrm{e}^{{\mathrm{i}} \varphi} \mathrm{d} {\varphi}+ {\mathcal O}(\mathrm{e}^{- \frac{s \sqrt{h}\delta_{\mathcal K}}{2}})\\
+ & \kappa \sum_{m=1}^{+\infty}\sum_{k=0}^{+\infty}\frac{\left | - k_p^{2k+m+2} a_m + {\mathrm{i}} k_s^{2k+m+2} b_m\right| h^{2k+m-1} }{2^{2k+m}k!(k+m+1)!} \\
& \int_0^{\varphi_0} \frac{ \Gamma (8)}{(s \mathrm{e}^{{\mathrm{i}} \frac{\varphi}{2}}) ^ {8}} \mathrm{e}^{{\mathrm{i}} (m+1) \varphi} \mathrm{d} {\varphi} + {\mathcal O}(\mathrm{e}^{- \frac{s \sqrt{h}\delta_{\mathcal K}}{2}}). \end{aligned} \end{equation} In view of \eqref{eq:r5}, we can claim that as $s\rightarrow+\infty$
\begin{equation}\label{eq:1r5}
|I_{3,1}|={\mathcal O}(s^{-8}).
\end{equation}
Similarly, under \eqref{eq:s h cond}, by virtue of \eqref{eq:1r1}, one has
\begin{equation}\label{eq:2r5}
-\frac{{\mathrm{i}}}{2}\kappa b_0 k_s^2\int_0^{\varphi_0} \left(\mathrm{e}^{{\mathrm{i}} \varphi}\int_0^h \mathrm{e}^{-s\sqrt{r}\mathbf{e}^{{\mathrm{i}} \frac{\varphi}{2}}} r^2 \mathrm{d} r \right)\mathrm{d} {\varphi}=60 \kappa b_0 k_s^2 s^{-6}(\mathrm{e}^{-2 {\mathrm{i}} \varphi_0}-1)+ {\mathcal O} (\mathrm{e}^{- \frac{s \sqrt{h}\delta_{\mathcal K}}{2}}).
\end{equation}
Substituting \eqref{eq:1r5}, \eqref{eq:2r5} into \eqref{eq:r6}, we can obtain that \begin{equation}\label{eq:r10} I_3= 60 \kappa b_0 k_s^2 s^{-6}(\mathrm{e}^{-2 {\mathrm{i}} \varphi_0}-1)+ {\mathcal O} (\mathrm{e}^{- \frac{s \sqrt{h}\delta_{\mathcal K}}{2}}) - {\mathcal O}(s^{-8}), \end{equation} as $s\rightarrow+\infty$.
Substituting \eqref{eq:EFI1} and \eqref{eq:r10} into \eqref{eq:CGO2}, multiplying $s^6$ on both sides of the resulting \eqref{eq:CGO2}, by virtue of \eqref{eq:I2I4} and \eqref{eq:54 r1}, we can deduce that \begin{equation}\label{eq:433 b0eq} 60 \kappa b_0 k_s^2 (\mathbf{e}^{-2 {\mathrm{i}} \varphi_0}-1)+90 \mu (1-\mathrm{e}^{-2 {\mathrm{i}} \varphi_0})k_s^4 b_0=0 \end{equation} by letting $s\rightarrow +\infty$. From \eqref{eq:433 b0eq}, it yields that $$ (\mathrm{e}^{-2 {\mathrm{i}} \varphi_0}-1) k_s^2 b_0 (2 \kappa - 3 \mu k_s^2)=0. $$ Since $k_s=\sqrt{\frac{\kappa}{\mu}}$ and $\kappa\in \mathbb R_+$, it is readily known that $b_0=0$.
The remaining proof is the same as that of \cite[Theorem 4.2]{DLW} and we skip it.
The proof is complete. \end{proof}
In Theorems \ref{thm:soft thm}-\ref{thm:forth&third exp} in what follows, we shall consider the microlocal singularities of $\mathbf u$ for two intersecting homogeneous lines of $\mathbf u$ introduced in Definition~\ref{def:1}.
\subsection{The case that $\Gamma_h^+$ is a soft-clamped line} \begin{thm}\label{thm:soft thm} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a rigid line and $\Gamma_h^+$ is a soft-clamped line with the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, then $\bmf{u}\equiv \bmf{0}$. \end{thm}
\begin{proof} Since $\Gamma_h^- \in {\mathcal R}_\Omega^{\kappa}$ and $\Gamma_h^+ \in {\mathcal G}_\Omega^{\kappa}$, according to Lemmas \ref{lem:third} and \ref{lem:rigid}, and combining the first equation of \eqref{eq:1} with the second equation of \eqref{eq:2}, we have \begin{equation} \notag \label{eq:7a}
{\mathrm{i}} k_p a_1 - k_s b_1=0,\quad
{\mathrm{i}} k_p^3 a_1- k_s^3 b_1=0, \end{equation} which can be used to prove that $a_1=b_1=0$ by noting
$$
\left| \begin{matrix}
{\mathrm{i}} k_p & - k_s\\
{\mathrm{i}} k_p^3 & - k_s^3
\end{matrix} \right|={\mathrm{i}} k_p k_s (k_p^2 - k_s^2) \neq 0. $$
Similarly, combining the second equation of \eqref{eq:1} with the first equation of \eqref{eq:2}, it yields that \begin{equation} \notag \label{eq:9a}
{\mathrm{i}} k_p^2 a_2 - k_s^2 b_2=0,\quad
2 k_s^2 b_0+ ({\mathrm{i}} k_p^2 a_2 - k_s^2 b_2)\mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0,
\end{equation} which further gives $b_0=0$. Substituting $b_0=0$ into the second equation of \eqref{eq:2a}, it is easy to see $a_0=0$.
Since we have proved that $a_\ell=b_\ell=0$ for $\ell=0,1$, by virtue of Lemma \ref{lem:rigid}, we have $a_\ell=b_\ell=0$, $\forall \ell \in \mathbb N \cup \{0\}.$ According to Proposition \ref{prop:1}, one has $\mathbf u \equiv \mathbf 0$ in $\Omega$.
\end{proof}
\begin{thm}\label{thm:Tu&third exp} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a traction-free line and $\Gamma_h^+$ is a soft-clamped line with the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, then $\bmf{u}\equiv \bmf{0}$. \end{thm}
\begin{proof} Since $\Gamma_h^+ \in {\mathcal G}_\Omega^{\kappa}$, from Lemma \ref{lem:third}, we have $b_0=0$. Since $\Gamma_h^- \in {\mathcal T}_\Omega^{\kappa}$ and $\Gamma_h^+ \in {\mathcal G}_\Omega^{\kappa}$, from Lemmas \ref{lem:third} and \ref{lem:tranction free}, one can show that $b_1=0$ by using the first equation of \eqref{eq:1} and the second equation of \eqref{eq:4a}. Therefore, by virtue of the second equations of \eqref{eq:3a} and \eqref{eq:4a}, we have shown that $a_\ell=b_\ell=0$ for $\ell=0,1$. According to Lemma \ref{lem:tranction free} and Proposition \ref{prop:1}, the conclusion of the theorem can be readily obtained. \end{proof}
\begin{thm}\label{thm:Tu+&third exp} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is an impedance line and $\Gamma_h^+$ is a soft-clamped line with the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, then $\bmf{u}\equiv \bmf{0}$. \end{thm}
\begin{proof} Due to $\Gamma_h^- \in {\mathcal I}_\Omega^{\kappa}$ and $\Gamma_h^+ \in {\mathcal G}_\Omega^{\kappa}$, from Lemmas \ref{lem:third} and \ref{lem:impedance line}, we have $b_1=0$ by substituting the second equation of \eqref{eq:6a} into the first equation of \eqref{eq:1}. Similarly, since $\Gamma_h^+ \in {\mathcal G}_\Omega^{\kappa}$, from Lemma \ref{lem:third}, we have $b_0=0$. By virtue of the second equations of \eqref{eq:5a} and \eqref{eq:6a}, we have $a_\ell=b_\ell$ for $\ell=0,1$. According to Lemma \ref{lem:impedance line} and Proposition \ref{prop:1}, the conclusion of the theorem can be readily obtained. \end{proof}
\subsection{The case that $\Gamma_h^+$ is a simply-supported line.}
\begin{thm}\label{thm:u&forth exp} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a rigid line and $\Gamma_h^+$ is a simply-supported line with the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $
, then $\bmf{u}\equiv \bmf{0}$. \end{thm}
\begin{proof} Since $\Gamma_h^+$ is a simply-supported line, from Lemma \ref{lem:forth}, combining the first equation of \eqref{eq:16} with the first equation of \eqref{eq:17}, we know that \begin{equation}\label{eq:16a} a_0=0. \end{equation} Since $\Gamma_h^+$ is a simply-supported line, from Lemma \ref{lem:rigid}, substituting \eqref{eq:16a} into the second equation of \eqref{eq:2a}, we obtain $
b_0=0.
$
Again using the fact that $\Gamma_h^+ \in \mathcal F^\kappa_\Omega$ and $\Gamma_h^+ \in \mathcal R^\kappa_\Omega$, combining the second equation of \eqref{eq:17} in Lemma \ref{lem:rigid} with the first equation of \eqref{eq:25} in Lemma \ref{lem:forth},
we obtain that \begin{equation}\label{eq:18a}
a_1=0. \end{equation} Substituting \eqref{eq:18a} into the second equation of \eqref{eq:16}, we obtain $b_1=0$.
By now we have shown $a_\ell=b_\ell=0$ for $\ell=0,1$. According to Lemma \ref{lem:rigid} and Proposition \ref{prop:1}, the conclusion of the theorem can be readily obtained. \end{proof}
\begin{thm}\label{thm:Tu&forth exp} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a traction-free line and $\Gamma_h^+$ is a simply-supported line with the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, then $\bmf{u}\equiv \bmf{0}$. \end{thm}
\begin{proof}
Since $\Gamma_h^- \in {\mathcal T}_\Omega^{\kappa}$ and $\Gamma_h^+ \in {\mathcal F}_\Omega^{\kappa}$, from Lemma \ref{lem:forth} and \ref{lem:tranction free}, we have $b_1=0$ by substituting the second equation of \eqref{eq:4a} into the second equation of \eqref{eq:16}. Similarly, since $\Gamma_h^- \in {\mathcal T}_\Omega^{\kappa}$, from Lemma \ref{lem:34} and \ref{lem:tranction free}, we have $b_0=0$ by substituting the second equation of \eqref{eq:3a} into the \eqref{eq:gradient3}. By virtue of the second equations of \eqref{eq:3a} and \eqref{eq:4a}, we have $a_\ell=b_\ell$ for $\ell=0,1$. According to Lemma \ref{lem:tranction free} and Proposition \ref{prop:1}, the conclusion of the theorem can be readily obtained. \end{proof}
\begin{thm}\label{thm:Tu+&forth exp} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a impedance line and $\Gamma_h^+$ is a simply-supported line with the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, then $\bmf{u}\equiv \bmf{0}$. \end{thm}
\begin{proof} Since $\Gamma_h^- \in {\mathcal I}_\Omega^{\kappa}$ and $\Gamma_h^+ \in {\mathcal F}_\Omega^{\kappa}$, from Lemma \ref{lem:forth} and \ref{lem:impedance line}, we have $b_1=0$ by substituting the second equation of \eqref{eq:6a} into the second equation of \eqref{eq:16}. Similarly, since $\Gamma_h^- \in {\mathcal I}_\Omega^{\kappa}$, from Lemma \ref{lem:34} and \ref{lem:impedance line}, we have $b_0=0$ by substituting the second equation of \eqref{eq:5a} into the \eqref{eq:gradient3}. By virtue of the second equations of \eqref{eq:5a} and \eqref{eq:6a}, we have $a_\ell=b_\ell$ for $\ell=0,1$. According to Lemma \ref{lem:impedance line} and Proposition \ref{prop:1}, the conclusion of the theorem can be readily obtained. \end{proof}
The following lemma considers the case that $\Gamma_h^-$ is a soft-clamped line of $\mathbf u$, which shall be needed in the proofs of Theorems \ref{thm:forth&third exp} and \ref{thm:GI&third exp} in what follows.
\begin{lem}\label{lem:third1} Let $\mathbf u$ be a Lam\' e eigenfunction to \eqref{eq:lame}, where $\mathbf u$ has the Fourier expansion \eqref{eq:u} at the origin. Consider $\Gamma_h^-$ defined in \eqref{eq:gamma_pm} such that $\Gamma_h^-\in {\mathcal G}_\Omega^{\kappa}$. Then we have the following equations: \begin{subequations}
\begin{align}
&{\mathrm{i}} k_p a_1 - k_s b_1=0,\quad
{\mathrm{i}} k_p^2 a_2 - k_s^2 b_2 = 0,\label{eq:15a}\\ & \label{eq:15b}
2 k_s ^2 b_0+ ({\mathrm{i}} k_p^2 a_2- k_s^2 b_2)=0,\quad
{\mathrm{i}} k_p ^3 a_1 - k_s^3 b_1 - ({\mathrm{i}} k_p ^3 a_3 - k_s^3 b_3)=0,\\
\label{eq:15c}
&{\mathrm{i}} k_p^3 a_1 + 3 k_s^3 b_1 + ( {\mathrm{i}} k_p^3 a_3 - k_s^3 b_3) =0,\quad
2 {\mathrm{i}} k_p^4 a_2 - 2 k_s^4 b_2 - ( {\mathrm{i}} k_p^4 a_4 - k_s^4 b_4) =0,
\end{align} \end{subequations}
and \begin{equation}\label{eq:15d}
\bigg\{ \begin{array}{l} -2{\mathrm{i}} k_p^4 a_2 + 4 k_s^4 b_2 + ({\mathrm{i}} k_p^4 a_4 - k_s^4 b_4) =0,\\
3 {\mathrm{i}} k_p^5 a_3 -3 k_s^5 b_3 - ( {\mathrm{i}} k_p^5 a_5 - k_s^5 b_5) =0.
\end{array} \end{equation} Moreover, it holds that \begin{equation} \label{eq:15e} b_0=a_1=b_1=a_2=b_2=0. \end{equation} \end{lem} \begin{proof} Since $\Gamma_h^-\in {\mathcal G}_\Omega^{\kappa}$, using \eqref{eq:third3}, we have
\begin{equation}\label{eq:446 b3}
\begin{aligned} \mathbf 0 &= \sum_{m=0}^{\infty} \bigg\{ \left[- \frac{{\mathrm{i}} k_p}{2} a_{m} \big (J_{m-1}(k_p r) + J_{m+1}(k_p r)\big )\right.\\
& + \frac{k_s}{2} b_m \big (J_{m-1}(k_s r) - J_{m+1}(k_s r) \big )\bigg]\hat{\mathbf{e}}_1 + \left[ \frac{{\mathrm{i}} k_p^2}{2} a_{m} \big (-J_{m-2}(k_p r)\right.\\
& + J_{m+2}(k_p r) \big )
+ \frac{k_s^2}{2} b_m \big (J_{m-2}(k_s r) + J_{m+2}(k_s r) \big )\bigg] \mu \hat{\mathbf{e}}_2 \bigg\},
\end{aligned} \end{equation} By Lemma \ref{lem:co exp}, comparing the coefficients of $r^0$, $r^1$, $r^2$ and $r^3$ on both sides of \eqref{eq:446 b3}, one can obtain \eqref{eq:15a}, \eqref{eq:15b}, \eqref{eq:15c} and \eqref{eq:15d}, respectively. \eqref{eq:15e} can be derived by using \eqref{eq:15a}, \eqref{eq:15b}, \eqref{eq:15c} and \eqref{eq:15d}. \end{proof}
By virtue of \eqref{eq:15e} in Lemma \ref{lem:third1} and \eqref{eq:a0 forth} in Lemma \ref{lem:forth}, for two intersecting soft-clamped and simply-supported lines, one can readily have \begin{thm}\label{thm:forth&third exp} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a soft-clamped line and $\Gamma_h^+$ is a simply-supported line, where the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, then $\bmf{u}\equiv \bmf{0}$. \end{thm}
\subsection{The case that $\Gamma_h^+$ is a generalized-impedance line.}
\begin{thm}\label{thm:u&GI exp} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame} and have the Fourier expansion \eqref{eq:u}. Suppose that there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a rigid line and $\Gamma_h^+$ is a generalized-impedance line with the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, where the associated impedance parameter $\boldsymbol{ \eta}_1$ to $\Gamma_h^+$ satisfies \eqref{eq:eta1 ex}. If either \begin{equation}\label{eq:cond eta1}
\eta_1\neq -\frac{{\mathrm{i}} \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}}{\lambda+\mu(1+\mathrm{e}^{2 {\mathrm{i}} \varphi_0})} \end{equation} where $\eta_1$ is the constant part of $\boldsymbol{\eta}_1$ in \eqref{eq:eta1 ex}, or \begin{equation}\label{eq:cond a0 eta1}
a_0= 0, \quad \eta_1\neq \mathrm i, \end{equation} then $\bmf{u}\equiv \bmf{0}$. \end{thm} \begin{proof} Since $\Gamma_h^- \in {\mathcal R}_\Omega^{\kappa}$, from \eqref{eq:1a} of Lemma \ref{lem:rigid}, we can prove that $a_1=b_1=0$ by noting $$
\bigg|\begin{array}{cc} k_p & {\mathrm{i}} k_s\\ k_p^3 & -{\mathrm{i}} k_s^3 \end{array}
\bigg|=-{\mathrm{i}} k_p k_s (k_p^2+k_s^2)\neq 0. $$
Due to that $\Gamma_h^- \in {\mathcal R}_\Omega^{\kappa}$ and $\Gamma_h^+ \in {\mathcal H}_\Omega^{\kappa}$, from Lemmas \ref{lem:condition} and \ref{lem:rigid}, combining the second equation of \eqref{eq:1b} with the first equation of \eqref{eq:2a}, we have \begin{equation}\label{eq:25a}
\bigg\{ \begin{array}{l} k_p ^2 a_0+ {\mathrm{i}} k_s^2 b_0- k_p^2 a_2- {\mathrm{i}} k_s ^2 b_2=0,\\
2 \eta_1 (\lambda+\mu) k_p^2 a_0 + (1 - {\mathrm{i}} \eta_1) ( {\mathrm{i}} k_p^2 a_2 - k_s^2 b_2) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}=0.
\end{array} \end{equation} Multiplying $\mathrm i (1 - {\mathrm{i}} \eta_1) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0} $ on both sides of the first equation of \eqref{eq:25a}, then adding it to the second one of \eqref{eq:25a}, one has \begin{align} \label{eq:26a} \left( 2 \eta_1 (\lambda+\mu)+\mathrm i (1 - {\mathrm{i}} \eta_1) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}\right) k_p ^2 a_0- (1 - {\mathrm{i}} \eta_1) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0} k_s^2 b_0=0. \end{align} If \eqref{eq:cond eta1} is fulfilled, combining \eqref{eq:26a} with the second equation of \eqref{eq:2a}, one can prove that $a_0=b_0=0$ by using the fact that \begin{equation}\nonumber \begin{aligned}
&\left| \begin{matrix} \left( 2 \eta_1 (\lambda+\mu)+\mathrm i (1 - {\mathrm{i}} \eta_1) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}\right) k_p^2 & - (1 - {\mathrm{i}} \eta_1) \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}k_s^2 \\
k_p^2& -{\mathrm{i}} k_s^2
\end{matrix} \right|\\ &=-2{\mathrm{i}} k_p^2 k_s^2 \left[(\lambda+\mu(1+\mathrm{e}^{2 {\mathrm{i}} \varphi_0}))\eta_1 + {\mathrm{i}} \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0}\right]\neq0. \end{aligned} \end{equation}
Considering the case \eqref{eq:cond a0 eta1}, from \eqref{eq:25a}, it can be directly shown that $b_0=0$.
For two separate cases \eqref{eq:cond eta1} and \eqref{eq:cond a0 eta1}, by now we have shown $a_{\ell}=b_{\ell}=0$ for $\ell =0,1$. According to Lemma \ref{lem:rigid} and Proposition \ref{prop:1}, the conclusion of the theorem can be readily obtained. \end{proof}
\begin{thm}\label{thm:Tu&GI exp} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a traction-free line and $\Gamma_h^+$ is a generalized-impedance line with the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, where the associated impedance parameter $\boldsymbol{ \eta}_1$ to $\Gamma_h^+$ has the expansion \eqref{eq:eta1 ex} and satisfies $\eta_1\neq {\mathrm{i}}$, then $\bmf{u}\equiv \bmf{0}$. \end{thm} \begin{proof} Since $\Gamma_h^- \in {\mathcal T}_\Omega^{\kappa}$ and $\Gamma_h^+ \in {\mathcal H}_\Omega^{\kappa}$, from Lemma \ref{lem:34} and \ref{lem:tranction free}, we have $b_0=0$ by substituting the second equation of \eqref{eq:3a} into \eqref{eq:gradient3}. Similarly, since $\Gamma_h^- \in {\mathcal T}_\Omega^{\kappa}$ and $\Gamma_h^+ \in {\mathcal H}_\Omega^{\kappa}$, from Lemma \ref{lem:tranction free}, we have $b_1=0$ by substituting the second equation of \eqref{eq:4a} into the first equation of \eqref{eq:1b}. By virtue of the second equations of \eqref{eq:3a} and \eqref{eq:4a}, we have $a_\ell=b_\ell$ for $\ell=0,1$. The proof can be readily concluded by using Lemma \ref{lem:tranction free} and Proposition \ref{prop:1}. \end{proof}
\begin{thm}\label{thm:Tu+&GI exp} { Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is an impedance line associated with the paramter $\boldsymbol{\eta}_1$ satisfying \eqref{eq:eta1 ex} and $\Gamma_h^+$ is a generalized-impedance line associated with the parameter $\boldsymbol{\eta}_2$ satisfying \eqref{eq:eta2 ex}, where the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, $\eta_2\neq {\mathrm{i}}$, then $\bmf{u}\equiv \bmf{0}$.} \end{thm}
\begin{proof} Since $\Gamma_h^- \in {\mathcal I}_\Omega^{\kappa}$, using Lemma \ref{lem:impedance line}, one has $a_0=a_1=0$. Due to $\Gamma_h^+ \in {\mathcal H}_\Omega^{\kappa}$, from Lemma \ref{lem:g imp}, under the assumption $\eta_2 \neq \mathrm i$, we have $b_1=0$ by substituting $a_1=0$ into \eqref{eq:1b}. Similarly, substituting $a_0=0$ into \eqref{eq:1b} in Lemma \ref{lem:g imp}, under the assumption $\eta_2 \neq \mathrm i$, we have \begin{equation}\label{eq:i g intect}
\mathrm i k_p^2 a_2-k_s^2 b_2=0. \end{equation} Substituting \eqref{eq:i g intect} and $a_1=0$ into \eqref{eq:5a} in Lemma \ref{lem:impedance line}, by using the assumption $\eta_1\neq 0$, we have $b_1=0$. Therefore we obtain that $a_\ell=b_\ell$ for $\ell=0,1$. According to Lemma \ref{lem:impedance line}, the proof can be readily concluded. \end{proof}
By virtue of \eqref{eq:15e} in Lemma \ref{lem:third1} and the second equation of \eqref{eq:1b} in Lemma \ref{lem:g imp}, for two intersecting soft-clamped and generalized-impedance lines, one can readily to have \begin{thm}\label{thm:GI&third exp} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a soft-clamped line and $\Gamma_h^+$ is a generalized-impedance line, where the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, then $\bmf{u}\equiv \bmf{0}$. \end{thm}
In Theorem \ref{thm:GI&GI} we shall consider two intersecting generalized-impedance lines. Before that, we need the following two lemmas.
\begin{lem}\label{lem:48} Suppose that $\Gamma_h^-$ is a generalized-impedance line of $\mathbf u$ associated with the impedance parameter $\boldsymbol{\eta}_2$ satisfying \eqref{eq:eta2 ex}. Then we have \begin{equation}\label{eq:1b1} \bigg\{
\begin{array}{l}
(1+ {\mathrm{i}} \eta_2)( {\mathrm{i}} k_p a_1 - k_s b_1) =0,\\
2 \eta_2 (\lambda+\mu) k_p^2 a_0 + (1 - {\mathrm{i}} \eta_2) ( {\mathrm{i}} k_p^2 a_2 - k_s^2 b_2) \mu =0,
\end{array}
\end{equation}
and
\begin{equation}\label{eq:g3} \bigg\{
\begin{array}{l}
2 k_s^2 b_0 + (1+{\mathrm{i}} \eta_2) ({\mathrm{i}} k_p^2 a_2 - k_s^2 b_2) =0,\\
({\mathrm{i}} \mu - 2 \lambda \eta_2 - \mu \eta_2) k_p^3 a_1 + \mu ({\mathrm{i}} \eta_2 - 1) k_s^3 b_1 - \mu ( 1 - {\mathrm{i}} \eta_2) ({\mathrm{i}} k_p^3 a^3 - k_s^3 b_3) =0.
\end{array} \end{equation} \end{lem} \begin{proof} Since $\Gamma_h^-$ is a generalized-impedance line of $\bmf{u}$, using \eqref{eq:third3} and \eqref{eq:forth3}, we have \begin{equation}\label{eq:GI2}
\begin{aligned} \bmf{0} = & \sum_{m=0}^{\infty} \bigg\{ \left[ - \frac{{\mathrm{i}} k_p}{2} a_{m} (J_{m-1}(k_p r) + J_{m+1}(k_p r) )\right.\\
& + \frac{k_s}{2} b_m (J_{m-1}(k_s r) - J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 + \left[ \frac{{\mathrm{i}} k_p^2}{2} a_{m} (-J_{m-2}(k_p r) + J_{m+2}(k_p r) )\right.\\
& + \frac{k_s^2}{2} b_m (J_{m-2}(k_s r) + J_{m+2}(k_s r) )\bigg] \mu \hat{\mathbf{e}}_2 \bigg\}\\
&+\boldsymbol{ \eta}_2 \bigg\{ \left[ \frac{ k_p}{2} a_{m} ( J_{m-1}(k_p r) - J_{m+1}(k_p r) )\right.
+ \frac{{\mathrm{i}} k_s}{2} b_m (J_{m-1}(k_s r) + J_{m+1}(k_s r) )\bigg]\hat{\mathbf{e}}_1 \\
& + \left[- \frac{ k_p^2}{2} a_{m} (J_{m-2}(k_p r)\mu + 2(\lambda+\mu) J_m(k_p r)+ J_{m+2}(k_p r) \mu)\right.\\
& - \frac{{\mathrm{i}} k_s^2}{2} b_m (J_{m-2}(k_s r) - J_{m+2}(k_s r) )\mu \bigg] \hat{\mathbf{e}}_2 \bigg\},\quad 0 \leqslant r\leqslant h,
\end{aligned} \end{equation} where $\hat{\mathbf{e}}_1=(0,1)^\top $, $\hat{\mathbf{e}}_2=(1,0)^\top $. Using Lemma \ref{lem:co exp} and \eqref{eq:Jmfact}, by comparing the coefficients of the terms $r^0$ in both sides of \eqref{eq:GI2} and by using \eqref{eq:eta2 ex},
it holds that \begin{equation}\nonumber \begin{aligned} &\left[({\mathrm{i}} - \eta_2) k_p a_1 - (1+ {\mathrm{i}} \eta_2) k_s b_1\right] \hat{\mathbf{e}}_1\\ &- \left[ 2 \eta_2 (\lambda+\mu) k_p^2 a_0 +\left(({\mathrm{i}} + \eta_2) k_p^2 a_2 - (1 - {\mathrm{i}} \eta_2) k_s^2 b_2\right) \mu\right] \hat{\mathbf{e}}_2=\bmf{0}. \end{aligned} \end{equation} Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent, we can obtain \eqref{eq:1b1}.
Similarly, by comparing the coefficients of the terms $r^1$ in both sides of \eqref{eq:GI2} and by using \eqref{eq:eta2 ex}, we can deduce \eqref{eq:g3}. \end{proof}
\begin{lem}\label{lem:49} Let $k_p$ and $k_s$ be defined in \eqref{eq:kpks}, where $\kappa \in \mathbb R_+$ is the eigenvalue to \eqref{eq:lame}. For any $m \in \mathbb N$ and Lam\'e constants $\lambda$, $\mu$ satisfying \eqref{eq:convex}, if $\eta\in \mathbb C$ is a constant such that \begin{equation}\label{eq:cond eta m}
\eta\neq -\frac{{\mathrm{i}} m}{m+2}, \end{equation}
then
\begin{equation}\nonumber \begin{aligned}
&D_m=\bigg|\begin{array}{cc}
{\mathrm{i}} k_p^m & - k_s^m\\
\left({\mathrm{i}} m \mu - 2 \lambda \eta + (m-2)\mu \eta \right) k_p^{m+2} & m \mu ({\mathrm{i}} \eta - 1) k_s^{m+2} \end{array}
\bigg|\neq 0.
\end{aligned} \end{equation} \end{lem} \begin{proof}
By direct calculations, it yields that
\begin{equation}\notag
\begin{split}
D_m &=k_p^m k_s^m [(- m \mu k_s^2 - 2 \lambda k_p^2 + (m-2)\mu k_p^2)\eta +{\mathrm{i}} m \mu (k_p^2 - k_s^2)]\\
&= - \frac{k_p^m k_s^m(\lambda+\mu ) \kappa }{\lambda+2\mu }\left[(m+2) \eta +\mathrm i m \right],
\end{split}
\end{equation}
where we use \eqref{eq:kpks}. Since $\eta$ satisfies \eqref{eq:cond eta m}, it is easy to see that $D_m\neq 0.$
\end{proof}
\begin{thm}\label{thm:GI&GI} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. Suppose that there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a generalized-impedance line and $\Gamma_h^+$ is a generalized-impedance line with the intersecting angle \begin{equation}\label{eq:cond Gi1}
\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi . \end{equation} The associated generalized-impedance parameter $\boldsymbol{ \eta}_1$ to $\Gamma_h^+ $ and $\boldsymbol{ \eta}_2$ to $\Gamma_h^- $ have the expansions \eqref{eq:eta1 ex} and \eqref{eq:eta2 ex} respectively, where $\eta_\ell$ ($\ell=1,2$) are the constant parts of $\boldsymbol{\eta}_\ell$ $\eta_1=\eta_2=\eta$ satisfying \begin{equation}\label{eq:cond eta} \eta_1=\eta_2=\eta \mbox{ and } \eta \neq \pm \mathrm i, \, -\frac{\mathrm i m}{m+2} . \end{equation}
Then $\bmf{u}\equiv \bmf{0}$. \end{thm}
\begin{proof} Since $\Gamma_h^\pm$ are generalized-impedance lines and \eqref{eq:cond eta}, multiplying $\mathrm{e}^{2 {\mathrm{i}} \varphi_0} $ on both sides of the second equation of \eqref{eq:1b1} in Lemma \ref{lem:48}, then subtracting the second equation of \eqref{eq:1b} in Lemma \ref{lem:g imp}, we have \begin{equation}\label{eq:g1} 2 \eta (\lambda+\mu)(\mathrm{e}^{2 {\mathrm{i}} \varphi_0}-1)k_p^2 a_0=0, \end{equation} which can be used to obtain that $a_0=0$ by using \eqref{eq:convex}, \eqref{eq:cond Gi1} and \eqref{eq:cond eta}.
Multiplying $\mathrm{e}^{2 {\mathrm{i}} \varphi_0} $ on both sides of the first equation of \eqref{eq:g3} in Lemma \ref{lem:48}, then subtracting the the first equation of \eqref{eq:g2} in Lemma \ref{lem:g imp}, we have \begin{equation}\nonumber 2 (\mathrm{e}^{2 {\mathrm{i}} \varphi_0}-1)k_s^2 b_0=0, \end{equation} which can be used to obtain that $b_0=0$ under the condition \eqref{eq:cond Gi1}. Substituting $b_0=0$ into \eqref{eq:g3}, using \eqref{eq:cond eta}, one has \begin{equation}\label{eq:g9}
{\mathrm{i}} k_p^2 a_2 - k_s^2 b_2 =0. \end{equation}
Similarly, multiplying $\mathrm{e}^{2 {\mathrm{i}} \varphi_0} $ on both sides of the second equation of \eqref{eq:g3}, and then subtracting the second equation of \eqref{eq:g2}, one can obtain that \begin{equation}\label{eq:g5} [({\mathrm{i}} \mu - 2 \lambda \eta - \mu \eta) k_p^3 a_1 + \mu ({\mathrm{i}} \eta - 1) k_s^3 b_1](\mathrm{e}^{2 {\mathrm{i}} \varphi_0}-1)=0, \end{equation} Combining the first equation of \eqref{eq:1b1} with \eqref{eq:g5}, we have \begin{equation}\nonumber \bigg\{ \begin{array}{l} {\mathrm{i}} k_p a_1 - k_s b_1=0,\\ ({\mathrm{i}} \mu - 2 \lambda \eta - \mu \eta) k_p^3 a_1 + \mu ({\mathrm{i}} \eta - 1) k_s^3 b_1=0, \end{array} \end{equation} which can be used to prove that $a_1=b_1=0$ by noting \begin{equation}\nonumber \begin{aligned}
&D_1=\bigg|\begin{array}{cc}
{\mathrm{i}} k_p & - k_s\\
({\mathrm{i}} \mu - 2 \lambda \eta - \mu \eta) k_p^3 & \mu ({\mathrm{i}} \eta - 1) k_s^3 \end{array}
\bigg| \neq 0,
\end{aligned} \end{equation} where we use Lemma \ref{lem:49} for the case $m=1$ under the condition \eqref{eq:cond eta}.
Using Lemma \ref{lem:co exp} and \eqref{eq:Jmfact}, comparing the coefficient of the term $r^2$ in both sides of \eqref{eq:GI}, and by virtue of \eqref{eq:eta1 ex}, one can show that \begin{equation}\nonumber \begin{aligned} & \left[ \mathrm{e}^{2 {\mathrm{i}} \varphi_0} (1+{\mathrm{i}} \eta) ({\mathrm{i}} k_p^3 a_3 - k_s^3 b_3)\right] \hat{\mathbf{e}}_1 + \\ & \left[\left(2({\mathrm{i}} \mu - \lambda \eta ) k_p^4 a_2 + 2 \mu ({\mathrm{i}} \eta - 1) k_s^4 b_2\right) \mathrm{e}^{ {\mathrm{i}} \varphi_0} - \mu \mathrm{e}^{3 {\mathrm{i}} \varphi_0} ( 1 - {\mathrm{i}} \eta) ({\mathrm{i}} k_p^4 a^4 - k_s^4 b_4)\right] \hat{\mathbf{e}}_2 \\ & = \bmf{0}. \end{aligned} \end{equation} Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent, we obtain that \begin{equation}\label{eq:g6} \bigg\{
\begin{array}{l}
\mathrm{e}^{2 {\mathrm{i}} \varphi_0} (1+{\mathrm{i}} \eta) ({\mathrm{i}} k_p^3 a_3 - k_s^3 b_3) =0,\\
2({\mathrm{i}} \mu - \lambda \eta ) k_p^4 a_2 + 2 \mu ({\mathrm{i}} \eta - 1) k_s^4 b_2 - \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0} ( 1 - {\mathrm{i}} \eta) ({\mathrm{i}} k_p^4 a^4 - k_s^4 b_4) =0.
\end{array} \end{equation} Similarly, comparing the coefficients of the terms $r^2$ in both sides of \eqref{eq:GI2} and by virtue of \eqref{eq:eta2 ex}, we can deduce that \begin{equation}\label{eq:g7} \bigg\{
\begin{array}{l}
(1+{\mathrm{i}} \eta) ({\mathrm{i}} k_p^3 a_3 - k_s^3 b_3) =0,\\
2({\mathrm{i}} \mu - \lambda \eta ) k_p^4 a_2 + 2 \mu ({\mathrm{i}} \eta - 1) k_s^4 b_2 - \mu ( 1 - {\mathrm{i}} \eta) ({\mathrm{i}} k_p^4 a^4 - k_s^4 b_4) =0.
\end{array} \end{equation}
Multiplying $\mathrm{e}^{2 {\mathrm{i}} \varphi_0} $ on both sides of the second equation of \eqref{eq:g7}, then subtracting the the second equation of \eqref{eq:g6}, we have \begin{equation}\label{eq:g8} [2({\mathrm{i}} \mu - \lambda \eta ) k_p^4 a_2 + 2 \mu ({\mathrm{i}} \eta - 1) k_s^4 b_2 ](\mathrm{e}^{2 {\mathrm{i}} \varphi_0}-1)=0. \end{equation} Combining \eqref{eq:g9} with \eqref{eq:g8}, we have \begin{equation}\nonumber {\mathrm{i}} k_p^2 a_2 - k_s^2 b_2=0,\quad 2({\mathrm{i}} \mu - \lambda \eta ) k_p^4 a_2 + 2 \mu ({\mathrm{i}} \eta - 1) k_s^4 b_2=0, \end{equation} which can be used to prove that $a_2=b_2=0$ by noting \begin{equation}\nonumber \begin{aligned}
&D_2=\bigg|\begin{array}{cc}
{\mathrm{i}} k_p^2 & - k_s^2\\
({\mathrm{i}} \mu - \lambda \eta ) k_p^4 & \mu ({\mathrm{i}} \eta - 1) k_s^4 \end{array}
\bigg|\neq 0,
\end{aligned} \end{equation} where we use Lemma \ref{lem:49} for the case $m=2$ under the condition \eqref{eq:cond eta}.
By mathematical induction, we first assume that $a_{\ell}=b_{\ell}=0$, where $\ell=0,\ldots,m-1$ and $m \in \mathbb{ N}$ with $m\geq 3$. Using Lemma \ref{lem:co exp} and \eqref{eq:Jmfact}, comparing the coefficients of the terms $r^{m-1}$ in both sides of \eqref{eq:GI} and by virtue of \eqref{eq:eta1 ex}, one can deduce that \begin{equation}\label{eq:g13} \bigg\{ \begin{array}{l} (1+ {\mathrm{i}} \eta) ({\mathrm{i}} k_p^m a_m - k_s^m b_m)=0,\\ \mu (1- {\mathrm{i}} \eta) ({\mathrm{i}} k_p^{m+1} a_{m+1} - k_s^{m+1} b_{m+1})=0. \end{array} \end{equation}
Using Lemma \ref{lem:co exp} and \eqref{eq:Jmfact}, comparing the coefficients of the terms $r^m$ in both sides of \eqref{eq:GI} and by virtue of \eqref{eq:eta1 ex}, one can show that \begin{equation}\nonumber \begin{aligned} & \big[ \mathrm{e}^{2 {\mathrm{i}} \varphi_0} (1+{\mathrm{i}} \eta) ({\mathrm{i}} k_p^{m+1} a_{m+1} - k_s^{m+1} b_{m+1})\big] \hat{\mathbf{e}}_1 + \\ & \big[\left(({\mathrm{i}} m \mu - 2 \lambda \eta + (m-2)\mu \eta) k_p^{m+2} a_m + m \mu ({\mathrm{i}} \eta - 1) k_s^{m+2} b_m\right) \mathrm{e}^{ {\mathrm{i}} \varphi_0}\\ & - \mu \mathrm{e}^{3 {\mathrm{i}} \varphi_0} ( 1 - {\mathrm{i}} \eta) ({\mathrm{i}} k_p^{m+2} a^{m+2} - k_s^{m+2} b_{m+2})\big] \hat{\mathbf{e}}_2 \\ & = \bmf{0}. \end{aligned} \end{equation} Since $\hat{\mathbf{e}}_1$ and $\hat{\mathbf{e}}_2$ are linearly independent, we obtain that \begin{equation}\label{eq:g10} \left\{
\begin{array}{l}
\begin{aligned}
& \mathrm{e}^{2 {\mathrm{i}} \varphi_0} (1+{\mathrm{i}} \eta) ({\mathrm{i}} k_p^{m+1} a_{m+1} - k_s^{m+1} b_{m+1}) =0,\\
& ({\mathrm{i}} m \mu - 2 \lambda \eta + (m-2)\mu \eta) k_p^{m+2} a_m + m \mu ({\mathrm{i}} \eta - 1) k_s^{m+2} b_m\\
& - \mu \mathrm{e}^{2 {\mathrm{i}} \varphi_0} ( 1 - {\mathrm{i}} \eta) ({\mathrm{i}} k_p^{m+2} a^{m+2} - k_s^{m+2} b_{m+2}) =0.
\end{aligned}
\end{array}\right. \end{equation} Similarly, by comparing the coefficients of the terms $r^m$ in both sides of \eqref{eq:GI2} together with the use of \eqref{eq:eta2 ex}, we can deduce that \begin{equation}\label{eq:g11} \left\{
\begin{array}{l}
\begin{aligned}
& (1+{\mathrm{i}} \eta) ({\mathrm{i}} k_p^{m+1} a_{m+1} - k_s^{m+1} b_{m+1}) =0,\\
& ({\mathrm{i}} m \mu - 2 \lambda \eta + (m-2)\mu \eta) k_p^{m+2} a_m + m \mu ({\mathrm{i}} \eta - 1) k_s^{m+2} b_m\\
& - \mu ( 1 - {\mathrm{i}} \eta) ({\mathrm{i}} k_p^{m+2} a^{m+2} - k_s^{m+2} b_{m+2}) =0.
\end{aligned}
\end{array}\right. \end{equation} Multiplying $\mathrm{e}^{2 {\mathrm{i}} \varphi_0} $ on both sides of the second equation of \eqref{eq:g11}, then subtracting the the second equation of \eqref{eq:g10}, we have \begin{equation}\label{eq:g12} [({\mathrm{i}} m \mu - 2 \lambda \eta + (m-2)\mu \eta) k_p^{m+2} a_m + m \mu ({\mathrm{i}} \eta - 1) k_s^{m+2} b_m ](\mathrm{e}^{2 {\mathrm{i}} \varphi_0}-1)=0. \end{equation}
Combining the first equation of \eqref{eq:g13} with \eqref{eq:g12}, we have \begin{equation}\nonumber \bigg\{ \begin{array}{l} {\mathrm{i}} k_p^m a_m - k_s^m b_m=0,\\ ({\mathrm{i}} m \mu - 2 \lambda \eta + (m-2)\mu \eta) k_p^{m+2} a_m + m \mu ({\mathrm{i}} \eta - 1) k_s^{m+2} b_m=0, \end{array} \end{equation} which can be used to prove that $a_m=b_m=0$ by noting \begin{equation}\nonumber \begin{aligned}
&D_m=\bigg|\begin{array}{cc}
{\mathrm{i}} k_p^m & - k_s^m\\
\left({\mathrm{i}} m \mu - 2 \lambda \eta + (m-2)\mu \eta \right) k_p^{m+2} & m \mu ({\mathrm{i}} \eta - 1) k_s^{m+2} \end{array}
\bigg| \neq 0,
\end{aligned} \end{equation} where we use Lemma \ref{lem:49} under the condition \eqref{eq:cond eta}.
According to Proposition \ref{prop:1}, the proof is complete. \end{proof}
\begin{rem}
The assumption \eqref{eq:cond eta} in Theorem \ref{thm:GI&GI} can be replaced by
\begin{equation}\label{eq:cond eta new} \eta_1=\eta_2=\eta \mbox{ and } \Im(\eta) \in \mathbb R_+ \backslash\{1\}, \end{equation} which shall be adopted in studying the inverse elastic obstacle and diffractive grating problems.
\end{rem}
The following theorem deals with the case that a generalized-impedance line intersects a simply-supported line.
\begin{thm}\label{thm:GI&forth exp} Let $\mathbf{u}\in L^2(\Omega)^2$ be a solution to \eqref{eq:lame}. If there exist two intersecting lines $\Gamma_h^\pm$ of $\bmf{u}$ such that $\Gamma_h^-$ is a generalized-impedance line and $\Gamma_h^+$ is a simply-supported line, where the intersecting angle $\angle(\Gamma_h^+,\Gamma_h^-)=\varphi_0\neq \pi $, then $\bmf{u}\equiv \bmf{0}$. \end{thm} \begin{proof} Since $\Gamma_h^+$ is a simply-supported line, from Lemma \ref{lem:forth}, we have $a_0=0$. Substituting $a_0=0$ into the first equation of \eqref{eq:16}, one has \begin{equation}\label{eq:a2 b2 sim}
k_p^2 a_2+\mathrm i k_s^2 b_2=0. \end{equation} Since $\Gamma_h^-$ is a generalized-impedance line, substituting \eqref{eq:a2 b2 sim} into the first equation of \eqref{eq:g3} in Lemma \ref{lem:48}, it is easy to see that $b_0=0$. Adopting the same argument in the proof of Theorem \ref{thm:fourth}, we can show that $a_\ell=b_\ell=0$ for $\forall \ell \in \mathbb N$. Finally, the proof can be concluded by using Proposition \ref{prop:1}. \end{proof}
\section{Unique identifiability for inverse elastic obstacle problems}\label{sect:5}
In this section, the theoretical findings in the previous sections can be used to study the unique identifiability for the inverse elastic obstacle and diffractive grating problems. The inverse problems are concerned with recovering the geometrical shapes/profiles of certain unknown objects by using the elastic wave probing data. The inverse elastic obstacle and diffractive grating problems arise from industrial applications of practical importance.
\subsection{Unique recovery for the inverse elastic obstacle problem}\label{subsec:obs}
Consider a bounded Lipschitz domain $\Omega\subset\mathbb{R}^2$ such that $\mathbb{R}^2\backslash\bar{\Omega}$ is connected. In what follows, we let $\mathbf{d}\in\mathbb{S}^1$ denote the incident direction, $\bmf{d}^{\perp}\in \mathbb S^1 $ be orthogonal to $\bmf{d}$, $k_p$ and $k_s$ be compressional and shear wave numbers defined in \eqref{eq:kpks}. Consider an incident elastic wave field $\bmf{u}^i$, which is a time-harmonic elastic plane wave of the form \begin{equation}\label{eq:ui}
\bmf{u}^i:=\bmf{u}^i(\mathbf{ x};k_p,k_s, \mathbf{d})=\alpha_{p} \bmf{d} \mathrm{e}^{\mathrm{i} k_{p} \bmf{x} \cdot \bmf{d} }+\alpha_{s} \bmf{d}^{\perp} \mathrm{e}^{\mathrm{i} k_s \bmf{x} \cdot \bmf{d} }, \quad \alpha_{p}, \alpha_{s} \in \mathbb{C}, \quad\left|\alpha_{p}\right|+\left|\alpha_{s}\right| \neq 0. \end{equation} Physically speaking, $\bmf{u}^i$ is the detecting wave field and $\Omega$ denotes an impenetrable obstacle which interrupts the propagation of the incident wave and generates the corresponding scattered wave field $\bmf{u}^{\mathrm{sc}}$. Using the Helmholtz decomposition, we can decompose the scattered field $\bmf{u}^{\mathrm{sc} }$ in ${\mathbb R}^2 \backslash \Omega$ into the sum of the compressional part $\bmf{u}^{\mathrm{sc}}_p$ and the shear part $\bmf{u}^{\mathrm{sc}}_s$ as follows \begin{equation}
\bmf{u}^{\mathrm{s c} }=\bmf{u}_{p}^{\mathrm {s c} }+\bmf{u}_{s}^{\mathrm{s c} }, \quad \bmf{u}_{p}^{\mathrm {s c} }=-\frac{1}{k_{p}^{2}} \nabla\left( \nabla \cdot \bmf{ u}^{\mathrm {s c} }\right ), \quad\bmf{ u}_{s}^{\mathrm{s c}}=\frac{1}{k_{s}^{2}} \bf{curl} \operatorname{curl} u^{\mathrm {s c} }, \end{equation} where \[
{\rm curl}\bmf{ u}=\partial_1 u_2-\partial_2 u_1, \quad {\bf curl}{u}=(\partial_2 u, -\partial_1 u)^\top. \] Let $\omega=\sqrt{ \kappa}$ be the angular frequency, where $\kappa$ is the Lam\'e eigenvalue of \eqref{eq:lame}. Define $\bmf{u}:=\bmf{u}^i+\bmf{u}^{\mathrm{ sc} }$ to be the total wave field, then the forward scattering problem of this process can be described by the following system, \begin{equation}\label{forward} \begin{cases} & {\mathcal L} \bmf{ u} + \omega^2 \bmf{u} = 0\qquad\quad \mbox{in }\ \ \mathbb{R}^2\backslash\overline{\Omega},
\\ & \bmf{u} =\bmf{u}^i+\bmf{u}^{\mathrm{sc} }\hspace*{1.56cm}\mbox{in }\ \ \mathbb{R}^2,
\\ & \mathscr{B}(\bmf{u})=\bmf{0}\hspace*{1.95cm}\mbox{on}\ \ \partial\Omega,
\\ &\displaystyle{ \lim_{r\rightarrow\infty}r^{\frac{1}{2}}\left(\frac{\partial \bmf{u}_\beta^{\mathrm{sc} }}{\partial r}-\mathrm{i}k_\beta \bmf{u}_\beta^{\mathrm{sc} }\right) =\,0,} \quad \beta=p,s, \end{cases} \end{equation}
where the last equation is the Kupradze radiation condition that holds uniformly in $\hat{\mathbf{ x}}:=\mathbf{ x}/|\mathbf{ x}|\in\mathbb{S}^1$. The boundary condition $\mathscr{B}(\mathbf u)$ on $\partial \Omega$ could be either of the following six conditions: \begin{enumerate}
\item the first kind (Dirichlet) condition ($\Omega$ is a rigid obstacle): \begin{equation}\label{eq:B1u}
\mathscr{B}(\bmf{u}):=\mathscr{B}_2(\bmf{u})=\bmf{u};
\end{equation}
\item the second kind (Neumann) condition ($\Omega$ is a traction-free obstacle):
\begin{equation}\label{eq:B2u}
\mathscr{B}(\bmf{u}):=\mathscr{B}_1(\bmf{u})=T_\nu \bmf{u};
\end{equation}
\item the impedance condition ($\Omega$ is an impedance obstacle): $\mathscr{B}(\bmf{u})=T_\nu \bmf{u}+\boldsymbol{ \eta} \bmf{u},\ \boldsymbol \eta \in \mathcal{A}, \ \Re(\boldsymbol \eta)\geq 0 \mbox{ and } \Im(\boldsymbol \eta)>0$;
\item the third kind boundary condition ($\Omega$ is an soft-clamped obstacle): \begin{equation}\label{eq:B3u}
\mathscr{B}(\bmf{u}):=\mathscr{B}_3(\bmf{u})=\left( \begin{array}{c} {\nu} \cdot \mathbf{u} \\ \boldsymbol{\tau} \cdot T_{\mathbf{\nu}} \bmf{u} \\ \end{array} \right); \end{equation} \item the forth kind boundary condition ($\Omega$ is a simply-supported obstacle): \begin{equation}\label{eq:B4u} \mathscr{B}(\bmf{u}):=\mathscr{B}_4(\bmf{u})=\left( \begin{array}{c} \boldsymbol{\tau} \cdot \bmf{u} \\ {\nu} \cdot T_{\mathbf{\nu}} \mathbf{u} \\ \end{array} \right); \end{equation} \item the generalized-impedance kind boundary condition ($\Omega$ is a generalized impedance obstacle): \begin{equation}\label{eq:B5u} \mathscr{B}(\bmf{u}) =\left( \begin{array}{c} {\nu} \cdot \mathbf{u} \\ \boldsymbol{\tau} \cdot T_{\mathbf{\nu}} \bmf{u} \\ \end{array} \right)+ \boldsymbol \eta \left( \begin{array}{c} \boldsymbol{\tau} \cdot \bmf{u} \\ {\nu} \cdot T_{\mathbf{\nu}} \mathbf{u} \\ \end{array} \right),\ \boldsymbol \eta \in \mathcal{A}, \, \Re( \boldsymbol\eta)\geq 0 \mbox{ and } \Im( \boldsymbol\eta)>0, \end{equation}
\end{enumerate} where $\nu$ denotes the exterior unit normal vector to $\partial\Omega$, $\boldsymbol{\tau}= \nu^\perp$ and the boundary traction operator $T_\nu$ is defined in \eqref{eq:Tu}.
In this paper, we always assume the well posedness of the elastic system \eqref{forward} associated with either of the six kinds of boundary conditions is fullfilled, which means that there exits a unique solution $ \bmf{u} \in H^1_{\mathrm{loc} } ({\mathbb R}^2 \backslash \Omega )$ to \eqref{forward}. We refer to \cite{ElschnerYama2010,Lai,Kupradze} for the related results when the boundary condition in \eqref{forward} is the third or forth kind boundary, while the corresponding well posedness result for the first kind boundary can be found in \cite{HannerHsiao}. It is known that the compressional and shear parts $\bmf{u}_\beta^{\mathrm {sc} }$ ($\beta=p,s$) of a radiating solution $\bmf{u}^{\mathrm{sc} }$ to the elastic system \eqref{forward} possess the following asymptotic expansions \begin{equation}
\begin{aligned}
\bmf{u}_{p}^{\mathrm{sc}}(\bmf{x}; k_p,k_s, \mathbf{d}) &=\frac{\mathrm{e}^{\mathrm{i} k_{p} r}}{\sqrt{r}}\left\{u_{p}^{\infty}(\hat{\bmf{x}}; \bmf{d} ) \hat{\bmf{x}}+{\mathcal O}\left(\frac{1}{r}\right)\right\} \\
\bmf{u}_{s}^{\mathrm{sc}}(\bmf{x};k_p,k_s, \mathbf{d}) &=\frac{\mathrm{e}^{\mathrm{i} k_{s} r}}{\sqrt{r}}\left\{u_{s}^{\infty}(\hat{\bmf{x}}; \bmf{d}) \hat{\bmf{x}}^{\perp}+{\mathcal O}\left(\frac{1}{r}\right)\right\}
\end{aligned} \end{equation}
as $r =|\bmf{x} | \rightarrow \infty$, where $u_{p}^{\infty}$ and $u_{s}^{\infty}$ are both scalar functions defined on $\mathbb S^1$. Hence, a Kupradze radiating solution has the asymptotic behavior $$ \bmf{u}^{\mathrm{sc}}(\bmf{x}; k_p,k_s, \mathbf{d})=\frac{\mathrm{e}^{\mathrm{i} k_{p} r}}{\sqrt{r}} u_{p}^{\infty}(\hat{\bmf{x} }; \bmf{d} ) \hat{\bmf{x}}+\frac{\mathrm{e}^{\mathrm{i} k_{s} r}}{\sqrt{r}} u_{s}^{\infty}(\hat{\bmf{x} }; \bmf{d} ) \hat{\bmf{x} }^{\perp}+{\mathcal O}\left(\frac{1}{r^{3 / 2}}\right) \quad \text { as } \quad r \rightarrow \infty $$ The far-field pattern $\bmf{u}^\infty$ of $\bmf{u}^{\mathrm{sc}} $ is defined as $$ \bmf{u}_t^{\infty}(\hat{\bmf{x}}; \bmf{d} ) :=u_{p}^{\infty}(\hat{\bmf{x}}; \bmf{d} ) \hat{\bmf{x}}+u_{s}^{\infty}(\hat{\bmf{x}}; \bmf{d} ) \hat{\bmf{x}}^{\perp}. $$ Obviously, the compressional and shear parts of the far-field are uniquely determined by $\bmf{u}^{\infty}$ as follows: $$ \bmf{u}_{p}^{\infty}(\hat{\mathbf{x}}; \bmf{d} )=\bmf{u}^{\infty}(\hat{\bmf{x}};\bmf{d} ) \cdot \hat{\bmf{x}}; \quad \bmf{u}_{s}^{\infty}(\hat{\bmf{x}},\bmf{d} )=\bmf{u}^{\infty}(\hat{\bmf{x}}; \bmf{d} ) \cdot \hat{\bmf{x}}^{\perp}. $$
The inverse elastic scattering problem corresponding to \eqref{forward} concerns the determination of the scatterer $\Omega$ (and $\boldsymbol \eta$ as well in the impedance case) by knowledge of the far-field pattern $\bmf{u}_\beta^\infty(\hat{\mathbf{ x}},\mathbf{d},k)$, where $\beta=t,p$ or $s$. We introduce the operator $\mathcal{F}$ which sends the obstacle to the corresponding far-field pattern and is defined by the forward scattering system \eqref{forward}, the aforementioned inverse problem can be formulated as \begin{equation}\label{inverse} \mathcal{F}(\Omega, \eta)=\bmf{u}_\beta^\infty(\hat{\mathbf{ x}}; \mathbf{d}),\quad \beta=t,p, \mbox{ or } s. \end{equation}
Next, we show that by using the generalized Holmgren's uniqueness principle, we can establish two novel unique identifiability results for \eqref{inverse} in determining an obstacle without knowing its a priori physical property as well as its possible surface impedance by at most six far-field patterns, namely $\bmf{u}_\beta^\infty(\hat{\mathbf{x}})$ corresponding to six distinct $\mathbf{d}$'s.
\begin{defn}\label{def:61} Let $Q\subset\mathbb{R}^2$ be a polygon in $\mathbb{R}^2$ such that \begin{equation}\label{eq:edge} \partial Q=\cup_{j=1}^\ell \Gamma_j, \end{equation} where each $\Gamma_j$ is an edge of $\partial Q$. $Q$ is said to be a generalized elastic obstacle associated with \eqref{forward} if there exists a Lipschitz dissection of $\Gamma_j$, $1\leq j\leq \ell$, \[ \Gamma_j=\cup_{i=1}^6 \Gamma_i^j \] such that \begin{equation}\label{eq:66} \begin{split} \mathscr{B}_i(\bmf{u})&=\mathbf{0} \quad \text { on } \Gamma_{i}^j, \quad i=1,2,3,4,\\ \mathscr{B}_1(\bmf{u})+\boldsymbol\eta \mathscr{B}_2(\bmf{u})&=\mathbf{0} \quad \text { on } \Gamma_{5}^j,\quad \mathscr{B}_3(\bmf{u})+\boldsymbol\eta \mathscr{B}_4(\bmf{u})=\mathbf{0} \quad \text { on } \Gamma_{6}^j,
\end{split} \end{equation} where $\boldsymbol \eta \in \mathcal{A}$ with $\Im( \boldsymbol \eta) \geq 0$, $\mathscr{B}_1(\mathbf{ u})$, $\mathscr{B}_2(\mathbf{ u})$, $\mathscr{B}_3(\mathbf{ u})$ and $\mathscr{B}_4(\mathbf{ u})$ are defined in \eqref{eq:B1u}, \eqref{eq:B2u}, \eqref{eq:B3u} and \eqref{eq:B4u}, respectively. \end{defn}
It is emphasized that in \eqref{eq:66}, either $\Gamma_1^j, \Gamma_2^j$, $\Gamma_3^j, \Gamma_4^j$, $\Gamma_5^j$ or $\Gamma_6^j$ could be an empty set, and hence a generalized-impedance obstacle could be purely a rigid obstacle, a traction-free obstacle, an impedance obstacle, a soft-clamped obstacle, a simply-supported obstacle, a generalized elastic obstacle or a mixed type. Moreover, one each edge of the polygonal obstacle, the impedance parameter can take different (complex) values. In order to simply notations, for $i\in \{1,3\}$ we formally write $\mathscr{B}_{i}(\bmf{u})+\boldsymbol\eta \mathscr{B}_{i+1}(\bmf{u})$ with $\boldsymbol \eta \equiv \infty$ to signify $\mathscr{B}_{i+1}(\bmf{u})=\bmf{0}$. In doing so, \eqref{eq:66} can be unified as $\mathscr{B}_{i}(\bmf{u})+\boldsymbol\eta \mathscr{B}_{i+1}(\bmf{u})=\bmf{0}$ on $\partial\Omega$ with \begin{equation}\label{eq:eta} \begin{split} \boldsymbol\eta=&0\cdot\chi_{\cup_{j=1}^\ell \Gamma_1^j}+0\cdot\chi_{\cup_{j=1}^\ell \Gamma_3^j}+\infty\cdot\chi_{\cup_{j=1}^\ell \Gamma_2^j}+\infty\cdot\chi_{\cup_{j=1}^\ell \Gamma_4^j}+\sum_{j=1}^\ell \boldsymbol \eta_j\cdot\chi_{\Gamma_5^j}\\ &+\sum_{j=1}^\ell \boldsymbol \eta_j\cdot\chi_{\Gamma_6^j}\quad (\boldsymbol \eta_j \in \mathcal A) \end{split} \end{equation} and $i \in \{1,3\}$. We write $(Q,\boldsymbol\eta)$ to denote a generalized polygonal impedance obstacle as describe above with $\boldsymbol\eta\in L^\infty(\partial Q)\cup\{\infty\}$. In what follows, $(\Omega , \boldsymbol\eta)$ is said to be an admissible complex obstacle if \begin{equation}\label{eq:p1} (\Omega, \boldsymbol\eta)=\cup_{j=1}^p (\Omega_j, \boldsymbol\eta_j), \end{equation} where each $(\Omega_j, \boldsymbol\eta_j)$ is a generalized polygonal impedance obstacle such that $\Omega_j, j=1, 2,\ldots, p$ are pairwise disjoint and \begin{equation}\label{eq:r2b} \boldsymbol\eta=\sum_{j=1}^p \boldsymbol\eta_j\chi_{\partial\Omega_j},\quad \boldsymbol\eta_j\in L^\infty(\partial\Omega_j)\cup\{\infty\}, \end{equation} the constant part $\eta_j$ of the variable function $\boldsymbol\eta_j \in \mathcal A$ is not equal to $$ \pm {\mathrm{i}} \mbox{ and } \frac{\pm \sqrt{(\lambda + 3\mu)(\lambda + \mu)} - \mu {\mathrm{i}}}{\lambda + 2 \mu}. $$
The following auxiliary lemma states the linear independence of the set of total wave fields $\{\bmf{u}(\bmf{x}; k_p,k_s, \mathbf{d}_\ell), \, \ell=1,\ldots, n\}$ associated with the incident wave \eqref{eq:ui}, where $\mathbf{d}_\ell$ are pairwise distinct. \begin{lem}\cite[Lemma 5.1]{DLW}\label{lem:51}
Let $\mathbf {\mathbf d}_{\ell}\in\mathbb{S}^1$, $\ell=1,\ldots, n$, be $n$ vectors
which are distinct from each other. Suppose that $\Omega$ is a bounded Lipschitz domain and $\mathbb R^2\backslash \overline{\Omega } $ is connected. Let the incident elastic wave filed $\bmf{u}^i(\mathbf{ x};k_p,k_s, \mathbf{d}_\ell)$ be defined in \eqref{eq:ui}. Furthermore, suppose that the total elastic wave filed $\bmf{u}(\bmf{x}; k_p,k_s, \mathbf{d}_\ell)$ associated with $\bmf{u}^i(\mathbf{ x};k_p,k_s, \mathbf{d}_\ell)$
satisfies \eqref{forward}. Then {the following set of functions is linearly independent:}
$$
\{\bmf{u}(\bmf{x}; k_p,k_s, \mathbf{d}_\ell);~\mathbf x \in D , \ \ \ell=1,2,\ldots, n \},
$$
where $D \subset \mathbb R^2 \backslash \overline \Omega $ is an open set. \end{lem}
\begin{thm}\label{thm:uniqueness1} Let $(\Omega, \boldsymbol \eta)$ and $(\widetilde\Omega, \widetilde{\boldsymbol\eta})$ be two admissible complex obstacles. Let $\omega\in\mathbb{R}_+$ be fixed and $\mathbf{d}_\ell$, $\ell=1, \ldots, 8$ be eight distinct incident directions from $\mathbb{S}^1$. Let $\bmf{u}_\beta^\infty$ and $\widetilde{\bmf{u}}^\infty_\beta$ be, respectively, the far-field patterns associated with $(\Omega, \boldsymbol\eta)$ and $(\widetilde\Omega, \widetilde{\boldsymbol\eta})$, where $\beta=t,p, \mbox{ or } s$. If \begin{equation}\label{eq:cond1} \bmf{u}_\beta^\infty(\hat{\mathbf{ x}}; \mathbf{d}_\ell )=\widetilde{\bmf{u}}_\beta^\infty(\hat{\mathbf{ x}}; \mathbf{d}_\ell), \ \ \hat{\mathbf x}\in\mathbb{S}^1, \ell=1, \ldots, 8, \end{equation} then one has that \begin{equation}\label{eq:u1n} \Omega =\widetilde{\Omega}\mbox{ and } \boldsymbol\eta=\widetilde {\boldsymbol\eta}. \end{equation}
\end{thm}
\begin{proof} By an absurdity argument, we first prove that if \eqref{eq:cond1} holds, one must have that $\Omega=\widetilde \Omega$. Suppose that $\Omega $ and $\widetilde{\Omega}$ are two different admissible complex obstacles such that $\Omega\neq \widetilde\Omega$ and \eqref{eq:cond1} holds. Let $\mathbf{G}$ denote the unbounded connected component of $\mathbb{R}^2\backslash\overline{(\Omega\cup\widetilde\Omega)}$. Then by a similar topological argument to that in \cite{Liu-Zou}, one can show that there exists a line segment $\Gamma_h\subset\partial\mathbf{G}\backslash\partial\Omega$ or $\Gamma_h\subset\partial\mathbf{G}\backslash\partial\widetilde\Omega$. Without loss of generality, we assume the former case.
Let $\bmf{u}$ and $\widetilde{\bmf{u}}$ respectively denote the total wave fields to \eqref{forward} associated with $(\Omega, \eta)$ and $(\widetilde\Omega, \widetilde\eta)$. By \eqref{eq:cond1} and the Rellich theorem (cf. \cite{CK}), we know that \begin{equation}\label{eq:aa3} \bmf{u} (\mathbf x; k_p,k_s, \mathbf{d}_\ell)=\widetilde{\bmf{u} }(\mathbf x; k_p,k_s,\mathbf{d}_\ell),\quad \mathbf{x}\in\mathbf{G},\ \ell=1, \ldots, 8. \end{equation}
By using \eqref{eq:aa3} as well as the generalized-impedance boundary condition on $\partial\widetilde\Omega$, we readily have \begin{equation}\label{eq:aa4}
\mathscr{B}_{i}(\bmf{u})+{\widetilde {\boldsymbol\eta}} \mathscr{B}_{i+1}(\bmf{u})=
\mathscr{B}_{i}( \widetilde {\bmf{u}} )+\widetilde{\boldsymbol \eta} \mathscr{B}_{i+1}(\widetilde{ \bmf{u}} )=\bmf{0}\quad\mbox{on}\ \ \Gamma_{h} \mbox{ for } i \in \{1,3\}. \end{equation}
Consider a fixed point $\mathbf x_0 \in \Gamma_h $. There exits a sufficient small positive number $\varepsilon \in {\mathbb R}_+$ such that $B_{2\varepsilon} (\mathbf x_0) \Subset \mathbf{G} $, where $B_{2\varepsilon }(\mathbf x_0)$ is a disk centered at $\mathbf x_0$ with the radius $2\varepsilon$. Without loss of generality, we may assume that $\mathbf x_0 $ is the origin. Let $\Gamma_\varepsilon = B_\varepsilon (\mathbf 0) \cap \Gamma_h$, where $B_{\varepsilon }(\mathbf 0)$ is a disk centered at $\mathbf{x_0}$ with the radius $\varepsilon$. It is also noted that $$ -{\mathcal L} \bmf{u}= \omega^2 \bmf{u} \mbox{ in } B_{2\varepsilon } (\mathbf 0). $$ Therefore due to \eqref{eq:u} one knows that $\bmf{u}(\mathbf x; k_p,k_s, \mathbf{d}_\ell)$ has the Fourier expansion around the origin as follows \begin{equation}\label{eq:u new}
\begin{aligned}
\mathbf{u}(\mathbf{x}; k_p,k_s, \mathbf{d}_\ell)= &\sum_{m=0}^{\infty} \left\{ \frac{k_p}{2} a_{\ell,m} \mathrm{e}^{{\mathrm{i}} m \varphi} \left\{J_{m-1}\left(k_{p} r\right)\mathrm{e}^{-{\mathrm{i}} \varphi}\mathbf{e}_1
-J_{m+1}\left(k_{p}r\right)\mathrm{e}^{{\mathrm{i}} \varphi}\mathbf{e}_2 \right\}\right.\\
& \, + \frac{{\mathrm{i}} k_s}{2} b_{\ell,m} \mathrm{e}^{{\mathrm{i}} m \varphi} \left\{J_{m-1}\left(k_{s} r\right)\mathrm{e}^{-{\mathrm{i}} \varphi}\mathbf{e}_1
+J_{m+1}\left(k_{s} r\right)\mathrm{e}^{{\mathrm{i}} \varphi}
\mathbf{e}_2
\right\} \bigg\} .
\end{aligned} \end{equation}
Recall that the unit normal vector $\nu $ and the tangential vector $\boldsymbol{\tau}$ to $\Gamma_h$ are defined in \eqref{eq:nutau}, respectively. Due to the linear dependence of eight $\mathbb{C}^7$-vectors, it is easy to see that there exist eight complex constants $a_\ell$ such that
\begin{equation}\notag
\sum_{\ell=1}^8 \alpha_{\ell } \begin{bmatrix}
\bmf{u} (\mathbf 0; k_p,k_s, \mathbf{d}_\ell)\cr
\boldsymbol{\nu}^\top \nabla \bmf{u} \boldsymbol{\nu}|_{ \mathbf x=\bmf{0} }\\
\boldsymbol{\tau}^\top \nabla \bmf{u} \boldsymbol{\nu} |_{ \mathbf x=\bmf{0} }\\
b_{\ell,0}\\
b_{\ell,1}\\
b_{\ell,2} \end{bmatrix}=\bmf{0},
\end{equation} where $b_{\ell,m}$ is given in \eqref{eq:u new}, $m=0,1,2$. Moreover, there exits at least one $\alpha_\ell $ is not zero. Let \begin{equation}\label{eq:u513} \begin{split}
\bmf{u}(\bmf{x};k_p,k_s)&=\sum_{\ell=1}^8 \alpha_{\ell } \bmf{u} (\mathbf x; k_p,k_s, \mathbf{d}_\ell)\\
&=\sum_{m=0}^{\infty} \left\{ \frac{k_p}{2} a_{m} \mathrm{e}^{{\mathrm{i}} m \varphi} \left\{J_{m-1}\left(k_{p} r\right)\mathrm{e}^{-{\mathrm{i}} \varphi}\mathbf{e}_1
-J_{m+1}\left(k_{p}r\right)\mathrm{e}^{{\mathrm{i}} \varphi}\mathbf{e}_2 \right\}\right.\\
& \, + \frac{{\mathrm{i}} k_s}{2} b_{m} \mathrm{e}^{{\mathrm{i}} m \varphi} \left\{J_{m-1}\left(k_{s} r\right)\mathrm{e}^{-{\mathrm{i}} \varphi}\mathbf{e}_1
+J_{m+1}\left(k_{s} r\right)\mathrm{e}^{{\mathrm{i}} \varphi}
\mathbf{e}_2
\right\} \bigg\},
\end{split} \end{equation} where $$ a_{m}=\sum_{\ell=1}^8 \alpha_\ell a_{\ell,m},\quad b_{m}=\sum_{\ell=1}^8 \alpha_\ell b_{\ell,m},\quad \forall m \in \mathbb N\cup\{0\}. $$ Then we have \begin{equation}\label{eq:611}
\bmf{u}(\bmf{0};k_p,k_s,\mathbf d_\ell )=\bmf{0} \mbox{ and }
\boldsymbol{\nu}^\top \nabla \bmf{u} \boldsymbol{\nu}|_{ \mathbf x=\bmf{0} }=\boldsymbol{\tau}^\top \nabla \bmf{u} \boldsymbol{\nu} |_{ \mathbf x=\bmf{0} }=b_m =0,\, m=0,1,2. \end{equation} Next we distinguish two separate cases. The first case is that $ \bmf{u}(\bmf{x};k_p,k_s)\equiv \bmf{0},\, \forall \bmf{x} \in \mathbf{G}$. In view of \eqref{eq:u513}, since $\alpha_\ell $ are not all zero and $\bmf{d}_\ell$ are distinct, we readily have a contradiction by Lemma \ref{lem:51}. For the second case, we suppose that $ \bmf{u}(\bmf{x};k_p,k_s)\equiv\hspace*{-3mm}\backslash\ \bmf{0}$.
In view of \eqref{eq:aa4} and \eqref{eq:611}, recalling Definitions \ref{def:2} and \ref{def:3}, we know that $\Gamma_\varepsilon $ is a singular line of $\bmf{u}$, which implies that $\Gamma_\varepsilon $ could be a singular rigid, or singular traction-free, or singular impedance, or singular soft-clamped, or singular simply-supported or singular generalized-impedance line of $\bmf{u}$ in Definition \ref{def:2} and \ref{def:3}. Therefore, by the generalized Holmgren's principle (cf. \cite[Theorems 3.1 and 3.2]{DLW}, Theorems \ref{thm:impedance line}, \ref{thm:third}, \ref{thm:fourth} and \ref{thm:g im}), we obtain that \begin{equation}\label{eq:613 con}
\bmf{u}\equiv \bmf{0} \mbox{ in } B_{2\varepsilon }(\bmf{0}), \end{equation} which is obviously a contradiction.
Next, we prove that by knowing $\Omega =\widetilde{\Omega}$, one must have that $\boldsymbol \eta \equiv \widetilde {\boldsymbol \eta}$. Assume contrarily that $\boldsymbol \eta \neq \widetilde{\boldsymbol \eta}$. We consider two separated cases. The first case is that the involved boundary condition of $\mathbf u$ given by $$ \mathscr{B}_1(\bmf{u})+\boldsymbol \eta \mathscr{B}_2(\bmf{u})=\mathscr{B}_1(\bmf{u})+\widetilde {\boldsymbol \eta} \mathscr{B}_2(\bmf{u})=\mathbf{0} \mbox{ on } \Sigma \Subset \partial \Omega=\partial \widetilde \Omega, $$ where $\mathscr{B}_1(\bmf{u})$ and $\mathscr{B}_2(\bmf{u})$ are defined in \eqref{eq:B1u} and \eqref{eq:B2u}, respectively. One can easily show that there exists an open subset $\Sigma$ of $\partial \Omega=\partial \widetilde \Omega$ such that \begin{equation}\label{eq:523}
\mathbf{u}=T_\nu \mathbf{u}=\mathbf{0} \mbox { on } \Sigma. \end{equation} The second case is that the involved boundary condition of $\mathbf u$ given by $$ \mathscr{B}_3(\bmf{u})+\boldsymbol \eta \mathscr{B}_4(\bmf{u})=\mathscr{B}_3(\bmf{u})+\widetilde {\boldsymbol \eta} \mathscr{B}_4(\bmf{u})=\mathbf{0} \mbox{ on } \Sigma \Subset \partial \Omega=\partial \widetilde \Omega, $$ where $\mathscr{B}_3(\bmf{u})$ and $\mathscr{B}_4(\bmf{u})$ are defined in \eqref{eq:B3u} and \eqref{eq:B4u}, respectively. Since $\boldsymbol \eta \neq \widetilde {\boldsymbol \eta}$, one can readily have $$ \mathscr{B}_3(\bmf{u})=\mathscr{B}_4(\bmf{u})=\mathbf 0. $$ By \eqref{eq:B3u} and \eqref{eq:B4u}, it is obvious that $$ \nu \cdot \mathbf u=\boldsymbol{\tau} \cdot \mathbf u=\nu \cdot T_\nu \mathbf u=\boldsymbol{\tau} \cdot T_\nu\mathbf u=0 \mbox{ on } \Sigma. $$ Since $\nu \perp \boldsymbol{\tau}$, $\nu \in \mathbb R^2$, $\boldsymbol{\tau} \in \mathbb R^2$, by distinguishing the real and imaginary parts of $\mathbf u$ and $T_\nu \mathbf u$, one can know that \eqref{eq:523} holds.
Therefore for these two cases by the classical Holmgren's principle, we know that $\mathbf{u}\equiv \mathbf{0}$ in ${\mathbb R}^2 \backslash \Omega $, which readily yields a contradiction.
The proof is complete.
\end{proof}
\begin{rem}
The shape of an admissible complex obstacle and its physical properties (in the case that $\boldsymbol \eta=0$ or $\boldsymbol \eta=\infty$) can be determined by at most eight far-field patterns. Furthermore, if it is of generalized-impedance type or impedance type, one can determine the surface generalized-impedance parameter as well. In Table \ref{table:results}, we summarize the unique identifiability result for a pure polygonal rigid obstacle, a pure polygonal traction-free obstacle, a pure polygonal impedance obstacle, a pure polygonal soft-clamped obstacle, a pure polygonal simply-supported obstacle and a pure polygonal generalized-impedance obstacle by analyzing the assumptions' quantities in Definitions \ref{def:2} and \ref{def:3}. \end{rem}
\begin{table} [!htbp]
\centering
{
\begin{tabular}{|c|c|}
\hline
Types of the elastic obstacle &\tabincell{c}{ The number of distinct incident directions \\ are needed for the unique identifiability}\\
\hline
\tabincell{c} {a pure polygonal \\rigid obstacle} &2 \\
\hline
\tabincell{c}{a pure polygonal\\ traction-free obstacle} &4 \\
\hline
\tabincell{c}{a pure polygonal\\ impedance obstacle} &4 \\
\hline
\tabincell{c}{a pure polygonal\\ soft-clamped obstacle} &2 \\
\hline
\tabincell{c}{a pure polygonal \\
simply-supported obstacle} &2 \\
\hline
\tabincell{c}{a pure polygonal \\ generalized-impedance obstacle }&6 \\
\hline
\end{tabular}
}
\caption{}
\label{table:results} \end{table}
Finally, we show that if fewer far-field patterns are used, one can establish a local uniqueness result in determining a generic class of admissible complex obstacles. In \cite[Theorem 5.2]{DLW}, the corresponding local uniqueness result are considered by imposing a certain condition on the degree of an admissible complex obstacle; please refer to the detailed discussions in \cite[Theorem 5.2]{DLW}.
To that end, we first introduce an admissible complex obstacle with the class $\mathcal C$. Let $\Omega$ be defined in \eqref{eq:p1} that consists of finitely many pairwise disjoint polygons. Let $\Gamma, \Gamma'\subset\partial\Omega$ be two adjacent edges of $\partial\Omega$. Moreover, we let $\boldsymbol \zeta$ and ${\boldsymbol \zeta}'$ respectively signify the values of $\boldsymbol \eta$ on $\Gamma$ and $\Gamma'$ around the vertex formed by those two edges. It is noted that $\boldsymbol \zeta$ and $\boldsymbol \zeta'$ may be $0, \infty$ or a variable function belonging to the class $\mathcal A$ introduced in Definition \ref{def:3}. If $\boldsymbol \zeta \in \mathcal A$, according to Definition \ref{def:3}, $\boldsymbol \zeta$ is given by the following absolutely convergent series at the intersecting point $\Gamma \cap \Gamma'$: \begin{equation}\label{eq:zeta1 ex}
\boldsymbol{\zeta}=\zeta+\sum_{j=1}^\infty \zeta_{1,j} r^j,
\end{equation}
where $\zeta\in\mathbb{C}\backslash\{0\}$, $\zeta_{1,j}\in \mathbb C$ and $r\in [0,h]$. Similarly, if $\boldsymbol \zeta' \in \mathcal A$, $\boldsymbol \zeta'$ is given by the following absolutely convergent series at the intersecting point $\Gamma \cap \Gamma'$: \begin{equation}\label{eq:zeta2 ex}
\boldsymbol{\zeta}'=\zeta'+\sum_{j=1}^\infty \zeta_{2,j} r^j,
\end{equation}
where $\zeta'\in\mathbb{C}\backslash\{0\}$, $\zeta_{2,j}\in \mathbb C$ and $r\in [0,h]$.
An admissible complex obstacle $(\Omega, \eta)$ is said to belong to the class $\mathcal{C}$ if \begin{equation}\label{eq:cond1n} \Im(\zeta) \in \mathbb R_+\backslash\{1\}, \quad \Im(\zeta') \in \mathbb R_+\backslash\{1\}, \end{equation} where $\zeta$ and $\zeta'$ are defined in \eqref{eq:zeta1 ex} and \eqref{eq:zeta2 ex} respectively, and \begin{equation}\label{eq:cond2n} \zeta=\zeta'\quad \mbox{if both $\boldsymbol \zeta$ and $\boldsymbol \zeta'$ are variable functions given by \eqref{eq:zeta1 ex} and \eqref{eq:zeta2 ex} }, \end{equation} for all two adjacent edges $\Gamma, \Gamma'$ of $\partial\Omega$.
\begin{thm}\label{thm:uniqueness2} Let $(\Omega, \boldsymbol \eta)$ and $(\widetilde\Omega, \widetilde{\boldsymbol \eta})$ be two admissible complex obstacles from the class $\mathcal{C}$ as described above. Let $\omega \in\mathbb{R}_+$ be fixed and $\mathbf{d}_\ell$, $\ell=1, \ldots, 5$ be five distinct incident directions from $\mathbb{S}^1$. Let $\mathbf{G}$ denote the unbounded connected component of $\mathbb{R}^2\backslash\overline{(\Omega\cup\widetilde\Omega)}$. Let $\bmf{u}_\beta^\infty$ and $\widetilde{\bmf{u}}^\infty_\beta$ be, respectively, the far-field patterns associated with $(\Omega, \eta)$ and $(\widetilde\Omega, \widetilde\eta)$, where $\beta=t,p, \mbox{ or } s$. If \begin{equation}\label{eq:cond1 corner} \bmf{u}_\beta^\infty(\hat{\mathbf{ x}},\mathbf{d}_\ell )=\widetilde{\bmf{u}}_\beta^\infty(\hat{\mathbf{ x}},\mathbf{d}_\ell), \ \ \hat{\mathbf x}\in\mathbb{S}^1, \ell=1, \ldots, 5, \end{equation} then one has that $$ \left(\partial \Omega \backslash \partial \overline{ \widetilde{\Omega }} \right )\cup \left(\partial \widetilde{\Omega } \backslash \partial \overline{ \Omega } \right) $$
cannot have a corner on $\partial \mathbf{G}$. \end{thm}
\begin{proof}
We prove the theorem by contradiction. Assume \eqref{eq:cond1 corner} holds but $ \left(\partial \Omega \backslash \partial \overline{ \widetilde{\Omega }} \right )\cup \left(\partial \widetilde{\Omega } \backslash \partial \overline{ \Omega } \right) $ has a corner $\mathbf x_c$ on $\partial \mathbf{G}$. Clearly, $\mathbf x_c$ is either a vertex of $\Omega$ or a vertex of $\widetilde\Omega$. Without loss of generality, we assume the latter case. Let $h\in\mathbb{R}_+$ be sufficiently small such that $B_h(\bmf{x}_c)\Subset\mathbb{R}^2\backslash\overline \Omega $. Moreover, since $\bmf{x}_c$ is a vertex of $\widetilde\Omega$, we can assume that \begin{equation}\label{eq:aa2} B_h(\mathbf x_c)\cap \partial\widetilde\Omega=\Gamma_h^\pm, \end{equation} where $\Gamma_h^\pm$ are the two line segments lying on the two edges of $\widetilde\Omega$ that intersect at $\mathbf x_c$. Furthermore, on $\Gamma_h^\pm$ the boundary conditions are given by \eqref{eq:66}.
By \eqref{eq:cond1 corner} and the Rellich theorem (cf. \cite{CK}), we know that \begin{equation}\label{eq:aa5} \bmf{u} (\mathbf x; k_p,k_s, \mathbf{d}_\ell)=\widetilde{\bmf{u} }(\mathbf x; k_p,k_s, \mathbf{d}_\ell),\quad x\in\mathbf{G},\ \ell=1,\ldots, 5. \end{equation} It is clear that $\Gamma_h^\pm\subset\partial\mathbf{G}$. Hence, by using \eqref{eq:aa3} as well as the generalized boundary condition \eqref{eq:66} on $\partial\widetilde\Omega$, we readily have \begin{equation}\label{eq:aa4new} \mathscr{B}_{i}(\bmf{u})+{\widetilde {\boldsymbol\eta}} \mathscr{B}_{i+1}(\bmf{u})=
\mathscr{B}_{i}( \widetilde {\bmf{u}} )+\widetilde{\boldsymbol \eta} \mathscr{B}_{i+1}(\widetilde{ \bmf{u}} )=\bmf{0}\quad\mbox{on}\ \ \Gamma_{h}^\pm \mbox{ for } i \in \{1,3\}. \end{equation} It is also noted that in $B_h(\mathbf x_c)$, $-{\mathcal L} \bmf{u}=\omega^2 \bmf{u}$.
Due to the linear dependence of five $\mathbb{C}^2$-vectors, we see that there exits five complex constants $\alpha_\ell $ such that $$ \sum_{\ell=1}^5 \alpha_\ell \bmf{u}(\bmf{x}_c;k_p,k_s,\bmf{d}_\ell)=\bmf{0}, $$ where there exits at least one $\alpha_\ell $ is not zero. Set $\bmf{u}(\bmf{x};k_p,k_s)=\sum_{\ell=1}^5 \alpha_\ell \bmf{u}(\bmf{x};k_p,k_s,\bmf{d}_\ell)$. Then we know that \begin{equation}\label{eq:611new}
\bmf{u}(\bmf{x}_c;k_p,k_s)=\bmf{0}. \end{equation} Similar to the proof of Theorem \ref{thm:uniqueness2}, we consider the following two cases. The first one is $ \bmf{u}(\bmf{x};k_p,k_s)\equiv \bmf{0},\, \forall \bmf{x}\in \bmf{G}$. Since there exits at one $\alpha_\ell$ such that $\alpha_\ell \neq 0$ and $\mathbf{d}_\ell$ are distinct, from Lemma \ref{lem:51}, we can arrive at a contradiction. The other case is that $ \bmf{u}(\bmf{x};k_p,k_s)\equiv\hspace*{-3mm}\backslash\ \bmf{0}$. By \eqref{eq:cond1n} and \eqref{eq:cond2n}, as well as the generalized Holmgren's principle in \cite[Theorems 4.1, 4.3]{DLW}, Theorems \ref{thm:54}--\ref{eq:impedance line exp}, Theorems \ref{thm:two traction}--\ref{thm:GI&forth exp}, one can show that \[ \mathbf{u} \equiv \mathbf{0} \mbox{ in } \mathbf{G}, \] which yields a contradiction again.
The proof is complete. \end{proof}
\subsection{Unique recovery for the inverse elastic diffraction grating problem}
In this subsection, we consider the unique recovery for the inverse elastic diffraction grating problem. First we give a brief review of the basic mathematical model for this inverse problem. Let the profile of a diffraction grating be described by the curve \begin{equation}\label{grating} \Lambda_f=\{(x_1,x_2) \in \mathbb{R}^2 ;~x_2 =f(x_1)\}, \end{equation} where $f$ is a periodic Lipschitz function with period $2\pi$. Let $$ \Omega_f=\{ \mathbf x\in \mathbb{R}^2; x_2 > f(x_1), x_1 \in \mathbb{R}\} $$ be filled with an elastic material whose mass density is equal to one. Suppose further that the incident wave is given by the pressure wave \begin{equation} \label{eq:p int} \mathbf u^i(\mathbf x;k_p, \mathbf d)=\mathbf d {\mathrm e}^{{\mathrm{i}} k_p \mathbf d \cdot \mathbf x},\quad \mathbf d=(\sin \theta, -\cos \theta)^\top ,\quad \theta \in \left( - \frac{\pi}{2} , \frac{\pi}{2} \right), \end{equation} or the shear wave \begin{equation}\label{eq:s int}
\mathbf u^i(\mathbf x;k_s, \mathbf d)=\mathbf d^\perp {\mathrm e}^{{\mathrm{i}} k_s \mathbf d \cdot \mathbf x},\quad \mathbf d^\perp=( \cos \theta,\sin \theta)^\top ,\quad \theta \in \left( - \frac{\pi}{2} , \frac{\pi}{2} \right), \end{equation} propagating to $\Lambda_f$ from the top, where $k_p$ and $k_s$ are defined in \eqref{eq:kpks}. Then the total wave field satisfies the following Navier system: \begin{equation}\label{eq:76}
\mathcal L \mathbf u+\omega ^2 \mathbf u=\mathbf 0\hspace*{0,5cm} \mbox{in}\ \ \Omega_f; \quad {\mathscr B}(\mathbf u)\big|_{\Lambda _f}=\mathbf 0 \hspace*{0,5cm} \mbox{on} \ \ \Lambda_f, \end{equation} with the generalized-impedance boundary condition \begin{equation}\label{eq:gbc} \mathscr{B}(\mathbf u)=\mathscr{B}_{i}(\bmf{u})+\boldsymbol\eta \mathscr{B}_{i+1}(\bmf{u})=\bmf{0}\quad\mbox{on}\ \ \Lambda_f \mbox{ for } i \in \{1,3\} \end{equation} where $\boldsymbol \eta$ can be $\infty$, $0$, or $\boldsymbol \eta \in \mathcal{A}$ with $\Im( \boldsymbol \eta) \geq 0$, and $\mathscr{B}_1(\mathbf{ u})$, $\mathscr{B}_2(\mathbf{ u})$, $\mathscr{B}_3(\mathbf{ u})$ and $\mathscr{B}_4(\mathbf{ u})$ are defined in \eqref{eq:B1u}, \eqref{eq:B2u}, \eqref{eq:B3u} and \eqref{eq:B4u}, respectively.
In what follows, we shall mainly consider the incident pressure wave \eqref{eq:p int}. To achieve the uniqueness of \eqref{eq:76}, the total wave field $\mathbf u$ should be $\alpha$-quasiperiodic in the $x_1$-direction, with $\alpha=k_p\sin \theta$ for the incident pressure wave \eqref{eq:p int},
which means that $$ \mathbf u(x_1+2\pi, x_2)=e^{2{\mathrm{i}} \alpha \pi}\cdot \mathbf u(x_1,x_2), $$ and the scattered field $u^s$ satisfies the Rayleigh expansion (cf.\cite{Arens,CGK}): \begin{align}\label{radiation} \mathbf u^s(\mathbf x;\omega, \mathbf d )&=\sum_{n\in \mathbb Z} u_{p,n}{\mathrm e}^{{\mathrm{i}} { \xi}_{p,n} (\theta ) \cdot \mathbf x } { \xi}_{p,n}(\theta )
+\sum_{n\in \mathbb Z} u_{s,n}{\mathrm e}^{{\mathrm{i}} { \xi}_{s,n} (\theta ) \cdot \mathbf x }\begin{bmatrix}
0& 1\cr-1&0 \end{bmatrix} { \xi}_{s,n} (\theta )
\end{align} for $ x_2 > \max_{x_1\in [0, 2\pi]} f(x_1)$, where $u_{p,n},\, u_{s,n}\in \mathbb{C}(n\in \mathbb Z)$ are called the Rayleigh coefficient of $\mathbf u^s$, and \begin{equation}\label{eq:839n} \begin{split} \xi_{p,n}(\theta )&=\left(\alpha_{n}(\theta), \beta_{p,n}(\theta)\right)^\top, \quad \alpha_{n}(\theta )=n+ \alpha, \\ \beta_{p,n}(\theta )&=\left\{\begin{array} {c}
\sqrt{k_p^2- \alpha_n^2 (\theta) }, \quad\mbox{ if } |\alpha_{n} (\theta )| \leq k_p\\ \\[1pt]
{\mathrm{i}} \sqrt{\alpha_{n}^2 (\theta)-k_p^2 }, \quad\mbox{ if } |\alpha_{n} (\theta)| > k_p \end{array}, \right. \\ \xi_{s,n}(\theta )&=\left(\alpha_{n}(\theta),\beta_{s,n}(\theta)\right)^\top, \\
\beta_{s,n}(\theta )&=\left\{\begin{array} {c}
\sqrt{k_s^2- \alpha_{n}^2 (\theta) }, \quad\mbox{ if } |\alpha_{n} (\theta )| \leq k_s\\ \\[1pt]
{\mathrm{i}} \sqrt{\alpha_{n}^2 (\theta)-k_s^2 }, \quad\mbox{ if } |\alpha_{n} (\theta)| > k_s \end{array}. \right. \end{split} \end{equation}
The existence and uniqueness of the $\alpha$-quasiperiodic solution to \eqref{eq:76} for the the first kind (Dirichlet) condition boundary condition can be found in \cite{Arens}. In our subsequent study, we assume the well-posedness of the forward scattering problem and focus on the study of the inverse grating problem.
Introduce a measurement boundary as $$
\Gamma_b:=\{ (x_1,b)\in \mathbb{R}^2 ;~ 0 \leq x_1 \leq 2 \pi, \, b> \max_{x_1\in [0, 2\pi]}|f(x_1)| \}. $$
The inverse diffraction grating problem is to determine $(\Lambda_f,\eta)$ from the knowledge of $\mathbf u(\mathbf x|_{\Gamma_b};\omega ,\mathbf d)$, and can be formulated as the operator equation: \begin{equation}\label{eq:iopd}
\mathcal{F}(\Lambda_f,\boldsymbol \eta )=\mathbf u(\mathbf x;k_p,\mathbf d), \quad \mathbf x\in \Gamma_b, \end{equation} where $\mathcal{F}$ is defined by the forward diffraction scattering system, and is nonlinear.
The unique recovery result on the inverse elastic diffraction grating problem with the the first kind boundary condition under a priori information about the height of the grating profile by a finite number of incident plane waves can be found in \cite{CGK}.
But the unique identifiability is still open for the impedance or generalized-impedance cases, which we shall resolve in the rest of the paper.
In doing so,
we introduce the following admissible polygonal gratings associated with
the inverse elastic diffraction grating problem.
\begin{defn}\label{def:dd1}
Let $(\Lambda_f, \boldsymbol \eta)$ be a periodic grating as described in \eqref{grating} with the generalized-impedance boundary condition \eqref{eq:gbc}. Suppose that $\Lambda_f$ restricted to the interval $[0,2\pi]$ consists of finitely many line segments $
\Gamma_j$ ($j=1,\ldots,\ell $), and $\Lambda_f$ is not a straight line parallel to the $x_1$-axis. $(\Lambda_f, \boldsymbol \eta)$ is said to be an admissible polygonal grating associated with \eqref{eq:iopd} if there exists a Lipschitz dissection of $\Gamma_j$, $1\leq j\leq \ell$, \[ \Gamma_j=\cup_{i=1}^6 \Gamma_i^j , \] such that \eqref{eq:66} is fulfilled, where the variable function $\boldsymbol \eta$ in \eqref{eq:66} satisfies
$ \boldsymbol{\eta}\in \mathcal{A}$ with $\Im( \boldsymbol \eta) \geq 0$, and the constant part of the variable function $\boldsymbol\eta \in \mathcal A$ is not equal to $$ \pm {\mathrm{i}} \mbox{ and } \frac{\pm \sqrt{(\lambda + 3\mu)(\lambda + \mu)} - \mu {\mathrm{i}}}{\lambda + 2 \mu}. $$
\end{defn}
We should emphasize that in \eqref{eq:66}, either $\Gamma_1^j, \Gamma_2^j$, $\Gamma_3^j, \Gamma_4^j$, $\Gamma_5^j$ or $\Gamma_6^j$ could be an empty set.
Next, we establish our uniqueness result in determining an admissible polygonal grating by at most eight incident waves. We first present a useful lemma.
\begin{lem}\cite[Lemma 8.1]{CDLZ} \label{lem:83}
Let $\mathbf \xi_{\ell}\in\mathbb{R}^2$, $\ell=1,\ldots, n$, be $n$ vectors
which are distinct from each other, $D$ be an open set in $\mathbb{R}^2$.
Then all the functions in the following set are linearly independent:
$$
\{\mathrm e^{{\mathrm{i}} \mathbf \xi_{\ell} \cdot \mathbf x} ;~\mathbf x \in D, \ \ \ell=1,2,\ldots, n \}
$$ \end{lem}
\begin{thm}\label{thm:uniqueness1g}
Let $(\Lambda_f, \boldsymbol \eta)$ and $(\Lambda_{\widetilde f}, \widetilde{\boldsymbol \eta})$ be two admissible polygonal gratings.
Let $\omega \in\mathbb{R}_+$ be fixed and $\mathbf{d}_\ell$, $\ell=1, \ldots, 8$ be eight distinct incident directions from $\mathbb{S}^1$, with
\begin{equation}\label{eq:8incident} \notag
\mathbf{d}_\ell=(\sin \theta_\ell, -\cos \theta_\ell)^\top ,\quad \theta_\ell \in \left( - \frac{\pi}{2} , \frac{\pi}{2} \right).
\end{equation}
Let $\mathbf u(\mathbf x;\omega ,\mathbf d_\ell )$ and $\widetilde {\mathbf u}(\mathbf x;\omega ,\mathbf d_\ell )$ denote the total fields associated with $(\Lambda_f, \boldsymbol \eta)$ and $(\Lambda_{\widetilde f}, \widetilde{\boldsymbol \eta})$ respectively and let $\Gamma_b$ be a measurement boundary given by
$$
\Gamma_b:=\left\{ (x_1,b)\in \mathbb{R}^2 ;~ 0 \leq x_1 \leq 2 \pi, \, b>\max\left\{ \max_{x_1\in [0, 2\pi]}|f(x_1)|,\, \max_{x_1\in [0, 2\pi]}|\widetilde {f}(x_1)| \right\} \right \},
$$
If it holds that
\begin{equation}\label{eq:836}
\mathbf u(\mathbf x;\omega ,\mathbf d_\ell )=\widetilde {\mathbf u} (\mathbf x; \omega ,\mathbf d_\ell ),\quad \ell=1,\ldots,8, \quad \mathbf x=(x_1, b) \in \Gamma_b,
\end{equation}
then $$ \Lambda_{f}=\Lambda_{\widetilde f} \mbox{ and } \boldsymbol \eta= \widetilde{\boldsymbol \eta}. $$
\end{thm}
\begin{proof}
The proof follows from a similar argument to that for Theorem~\ref{thm:uniqueness1}, and
we only sketch the necessary modifications in this new setup.
By contradiction and without loss of generality, we assume that there exists a line segment $\Gamma_h$ of $\Lambda_{\widetilde f}$ which lies on $\partial\mathbf{G}\backslash\Lambda_{ f}$, where $\mathbf{G}$ be the unbounded connected component of $\Omega_f\cap \Omega_{\widetilde f}$.
{ First, by the well-posedness of the diffraction grating problem \eqref{eq:76}-\eqref{radiation} as well as the unique continuation, we show that $\mathbf u(\mathbf x;\omega ,\mathbf d_\ell )=\widetilde {\mathbf u}(\mathbf x;\omega ,\mathbf d_\ell )$ for $\mathbf{x}\in\mathbf{G}$. In fact, by introducing $\mathbf w(\mathbf{ x}; \omega , \mathbf d_\ell):=\mathbf u(\mathbf{ x}; \omega , \mathbf d_\ell)-\widetilde{\mathbf u}(\mathbf x;\omega ,\mathbf d_\ell)$, $\ell=1,\ldots, 8$,
{we see from \eqref{eq:836}} that $\mathbf w$ fulfils
\[
{\mathcal L}\mathbf w+\omega ^2 \mathbf w=0\mbox{ in }\mathbf{U}; \ \mathbf w=\mathbf 0 \mbox{ on } \Gamma_b\ \ \mbox{and}\ \ \mathbf w\ \mbox{satisfies the Rayleigh expansion \eqref{radiation}, }
\]
where $ {\mathbf U}:=\mathbf{G} \cap \{ \mathbf x\in \mathbb{R}^2; x_2 > b, x_1 \in \mathbb{R}\}$ with $\partial \mathbf{U}=\Gamma_b$. Hence, by the uniqueness of the solution to the diffraction grating problem, we readily know $\mathbf w= \mathbf 0$ in $\mathbf{U}$. On the other hand, since $\mathbf u(\mathbf x;\omega ,\mathbf d_\ell )$ and $\widetilde {\mathbf u}(\mathbf x;\omega ,\mathbf d_\ell )$ are analytic in $\mathbf G$,
{we know $\mathbf w(\mathbf x;\omega ,\mathbf d_\ell )=0$ in $\mathbf{G}$ by means of the analytic continuation, that is,}
$\mathbf u(\mathbf x;\omega ,\mathbf d_\ell )=\widetilde {\mathbf u}(\mathbf x;\omega ,\mathbf d_\ell )$ for $\mathbf{x}\in\mathbf{G}$.}
Next, using a similar argument to the proof of Theorem \ref{thm:uniqueness1}, we can prove that
\begin{equation}\label{eq:542}
\sum_{\ell=1}^8 \alpha_\ell \mathbf u(\mathbf x; \omega ,\mathbf d_\ell) = \mathbf 0 \quad \mbox{for} \quad x_2 > \max_{x_1\in [0, 2\pi]} f(x_1),
\end{equation}
where $\alpha_\ell$ are nonzero complex constant, $\ell=1,\ldots, 8$.
Next, when $x_2 > \max_{x_1\in [0, 2\pi]}|f(x_1)| $, $\mathbf u(\mathbf x;\omega , \mathbf d_\ell )$ has the Rayleigh expansion (cf.\cite{Arens}):
\begin{align}\label{eq:839}
\mathbf u(\mathbf x;\omega ,\mathbf d_\ell )&={\mathbf d}_\ell {\mathrm e}^{{\mathrm{i}} k_p \mathbf d_\ell \cdot \mathbf x}+ \sum_{n\in \mathbb Z} u_{p,n}(\theta_\ell ){\mathrm e}^{{\mathrm{i}} { \xi}_{p,n} (\theta_\ell ) \cdot \mathbf x } { \xi}_{p,n}(\theta_\ell )
\\ &\quad +\sum_{n\in \mathbb Z} u_{s,n}(\theta_\ell ){\mathrm e}^{{\mathrm{i}} { \xi}_{s,n} (\theta_\ell ) \cdot \mathbf x }\begin{bmatrix}
0& 1\cr-1&0 \end{bmatrix} { \xi}_{s,n} (\theta_\ell ) \notag
\end{align} for $x_2 > \max_{x_1\in [0, 2\pi]} f(x_1)$, where $\xi_n(\theta_\ell )$, $\alpha_n(\theta_\ell)$, $\beta_{p,n}(\theta_\ell )$ and $\beta_{s,n}(\theta_\ell )$ are defined in \eqref{eq:839n}. Using the definition of $\alpha_0(\theta_\ell)$ and $\beta_0(\theta_\ell) $ in \eqref{eq:839n}, we can readily show that
\begin{equation}\label{eq:840}
k_p\mathbf d_{\ell}=(\alpha_0(\theta_\ell), -\beta_0(\theta_\ell) )^\top .
\end{equation}
Substituting \eqref{eq:839} into \eqref{eq:542}, it holds that
\begin{align}\label{eq:842} \mathbf 0&= \sum_{\ell=1}^8 \alpha_\ell \mathbf d_\ell {\mathrm e}^{{\mathrm{i}} k \mathbf d_\ell \cdot \mathbf x}+ \sum_{n\in \mathbb Z} \sum_{\ell=1}^8 \alpha_\ell u_{p,n} (\theta_\ell ) {\mathrm e}^{{\mathrm{i}} { \xi}_{p,n} (\theta_\ell ) \cdot \mathbf x }{ \xi}_{p,n} (\theta_\ell )\\ &+\quad \quad \sum_{n\in \mathbb Z} \sum_{\ell=1}^8 \alpha_\ell u_{s,n}(\theta_\ell ){\mathrm e}^{{\mathrm{i}} { \xi}_{s,n} (\theta_\ell ) \cdot \mathbf x }\begin{bmatrix}
0& 1\cr-1&0 \end{bmatrix} { \xi}_{s,n} (\theta_\ell ) \notag\quad \mbox{for} \quad x_2 > \max_{x_1\in [0, 2\pi]} f(x_1),
\end{align}
where $u_{p,n}(\theta_\ell), \, u_{s,n}(\theta_\ell) \in \mathbb{C}(n\in \mathbb Z)$ are the Rayleigh coefficients of $\mathbf u^s(\mathbf x; \omega , \mathbf d_\ell)$ associated with the incident wave $\mathbf d_\ell {\mathrm e}^{{\mathrm{i}} k_p \mathbf d_\ell \cdot \mathbf x}$. In \eqref{eq:842}, taking inner product with $\hat{\mathbf e}_2$, then letting $x_2=l x_1$ ($l>0$), one has
\begin{equation}\notag
\begin{split}
S_1(x_1,lx_1 )=S_2(x_1,lx_1 ), \quad \forall x_1 \in \mathbb R_+,
\end{split}
\end{equation}
where
\begin{equation}\notag
\begin{split}
S_1(x_1,lx_1 )= &\sum_{\ell=1}^8 \alpha_\ell \cos(\theta_\ell) {\mathrm e}^{{\mathrm{i}} k \mathbf d_\ell \cdot (1,l) x_1}- \sum_{|\alpha_n (\theta_\ell)| \leq k_p} \sum_{\ell=1}^8 \alpha_\ell u_{p,n} (\theta_\ell ) {\mathrm e}^{{\mathrm{i}} { \xi}_{p,n} (\theta_\ell ) \cdot (1,l) x_1 }{ \beta }_{p,n} (\theta_\ell )\\
&\quad +\sum_{ |\alpha_n(\theta_\ell )| \leq k_s } \sum_{\ell=1}^8 \alpha_\ell u_{s,n}(\theta_\ell ){\mathrm e}^{{\mathrm{i}} { \xi}_{s,n} (\theta_\ell ) \cdot (1,l) x_1 } { \alpha }_{n} (\theta_\ell ),\\
S_2(x_1,lx_1 )&= \sum_{|\alpha_n (\theta_\ell)| >k_p} \sum_{\ell=1}^8 \alpha_\ell u_{p,n} (\theta_\ell ) {\mathrm e}^{{\mathrm{i}} { \xi}_{p,n} (\theta_\ell ) \cdot (1,l) x_1 }{ \beta }_{p,n} (\theta_\ell )\\
&\quad-\sum_{ |\alpha_n(\theta_\ell )| >k_s } \sum_{\ell=1}^8 \alpha_\ell u_{s,n}(\theta_\ell ){\mathrm e}^{{\mathrm{i}} { \xi}_{s,n} (\theta_\ell ) \cdot (1,l) x_1 } { \alpha }_{n} (\theta_\ell ).
\end{split}
\end{equation} Noting that $S_1(x_1,lx_1 )$ is an almost periodic function on $\mathbb R_+$ and $S_2(x_1,lx_1 )$ is exponentially decaying functions as $x_1 \rightarrow +\infty$, from \cite[(d) in Page 493]{Bottcher} we can prove that \begin{equation}\notag
\begin{split}
\max_{ x_1 \in \mathbb R_+ } |S_1(x_1,lx_1 )|=\limsup_{x_1\rightarrow +\infty } |S_1(x_1,lx_1 )|=\limsup_{x_1\rightarrow +\infty } |S_2(x_1,lx_1 )| \leq
\varepsilon
\end{split} \end{equation} for any $\varepsilon \in \mathbb R_+$. Therefore it yields that \begin{equation}\label{eq:548} S_1(x_1,lx_1 ) \equiv 0,\quad \forall x_1 \in \mathbb R_+. \end{equation}
Clearly, any two vectors of the set
$$
\bigcup_{\ell=1}^8 \{ k_p \mathbf d_\ell \}\bigcup_{\ell=1}^8 \bigcup \{\xi_{p,n} (\theta_\ell )~|~n\in \mathbb Z\}\bigcup_{\ell=1}^8 \bigcup \{\xi_{s,n} (\theta_\ell )~|~n\in \mathbb Z\}
$$
are distinct since $|\theta_\ell| < \pi/2$ and \eqref{eq:840}.
Using Lemma \ref{lem:83} and
\eqref{eq:548}, we can see $\alpha_\ell=0$ for $\ell=1,\ldots, 8$ by noting $\cos \theta_\ell \in \mathbb R_+$, which is a contradiction to the fact there exits an index $\ell_0\in \{ 1,\ldots, 8\}$ such that $\alpha_{\ell_0} \neq 0$. \end{proof}
Similar to Theorem \ref{thm:uniqueness2}, we can establish a local uniqueness result in determining a generic class of admissible polygonal grating by using fewer scattering measurements, which we choose not to discuss the details in this paper.
\vspace*{-.3cm}
\section*{Appendix}
\begin{proof}[Proof of Lemma~\ref{lem:condition}] We first prove \eqref{eq:gradient1}. Using \eqref{eq:u1} and \eqref{eq:u2}, it is directly shown that \begin{eqnarray}
\frac{\partial u_1}{\partial r} &=& \sum_{m=0} ^\infty \left\{ \frac{k_p^2}{4}a_m \left\{\mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}\left(k_p r\right) - \left[\mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi}+\mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi}\right] J_m \left(k_p r\right) \right\}\right.\notag \\ && \hspace*{-3mm} + \frac{{\mathrm{i}} k_s^2}{4}b_m \left\{\mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}\left(k_s r\right) - \left[\mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi}-\mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi}\right] J_m \left(k_s r\right) \right\}\notag \\ && \hspace*{-3mm} + \frac{k_p^2}{4}a_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2} \left(k_p r\right)-\frac{{\mathrm{i}} k_s^2}{4}b_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2} \left(k_s r\right)
\bigg\},\notag \\
\frac{\partial u_1}{\partial \varphi}
& =& \sum_{m=0} ^\infty \left\{\frac{{\mathrm{i}}\left(m-1\right)}{2} k_p \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-1} \left(k_p r\right) a_m - \frac{{\mathrm{i}}\left(m+1\right)}{2} k_p \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+1} \left(k_p r\right) a_m \right. \notag \\
&& \hspace*{-3mm} - \frac{\left(m-1\right)}{2} k_s \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-1} \left(k_s r\right) b_m - \frac{\left(m+1\right)}{2} k_s \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+1} \left(k_s r\right) b_m\bigg\}, \label{eq:u1 par}
\end{eqnarray} and \begin{eqnarray}
\frac{\partial u_2}{\partial r} &=& \sum_{m=0} ^\infty \left\{\frac{{\mathrm{i}} k_p^2}{4}a_m \left\{\mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}\left(k_p r\right) - \left[\mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi}-\mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi}\right] J_m \left(k_p r\right)\right\}\right.\notag \\ &&\hspace*{-3mm} + \frac{k_s^2}{4}b_m \left\{-\mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}\left(k_s r\right) + \left[\mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi}+\mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi}\right] J_m \left(k_s r\right) \right\}\notag \\ &&\hspace*{-3mm} - \frac{{\mathrm{i}} k_p^2}{4}a_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2} \left(k_p r\right) -\frac{k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2} \left(k_s r\right) \bigg\},\notag \\
\frac{\partial u_2}{\partial \varphi} &=& \sum_{m=0} ^\infty \left\{-\frac{\left(m-1\right)}{2} k_p \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-1} \left(k_p r\right) a_m - \frac{\left(m+1\right)}{2} k_p \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+1} \left(k_p r\right) a_m \right.\notag \\
&& \hspace*{-3mm}- \frac{{\mathrm{i}} \left(m-1\right)}{2} k_s \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-1} \left(k_s r\right) b_m + \frac{{\mathrm{i}} \left(m+1\right)}{2} k_s \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+1} \left(k_s r\right) b_m\bigg\}. \label{eq:u2 par}
\end{eqnarray} For $i=1,2$, one has \begin{equation}\label{eq:u3 par}
\begin{split}
\frac{\partial u_i}{\partial x_1}&=\cos\varphi \cdot \frac{\partial u_i}{\partial r}- \frac{\sin \varphi}{r} \cdot \frac{\partial u_i}{\partial \varphi},\quad
\frac{\partial u_i}{\partial x_2}=\sin \varphi \cdot \frac{\partial u_i}{\partial r}+ \frac{\cos \varphi}{r} \cdot \frac{\partial u_i}{\partial \varphi}.
\end{split} \end{equation}
Combining \eqref{eq:u1 par}, \eqref{eq:u2 par} with \eqref{eq:u3 par}, after tedious but straightforward calculations, one can obtain that \begin{eqnarray}
& &\partial_1 u_1 \cdot \left(-\sin \varphi_0\right)+\partial_1 u_2 \cdot \left(\cos \varphi_0\right) \notag \\ && = \sum_{m=0} ^\infty \Bigg \{ \cos{\varphi}\Big[\frac{{\mathrm{i}} k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0} - \frac{{\mathrm{i}} k_p^2}{4} a_m J_m(k_p r)\mathrm{e}^{{\mathrm{i}} (m-1)\varphi}(\mathrm{e}^{{\mathrm{i}} \varphi_0} -\mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} )\notag \\ && \quad - \frac{{\mathrm{i}} k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} - \frac{ k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \notag \\ && \quad + \frac{ k_s^2}{4} b_m J_m(k_s r) \mathrm{e}^{{\mathrm{i}} (m-1) \varphi} (\mathrm{e}^{{\mathrm{i}} \varphi_0} +\mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} ) - \frac{ k_s^2}{4} b_m J_{m+2}(k_s r) \mathrm{e}^{{\mathrm{i}} (m+1) \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} \Big ] \notag \\ &&\quad + \frac{ \sin{\varphi} }{r} \Big [\frac{(m-1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0}\notag \\ && \quad + \frac{ (m+1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} + \frac{{\mathrm{i}} (m-1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0}\notag \\
&&\quad - \frac{{\mathrm{i}} (m+1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_s r) \mathrm{e}^{-{\mathrm{i}} \varphi_0}\Big] \Bigg \}, \notag \\ && \partial_2 u_1 \cdot \left(-\sin \varphi_0\right)+\partial_2 u_2 \cdot \left(\cos \varphi_0\right)\notag \\ && = \sum_{m=0} ^\infty \Bigg \{ \sin{\varphi } \Big [ \frac{{\mathrm{i}} k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0} - \frac{{\mathrm{i}} k_p^2}{4} a_m J_m(k_p r) \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} (\mathrm{e}^{{\mathrm{i}} \varphi_0} -\mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} ) \notag \\ &&\quad - \frac{{\mathrm{i}} k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} - \frac{ k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_s r)\mathrm{e}^{{\mathrm{i}} \varphi_0}\notag \\ &&\quad + \frac{ k_s^2}{4} b_m J_m(k_s r) \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} (\mathrm{e}^{{\mathrm{i}} \varphi_0} +\mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} ) - \frac{ k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2}(k_s r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} \Big] \notag \\ && \quad + \frac{\cos{\varphi} }{r} \Big[ \frac{-(m-1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0}\notag \\ && \quad - \frac{(m+1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} - \frac{{\mathrm{i}}(m-1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \notag \\
&&\quad + \frac{{\mathrm{i}}(m+1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_s r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} \Big] \Bigg \}, \label{eq:u4a par}
\end{eqnarray} and \begin{eqnarray}
&& \partial_1 u_1 \cdot \left(-\cos \varphi_0\right)+\partial_1 u_2 \cdot \left(-\sin \varphi_0\right) \notag \\ && \quad = \sum_{m=0} ^\infty \Bigg \{ \cos{\varphi}\Big[\frac{- k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0} + \frac{ k_p^2}{4} a_m J_m(k_p r)\mathrm{e}^{{\mathrm{i}} (m-1)\varphi}(\mathrm{e}^{{\mathrm{i}} \varphi_0} +\mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} ) \notag \\ &&\quad - \frac{ k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} - \frac{{\mathrm{i}} k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \notag \\ &&\quad + \frac{{\mathrm{i}} k_s^2}{4} b_m J_m(k_s r) \mathrm{e}^{{\mathrm{i}} (m-1) \varphi} (\mathrm{e}^{{\mathrm{i}} \varphi_0} -\mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} ) + \frac{{\mathrm{i}} k_s^2}{4} b_m J_{m+2}(k_s r) \mathrm{e}^{{\mathrm{i}} (m+1) \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} \Big ] \notag \\ &&\quad + \frac{ \sin{\varphi} }{r} \Big [\frac{(m-1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \notag \\ && \quad - \frac{{\mathrm{i}} (m+1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} - \frac{(m-1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \notag \\
&&\quad - \frac{ (m+1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_s r) \mathrm{e}^{-{\mathrm{i}} \varphi_0}\Big] \Bigg \}, \notag \\ && \partial_2 u_1 \cdot \left(-\cos \varphi_0\right)+\partial_2 u_2 \cdot \left(-\sin \varphi_0\right) \notag
\\ &&\quad = \sum_{m=0} ^\infty \Bigg \{ \sin{\varphi } \Big [ \frac{- k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0} + \frac{ k_p^2}{4} a_m J_m(k_p r) \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} (\mathrm{e}^{{\mathrm{i}} \varphi_0} + \mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} ) \notag \\ &&\quad - \frac{ k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} - \frac{{\mathrm{i}} k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_s r)\mathrm{e}^{{\mathrm{i}} \varphi_0} \notag \\ &&\quad + \frac{{\mathrm{i}} k_s^2}{4} b_m J_m(k_s r) \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} (\mathrm{e}^{{\mathrm{i}} \varphi_0} - \mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} ) + \frac{ {\mathrm{i}} k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2}(k_s r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} \Big] \notag\\ & &\quad + \frac{\cos{\varphi} }{r} \Big[ \frac{- {\mathrm{i}} (m-1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \notag \\ && \quad + \frac{{\mathrm{i}} (m+1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} + \frac{ (m-1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \notag \\ & &\quad + \frac{ (m+1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_s r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} \Big] \Bigg \}.\label{eq:u4b par}
\end{eqnarray}
Furthermore, we have \begin{eqnarray}
&& \partial_1 u_1 \cdot \left(-\sin \varphi_0\right)+\partial_2 u_1 \cdot \left(\cos \varphi_0\right) \notag \\ && = \sum_{m=0} ^\infty \bigg\{\frac{{\mathrm{i}} k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} (m-2)\varphi} J_{m-2}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0} + \frac{ k_p^2}{2} a_m \mathrm{e}^{{\mathrm{i}} m \varphi} J_{m}(k_p r) \sin \varphi_0 \notag\\ & &- \frac{{\mathrm{i}} k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} (m+2)\varphi} J_{m+2}(k_p r) \mathrm{e}^{ - {\mathrm{i}} \varphi_0} - \frac{ k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} (m-2)\varphi} J_{m-2}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0}\notag \\ & &- \frac{ k_s^2}{2} b_m \mathrm{e}^{{\mathrm{i}} m \varphi} J_{m}(k_s r) \cos \varphi_0 - \frac{ k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} (m+2) \varphi} J_{m+2}(k_s r) \mathrm{e}^{ - {\mathrm{i}} \varphi_0} \bigg \}.\notag\\ && \partial_1 u_2 \cdot \left(-\sin \varphi_0\right)+\partial_2 u_2 \cdot \left(\cos \varphi_0\right)\notag\\ & &= \sum_{m=0} ^\infty \bigg\{- \frac{ k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} (m-2)\varphi} J_{m-2}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0} - \frac{ k_p^2}{2} a_m \mathrm{e}^{{\mathrm{i}} m \varphi} J_{m}(k_p r) \cos \varphi_0 \notag\\ && - \frac{ k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} (m+2)\varphi} J_{m+2}(k_p r) \mathrm{e}^{ - {\mathrm{i}} \varphi_0} - \frac{ {\mathrm{i}} k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} (m-2)\varphi} J_{m-2}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0}\notag \\ & &- \frac{ k_s^2}{2} b_m \mathrm{e}^{{\mathrm{i}} m \varphi} J_{m}(k_s r) \sin \varphi_0 + \frac{ {\mathrm{i}} k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} (m+2) \varphi} J_{m+2}(k_s r) \mathrm{e}^{ - {\mathrm{i}} \varphi_0} \bigg \}.\label{eq:u4c par}
\end{eqnarray}
According to \eqref{eq:u4a par}, we can obtain that \begin{eqnarray}
& &-\sin \varphi_0 (-\sin \varphi_0 \cdot \partial_1 u_1 +\cos \varphi_0 \cdot \partial_1 u_2 ) + \cos \varphi_0 ( -\sin \varphi_0 \cdot \partial_2 u_1 + \cos \varphi_0 \cdot \partial_2 u_2) \notag \\ && = \sum_{m=0} ^\infty \Bigg \{ \sin(\varphi_0-\varphi) \Big[\frac{-{\mathrm{i}} k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \notag \\ & &\quad + \frac{{\mathrm{i}} k_p^2}{4} a_m J_m(k_p r)\mathrm{e}^{{\mathrm{i}} (m-1)\varphi}(\mathrm{e}^{{\mathrm{i}} \varphi_0} -\mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} )+ \frac{{\mathrm{i}} k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} \notag \\
&& \quad + \frac{ k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0} - \frac{ k_s^2}{4} b_m J_m(k_s r) \mathrm{e}^{{\mathrm{i}} (m-1) \varphi} (\mathrm{e}^{{\mathrm{i}} \varphi_0} +\mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} )\notag \\
&& \quad + \frac{ k_s^2}{4} b_m J_{m+2}(k_s r) \mathrm{e}^{{\mathrm{i}} (m+1) \varphi} J_{m+2}(k_s r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} \Big ] \notag \\ &&\quad + \frac{ \cos (\varphi_0-\varphi) }{r} \Big [\frac{-(m-1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0}\notag \\ & &\quad - \frac{ (m+1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} - \frac{{\mathrm{i}} (m-1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \notag \\
&&\quad + \frac{{\mathrm{i}} (m+1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_s r) \mathrm{e}^{-{\mathrm{i}} \varphi_0}\Big] \Bigg \}. \label{eq:u5 par}
\end{eqnarray}
By directly calculations, one has \begin{equation}\label{eq:u6 par} \begin{aligned}
&{\nu}^\top \nabla \bmf{u} {\nu}|_{\bmf x \in \Gamma_h^+ }
=\left(
\begin{array}{cc}
\nu_1 & \nu_2
\end{array}
\right)
\left(
\begin{array}{cc}
\partial_1 u_1 & \partial_2 u_1\\
\partial_1 u_2 & \partial_2 u_2\\
\end{array}
\right)
\left(
\begin{array}{c}
\nu_1 \\
\nu_2 \\
\end{array}
\right)\\
& = -\sin \varphi_0 (-\sin \varphi_0 \cdot \partial_1 u_1 +\cos \varphi_0 \cdot \partial_1 u_2 ) + \cos \varphi_0 ( -\sin \varphi_0 \cdot \partial_2 u_1 + \cos \varphi_0 \cdot \partial_2 u_2). \end{aligned} \end{equation} Substituting \eqref{eq:u5 par} into \eqref{eq:u6 par}, letting $\varphi=\varphi_0$, according to the assumption in \eqref{eq:gradient1}, we can deduce that \begin{equation}\label{eq:u7 par} \begin{aligned} 0=& \frac{1}{r} \sum_{m=0} ^\infty \Bigg\{ \frac{-(m-1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m-1}(k_p r) - \frac{ (m+1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m+1}(k_p r) \\ & - \frac{{\mathrm{i}} (m-1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m-1}(k_s r) + \frac{{\mathrm{i}} (m+1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m+1}(k_s r) \Bigg\}\\
= & \sum_{m=0} ^\infty \Bigg\{ \frac{-k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m-2}(k_p r) - \frac{k_p^2}{2} a_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_m (k_p r) - \frac{k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m+2}(k_p r)\\ & - \frac{{\mathrm{i}} k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m-2}(k_s r) + \frac{{\mathrm{i}} k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m+2}(k_s r) \Bigg\}, \end{aligned} \end{equation} where we use the property (cf.\cite{Abr}) \begin{align}\label{eq:property}
J_m\left(t\right)=\frac{t\left(J_{m-1}\left(t\right)+J_{m+1}\left(t\right)\right)}{2m}. \end{align}
Comparing the coefficients of $r^0$ on both sides of \eqref{eq:u7 par}, we can obtain \eqref{eq:gradient1}.
Next we shall prove \eqref{eq:gradient2}. According to \eqref{eq:u4b par}, we can obtain that \begin{eqnarray}
\begin{aligned} & -\sin \varphi_0 (-\cos \varphi_0 \cdot \partial_1 u_1 -\sin \varphi_0 \cdot \partial_1 u_2 ) + \cos \varphi_0 ( -\cos \varphi_0 \cdot \partial_2 u_1 - \sin \varphi_0 \cdot \partial_2 u_2) \\ & = \sum_{m=0} ^\infty \Bigg \{ \sin(\varphi_0-\varphi) \Big[\frac{ k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \\ & \quad - \frac{ k_p^2}{4} a_m J_m(k_p r)\mathrm{e}^{{\mathrm{i}} (m-1)\varphi}(\mathrm{e}^{{\mathrm{i}} \varphi_0} + \mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} )+ \frac{ k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} \left(m+1\right) \varphi} J_{m+2}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} \\
& \quad + \frac{{\mathrm{i}} k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} \left(m-1\right) \varphi} J_{m-2}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0} - \frac{{\mathrm{i}} k_s^2}{4} b_m J_m(k_s r) \mathrm{e}^{{\mathrm{i}} (m-1) \varphi} (\mathrm{e}^{{\mathrm{i}} \varphi_0} - \mathrm{e}^{2 {\mathrm{i}} \varphi} \mathrm{e}^{-{\mathrm{i}} \varphi_0} ) \\
& \quad - \frac{{\mathrm{i}} k_s^2}{4} b_m J_{m+2}(k_s r) \mathrm{e}^{{\mathrm{i}} (m+1) \varphi} J_{m+2}(k_s r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} \Big ] \\ &\quad + \frac{ \cos (\varphi_0-\varphi) }{r} \Big [\frac{-{\mathrm{i}} (m-1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_p r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \\ & \quad + \frac{ {\mathrm{i}} (m+1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_p r) \mathrm{e}^{-{\mathrm{i}} \varphi_0} + \frac{(m-1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m-1)\varphi} J_{m-1}(k_s r) \mathrm{e}^{{\mathrm{i}} \varphi_0} \\
&\quad + \frac{ (m+1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} (m+1)\varphi} J_{m+1}(k_s r) \mathrm{e}^{-{\mathrm{i}} \varphi_0}\Big] \Bigg \}. \label{eq:u8 par} \end{aligned}
\end{eqnarray}
One can readily deduce that \begin{equation}\label{eq:u9 par} \begin{aligned}
&\boldsymbol{\tau}^\top \nabla \bmf{u} {\nu}|_{\mathbf x \in \Gamma_h^+ }
=\left(
\begin{array}{cc}
\tau_1 & \tau_2
\end{array}
\right)
\left(
\begin{array}{cc}
\partial_1 u_1 & \partial_2 u_1\\
\partial_1 u_2 & \partial_2 u_2\\
\end{array}
\right)
\left(
\begin{array}{c}
\nu_1 \\
\nu_2 \\
\end{array}
\right)\\
& = -\sin \varphi_0 (-\cos \varphi_0 \cdot \partial_1 u_1 - \sin \varphi_0 \cdot \partial_1 u_2 ) + \cos \varphi_0 ( -\cos \varphi_0 \cdot \partial_2 u_1 - \sin \varphi_0 \cdot \partial_2 u_2). \end{aligned} \end{equation} Substituting \eqref{eq:u8 par} into \eqref{eq:u9 par}, letting $\varphi=\varphi_0$, from \eqref{eq:property}, according to the assumption in \eqref{eq:gradient2}, it yields that \begin{equation}\label{eq:u10 par} \begin{aligned}
0=& \frac{1}{r} \sum_{m=0} ^\infty \Bigg\{ \frac{- {\mathrm{i}} (m-1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m-1}(k_p r) + \frac{ {\mathrm{i}} (m+1)k_p}{2} a_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m+1}(k_p r) \\ & + \frac{ (m-1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m-1}(k_s r) + \frac{ (m+1)k_s}{2} b_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m+1}(k_s r) \Bigg\}\\
= & \sum_{m=0} ^\infty \Bigg\{ \frac{- {\mathrm{i}} k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m-2}(k_p r) + \frac{ {\mathrm{i}} k_p^2}{4} a_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m+2}(k_p r)\\ & + \frac{ k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m-2}(k_s r) + \frac{ k_s^2}{2} b_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m+2}(k_s r) + \frac{ k_s^2}{4} b_m \mathrm{e}^{{\mathrm{i}} m \varphi_0} J_{m+2}(k_s r) \Bigg\}. \end{aligned} \end{equation} Comparing the coefficients of $r^0$ on both sides of \eqref{eq:u10 par}, we can obtain \eqref{eq:gradient2}. \end{proof}
\end{document} | arXiv |
# Understanding convergence and divergence
Convergence and divergence are fundamental concepts in mathematics and computer science. Understanding these concepts is crucial for analyzing and solving various problems in these fields.
In this section, we will explore the definitions and properties of convergence and divergence. We will also discuss the importance of these concepts in various applications, such as numerical analysis, computer graphics, and optimization.
Consider the following series:
$$\sum_{n=1}^{\infty} \frac{1}{n^2}$$
This series converges because the terms decrease in size as $n$ increases, and the sum of the terms is finite. On the other hand, the series:
$$\sum_{n=1}^{\infty} \frac{1}{n}$$
diverges because the terms do not decrease in size, and the sum of the terms is infinite.
## Exercise
Instructions:
1. Write a JavaScript function that checks whether a given series converges or diverges. The function should take an array of numbers as input and return a string "Convergent" or "Divergent".
2. Test the function using the examples from the text.
Correct answer:
```javascript
function checkConvergence(series) {
// Your code here
}
let series1 = [1, 1/4, 1/9, 1/16, ...];
let series2 = [1, 1/2, 1/3, 1/4, ...];
console.log(checkConvergence(series1)); // Output: Convergent
console.log(checkConvergence(series2)); // Output: Divergent
```
# Iterative processes in JavaScript
Iterative processes are repetitive processes that perform a set of instructions multiple times until a certain condition is met. In JavaScript, you can use loops to implement iterative processes.
In this section, we will learn about different types of loops in JavaScript, such as `for`, `while`, and `do-while` loops. We will also discuss how to use these loops to implement iterative processes for solving mathematical problems.
Here is an example of a `for` loop in JavaScript:
```javascript
for (let i = 0; i < 5; i++) {
console.log(i);
}
```
This loop iterates over the numbers 0 to 4 and prints each number to the console.
## Exercise
Instructions:
1. Write a JavaScript function that calculates the sum of the first `n` terms of a geometric series. The function should take two arguments: the first term `a` and the common ratio `r`.
2. Test the function using the geometric series with first term 2 and common ratio 3/2.
Correct answer:
```javascript
function sumGeometricSeries(a, r, n) {
let sum = 0;
for (let i = 0; i < n; i++) {
sum += a * Math.pow(r, i);
}
return sum;
}
let a = 2;
let r = 3/2;
let n = 5;
console.log(sumGeometricSeries(a, r, n)); // Output: 13.875
```
# Recursive processes in JavaScript
Recursive processes are processes that call themselves to solve a problem. In JavaScript, you can use recursion to solve problems that can be broken down into smaller, similar problems.
In this section, we will learn how to implement recursive processes in JavaScript. We will also discuss how to use recursion to solve mathematical problems, such as calculating the factorial of a number.
Here is an example of a recursive function in JavaScript:
```javascript
function factorial(n) {
if (n === 0) {
return 1;
} else {
return n * factorial(n - 1);
}
}
```
This function calculates the factorial of a number `n` by calling itself with the argument `n - 1` until `n` is 0.
## Exercise
Instructions:
1. Write a JavaScript function that calculates the sum of the first `n` terms of a geometric series using recursion. The function should take two arguments: the first term `a` and the common ratio `r`.
2. Test the function using the geometric series with first term 2 and common ratio 3/2.
Correct answer:
```javascript
function sumGeometricSeries(a, r, n) {
if (n === 0) {
return 0;
} else {
return a * Math.pow(r, n - 1) + sumGeometricSeries(a, r, n - 1);
}
}
let a = 2;
let r = 3/2;
let n = 5;
console.log(sumGeometricSeries(a, r, n)); // Output: 13.875
```
# Visualizing convergence and divergence
In this section, we will also discuss how to use these visualizations to analyze the behavior of iterative and recursive processes.
Here is an example of a line chart created using Plotly.js:
```javascript
let trace1 = {
x: [0, 1, 2, 3],
y: [0, 2, 4, 6],
mode: 'lines',
name: 'Convergent'
};
let trace2 = {
x: [0, 1, 2, 3],
y: [1, 3, 5, 7],
mode: 'lines',
name: 'Divergent'
};
let layout = {
title: 'Convergence and Divergence',
xaxis: {title: 'Iterations'},
yaxis: {title: 'Values'}
};
Plotly.newPlot('plot', [trace1, trace2], layout);
```
This code creates a line chart with two lines representing convergent and divergent processes.
## Exercise
Instructions:
1. Create a line chart using D3.js or Plotly.js to visualize the convergence and divergence of the series from the previous section.
2. Add a slider to the chart that allows the user to change the number of terms in the series.
# Using visualizations to analyze convergence and divergence
We will also discuss how to use visualizations to solve mathematical problems, such as finding the limit of a sequence or determining the convergence or divergence of a series.
Here is an example of a line chart created using Plotly.js:
```javascript
let trace1 = {
x: [0, 1, 2, 3],
y: [0, 2, 4, 6],
mode: 'lines',
name: 'Convergent'
};
let trace2 = {
x: [0, 1, 2, 3],
y: [1, 3, 5, 7],
mode: 'lines',
name: 'Divergent'
};
let layout = {
title: 'Convergence and Divergence',
xaxis: {title: 'Iterations'},
yaxis: {title: 'Values'}
};
Plotly.newPlot('plot', [trace1, trace2], layout);
```
This code creates a line chart with two lines representing convergent and divergent processes.
## Exercise
Instructions:
1. Create a line chart using D3.js or Plotly.js to visualize the convergence and divergence of the series from the previous section.
2. Add a slider to the chart that allows the user to change the number of terms in the series.
# Creating interactive visualizations in JavaScript
In this section, we will also discuss how to use interactive visualizations to analyze convergence and divergence in mathematical problems.
Here is an example of a line chart created using Plotly.js:
```javascript
let trace1 = {
x: [0, 1, 2, 3],
y: [0, 2, 4, 6],
mode: 'lines',
name: 'Convergent'
};
let trace2 = {
x: [0, 1, 2, 3],
y: [1, 3, 5, 7],
mode: 'lines',
name: 'Divergent'
};
let layout = {
title: 'Convergence and Divergence',
xaxis: {title: 'Iterations'},
yaxis: {title: 'Values'}
};
Plotly.newPlot('plot', [trace1, trace2], layout);
```
This code creates a line chart with two lines representing convergent and divergent processes.
## Exercise
Instructions:
1. Create a line chart using D3.js or Plotly.js to visualize the convergence and divergence of the series from the previous section.
2. Add a slider to the chart that allows the user to change the number of terms in the series.
# Applications of convergence and divergence in computer graphics
Convergence and divergence are fundamental concepts in computer graphics. They are used to analyze and optimize various algorithms and data structures used in computer graphics.
In this section, we will discuss the applications of convergence and divergence in computer graphics, such as ray tracing, image rendering, and mesh simplification.
Here is an example of a line chart created using Plotly.js:
```javascript
let trace1 = {
x: [0, 1, 2, 3],
y: [0, 2, 4, 6],
mode: 'lines',
name: 'Convergent'
};
let trace2 = {
x: [0, 1, 2, 3],
y: [1, 3, 5, 7],
mode: 'lines',
name: 'Divergent'
};
let layout = {
title: 'Convergence and Divergence',
xaxis: {title: 'Iterations'},
yaxis: {title: 'Values'}
};
Plotly.newPlot('plot', [trace1, trace2], layout);
```
This code creates a line chart with two lines representing convergent and divergent processes.
## Exercise
Instructions:
1. Create a line chart using D3.js or Plotly.js to visualize the convergence and divergence of the series from the previous section.
2. Add a slider to the chart that allows the user to change the number of terms in the series.
# Implementing convergence and divergence in JavaScript
Here is an example of a line chart created using Plotly.js:
```javascript
let trace1 = {
x: [0, 1, 2, 3],
y: [0, 2, 4, 6],
mode: 'lines',
name: 'Convergent'
};
let trace2 = {
x: [0, 1, 2, 3],
y: [1, 3, 5, 7],
mode: 'lines',
name: 'Divergent'
};
let layout = {
title: 'Convergence and Divergence',
xaxis: {title: 'Iterations'},
yaxis: {title: 'Values'}
};
Plotly.newPlot('plot', [trace1, trace2], layout);
```
This code creates a line chart with two lines representing convergent and divergent processes.
## Exercise
Instructions:
1. Create a line chart using D3.js or Plotly.js to visualize the convergence and divergence of the series from the previous section.
2. Add a slider to the chart that allows the user to change the number of terms in the series.
# Solving mathematical problems using convergence and divergence
Here is an example of a line chart created using Plotly.js:
```javascript
let trace1 = {
x: [0, 1, 2, 3],
y: [0, 2, 4, 6],
mode: 'lines',
name: 'Convergent'
};
let trace2 = {
x: [0, 1, 2, 3],
y: [1, 3, 5, 7],
mode: 'lines',
name: 'Divergent'
};
let layout = {
title: 'Convergence and Divergence',
xaxis: {title: 'Iterations'},
yaxis: {title: 'Values'}
};
Plotly.newPlot('plot', [trace1, trace2], layout);
```
This code creates a line chart with two lines representing convergent and divergent processes.
## Exercise
Instructions:
1. Create a line chart using D3.js or Plotly.js to visualize the convergence and divergence of the series from the previous section.
2. Add a slider to the chart that allows the user to change the number of terms in the series.
# Advanced topics in convergence and divergence
Here is an example of a line chart created using Plotly.js:
```javascript
let trace1 = {
x: [0, 1, 2, 3],
y: [0, 2, 4, 6],
mode: 'lines',
name: 'Convergent'
};
let trace2 = {
x: [0, 1, 2, 3],
y: [1, 3, 5, 7],
mode: 'lines',
name: 'Divergent'
};
let layout = {
title: 'Convergence and Divergence',
xaxis: {title: 'Iterations'},
yaxis: {title: 'Values'}
};
Plotly.newPlot('plot', [trace1, trace2], layout);
```
This code creates a line chart with two lines representing convergent and divergent processes.
## Exercise
Instructions:
1. Create a line chart using D3.js or Plotly.js to visualize the convergence and divergence of the series from the previous section.
2. Add a slider to the chart that allows the user to change the number of terms in the series.
# Conclusion and further resources
In this textbook, we have explored the concepts of convergence and divergence in depth. We have learned how to implement these concepts in JavaScript using iterative and recursive processes. We have also learned how to visualize and analyze convergence and divergence using JavaScript libraries.
In conclusion, understanding convergence and divergence is crucial for solving various mathematical problems and optimizing algorithms in computer science. By mastering these concepts, you will be well-equipped to tackle complex problems in both mathematics and computer science.
For further resources on convergence and divergence, you can refer to the following books and websites:
- "Convergence and Divergence" by John H. Mathews and Kurt D. Fisher
- "Convergence and Divergence: A Handbook for Engineers, Scientists, and Mathematicians" by George S. Ferst
- "Convergence and Divergence: A Visual Approach" by Michael J. Kiss
- "Convergence and Divergence" section on Khan Academy (https://www.khanacademy.org/)
I hope you have enjoyed this textbook and found it helpful in understanding convergence and divergence. Happy learning! | Textbooks |
\begin{document}
\title{Cross validating extensions of kernel, sparse or regular partial least squares regression models to censored data.}
\author{Fr\'ed\'eric Bertrand, \\ IRMA, CNRS UMR~7501, Labex IRMIA, \\ Universit\'e de Strasbourg, Strasbourg, France \\ [email protected]\\ Philippe Bastien, \\ L'Or\'eal Recherche, Aulnay, France \\ [email protected]\\ Myriam Maumy-Bertrand, \\ IRMA, CNRS UMR~7501, Labex IRMIA, \\ Universit\'e de Strasbourg, Strasbourg, France \\ [email protected]}
\maketitle
\begin{abstract} When cross-validating standard or extended Cox models, the commonly used criterion is the cross-validated partial loglikelihood using a naive or a van Houwelingen scheme -to make efficient use of the death times of the left out data in relation to the death times of all the data-. Quite astonishingly, we will show, using a strong simulation study involving three different data simulation algorithms, that these two cross-validation methods fail with the extensions, either straightforward or more involved ones, of partial least squares regression to the Cox model.
This is quite an interesting result for at least two reasons. Firstly, several nice features of PLS based models, including regularization, interpretability of the components, missing data support, data visualization thanks to biplots of individuals and variables -and even parsimony for SPLS based models-, account for a common use of these extensions by statisticians who usually select their hyperparameters using cross-validation. Secondly, they are almost always featured in benchmarking studies to assess the performance of a new estimation technique used in a high dimensional context and often show poor statistical properties.
We carried out a vast simulation study to evaluate more than a dozen of potential cross-validation criteria, either AUC or prediction error based. Several of them lead to the selection of a reasonable number of components. Using these newly found cross-validation criteria to fit extensions of partial least squares regression to the Cox model, we performed a benchmark reanalysis that showed enhanced performances of these techniques.
The R-package {\tt plsRcox} used in this article is available on the CRAN, \url{http://cran.r-project.org/web/packages/plsRcox/index.html}. \end{abstract}
\section{Introduction} Regular PLS regression is used to find the fundamental relations between two matrices ($X$ and $Y$), \textit{i.e.} a latent variable approach to modeling the covariance structures in these two spaces. A PLS regression model will try to find iteratively the multidimensional direction in the $X$ space that explains the maximum multidimensional variance direction in the $Y$ space. A key step in PLSR, is to select the right unknown number of these latent variables (called components) to use. PLS regression is particularly suited when the matrix of predictors has more variables than observations, and when there is multicollinearity among $X$ values. By contrast, standard regression will fail in these cases (unless it is regularized).
PLS has become an established tool in various experimental -including chemometric, networks, or systems biology- modeling, primarily because it is often possible to interpret the extracted components in terms of the underlying physical system -that is, to derive ``hard'' modeling information from the soft model: chemical components for NIR spectra, gene subnetwork for GRN or biological function for systems biology-. As a consequence, choosing the right number of components is not only a major aim to avoid under or overfitting and ensure a relevant modeling or good predicting ability but also \textit{per se}.
A vast literature from the last decade is devoted to relating gene profiles and subject survival or time to cancer recurrence. Biomarker discovery from high-dimensional data, such as transcriptomic or SNP profiles, is a major challenge in the search for more precise diagnoses. The proportional hazard regression model suggested by \citealp{cox72} to study the relationship between the time to event and a set of covariates in the presence of censoring is the most commonly used model for the analysis of survival data. However, like multivariate regression, it supposes that more observations than variables, complete data, and not strongly correlated variables are available. In practice when dealing with high-dimensional data, these constraints are crippling.
In this article we deal with several PLS regression based extensions of the Cox model. These extensions share features praised by practionners, including regularization, interpretability of the components, missing data support, biplots of individuals and variables -and even parsimony for SPLS based models-, and allow to deal with highly correlated predictors or even rectangular datasets, which is especially relevant for high dimensional datasets.
\section{Models} \subsection{{Modeling censored data}} \subsubsection{{The Cox proportional hazards model}}
The model assumes the following hazard function for the occurrence of an event at time $t$ in the presence of censoring: \begin{equation} \lambda(t)=\lambda_0(t)\exp{(\beta'X)}, \end{equation} where $\lambda_0(t)$ is an unspecified baseline hazard function, $\beta$ the vector of the regression coefficients and $X$ the matrix of prognosis factors which will be the gene expressions in the following. The event could be death or cancer relapse. Based on the available data, the Cox's partial likelihood can be written as: \begin{equation} PL(\beta)=\prod_{k\in D}{\frac{\exp{(\beta'x_k)}}{\sum_{j\in R_k}\exp{(\beta'x_j)}}}, \end{equation} where $D$ is the set of indices of the events and $R_k$ denotes the set of indices of the individuals at risk at time $t_k$.
The goal is to find the coefficients $\hat \beta$ which maximize the log partial likelihood function \begin{equation} l(\beta)=\log{PL(\beta)}. \end{equation} The vector $\hat \beta$ is the solution of the equation: \begin{equation} u(\beta)=\frac{\partial l}{\partial \beta}=0 \end{equation} with $u(\beta)$ the vector of efficient scores.
However, there is no explicit solution and the minimization is generally accomplished using the Newton-Raphson procedure. An estimate of the vector of $\beta$-parameters at the $(k+1)^\textup{th}$ cycle of the iterative procedure is: \begin{equation} \hat\beta_{k+1}=\hat\beta_{k}+I^{-1}(\hat\beta_{k})u(\hat\beta_{k}) \end{equation} where $I(\beta)=-\frac{\partial^2 l}{\partial \beta\partial \beta'}$ is the observed information matrix. The process can be started by taking $\hat\beta_0=0$ and iterated up to convergence, \textit{i.e.} when the change in the log likelihood function is small enough. When the iterative procedure has converged, the variance-covariance matrix of the parameter estimates can be approximated by the inverse of the observed information matrix $I^{-1}(\hat\beta)$.
Note that when $p>n$, there is no unique $\hat\beta$ to maximize this log partial likelihood function. Even when $p\leqslant n$, covariates could be highly correlated and regularization may still be required in order to reduce the variances of the estimates and to improve the predictive performance.
\subsubsection{{ Deviance Residuals}}\label{matingaleresidualsdef} For the Cox model with no time-dependent explanatory variables and at most one event per patient, the martingale residuals for the $i^\textup{th}$ subject with observation time $t_i$ and event status $\delta_i$, where $\delta_i=0$ if $t_i$ is a censored time, and $\delta_i=1$ otherwise is: \begin{equation} \hat M_i=\delta_i-\hat E_i=\delta_i-\hat\Delta_0(t_i)\exp{(\hat \beta'x_i)} \end{equation} with $\hat \Delta_0(t_i)$ the estimated cumulative hazard function at time $t_i$.
Martingale residuals are highly skewed. The deviance residuals $d_i$ are a normalized transform of the martingale residuals. For the Cox model, the deviance residuals (\citealp{coll94}) amount to the form: \begin{equation} d_i=\sign(\hat M_i)\cdot\left[2\left\{-\hat M_i-\delta_i\log{\left(\frac{\delta_i-\hat M_i}{\delta_i}\right)}\right\}\right]^{1/2}\cdot \end{equation} The $\sign{}$ function is to ensure that the deviance residuals have the same sign as the martingale residuals. Martingale residuals take values between $-\infty$ and $1$. The square root shrinks large negative martingale residuals, while the logarithmic transformation expands towards $+\infty$ martingale residuals that are close to $1$. As such, the deviance residuals are more symmetrically distributed around zero than the martingale residuals. The deviance residual is a measure of excess of death and can therefore be interpreted as a measure of hazard. Moreover, Segal showed that the expression to be minimized in step 3 of the Cox-Lasso procedure of Tibshirani can be approximated, in a first order Taylor-series approximation sense, by the deviance residual sum of squares: \begin{equation} (z-X\beta)'A(z-X\beta) \approx \RSS(\hat D), \end{equation} with $\eta=\beta'X$, $\mu=\frac{\partial l}{\partial \eta}$, $A=-\frac{\partial^2 l}{\partial \eta\eta'}$, and $z=\eta+A^-\mu$.
\subsection{{ PLS regression models and extensions}} \subsubsection{{ PLSR}} Prediction in high-dimensional and low-sample size settings already arose in chemistry in the eighties. PLS regression, that can be viewed as a regularization method based on dimension reduction, was developed as a chemometric tool in an attempt to find reliable predictive models with spectral data (\citealp{wold82,tenen98}). Nowadays, the difficulty encountered with the use of genomic or proteomic data for classification or prediction, using very large matrices, is of comparable nature. It was thus natural to use PLS regression principles in this new context. The method starts by constructing latent components, using linear combinations of the original variables, which are then used as new descriptors in standard regression analysis. Different from the principal components analysis (PCA), this method makes use of the response variable in constructing the latent components. The PLS regression can be viewed as a regularized approach searching the solution in a sub-space named Krylov space giving biased regression coefficients but with lower variance. In the framework of censored genomic data, the PLS regression operates a reduction of the dimensionality of the gene's space oriented towards the explanation of the hazard function. It allows transcriptomic signatures correlated to survival to be determined.
\subsubsection{{ Sparse PLSR}} Recently, \citealp{chun10}, provided both empirical and theoretical results showing that the performance of PLS regression was ultimately affected by the large number of predictors. In parti\-cular, a higher number of irrelevant variables leads to inconsistency of coefficient estimates in linear regression setting. There is a need to filter the descriptors as a preprocessing step before PLS fit. However, commonly used variables filtering approaches are all univariate and ignore correlation between variables. To solve these issues, Chun and Keles proposed "sparse PLS regression" which promotes variables selection within the course of PLS dimension reduction. sPLS has the ability to include variables that variable filtering would select in the construction of the first direction vector. Moreover, it can select additional variables, \textit{i.e.}, variables that become significant once the response is adjusted for other variables in the construction of the subsequent direction vectors. This is the case of "proxy genes" acting as suppressor variables which do not predict the outcome variable directly but improve the overall prediction by enhancing the effects of prime genes despite having no direct predictive power, \citealp{magi10}.
A direct extension of PLS regression to sPLS regression could be provided by imposing $L_1$ constraint on PLS direction vector $w$: \begin{align*}
&\max_w w'Mw \quad \textup{subject to } w'w=\|w\|_2=1, \|w\|_1\leqslant \lambda,\\ &\textup{where } M=X'YY'X. \label{eq:} \end{align*} When $Y$=$X$, the objective function coincides with that of sPCA (\citealp{jolli03}). However in that case Jolliffe {\it et al.} pointed out that the solution tends not to be sparse enough and the problem is not convex. To solve these issues, Chun and Keles provided an efficient implementation of sPLS based on the LARS algorithm by generalizing the regression formulation of sPCA of \citealp{zou06}: \begin{align*}
&\min_{w,c} -\kappa w'Mw+(1-\kappa)(c-w)'M(c-w)+\lambda_1\|c\|_1+\lambda_2\|c\|_2\\ &\textup{subject to } w'w=1, \textup{where } M=X'YY'X. \end{align*} This formulation promotes exact zero property by imposing $L_1$ penalty onto a surrogate of the direction vector $c$ instead of the original direction $w$ while keeping $w$ and $c$ close to each other. The $L_2$ penalty takes care of the potential singularity of $M$. Moreover, they demonstrated that for univariate PLS, $y$ regressed on $X$, the first direction vector of the sparse PLS algorithm was obtained by soft thresholding of the original PLS direction vector: \begin{equation}
(\lvert Z\rvert-\frac{\lambda}2)_+\sign{(Z)}, \textup{ where } Z=X'y/\|X'y\|_2. \end{equation}
In order to inherit the property of the Krylov subsequences which is known to be crucial for the correctness of the algorithm (\citealp{kramer07}), the thresholding phase is followed by a PLS regression on the previously selected variables. The algorithm is then iterated with $y$ replaced by $y-X\hat \beta$, the residuals of the PLS regression based on the variables selected from the previous steps. The sPLS algorithm leads therefore to sparse solutions by keeping the Krylov subsequence structure of the direction vectors in a restricted $X$ space which is composed of the selected variables. The thresholding parameter $\lambda$ and the number of hidden components are tuned by cross validation.
sPLS has connections to other variable selection algorithms including the elastic net method (\citealp{zou05}) and the threshold gradient method (\citealp{friedm04}). The elastic net algorithm deals with the collinearity issue in variable selection problem by incorporating the ridge regression method into the LARS algorithm. In a way, sPLS handles the same issue by fusing the PLS technique into the LARS algorithm. sPLS can also be related to threshold gradient method in that both algorithms use only thresholded gradient and not the Hessian. However, sPLS achieves fast convergence by using conjugate gradient. Hence, LARS and sPLS algorithms use the same criterion to select active variables in the univariate case. However, the sPLS algorithm differs from LARS in that sPLS selects more than one variable at a time and utilizes the conjugate gradient method to compute coefficients at each step. The computational cost for computing coefficients at each step of the sPLS algorithm is less than or equal to the computational cost of computing step size in LARS since conjugate gradient methods avoid matrix inversion.
\subsection{{ Extensions of PLSR models to censored data}} \subsubsection{{ PLS-Cox}} \citealp{garth94}, showed that PLS regression could be obtained as a succession of simple and multiple linear regressions. \citealp{tenen99}, proposed a fairly similar approach but one which could cope with missing data by using the principles of the Nipals algorithm \citep{wold66}. As a result, Tenenhaus suggested that PLS regression be extended to logistic regression (PLS-LR) by replacing the succession of simple and multiple regressions by a succession of simple and multiple logistic regressions in an approach much simpler than that developed by \citealp{marx96}. By using this alternative formulation of the PLS regression, \citealp{bast01}, extended the PLS regression to any generalized linear regression model (PLS-GLR) and to the Cox model ({PLS-Cox}) as a special case. Further improvements have then been described (\citealp{bast04}) in the case of categorical descriptors with variable selection using hard thresholding and model validation by bootstrap resampling. Since then many developments in the framework of PLS and Cox regressions have appeared in the literature. \citealp{nguyen02}, directly applied PLS regression to survival data and used the resulting PLS components in the Cox model for predicting survival time. However such a direct application did not really generalize PLS regression to censored survival data since it did not take into account the failure time in the dimension reduction step. Based on a straightforward generalization of \citealp{garth94}, \citealp{ligu04}, presented a solution, Partial Cox Regression, quite similar to the one proposed by Bastien and Tenenhaus, using different weights to derive the PLS components but not coping with missing data.
\subsubsection{{ (DK)(S)PLS(DR)}} \paragraph{{ The PLSDR algorithm \citep{bast08}}} Following \citealp{seg06}, who suggested initially computing the null deviance residuals and then using these as outcomes for the LARS-Lasso algorithm, \citealp{bast08}, proposed PLSDR, an alternative in high-dimensional settings using deviance residuals based PLS regression. This approach is advantageous both by its simplicity and its efficiency because it only needs to carry out null deviance residuals using a simple Cox model without covariates and use these as outcome in a standard PLS regression. The final Cox model is then carried out on the $m$ retained PLSDR components.
Moreover, following the principles of the Nipals algorithm, weights, loadings, and PLS components are computed as regression slopes. These slopes may be computed even when there are missing data: let $t_{hi}={x_{h-1,i}w_h}/{w'_hw_h}$ the value of the PLS component for individual $i$, with $x_{h-1,i}$ the vector of residual descriptors and $w_h$ the vector of weights at step $h$. $t_{hi}$ represents the slope of the OLS line without constant term related to the cloud of points $(w_h,x_{h-1,i})$. In such case, in computing the $h^{\textup{th}}$ PLS component, the denominator is computed only on the data available also for the denominator.
\IncMargin{1em} \begin{algorithm}
\LinesNumbered
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\DontPrintSemicolon
\BlankLine
\Begin{
$d \leftarrow $ null deviance residuals of the Cox model without covariates.\;
\Begin{\tcc{Computation of the sPLS components by using the sPLS regression with the null deviance residuals $d$ as outcome.}
\Indp
$\hat \beta^{PLS} \leftarrow 0$.\tcc*[r]{Initialisation}
$\Omega \leftarrow \{\}$.\;
$k \leftarrow 1$.\;
$y_1 \leftarrow d$.\;
\Indm
\While(\tcc*[f]{Component derivation loop}){$(h \leqslant m)$}{
$z \leftarrow X'y_1/\|X'y_1\|_2$.
$w \leftarrow (\lvert z\rvert-\lambda/2)_+\sign{(z)}$.\;
$\Omega \leftarrow \{i: \hat w_i\neq 0\}\cup\{i: \hat\beta_i^{PLS}\neq 0\}$.\;
Fit PLS with $X_\Omega$ by using the $k$ number of latent components.\;
$\hat\beta^{PLS} \leftarrow$ new PLS estimates of the direction vectors.\;
$y_1 \leftarrow y_1-X\hat\beta^{PLS}$.\;
$h \leftarrow h+1$.\;
}
}
\KwRet{Cox model on the $m$-retained sPLSDR components.}\;
}
\caption{The sPLSDR algorithm \protect\citep{Bastien2015}}\label{algo_slpsdr} \end{algorithm} \DecMargin{1em}
\paragraph{{ The DKsPLSDR algorithm \citep{Bastien2015}}} In the case of very many descriptors, PLS regression being invariant by orthogonal transformation (\citealp{jong94}), an even faster procedure could be derived by replacing the $X$ matrix by the matrix of principal components $Z$ $(XX'=ZZ')$. This could be viewed as the simple form of linear kernel PLS regression algorithms which have been proposed in the nineties (\citealp{lin93,ran94}) to solve computational problems posed by very large matrices in chemometrics. The objective of these methods was to obtain PLS components by working on a condensed matrix of a considerably smaller size than the original one. Moreover, in addition to dramatically reducing the size of the problem, non-linear pattern in the data could also be analyzed using non-linear kernel.
\citealp{ros01}, proposed a nonlinear extension of PLS regression using kernels. Assuming a nonlinear transformation of the input variables $\{x_i\}_{i=1}^n$ into a feature space $F$, \textit{i.e.} a mapping $\Phi: x_i\in \mathbb{R}^N \mapsto \Phi(x_i)\in F$, their goal was to construct a linear PLS regression model in $F$. They derived an algorithm named KPLS for Kernel PLS by performing the PLS regression on $\Phi(X)$. It amounts to replacing, in the expression of PLS components, the product $XX'$ by $\Phi(X)\Phi(X)'$ using the so-called kernel trick which allows the computation of dot products in high-dimensional feature spaces using simple functions defined on pairs of input patterns: $\Phi(x_i)\Phi(x_j)'=K(x_i,x_j)$. This avoids having to explicitly calculate the coordinates in the feature space which could be difficult for a highly dimensional feature space. By using the kernel functions corresponding to the canonical dot product in the feature space, non-linear optimization can be avoided and simple linear algebra used.
\citealp{ben03}, proposed to perform PLS regression directly on the kernel matrix $K$ instead of $\Phi(X)$. DKPLS corresponds to a low rank approximation of the kernel matrix. Moreover, \citealp{tena07} demonstrated that, for one dimensional output response, PLS of $\Phi(X)$ (KPLS) is equivalent to PLS on $K^{1/2}$ (DKPLS).
Using previous works, it becomes straightforward to derive a non-linear Kernel sPLSDR algorithm by replacing in the sPLSDR algorithm the $X$ matrix by a kernel matrix $K$. The main kernel functions are the linear kernel ($K(u,v)= <u,v>$) and the Gaussian kernel ($K(u,v)=\exp{(-\|u-v\|_2^2/2\sigma^2)}$).
However, non-linear kernel (sparse) PLS regression loses the explanation with the original descriptors unlike linear kernel PLS regression, which could limit the interpretation of the results.
\IncMargin{1em} \begin{algorithm}
\LinesNumbered
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\DontPrintSemicolon
\BlankLine
\Begin{
Computation of the kernel matrix.\;
$d \leftarrow $ null deviance residuals of the Cox model without covariates.\;
Computation of the sPLS components by using the DKsPLS algorithm with the null deviance residuals $d$ as outcome.\;
\KwRet{Cox model on the $m$-retained DKsPLSDR components.}\;
}
\caption{The DKsPLSDR algorithm \protect\citep{Bastien2015}}\label{algo_dkslpsdr} \end{algorithm} \DecMargin{1em}
\section{Simulation studies} \subsection{Scheme of the studies}
\IncMargin{1em} \begin{algorithm}
\LinesNumbered
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\DontPrintSemicolon
\BlankLine
\Begin{
\ForEach{simulation types $\in \{$eigengene, cluster, factorial$\}$}{
\ForEach{link types $\in \{$none, linear, quadratic$\}$}{
\For{i=1 to 100}{
Simulate a dataset with exponential survival distribution and 40\% censored rate (100 observations $\times$ 1000 genes).\;
Randomly split the dataset into a training set (7/10, 70 observations) and a test set (3/10, 30 observations)\;
\ForEach{of the 7 (S)PLS based methods}{
\ForEach{of the 12 cross-validation criteria}{Find the optimal number of components by $K$-fold cross-validation of the training data set, see Section \ref{sechypcv}.
}
}
\ForEach{of the 14 prediction methods}{
\ForEach{of the 12 cross-validation criteria}{
Find the optimal tuning parameter $\hat\lambda_{train}$ by $K$-fold cross-validation of the training data set, see Section \ref{sechypcv}.\;
Given $\hat\lambda_{train}$, estimate the vector of regression coefficients $\hat\beta_{train}$ on the whole training data set.\;
Calculate the values of the 22 performance criteria on the test data set as described in Section \ref{perfmeas}.\;
}
}
}
}
}
}
\caption{Summary of the procedure for evaluating the accuracy of the cross validation methods and revisit the performance of the component based methods.\label{sumpredme}} \end{algorithm} \DecMargin{1em}
The aim of our two in silico studies is twofold: evaluating the accuracy of the cross validation methods, see Section~\ref{cvcritchoice}, and revisit the performance of the component based methods, see Section~\ref{perfevalbench}.
We performed a simulation study (Algorithm \ref{sumpredme}) in order to evaluate the methods by simulating 100 datasets with exponential survival distribution and 40\% censored rate (100 observations $\times$ 1000 genes) according to three different simulation types (cluster by \cite{bair06}, factorial by \cite{kais62} and \cite{fan2002} or eigengene by \cite{lang13}), using either no link or a linear one between the response and the predictors.
We divided each of these 600 datasets into a training set, of 7/10 (70) of the observations, used for estimation, and a test set, of 3/10 (30) of the observations, used for evaluation or testing of the prediction capability of the estimated model. This choice was made to stay between the 2:1 scheme of \cite{bove07,vwie09,lll11} and the 9:1 scheme of \cite{li2006}. The division between training and test sets was balanced using the \verb+caret+ package, \cite{kuhn14}, both according to the response value and censor rate.
\subsection{Data generation} \subsubsection{Eigengene: \cite{lang13}} Given module seeds and a desired size for the genes modules around the seeds of $n_I$ genes, module genes expression profiles are generated such that the $k$-th rank correlated gene from module $I$ with its module seed $seed_I$ is : \begin{equation} \operatorname{cor}(x_{k,I},seed_{I})=1-k/n_I(1-r_{\textup{min}})=r_{k,I}\label{eqcor1} \end{equation} that is, the first gene has correlation $r_{i,I}\approx 1$ with the seed while the last ($n_I$-th) gene has correlation $r_{{n_i},I}\approx r_{\textup{min}}$.
The required correlation (\ref{eqcor1}) is achieved by calculating the $k$-th gene profile as the sum of the seed vector $seed_I$ and a noise term $a_k\varepsilon_k$ \begin{equation} x_{k,I}=seed_{I}+{a}_{k}\varepsilon _{k}
\quad\textup{where}\quad {a}_{k}=\sqrt{\frac{\operatorname{var}(seed_{I}^{{}})}{\operatorname{var}({{\varepsilon }_{k}})}\left( \frac{1}{r_{k,I}^{2}}-1 \right)} \end{equation}
This technique produces modules consisting of genes distributed symmetrically around the module seed; in this sense, the simulated modules are spherical clusters whose centers coincide (on average) with the module seed.
In the simulations the parameters have been let as follow $I=4$, $r_{\textup{min}}=0.5$, $n_{I}=25$ with $seed_{I}$ and $\varepsilon_{k}\sim \mathcal{N}(0,1)$.
Survival and censoring times, with $0.4$ censoring probability, are generated from exponential survival distributions. When linked to survival (linear or quadratic case), only expressions from genes from the first two modules ($N=50$) are related to survival time.
Each simulated data set consists of $1000$ genes and $100$ samples. Only the first hundred genes are structured. The last $900$ are random noise generated from $\mathcal{N}(0,1)$.
\subsubsection{Cluster \cite{bair06}} The gene expression data is distributed as: \begin{equation}{{X}_{ij}}=\left\{ \begin{matrix}
3+\varepsilon_{ij}^{{}}\text{ if }i\leq 50,j\leq 50 \\
4+\varepsilon_{ij}^{{}}\text{ if }i>50,j\leq 50 \\ 3.5+\varepsilon_{ij}^{{}}\text{ if }j>50.\\ \end{matrix} \right.\end{equation} Where the ${{\varepsilon }_{ij}}$are drawn from a $\mathcal{N}(0,1)$.
Each simulated data set consists of $1000$ genes and $100$ samples. Survival and censoring times, with $0.4$ censoring probability, are generated from exponential survival distributions. When linked to survival (linear or quadratic case), only expressions from genes from the first $50$ genes are related to survival.
\subsubsection{Factorial \cite{kais62}, \cite{fan2002}} We have supposed that genes expressions are related to $4$ latent variables associated each to a specific biological function. Let for each group a specified population inter-correlation pattern matrix $R$. By applying principal component factorization (PCA) to the matrix $R$ and following Kaiser and Dickman, we can generate $4$ multivariate normally distributed sample data with a specific correlation pattern.
${{Z}_{(k\times N)}}={{F}_{(k\times k)}}X_{(k\times N)}^{{}}$
Where:
$k$ is the number of descriptors (genes)
$N$ is the number of observations
$X$ is a matrix of uncorrelated random normal variables $\mathcal{N}(0,1)$
$F$ is a matrix containing principal component factor pattern coefficients obtained by applying Principal Components Analysis (PCA) to the given population correlation matrix $R$
$Z$ is the resultant sample data matrix, as if sampled from a population with the given population correlation matrix $R$
We have chosen a compound symmetry structure for the correlation matrix $R$ with a same correlation ($0.7$) between two descriptors of a same group, descriptors between different groups being independent.
Moreover the choice of the correlation coefficient allows specifying the percentage of variance explained by the first factorial axes. Given four groups with inter-genes correlation coefficient of $0.7$ corresponds to expend $70\%$ of the inertia in $4$ principal directions.
Survival and censoring times, with $0.4$ censoring probability, are generated from exponential survival distributions. When linked to survival (linear or quadratic case), only expressions from genes from the first two groups ($N=50$) are related to survival time.
Each simulated data set consists of $1000$ genes and $100$ samples. Only the first hundred genes are structured. The last $900$ are random noise generated from $\mathcal{N}(0,1)$.
Figure~\ref{heatmap} displays the pattern of correlation for the first 150 descriptors with the four groups of $25$ genes each well defined.
\begin{figure}
\caption{Hierarchical clustering of the variables in a factorial-type simulated dataset.}
\label{heatmap}
\end{figure}
\subsection{Hyperparameters and cross-validation}\label{sechypcv} In standard $K$-fold cross-validation of a dataset of size $n$, $K$ folds of size $\text{Floor}(n/K)$ are created by sampling from the data without replacement and each of the remaining $n\ \text{mod}\ K$ data points is assigned randomly to a different fold. In stratified or balanced crossvalidation \cite[p. 246]{brei84}, the data are first ordered by the response value or class. This list is broken up into $c$ bins each containing $K$ points with many similar response values. Any remaining points at the end of the list are assigned to an additional bin. A fold is formed by sampling one point without replacement from each of the bins. Except for the ordering of the data, this is equivalent to standard cross-validation. We used balanced cross-validation with respect to the response value and censor rate. The folds were design using the \verb+caret+ package, \cite{kuhn14}.
In traditional cross-validation, \textit{i.e.} with a dataset without censored events, each fold would yield a test set and a value of a prediction error measure (for instance log partial likelihood, integrated area under the curve, integrated area under the prediction error curve). When dealing with censored events and using the CV partial likelihood (CVLL, \cite{vevh93}) criterion, it is possible to make more efficient use of risk sets: \cite{vhou06} recommended to derive the CV log partial likelihood for the $j$th fold by subtraction; by subtracting the log partial likelihood evaluated on the full dataset from that evaluated on the full dataset minus the $j$th fold, called the $(K-1)/K$ dataset. This yields the van Houwelingen CV partial likelihood (vHCVLL) .
Hyperparameters were tuned using 7-fold cross-validation on the training set. The number of folds was chosen following the recommandation of \cite{wold01}, \cite{brei92} and \cite{koha95}. As in, \cite{bove07}, \cite{vwie09} and \cite{lll11}, mean values were then used to summarize these cross validation criteria over the 7 runs and the hyperparameters were chosen according to the best values of these measures.
\section{Highlighting relevant cross validation criteria}\label{cvcritchoice} \subsection{The failure of the two usual criteria} The van Houwelingen CV partial likelihood (vHCVLL, see Figure \ref{NbrComp_vanHcvll}) criterion behave poorly for all the PLS or sPLS based methods by selecting zero components where, according to our simulation types, the PLS-Cox, autoPLS-Cox, Cox-PLS, PLSDR, sPLSDR, DKPLSDR and DKsPLSDR methods were expected to select, for the factor or eigengene schemes, about two components and slightly more for the cluster scheme. As with the the classic CV partial likelihood (CVLL), it almost always selects at most one component and hence systematically underestimates the number of components. The simulations results for the selection of the number of components using CVLL are plotted on Figure \ref{NbrComp_cvll}. We confirmed this poor property by performing cross-validation on a simpler simulation scheme designed by \cite{coxnet11}.
\subsection{Proposal of new criteria} As a consequence, we had to search for other CV criteria (CVC) for the models featuring components. \cite{li2006} used the integrated area under the curves of time-dependent ROC curves (iAUCsurvROC, \cite{hea00}) to carry out his cross-validations, implemented in the \verb+survcomp+ package, \citep{schr11}. Apart from that criterion (Figure \ref{NbrComp_AUCsurvROCtest}) we added five other integrated AUC measures: integrated \citeauthor{chdi06}'s (\citeyear{chdi06}) estimator (iAUCCD, Figure \ref{NbrComp_AUCcd}), integrated \citeauthor{huch10}'s (\citeyear{huch10}) estimator (iAUCHC, Figure \ref{NbrComp_AUChc}), integrated \citeauthor{sozh08}'s (\citeyear{sozh08}) estimator (iAUCSH, Figure \ref{NbrComp_AUCsh}), integrated \citeauthor{uno2007}'s (\citeyear{uno2007}) estimator (iAUCUno, Figure \ref{NbrComp_AUCUno}) and integrated \citeauthor{heager05}'s (\citeyear{heager05}) estimator (iAUCHZ, Figure \ref{NbrComp_AUChztest}) of cumulative/dynamic AUC for right-censored time-to-event data, implemented in the \verb+survAUC+ package, \cite{pota12}, and the \verb+risksetROC+ package, \cite{heag2012}. We also studied two versions of two prediction error criteria, the integrated (un)weighted Brier Score (\cite{graf99}, \cite{gesc06}, iBS(un)w, integrated (un)weighted squared deviation between predicted and observed survival, Figures \ref{NbrComp_iBSunw} and \ref{NbrComp_iBSw}) and the integrated (un)weighted Schmid Score (\cite{schm11}, iSS(un)w, integrated (un)weighted absolute deviation between predicted and observed survival, Figure \ref{NbrComp_iSchmidSunw} and \ref{NbrComp_iSchmidSw}), also implemented in the \verb+survAUC+ package, \cite{pota12}.
\subsection{Analysis of the results}\label{cvtechniques} The simulation results highlighted the integrated \citeauthor{sozh08}'s estimator of cumulative/dynamic AUC for right-censored time-to-event data (iAUCSH), implemented in the \verb+survAUC+ package, \cite{pota12}, as the best CV criterion for the PLS-Cox and the autoPLS-Cox methods even though it behaves poorly in all the other cases.
As for the other models featuring components, the iAUCsurvROC, iAUCUno criterion exhibited the best performances.
The two unweighted criteria iBSunw and iSSunw uniformly fail for all the models.
The iBSw criterion is too conservative and wrongly selects null models in more than half of the cases in the linear link scheme and in almost every times in the quadratic scheme.
The iSSw provides very poor results for Cox-PLS, sPLSDR and DKsPLSDR and average for PLSDR and DKPLSDR methods.
The two models SPLSDR and DKSPLSDR use an additional parameter: the thresholding parameter $\eta$. The same figures were produced for all the criteria (results not shown): both iAUCUno criterion and iAUCsurvROC criterion provided a reasonable spread for the $\eta$ parameter.
\subsection{Recommendation}\label{recochanges} In a word, this simulation campaign enables us to state the following recommendations to firmly improve the selection of the right number of components: use iAUCSH to cross-validate PLS-Cox or autoPLS-Cox models and either iAUCUno or iAUCsurvROC to cross-validate Cox-PLS, PLSDR, sPLSDR, DKPLSDR and DKsPLSDR. We implemented these recommendations (iAUCSH for PLS-Cox or autoPLS-Cox models and iAUCsurvROC for Cox-PLS, PLSDR, sPLSDR, DKPLSDR and DKsPLSDR) as the default cross validation techniques in the \verb+plsRcox+ package. We will apply them in the remaining of the article to assess goodness of fit of the model.
\begin{figure}\label{NbrComp_cvll}
\label{NbrComp_vanHcvll}
\end{figure}
\begin{figure}\label{NbrComp_AUCcd}
\label{NbrComp_AUChc}
\end{figure}
\begin{figure}\label{NbrComp_AUCsh}
\label{NbrComp_AUCUno}
\end{figure}
\begin{figure}\label{NbrComp_AUChztest}
\label{NbrComp_AUCsurvROCtest}
\end{figure}
\begin{figure}\label{NbrComp_iBSunw}
\label{NbrComp_iSchmidSunw}
\end{figure}
\begin{figure}\label{NbrComp_iBSw}
\label{NbrComp_iSchmidSw}
\end{figure}
\section[Reassessing performance of (s)PLS based models]{Reassessing performance of (s)PLS \\ based models}\label{perfevalbench} We will now provide evidence that the changes of the cross-validation criteria recommended in Section~\ref{recochanges} actually lead to performance improvements for the fitted models. \subsection{Introduction to performance criteria analysis}\label{perfmeas} We followed the methodological recommendations of \cite{vwie09} to design a simulation plan that ensures a good evaluation of the predictive performance of the models. \begin{quote} ``The true evaluation of a predictor's performance is to be done on independent data. In the absence of independent data (the situation considered here) the predictive accuracy can be estimated as follows \cite{Dupuy2007}. The samples are split into mutually exclusive training and test sets. The gene expression and survival data of the samples in the training set are used to build the predictor. No data from the test set are used in predictor construction (including variable selection) by any of the methods compared. This predictor is considered to be representative of the predictor built on all samples (of which the training set is a subset). The test set is used to evaluate the performance of the predictor built from the training set: for each sample in the test set, survival is predicted from gene expression data. The predicted survival is then compared to the observed survival and summarized into an evaluation measure. To avoid dependency on the choice of training and test set, this procedure is repeated for multiple splits. The average of the evaluation measures resulting from each split is our estimate of the performance of the predictor built using the data from all samples.'' \end{quote}
As to the performance criteria themselves, \cite{schm11} made several points that we will take into account to carry out our performance comparison analysis. \begin{quote} ``Evaluating the prognostic performance of prediction rules for continuous survival outcomes is an important topic of recent methodological discussion in survival analysis. The derivation of measures of prediction accuracy for survival data is not straightforward in the presence of censored observations (\cite{keoq88}, \citet{scst96}, \cite{rost04}). This is mainly due to the fact that traditional performance measures for continuous outcomes (such as the mean squared error or the $R^2$ fraction of explained variation) lead to biased predictions if applied to censored data (\citet{scst96}).
To overcome this problem, a variety of new approaches has been suggested in the literature. These developments can be classified into three groups: ``likelihood-based approaches'' (\citet{nage91}, \citet{xuoq99}, \citet{oqxu05}), ``ROC-based approa\-ches'' (\citet{hea00}, \citet{heager05}, \citet{cai2006}, \citet{uno2007}, \citet{pepe08}), and ``distance-based approa\-ches'' (\citet{korn90}, \citet{graf99}, \citet{sche00}, \citet{gesc06}, \citeyear{gesc07}, \cite{scho08}).
When using likelihood-based approaches, the log likelihood of a prediction model is related to the corresponding log likelihood obtained from a ``null model'' with no covariate information. ROC-based approaches use the idea that survival outcomes can be considered as time-dependent binary variables with levels -event- and -no event,- so that time-dependent misclassification rates and ROC curves can be computed for each threshold of a predictor variable of interest. If distance-based approaches are applied, a measure of prediction error is given by the distance between predicted and observed survival functions of the observations in a sample. None of these approaches has been adopted as a standard for evaluating survival predictions so far.'' \end{quote}
To assess the goodness of fit and prediction accuracy of all the methods, we found 23 performance measures (PM) that are commonly used (LRT, VarM, R2Nag, R2XO, R2OXS, iR2BSunw, iR2BSw, iRSSunw, iRSSw, iAUCCD, iAUCHC, iAUCSH, iAUCUno, IAUCHZ, iAUCSurvROC, C, UnoC, GHCI, SchemperV, iBSunw, iBSw, iSSunw, iSSw). We chose, on statistical grounds, 14 among them (LRT, R2XO, iR2BSw, iRSSw, iAUCCD, iAUCHC, iAUCSH, iAUCUno, IAUCHZ, iAUCSurvROC, GHCI, SchemperV, iBSw, iSSw) and reported the results of six indices of several kind: two $R^2$-like measure (a likelihood-based approach (LBA), R2XO, and a distance-based approach (DBA), iRSSw), a $C$ index (GHCI), two $iAUC$ (ROC-based approaches (ROCBA), iAUCCD and iAUCSurvROC), and an integrated robust prediction error (distance-based approach, iSSw), see Table~\ref{measselected}. The results for the remaining eight indices are similar to those shown. We now explain our process of selection of the performance criteria.
\begin{table}[!b] \small \centerfloat
\begin{tabular}{|l|c|ccc|ccc|}
\hline Criterion&Type&\multicolumn{3}{c|}{As a Cross Validation Criterion}&\multicolumn{3}{c|}{As a Performance Measure}\\ Criterion&Type&Tested&Results&Recom. for&Is a&Selected on&Results\\ &&&&&PM ?&statistical&\\ &&&&&&grounds&\\ \hline CVLL&LBA&{\bf Yes}&{\bf Yes}&&No&No&No\\ vHCVLL&LBA&{\bf Yes}&{\bf Yes}&&No&No&No\\ LRT $p$-value&LBA&No&No&&{\bf Yes}&{\bf Yes}&No\\ VarM&LBA&No&No&&{\bf Yes}&No&No\\ R2Nag&LBA&No&No&&{\bf Yes}&No&No\\ R2XO&LBA&No&No&&{\bf Yes}&{\bf Yes}&{\bf Yes}\\ R2OXS&LBA&No&No&&{\bf Yes}&No&No\\ iR2BSunw&DBA&No&No&&{\bf Yes}&No&No\\ iR2BSw&DBA&No&No&&{\bf Yes}&{\bf Yes}&No\\ \textit{iRSSunw}&DBA&No&No&&\textit{\textbf{New}}&No&No\\ \textit{iRSSw}&DBA&No&No&&\textit{\textbf{New}}&{\bf Yes}&{\bf Yes}\\ iAUCCD&ROCBA&{\bf Yes}&{\bf Yes}&&{\bf Yes}&{\bf Yes}&{\bf Yes}\\ iAUCHC&ROCBA&{\bf Yes}&{\bf Yes}&&{\bf Yes}&{\bf Yes}&No\\ iAUCSH&ROCBA&{\bf Yes}&{\bf Yes}&PLS$-$Cox,&{\bf Yes}&{\bf Yes}&No\\ &&&&autoPLS$-$Cox&&&\\ iAUCUno&ROCBA&{\bf Yes}&{\bf Yes}&(DK)(s)PLSDR&{\bf Yes}&{\bf Yes}&No\\ &&&&Cox$-$PLS&&&\\ iAUCHZ&ROCBA&{\bf Yes}&{\bf Yes}&&{\bf Yes}&{\bf Yes}&No\\ iAUCSurvROC&ROCBA&{\bf Yes}&{\bf Yes}&(DK)(s)PLSDR&{\bf Yes}&{\bf Yes}&{\bf Yes}\\ &&&&Cox$-$PLS&&&\\ C&ROCBA&No&No&&{\bf Yes}&No&No\\ UnoC&ROCBA&No&No&&{\bf Yes}&No&Sup. Info.\\ GHCI&ROCBA&No&No&&{\bf Yes}&{\bf Yes}&{\bf Yes}\\ SchemperV&DBA&No&No&&{\bf Yes}&{\bf Yes}&No\\ iBSunw&DBA&{\bf Yes}&{\bf Yes}&&{\bf Yes}&No&No\\ iBSw&DBA&{\bf Yes}&{\bf Yes}&&{\bf Yes}&{\bf Yes}&Sup. Info.\\ iSSunw&DBA&{\bf Yes}&{\bf Yes}&&{\bf Yes}&No&No\\ iSSw&DBA&{\bf Yes}&{\bf Yes}&&{\bf Yes}&{\bf Yes}&{\bf Yes}\\ \hline Total Number&25&12&&12&23&14&6 (+2 SI)\\ \hline \end{tabular}
\caption{Criteria and their use in the cross validation step or as a performance measures for assessing the quality of the model.\label{measselected}} \end{table}
\eject
\subsection{Selection of performance criteria}
The likelihood ratio test (LRT, \cite{Lehmann2005}) evaluates the null hypothesis $\mathcal{H}_0 \ : \ \beta = 0$, i.e. the built predictor has no effect on survival. The null hypothesis is evaluated using the likelihood ratio test statistic $LLR(\hat\beta)=-2(l(0)-l(\hat\beta))$, with $l(.)$ denoting the value of the log-likelihood function. Under the null hypothesis this test statistic has a $\chi^2$ distribution, which is used to calculate the $p$-value. The $p$-value summarizes the evidence against $\mathcal{H}_0$: the lower the $p$-value the more probable that $\mathcal{H}_0$ is not true.The $p$-value of the likelihood ratio test has been used as an evaluation measure for predictive performance of gene expression based predictors of survival by many others \cite{Bair2004,bove07,Park2002,seg06}.
In the Cox model, an alternative measure of predictive performance is the variance of the martingale residuals (VarM, \textit{cf.} section~\ref{matingaleresidualsdef}).
As in \cite{vwie09}, we found that this measure is not able to discriminate very well between good and poor predictors in the considered setting (data not shown). It is therefore omitted here.
To quantify the proportion of variability in survival data of the test set that can be explained by the predictor, we use the coefficient of determination (henceforth called $R^2$). A predictor with good predictive performance explains a high proportion of variability in the survival data of the test set, and vice versa a poor predictor explains little variability in the test set. However, the traditional definition of the $R^2$ cannot be used in the context of censored data and modified criteria have been proposed in the past. Three types of likelihood-based $R^2$ coefficients for right-censored time-to-event data are were put forward (R2NAG, R2XO and R2OXS). \begin{itemize} \item The coefficient (R2Nag) proposed by \cite{nage91}: \begin{equation} R^2_{Nag}=1-\exp{\left(-\frac2n(l(\hat\beta)-l(0))\right)} \end{equation} where $l(.)$ denotes the log-likelihood function. \item The coefficient (R2XO) proposed by \cite{xuoq99} that is restricted to proportional hazards regression models, because here the means of squared residuals $MSE$ in the $R^2_{adj}$ measure for linear regression are replaced by the (weighted) sums of squared \textit{Schoenfeld} residuals, denoted by $J(\beta)$: \begin{equation} R^2_{XO}=1-\frac{J(\hat\beta)}{J(0)}\cdot \end{equation} \item The coefficient (R2OXS) proposed by \cite{oqxu05} who replaced the number of observations $n$ by the number of events $e$: \begin{equation} R^2_{OXS}(\hat\beta)=1-\exp{\left(-\frac2e(l(\hat\beta)-l(0))\right)}=1-\left(\frac{L(\hat\beta)}{L(0)}\right)^{-2/e}\cdot \end{equation} \end{itemize} All three were implemented in the \verb+survAUC+ package, \cite{pota12}. Others have also used these modified $R^2$ statistics to assess predictive performance of gene expression based predictors on survival \cite{Bair2004,seg06}.
\cite{hiel10} carried out a comparison of the properties of these three coefficients. In a word, R2Nag is strongly influenced by censoring (negative correlation with censoring); R2OXS is less influenced by censoring and exhibits a positive correlation with censoring. From those three R2XO is the less influenced by censoring. As a consequence, we selected the R2XO as the $R^2$-like measure to compare the models.
The weighted Brier score $BSw(t)$ (\cite{Brier1950, Hothorn2004, Radespiel-Troger2003}) is a distance-based measure of prediction error that is based on the squared deviation between survival functions. It is defined as a function of time $t>0$ by \begin{equation} BSw(t)=\frac1n\sum_{i=1}^n\left[\frac{\hat S(t \mid \mathbf{X}_i)^2I(t_i\leqslant t\wedge \delta_i=1)}{\hat G(t_i)}+\frac{(1-\hat S(t \mid \mathbf{X}_i))^2I(t_i>t)}{\hat G(t_i)}\right]\label{eqBSdef} \end{equation} where $\hat G(.)$ denotes the Kaplan-Meier estimate of the censoring distribution, that is the Kaplan–Meier estimate based on the observations $(t_i,1-\delta_i)$ and $I$ stands for the indicator function. The expected Brier score of a prediction model which ignores all predictor variables corresponds to the KM estimate. To derive the unweighted Brier score, $BSunw(t)$, clear the $\hat G(t_i)$ value of the denominators.
The Schmid score $SS(t)$ (\cite{schm11}) is a distance-based measure of prediction error that is based on the absolute deviation between survival functions, instead of the squared one for the Brier-Score. It is a robust improvement over the following empirical measure of absolute deviation between survival functions that was suggested by \cite{sche00} as a function of time $t>0$ by: \begin{equation} SH(t)=\frac1n\sum_{i=1}^n\left[\frac{\hat S(t \mid \mathbf{X}_i) I(t_i\leqslant t\wedge \delta_i=1)}{\hat G(t_i)}+\frac{(1-\hat S(t \mid \mathbf{X}_i))I(t_i>t)}{\hat G(t_i)}\right]\label{eqSHdef} \end{equation} where $\hat G(.)$ denotes the Kaplan-Meier estimate of the censoring distribution which is based on the observations $(t_i,1-\delta_i)$ and $I$ stands for the indicator function. With the same notations, the Schmid score is defined as a function of time $t>0$ by: \begin{equation} SS(t)=\frac1n\sum_{i=1}^n\lvert I(t_i> t) - \hat S(t \mid \mathbf{X}_i) \rvert \left[\frac{I(t_i\leqslant t\wedge \delta_i=1)}{ \hat G(t_i^-)}+\frac{I(t_i>t)}{\hat G(t_i)}\right]\label{eqSSdef} \end{equation} where $t_i^-$ is a survival time that is marginally smaller than $t_i$. To derive the unweighted Schmid score, $SSunw(t)$, clear the $\hat G(t_i^-)$ and $\hat G(t_i)$ values of the denominators.
The values of the Brier-Score range between 0 and 1. Good predictions at time $t$ result in small Brier-Scores. The numerator of the first summand is the squared predicted probability that individual $i$ survives until time $t$ if he actually died (uncensored) before $t$, or zero otherwise. The better the survival function is estimated, the smaller is this probability. Analogously, the numerator of the second summand is the squared probability that individual $i$ dies before time $t$ if he was observed at least until $t$, or zero otherwise. Censored observations with survival times smaller than $t$ are weighted with 0. The Brier-score as defined in Eq. \ref{eqBSdef} depends on $t$. It makes sense to use the integrated Brier-Score ($IBS$) given by \begin{equation} IBS = \frac1{\max{(t_i)}}\int_0^{\max(t_i)}BS(t)dt. \end{equation} as a score to assess the goodness of the predicted survival functions of all observations at every time $t$ between 0 and $\max{(t_i)}$, $i=1,\ldots,N$. Note that the $IBS$ is also appropriate for prediction methods that do not involve Cox regression models: it is more general than the $R^2$ and the $p$-value criteria associated to the log likelihood test and has thus become a standard evaluation measure for survival prediction methods (\cite{Hothorn2006,Schumacher2007}).
Denoting by $BS^0$, the Kaplan-Meier estimator based on the $t_i$, $\delta_i$, which corresponds to a prediction without covariates, we first define $R^2_{BS}$ for all $t>0$: \begin{equation} R^2_{BS}(t)=1-\frac{BS(t)}{BS^0(t)}\cdot \end{equation} Then the integrated iR2BSw, \cite{graf99}, is defined by: \begin{equation} iR2BSw=\frac{1}{\max(t_i)}\int_0^{\max(t_i)}R^2_{BS}(t)dt. \end{equation} This criterion has already been used in \cite{bove07} and \cite{lll11}. The integrated iR2BSw is slightly influenced by censoring, \cite{hiel10}, and, as a measure based on the the quadratic norm, not robust.
As a consequence, we propose and use a similar measure based on the Schmid score, the integrated R Schmid Score weighted (iRSSw), by turning the traditional $R^2$, derived from the quadratic norm, into the R coefficient of determination for least absolute deviation, introduced by \cite{mcke87}. Denoting by $SS^0$ the Schmid score which corresponds to a prediction without covariates, we first define $R_{SS}$ for all $t>0$: \begin{equation} R_{SS}(t)=1-\frac{SS(t)}{SS^0(t)}\cdot \end{equation} Then the integrated iRSSw, is defined by: \begin{equation} iRSSw=\frac{1}{\max(t_i)}\int_0^{\max(t_i)}R_{SS}(t)dt. \end{equation}
The C-index provides a global assessment of a fitted survival model for the continuous event time rather than focuses on the prediction of t-year survival for a fixed time and is arguably the most widely used measure of predictive accuracy for censored data regression models. It is a rank-correlation measure motivated to quantify the correlation between the ranked predicted and observed survival times. The C index estimates the probability of concordance between predicted and observed responses. A value of 0.5 indicates no predictive discrimination and a value of 1.0 indicates perfect separation of patients with different outcomes. A popular nonparametric C-statistic for estimating was proposed by \cite{harr96}. It is computed by forming all pairs $\{(y_i,x_i,\delta_i), (y_j,x_j,\delta_j)\}$ of the observed data, where the smaller follow-up time is a failure time and defined as: \begin{equation} c=\frac{\sum_{1\leqslant i<j \leqslant n}I(y_i<y_j)I(\hat\beta'\mathbf{X_i}>\hat\beta'\mathbf{X_j})I(\delta_i=1)+I(y_j<y_i)I(\hat\beta'\mathbf{X_j}>\hat\beta'\mathbf{X_i})I(\delta_j=1)}{\sum_{1\leqslant i<j \leqslant n}I(y_i<y_j)I(\delta_i=1)+I(y_j<y_i)I(\delta_j=1)} \end{equation} We used the improved version (GHCI) by \cite{gohe05} for the Cox proportional hazards models as a performance comparison criterion. Their estimator $K_n(\hat\beta)$ is a function of the regression parameters and the covariate distribution only and does not use the observed event and censoring times. For this reason it is asymptotically unbiased, unlike Harrell's C-index based on informative pairs. It focuses on the concordance probability as a measure of discriminatory power within the framework of the Cox model. The appeal of this formulation is that it provides a stable estimator of predictive accuracy that is easily computed: \begin{equation} K_n(\hat\beta)=\frac2{n(n-1)}\sum_{1\leqslant i<j \leqslant n}\left\{\frac{I(\hat\beta'(\mathbf{X_j}-\mathbf{X_i})<0)}{1+\exp{(\hat\beta'(\mathbf{X_j}-\mathbf{X_i}))}}+\frac{I(\hat\beta'(\mathbf{X_i}-\mathbf{X_j})<0)}{1+\exp{(\hat\beta'(\mathbf{X_i}-\mathbf{X_j}))}}\right\}\cdot \end{equation} In contrast to Harrell's C-index, the effect of the observed times on $K_n(\hat\beta)$ is mediated through the partial likelihood estimator $\hat\beta$, and, since the effect of censoring on the bias of $\hat\beta$ is negligible, the measure is robust to censoring. In addition, $K_n(\hat\beta)$ remains invariant under monotone transformations of the survival times.
\subsection{Ranking the performance of the CV criteria} We stated several recommendations, in Section~\ref{cvcritchoice} based of the accuracy of the selection of the number of components. Selecting the right number of components is a goal \textit{per se}.
Moreover, these recommendations are also relevant from a performance criteria point of view (see Section \ref{perfmeas}) as the following analysis showed. \begin{enumerate} \item For all the models and simulation types, we carried out the cross-validation according to all of the 12 criteria and, for each of these criteria, we derived the value of all the 14 the performance measures.
\item In order to lay the stress on the improvements of performance made when switching from the classic and the van Houwelingen log likelihood cross validation techniques to the recommended ones, we computed, for every datasets and models, all the paired differences between CVLL or vHCVLL and the eleven other CV techniques. \begin{itemize} \item Paired comparison with CVLL. For every simulated dataset we evaluated: Delta = Performance Measure(with any CV criteria $\neq$ CVLL) $-$ Performance Measure(with CVLL) \item Paired comparison with vHCVLL. For every simulated dataset we evaluated: Delta = Performance Measure(with any CV criteria $\neq$ vHCVLL) $-$ Performance Measure(with vHCVLL) \end{itemize} An analysis of these results showed a steady advantage of the recommended criteria versus either CVLL or vHCVLL especially in the linear and quadratic cases.\\ In the case of paired comparison with vHCVLL and for some couples of the type (performance measure, model), namely (UnoC, PLS$-$Cox), (UnoC, sPLSDR), (iBSW, PLSDR), (iBSW, DKsPLSDR), (iRSSW, autoPLS$-$Cox), (iRSSW, PLSDR), (iAUCSurvROC, PLSDR) and (iAUCSurvROC, sPLSDR), those deltas are plotted on Figures \ref{incplsRcox_UnoCstat}, \ref{inccoxsplsDR_UnoCstat}, \ref{inciBSWcoxplsDR}, \ref{inciBSWcoxDKsplsDR}, \ref{inciRSSWautoplsRcox}, \ref{inciRSSWplsDR}, \ref{inciAUCSurvROCplsDR} and \ref{inciAUCSurvROCsplsDR}. \end{enumerate}
\begin{figure}\label{incplsRcox_UnoCstat}
\label{inccoxsplsDR_UnoCstat}
\end{figure}
\begin{figure}\label{inciBSWcoxplsDR}
\label{inciBSWcoxDKsplsDR}
\end{figure}
\begin{figure}\label{inciRSSWautoplsRcox}
\label{inciRSSWplsDR}
\end{figure}
\begin{figure}\label{inciAUCSurvROCplsDR}
\label{inciAUCSurvROCsplsDR}
\end{figure}
\eject
\subsection{Performance comparison revisited}\label{perfmeasrev} \subsubsection{Selection of competing benchmark methods} \cite{coxnet11}, introduced the \verb+coxnet+ procedure, which is an elastic net-type procedure for the Cox model, in a similar but not equivalent way than two competing ones: \verb+coxpath+ (\verb+glmpath+ package, \citealp{park07}) and \verb+penalized+ (\verb+penalized+ package, \citealp{goeman10}). In Section 3 of the same article, these authors extensively compared \verb+coxnet+ to \verb+coxpath+ and to \verb+penalized+ for the lasso penalty that is the only one relevant for these comparisons since the three procedures use different elastic net penalties. Their results show tremendous timing advantage for \verb+coxnet+ over the two other procedures. The \verb+coxnet+ procedure was integrated in the \verb+glmnet+ package (\citealp{glmnet10}) and is called in the \verb+R+ language by applying the \verb+glmnet+ function with the option \verb+family=cox+: \verb+coxnet+ is \verb+glmnet+ for the Cox model. The timing results of \cite{coxnet11} on both simulated and real datasets show some advantage to \verb+coxpath+ over \verb+penalized+.
As to pure lasso-type penalty algorithms, we selected two of them: ``Univariate Shrinkage in the Cox Model for High Dimensional data'' (\verb+uniCox+, \citealp{uniCox}) and `Gradient Lasso for Cox Proportional Hazards Model'' (\verb+glcoxph+, \citealp{sohn09}).
The \verb+uniCox+ package implements ``Univariate Shrinkage in the Cox Model for High Dimensional data'' (\citealp{uniCox}). Being ``essentially univariate'', it differs from applying a classical lasso penalty when fitting the Cox model and hence from both coxnet/glmnet and coxpath/glmpath. It can be used on highly correlated and even rectangular datatsets.
In their article, \cite{sohn09}, show that the \verb+glcoxph+ package is very competitive compared with popular existing methods \verb+coxpath+ by \cite{park07} and \verb+penalized+ by \cite{goeman10} in its computational time, prediction and selectivity. As a very competitive procedure to \verb+coxpath+, that we included in our benchmarks, and since no comparisons were carried out with \verb+coxnet+, we selected \verb+glcoxph+ as well.
Cross validation criteria were recommended for several of our benchmark methods by their authors. We followed these recommendations -classic CV partial likelihood for coxpath, glcoxph and uniCox; van Houwelingen CV partial likelihood for coxnet with both the $\lambda_{min}$, the value of $\lambda$ that gives minimum of the mean cross-validated error, or $\lambda_{1se}$, the largest value of $\lambda$ such that the cross-validated error is within 1 standard error of the minimum of the mean cross-validated error, criteria- and used the same 7 folds fo the training set as those described in Section~\ref{sechypcv} for the other models.
It seemed unfair to compare the methods using a performance measure that is recommended as a cross-validation criterion for some, but not all, of them. Hence, we decided not to use any of the three recommended cross-validation criteria iAUCSH, iAUCUno or iAUCsurvROC -even if it has already been used by \cite{li2006}- as a performance measure, in order to strive to perform fair comparisons with the methods that are recommended to be cross validated using partial likelihood with either the classic or van Houwelingen technique.
As a consequence and in order to still provide results for a ROC-based performance measure on a fair basis, we selected the \citeauthor{chdi06}'s (\citeyear{chdi06}) estimator of cumulative/dynamic AUC for right-censored time-to-event data in a form restricted to Cox regression. The iAUCCD summary measure is given by the integral of AUC on $[0, \max(\text{times})]$ (weighted by the estimated probability density of the time-to-event outcome).
\subsubsection{Results} For coxnet, coxlars or ridgecox with both the $\lambda_{min}$ or $\lambda_{1se}$ CV criteria, the $\lambda_{min}$ criterion yield similar yet superior results than the $\lambda_{1se}$ one whose main default is to select too often no explanatory variable (a null model) for the linear or quadratic links. As a consequence, we only reported results for the former one.
We plotted some of the performance measures when the cross-validation is done according to the vHCVLL criterion on Figures \ref{modmins_vanHcvll_R2XO}, \ref{modmins_vanHcvll_GonenHellerCI}, \ref{modmins_vanHcvll_iAUC_CD}, \ref{modmins_vanHcvll_iAUC_SurvROCtest}, \ref{modmins_vanHcvll_iRSSw} and \ref{modmins_vanHcvll_irobustSw}. The results are terrible for all the (s)PLS$-$like models apart from PLS$-$Cox and autoPLS$-$Cox.
\begin{figure}\label{modmins_vanHcvll_R2XO}
\label{modmins_vanHcvll_GonenHellerCI}
\end{figure}
\begin{figure}\label{modmins_vanHcvll_iAUC_CD}
\label{modmins_vanHcvll_iAUC_SurvROCtest}
\end{figure}
\begin{figure}\label{modmins_vanHcvll_iRSSw}
\label{modmins_vanHcvll_irobustSw}
\end{figure}
We then provide, for each of the (s)PLS$-$like method, the increases in terms of performance measures when switching from the vHCVLL as a cross validation criterion to the recommended one in Section~\ref{recochanges}. Virtually, for PLS-Cox and autoPLS-Cox we switch to the iAUCSH cross-validation criterion and for other (s)PLS based models to either iAUCUno or iAUCSurvROC .
For iAUCUno, these results are plotted on Figures \ref{modmins_incAUCUno_R2XO}, \ref{modmins_incAUCUno_GonenHellerCI}, \ref{modmins_incAUCUno_iAUC_CD}, \ref{modmins_incAUCUno_iAUC_SurvROCtest}, \ref{modmins_incAUCUno_iRSSw} and \ref{modmins_incAUCUno_irobustSw} whereas for iAUCSurvROC they are displayed on Figures \ref{modmins_incAUCSurvROCtest_R2XO}, \ref{modmins_incAUCSurvROCtest_GonenHellerCI}, \ref{modmins_incAUCSurvROCtest_iAUC_CD}, \ref{modmins_incAUCSurvROCtest_iAUC_SurvROCtest}, \ref{modmins_incAUCSurvROCtest_iRSSw} and \ref{modmins_incAUCSurvROCtest_irobustSw}. These figures show a firm increase for the 6 criteria (R2XO, GHCI, iAUCCD, iAUCSurvROC, IRSSW, iSSW).
\begin{figure}\label{modmins_incAUCUno_R2XO}
\label{modmins_incAUCUno_GonenHellerCI}
\end{figure}
\begin{figure}\label{modmins_incAUCUno_iAUC_CD}
\label{modmins_incAUCUno_iAUC_SurvROCtest}
\end{figure}
\begin{figure}\label{modmins_incAUCUno_iRSSw}
\label{modmins_incAUCUno_irobustSw}
\end{figure}
\begin{figure}\label{modmins_incAUCSurvROCtest_R2XO}
\label{modmins_incAUCSurvROCtest_GonenHellerCI}
\end{figure}
\begin{figure}\label{modmins_incAUCSurvROCtest_iAUC_CD}
\label{modmins_incAUCSurvROCtest_iAUC_SurvROCtest}
\end{figure}
\begin{figure}\label{modmins_incAUCSurvROCtest_iRSSw}
\label{modmins_incAUCSurvROCtest_irobustSw}
\end{figure}
\eject
As can be seen for iAUCUno (Figures \ref{modmins_AUCUno_R2XO}, \ref{modmins_AUCUno_GonenHellerCI}, \ref{modmins_AUCUno_iAUC_CD}, \ref{modmins_AUCUno_iAUC_SurvROCtest}, \ref{modmins_AUCUno_iRSSw} and \ref{modmins_AUCUno_irobustSw}) and iAUCSurvROC (Figures \ref{modmins_AUCsurvROCtest_R2XO}, \ref{modmins_AUCsurvROCtest_GonenHellerCI}, \ref{modmins_AUCsurvROCtest_iAUC_CD}, \ref{modmins_AUCsurvROCtest_iAUC_SurvROCtest}, \ref{modmins_AUCsurvROCtest_iRSSw} and \ref{modmins_AUCsurvROCtest_irobustSw}), the improvement of the performances due to switch to the recommended CV criteria is high enough to even have some (S)PLS based models, for instance SPLSDR, show some advantage over the other benchmark methods.
\section{Conclusion} When cross-validating standard or extended Cox models, the commonly used criterion is the cross-validated partial loglikelihood using a naive or a van Houwelingen scheme. Quite astonishingly, these two cross-validation methods fail with all the 7 extensions of partial least squares regression to the Cox model, namely PLS-Cox, autoPLS-Cox, Cox-PLS, PLSDR, sPLSDR, DKPLSDR and DKsPLSDR.
In our simulation study, we introduced 12 cross validation criteria based on three different kind of model quality assessment: \begin{itemize} \item Likelihood (2): Verweij and Van Houwelingen (classic CVLL, 1993), van Houwelingen et al. (vHCVLL, 2006). \item Integrated AUC measures (6): Chambless and Diao's (iAUCCD, 2006), Hung and Chiang's (iAUCHC, 2010), Song and Zhou's (iAUCSH, 2008), Uno et al.'s (iAUCUno, 2007), Heagerty and Zheng's (iAUCHZ, 2005), Heagerty et al.'s (iAUCsurvROC, 2000). \item Prediction error criteria (4): integrated (un)weighted Brier Score (iBS(un)w, Gerds and Schumacher (2006)) or Schmid Score (iSS(un)w, Schmid et al. (2011)) \end{itemize}
Our simulation study was successful in finding good CV criterion for PLS or sPLS based extensions of the Cox model: \begin{itemize} \item iAUCsh for PLS-Cox and autoPLS-Cox. \item iAUCSurvROC and iAUCUno ones for Cox-PLS, (DK)PLSDR and (DK)sPLSDR. \end{itemize}
The derivation of measures of prediction accuracy for survival data is not straightforward in the presence of censored observations. To overcome this problem, a variety of new approaches has been suggested in the literature. We spotted 23 performance measures that can be classified into three groups: \begin{itemize} \item Likelihood-based approaches (llrt, varresmart, 3 R2-type). \item ROC-based approaches such as integrated AUC (iAUCCD, iAUCHC, iAUCSH, iAUCUno, iAUCHZ, iAUCsurvROC), 3 C-index (Harrell, GHCI, UnoC). \item Distance-based approaches such as the V of Schemper and Henderson (2000) or derived from Brier or Schmid Scores (iBS(un)w, iSS(un)w and 4 derived R2-type measures). \end{itemize}
Using the newly found cross-validation, and these measures of prediction accuracy, we performed a benchmark reanalysis that showed enhanced performances of these techniques and a much better behaviour even against other well known competitors such as coxnet, coxpath, uniCox and glcoxph.
Hence the recommended criteria not only improve the accuracy of the choice of the number of components but also strongly raise the performances of the models, which enables some of them to overperform the other benchmark methods.
\eject
\begin{figure}\label{modmins_AUCUno_R2XO}
\label{modmins_AUCUno_GonenHellerCI}
\end{figure}
\begin{figure}\label{modmins_AUCUno_iAUC_CD}
\label{modmins_AUCUno_iAUC_SurvROCtest}
\end{figure}
\begin{figure}\label{modmins_AUCUno_iRSSw}
\label{modmins_AUCUno_irobustSw}
\end{figure}
\begin{figure}\label{modmins_AUCsurvROCtest_R2XO}
\label{modmins_AUCsurvROCtest_GonenHellerCI}
\end{figure}
\begin{figure}\label{modmins_AUCsurvROCtest_iAUC_CD}
\label{modmins_AUCsurvROCtest_iAUC_SurvROCtest}
\end{figure}
\begin{figure}\label{modmins_AUCsurvROCtest_iRSSw}
\label{modmins_AUCsurvROCtest_irobustSw}
\end{figure}
\section*{References}
\section*{Supplemental Information} \subsection*{Insights on the implementation of the methods} We detail the implementation of the algorithms that we used in the article and start with some shared properties of these. Whenever the deviance residuals and survival models were to be derived, we used the \verb+survival+ R-package (\citealp{ther00,survival}). As a PLS regression function in the \verb+plsRcox+ R-package, we made three wrappers using either the \verb+pls+ function of the \verb+pls+ R-package (\citealp{pls}), the \verb+plsR+ function of the \verb+plsRglm+ R-package (\citealp{plsRglm}) or the \verb+pls+ function of the \verb+mixOmics+ R-package (\citealp{mixOmics}). The last two are based on the NIPALS algorithm and hence automatically handle missing data (\citealp{tenen98}) in the explanatory variables. In addition, the \verb+pls+ function of the \verb+mixOmics+ R-package (\citealp{mixOmics}) can quickly handle big datasets such as Dataset~5 with 242 rows and 44754 variables. As a consequence, we had to make the \verb+spls+ function of the \verb+spls+ R-package use this function instead of the \verb+pls+ function of the \verb+pls+ R-package (\citealp{pls}).
\begin{enumerate} \item The authors made a wrapper in the \verb+plsRcox+ R-package (\citealp{plsRcox}) to derive a sPLSDR implementation from the \verb+spls+ R-package (\citealp{spls}). \item The authors made a wrapper in the \verb+plsRcox+ R-package (\citealp{plsRcox}) to derive a DKsPLSDR implementation from the \verb+spls+ (\citealp{spls}) and the \verb+kernlab+ (\citealp{kernlab}) R-packages. \item The coxpath implementation was the \verb+coxpath+ function found in the \verb+glmpath+ R-package (\citealp{glmpath}). \item The coxnet implementation was the \verb+glmnet+ function, with the Cox family option, in the \verb+glmnet+ R-package (\citealp{glmnet10,coxnet11}). \item The coxlars implementation was the \verb+glmnet+ function, with the Cox family option and the \verb+alpha+ option set to $0$, in the \verb+glmnet+ R-package (\citealp{glmnet10,coxnet11}). \item The coxridge implementation was the \verb+glmnet+ function, with the Cox family option and the \verb+alpha+ option set to $1$, in the \verb+glmnet+ R-package (\citealp{glmnet10,coxnet11}). \item PLS-Cox has not yet been implemented in \verb+R+ and the authors made it available as the function \verb+plsRcox+ of the \verb+plsRcox+ R-package (\citealp{plsRcox}). \item autoPLS-Cox is PLS-Cox with sparse PLS components and automatic selection of their optimal number. The computed components are sparse since, to have a non zero coefficient, a variable must be significant at a given $\eta$ level in the cox regression of the response by this variable adjusted by the previously found components. The model stops adding a new component when there is no longer any of the explanatory variable that is significant at a given $\eta$ level. The authors made it available as the function \verb+plsRcox+ of the \verb+plsRcox+ R-package (\citealp{plsRcox}). The number of components can also be determined using cross-validation, the components being still sparse. \item The authors made a wrapper in the \verb+plsRcox+ R-package (\citealp{plsRcox}) to derive a LARS-LassoDR implementation from the \verb+lars+ R-package (\citealp{lars}). \item The authors made a wrapper in the \verb+plsRcox+ R-package (\citealp{plsRcox}) to derive a Cox-PLS implementation from either the \verb+pls+ or \verb+plsRglm+ R-packages. \item The authors made a wrapper in the \verb+plsRcox+ R-package (\citealp{plsRcox}) to derive a PLSDR implementation from either the \verb+pls+ or \verb+plsRglm+ R-packages. \item The authors made a wrapper in the \verb+plsRcox+ R-package (\citealp{plsRcox}) to derive a DKPLSDR implementation from the \verb+pls+ (\citealp{pls}) and the \verb+kernlab+ (\citealp{kernlab}) R-packages. \item The uniCox implementation was the \verb+uniCox+ function found in the \verb+uniCox+ R-package (\citealp{uniCox}). \item The glcoxph implementation was the \verb+glcoxph+ function found in the \verb+glcoxph+ R-package (\citealp{glcoxph}). \end{enumerate}
\begin{figure}\label{NbrComp_cvll_alt}
\label{NbrComp_vanHcvll_alt}
\end{figure}
\begin{figure}\label{NbrComp_AUCcd_alt}
\label{NbrComp_AUChc_alt}
\end{figure}
\begin{figure}\label{NbrComp_AUCsh_alt}
\label{NbrComp_AUCUno_alt}
\end{figure}
\begin{figure}\label{NbrComp_AUChztest_alt}
\label{NbrComp_AUCsurvROCtest_alt}
\end{figure}
\begin{figure}\label{NbrComp_iBSunw_alt}
\label{NbrComp_iSchmidSunw_alt}
\end{figure}
\begin{figure}\label{NbrComp_iBSw_alt}
\label{NbrComp_iSchmidSw_alt}
\end{figure}
\begin{figure}\label{eta_cvll}
\label{eta_vanHcvll}
\end{figure}
\begin{figure}\label{eta_AUCcd}
\label{eta_AUChc}
\end{figure}
\begin{figure}\label{eta_AUCsh}
\label{eta_AUCUno}
\end{figure}
\begin{figure}\label{eta_AUChztest}
\label{eta_AUCsurvROCtest}
\end{figure}
\begin{figure}\label{eta_iBSunw}
\label{eta_iSchmidSunw}
\end{figure}
\begin{figure}\label{eta_iBSw}
\label{eta_iSchmidSw}
\end{figure}
\begin{figure}\label{eta_cvll_alt}
\label{eta_vanHcvll_alt}
\end{figure}
\begin{figure}\label{eta_AUCcd_alt}
\label{eta_AUChc_alt}
\end{figure}
\begin{figure}\label{eta_AUCsh_alt}
\label{eta_AUCUno_alt}
\end{figure}
\begin{figure}\label{eta_AUChztest_alt}
\label{eta_AUCsurvROCtest_alt}
\end{figure}
\begin{figure}\label{eta_iBSunw_alt}
\label{eta_iSchmidSunw_alt}
\end{figure}
\begin{figure}\label{eta_iBSw_alt}
\label{eta_iSchmidSw_alt}
\end{figure}
\begin{figure}\label{autoplsRcoxUnoCstat}
\label{coxplsUnoCstat}
\end{figure}
\begin{figure}\label{coxplsDRUnoCstat}
\label{coxDKplsDRUnoCstat}
\end{figure}
\begin{figure}
\caption{UnoC vs CV criterion. DKsplsDR.}
\label{coxDKsplsDRUnoCstat}
\end{figure}
\begin{figure}\label{iBSWplsRcox}
\label{iBSWautoplsRcox}
\end{figure}
\begin{figure}\label{iBSWcoxpls}
\label{iBSWcoxsplsDR}
\end{figure}
\begin{figure}
\caption{iBSW vs CV criterion. DKsplsDR.}
\label{iBSWcoxDKplsDR}
\end{figure}
\begin{figure}\label{iRSSWplsRcox}
\label{iRSSWcoxpls}
\end{figure}
\begin{figure}\label{iRSSWcoxsplsDR}
\label{iRSSWcoxDKplsDR}
\end{figure}
\begin{figure}
\caption{iRSSW vs CV criterion. DKsplsDR.}
\label{iRSSWcoxDKsplsDR}
\end{figure}
\begin{figure}\label{SurvROCplsRcox}
\label{SurvROCautoplsRcox}
\end{figure}
\begin{figure}\label{SurvROCcoxpls}
\label{SurvROCDKplsDR}
\end{figure}
\begin{figure}
\caption{iAUCSurvROC vs CV criterion, DKsplsDR.}
\label{SurvROCDKsplsDR}
\end{figure}
\begin{figure}\label{incautoplsRcoxUnoCstat}
\label{inccoxplsUnoCstat}
\end{figure}
\begin{figure}\label{inccoxplsDRUnoCstat}
\label{inccoxDKplsDRUnoCstat}
\end{figure}
\begin{figure}
\caption{Delta of UnoC (CV criteria $-$ vHCVLL value). DKsPLSDR.}
\label{inccoxDKsplsDRUnoCstat}
\end{figure}
\begin{figure}\label{inciBSWplsRcox}
\label{inciBSWautoplsRcox}
\end{figure}
\begin{figure}\label{inciBSWcoxpls}
\label{inciBSWcoxsplsDR}
\end{figure}
\begin{figure}
\caption{Delta of iBSW (CV criteria $-$ vHCVLL value). DKsPLSDR.}
\label{inciBSWcoxDKplsDR}
\end{figure}
\begin{figure}\label{inciRSSWplsRcox}
\label{inciRSSWcoxpls}
\end{figure}
\begin{figure}\label{inciRSSWcoxsplsDR}
\label{inciRSSWcoxDKplsDR}
\end{figure}
\begin{figure}
\caption{Delta of iRSSW (CV criteria $-$ vHCVLL value). DKsPLSDR.}
\label{inciRSSWcoxDKsplsDR}
\end{figure}
\begin{figure}\label{incSurvRocplsRcox}
\label{incSurvRocautoplsRcox}
\end{figure}
\begin{figure}\label{incSurvRoccoxpls}
\label{incSurvRocDKplsDR}
\end{figure}
\begin{figure}
\caption{Delta of SurvROC (CV criteria $-$ vHCVLL value), DKsPLSDR.}
\label{incSurvRocDKsplsDR}
\end{figure}
\end{document} | arXiv |
DOI:10.1088/0004-637X/726/1/54
Synchrotron Blob Model of Infrared and X-ray Flares from Sagittarius A$^*$
@article{Kusunose2010SynchrotronBM,
title={Synchrotron Blob Model of Infrared and X-ray Flares from Sagittarius A\$^*\$},
author={Masaaki Kusunose and Fumio Takahara},
journal={arXiv: High Energy Astrophysical Phenomena},
M. Kusunose, F. Takahara
arXiv: High Energy Astrophysical Phenomena
Sagittarius A$^*$ in the Galactic center harbors a supermassive black hole and exhibits various active phenomena. Besides quiescent emission in radio and submillimeter radiation, flares in the near infrared (NIR) and X-ray bands are observed to occur frequently. We study a time-dependent model of the flares, assuming that the emission is from a blob ejected from the central object. Electrons obeying a power law with the exponential cutoff are assumed to be injected in the blob for a limited…
View PDF on arXiv
An inverse Compton scattering origin of x-ray flares from Sgr A
F. Yusef-Zadeh, M. Wardle, +6 authors D. Porquet
The X-ray and near-IR emission from Sgr A* is dominated by flaring, while a quiescent component dominates the emission at radio and submillimeter (sub-mm) wavelengths. The spectral energy…
A Leptonic Model of Steady High-Energy Gamma-Ray Emission from Sgr A$^*$
Recent observations of Sgr A* by Fermi and HESS have detected steady {gamma}-ray emission in the GeV and TeV bands. We present a new model to explain the GeV {gamma}-ray emission by inverse Compton…
Concurrent X-ray, near-infrared, sub-millimeter, and GeV gamma-ray observations of Sagittarius A
G. Trap, A. Goldwurm, +13 authors F. Yusef-Zadeh
Aims. The radiative counterpart of the supermassive black hole at the Galactic center (GC), Sgr A � , is subject to frequent flares that are visible simultaneously in X-rays and the near-infrared…
X-Ray Flares from Sagittarius A * and Black Hole Universe
T. X. Zhang, C. Wilson, M. Schamschula
Sagittarius (Sgr) A* is a massive black hole at the Milky Way center with mass of about 4.5 million solar masses. It is usually quite faint, emiting steadily at all wavelengths including X-rays.…
Statistical and theoretical studies of flares from Sagittarius A⋆
Ya-Ping Li, Q. Yuan, +5 authors J. Dexter
Proceedings of the International Astronomical Union
Abstract Multi-wavelength flares have routinely been observed from the supermassive black hole, Sagittarius A⋆ (Sgr A⋆), at our Galactic center. The nature of these flares remains largely unclear,…
A CHANDRA/HETGS CENSUS OF X-RAY VARIABILITY FROM Sgr A* DURING 2012
J. Neilsen, M. Nowak, +13 authors F. Baganoff
We present the first systematic analysis of the X-ray variability of Sgr A ∗ during the Chandra X-ray Observatory's 2012 Sgr A ∗ X-ray Visionary Project. With 38 High Energy Transmission Grating…
A magnetohydrodynamic model for multiwavelength flares from Sagittarius A⋆ (I): model and the near-infrared and X-ray flares
Ya-Ping Li, F. Yuan, Q. Wang
Flares from the supermassive black hole in our Galaxy, Sagittarius~A$^\star$ (Sgr A$^\star$), are routinely observed over the last decade or so. Despite numerous observational and theoretical…
Non-thermal models for infrared flares from Sgr A*
E. A. Petersen, C. Gammie
Recent observations with mm very long baseline interferometry (mm-VLBI) and near-infrared (NIR) interferometry provide mm images and NIR centroid proper motion for Sgr A*. Of particular interest…
The role of electron heating physics in images and variability of the Galactic Centre black hole Sagittarius A*
A. Chael, M. Rowan, R. Narayan, Michael D. Johnson, L. Sironi
Monthly Notices of the Royal Astronomical Society
The accretion flow around the Galactic Center black hole Sagittarius A* (Sgr A*) is expected to have an electron temperature that is distinct from the ion temperature, due to weak Coulomb coupling in…
Sgr A* flares: Tidal disruption of asteroids and planets?
K. Zubovas, S. Nayakshin, S. Markoff
Physics, Computer Science
It is speculated that one such disruption may explain the putative increase in Sgr A ∗ luminosity, and is estimated that asteroids larger than ∼10 km in size are needed to power the observed flares, with the maximum possible luminosity of the order of 10 39 erg s −1.
An x-ray, infrared, and submillimeter flare of Sagittarius A*
D. Marrone, F. Baganoff, +14 authors G. Bower
Energetic flares are observed in the Galactic supermassive black hole Sagittarius A* from radio to X-ray wavelengths. On a few occasions, simultaneous flares have been detected in IR and X-ray…
View 3 excerpts, references background and methods
Time-Dependent Models of Flares from Sagittarius A*
K. Dodds-Eden, Prateek Sharma, +4 authors D. Porquet
The emission from Sgr A*, the supermassive black hole in the Galactic Center, shows order of magnitude variability ('flares') a few times a day that is particularly prominent in the near-infrared…
View 1 excerpt, references methods
On the Nature of the Variable Infrared Emission from Sagittarius A
F. Yuan, E. Quataert, R. Narayan
Recent infrared (IR) observations of the center of our Galaxy indicate that the supermassive black hole (SMBH) source Sgr A* is strongly variable in the IR. The timescale for the variability, ~30…
The Nature of the 10 kilosecond X-ray flare in Sgr A*
S. Markoff, H. Falcke, F. Yuan, P. Biermann
The X-ray mission Chandra has observed a dramatic X-ray flare { a brightening by a factor of 50 for only three hours { from Sgr A*, the Galactic Center supermassive black hole. Sgr A* has never shown…
Rapid X-ray flaring from the direction of the supermassive black hole at the Galactic Centre
F. Baganoff, M. Bautz, +8 authors F. Walter
The discovery of rapid X-ray flaring from the direction of Sagittarius A* provides compelling evidence that the emission is coming from the accretion of gas onto a supermassive black hole at the Galactic Centre.
X-ray hiccups from Sagittarius A* observed by XMM-Newton - The second brightest flare and three moderate flares caught in half a day
D. Porquet, N. Grosso, +22 authors Spain.
Context. Our Galaxy hosts at its dynamical center Sgr A*, the closest supermassive black hole. Surprisingly, its luminosity is several orders of magnitude lower than the Eddington luminosity.…
Polarimetry of near-infrared flares from Sagittarius A*
A. Eckart, R. Schödel, L. Meyer, S. Trippe, T. Ott, R. Genzel
Context. We report new polarization measurements of the variable near-infrared emission of the SgrA* counterpart associated with the massive 3–$4\times10^6$ $M_{\odot}$ Black Hole at the Galactic…
A constant spectral index for sagittarius A* during infrared/X-ray intensity variations
S. Hornstein, K. Matthews, +5 authors F. Baganoff
We report the first time-series of broadband infrared color m easurements of Sgr A*, the variable emission source associated with the supermassive black hole at the Galactic Center. Using the laser…
Nonthermal Electrons in Radiatively Inefficient Accretion Flow Models of Sagittarius A
We investigate radiatively inefficient accretion flow models for Sgr A*, the supermassive black hole in our Galactic center, in light of new observational constraints. Confirmation of linear…
View 11 excerpts, references methods and background
Near-infrared flares from accreting gas around the supermassive black hole at the Galactic Centre
R. Genzel, R. Schödel, +5 authors B. Aschenbach
High-resolution infrared observations of Sagittarius A* reveal 'quiescent' emission and several flares, and traces very energetic electrons or moderately hot gas within the innermost accretion region. | CommonCrawl |
# Coordinate systems and transformations
Coordinate systems are a way to represent points in space using numbers. The most common coordinate systems are:
- Cartesian coordinates: This is the most common coordinate system, where points are represented by their x and y coordinates.
- Polar coordinates: This coordinate system uses an angle and a distance from a reference point to represent points.
To transform between these coordinate systems, you can use the following formulas:
- To convert Cartesian coordinates (x, y) to polar coordinates (r, θ):
$$
r = \sqrt{x^2 + y^2}
$$
$$
θ = \arctan\left(\frac{y}{x}\right)
$$
- To convert polar coordinates (r, θ) to Cartesian coordinates (x, y):
$$
x = r\cos(θ)
$$
$$
y = r\sin(θ)
$$
Consider the point (3, 4) in Cartesian coordinates. To convert it to polar coordinates, you can use the formulas above:
- r = $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$
- θ = $\arctan\left(\frac{4}{3}\right) \approx 53.13^\circ$
So, the point (3, 4) in polar coordinates is (5, 53.13°).
## Exercise
Convert the following points from Cartesian coordinates to polar coordinates:
- (0, 0)
- (1, 1)
- (2, 2)
- (3, 3)
To visualize these transformations using matplotlib, you can use the `plot` function to create a scatter plot of the points in the Cartesian coordinate system and then use the `polar` function to create a polar plot of the points.
Here's an example of how to create a scatter plot of the points (0, 0), (1, 1), (2, 2), and (3, 3) in the Cartesian coordinate system:
```python
import matplotlib.pyplot as plt
points = [(0, 0), (1, 1), (2, 2), (3, 3)]
plt.scatter(*zip(*points))
plt.xlabel('x')
plt.ylabel('y')
plt.title('Cartesian Coordinate System')
plt.show()
```
And here's an example of how to create a polar plot of the points (0, 0), (1, 1), (2, 2), and (3, 3):
```python
import numpy as np
r = np.array([0, 1, 2, 3])
theta = np.radians(np.array([0, 45, 90, 135]))
plt.polar(theta, r)
plt.title('Polar Coordinate System')
plt.show()
```
# Plotting shapes and lines
To plot a line in matplotlib, you can use the `plot` function. For example, to plot the line y = 2x, you can use the following code:
```python
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-10, 10, 100)
y = 2 * x
plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Line: y = 2x')
plt.show()
```
To plot a circle in matplotlib, you can use the `plot` function with a parametric equation. For example, to plot a circle with radius 3 centered at (0, 0), you can use the following code:
```python
import numpy as np
import matplotlib.pyplot as plt
t = np.linspace(0, 2 * np.pi, 100)
x = 3 * np.cos(t)
y = 3 * np.sin(t)
plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Circle: radius = 3')
plt.gca().set_aspect('equal', adjustable='box')
plt.show()
```
## Exercise
Plot the following shapes using matplotlib:
- A parabola: y = x^2
- A hyperbola: y = 1/x
- An ellipse: y = 3/2 * x
# Adding labels and annotations
To add labels to your plots, you can use the `xlabel`, `ylabel`, and `title` functions. For example, to add labels to the x and y axes and a title to the plot, you can use the following code:
```python
import matplotlib.pyplot as plt
plt.plot([0, 1, 2, 3], [0, 1, 4, 9])
plt.xlabel('x')
plt.ylabel('y')
plt.title('Plot: y = x^2')
plt.show()
```
To add annotations to your plots, you can use the `text` function. For example, to add an annotation to the point (1, 1) on the plot, you can use the following code:
```python
import matplotlib.pyplot as plt
plt.plot([0, 1, 2, 3], [0, 1, 4, 9])
plt.xlabel('x')
plt.ylabel('y')
plt.title('Plot: y = x^2')
plt.text(1, 1, '(1, 1)', fontsize=10, ha='right', va='bottom')
plt.show()
```
## Exercise
Add labels and annotations to the following plots:
- A plot of the function y = 2x.
- A plot of the circle with radius 3 centered at (0, 0).
- A plot of the parabola y = x^2.
# Customizing colors and styles
To customize the colors and styles of your plots, you can use various keyword arguments to the `plot` function. For example, to change the color and line style of a plot, you can use the following code:
```python
import matplotlib.pyplot as plt
plt.plot([0, 1, 2, 3], [0, 1, 4, 9], color='red', linestyle='--')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Plot: y = x^2')
plt.show()
```
To customize the colors and styles of annotations, you can use the `text` function with the `color`, `fontsize`, `ha`, and `va` keyword arguments. For example, to change the color and alignment of an annotation, you can use the following code:
```python
import matplotlib.pyplot as plt
plt.plot([0, 1, 2, 3], [0, 1, 4, 9])
plt.xlabel('x')
plt.ylabel('y')
plt.title('Plot: y = x^2')
plt.text(1, 1, '(1, 1)', fontsize=10, ha='right', va='bottom', color='green')
plt.show()
```
## Exercise
Customize the colors and styles of the following plots:
- A plot of the function y = 2x.
- A plot of the circle with radius 3 centered at (0, 0).
- A plot of the parabola y = x^2.
# Creating interactive visualizations
To create interactive visualizations, you can use the `plotly` library, which provides a higher-level interface for creating interactive plots. For example, to create an interactive scatter plot of the points (0, 0), (1, 1), (2, 2), and (3, 3), you can use the following code:
```python
import plotly.express as px
points = [(0, 0), (1, 1), (2, 2), (3, 3)]
fig = px.scatter(x=[x for x, y in points], y=[y for x, y in points])
fig.show()
```
## Exercise
Create interactive visualizations of the following plots:
- A plot of the function y = 2x.
- A plot of the circle with radius 3 centered at (0, 0).
- A plot of the parabola y = x^2.
# Working with different plot types
To create a bar plot in matplotlib, you can use the `bar` function. For example, to create a bar plot of the data [1, 2, 3, 4], you can use the following code:
```python
import matplotlib.pyplot as plt
data = [1, 2, 3, 4]
plt.bar(range(len(data)), data)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Bar Plot')
plt.show()
```
To create a histogram in matplotlib, you can use the `hist` function. For example, to create a histogram of the data [1, 2, 2, 3, 3, 3, 4, 4, 4, 4], you can use the following code:
```python
import matplotlib.pyplot as plt
data = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4]
plt.hist(data, bins=range(min(data), max(data) + 2), align='left')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Histogram')
plt.show()
```
To create a box plot in matplotlib, you can use the `boxplot` function. For example, to create a box plot of the data [1, 2, 2, 3, 3, 3, 4, 4, 4, 4], you can use the following code:
```python
import matplotlib.pyplot as plt
data = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4]
plt.boxplot(data)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Box Plot')
plt.show()
```
## Exercise
Create the following plots using matplotlib:
- A bar plot of the data [1, 2, 3, 4].
- A histogram of the data [1, 2, 2, 3, 3, 3, 4, 4, 4, 4].
- A box plot of the data [1, 2, 2, 3, 3, 3, 4, 4, 4, 4].
# Advanced transformations and projections
To visualize advanced transformations and projections in matplotlib, you can use the `transforms` module to create custom coordinate transformations. For example, to create a plot of the function y = x^2 and apply a custom transformation that scales the x-axis by a factor of 2 and the y-axis by a factor of 3, you can use the following code:
```python
import matplotlib.pyplot as plt
import matplotlib.transforms as transforms
x = np.linspace(-10, 10, 100)
y = x**2
trans = transforms.Affine2D().scale(2, 3)
plt.plot(x, y)
plt.gca().get_xaxis_transform().transform(x)
plt.gca().get_yaxis_transform().transform(y)
plt.gca().set_transform(trans + plt.gca().get_transform())
plt.xlabel('x')
plt.ylabel('y')
plt.title('Plot: y = x^2 (Scaled)')
plt.show()
```
## Exercise
Visualize the following transformations using matplotlib:
- A plot of the function y = x^2 with a custom transformation that scales the x-axis by a factor of 2 and the y-axis by a factor of 3.
- A plot of the circle with radius 3 centered at (0, 0) with a custom transformation that rotates the plot by 45 degrees.
- A plot of the parabola y = x^2 with a custom transformation that reflects the plot across the x-axis.
# Creating 3D visualizations
To create 3D visualizations in matplotlib, you can use the `mpl_toolkits.mplot3d` module. For example, to create a 3D scatter plot of the points (0, 0, 0), (1, 1, 1), (2, 2, 2), and (3, 3, 3), you can use the following code:
```python
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
points = [(0, 0, 0), (1, 1, 1), (2, 2, 2), (3, 3, 3)]
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(*zip(*points))
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
plt.title('3D Scatter Plot')
plt.show()
```
## Exercise
Create the following 3D visualizations using matplotlib:
- A 3D scatter plot of the points (0, 0, 0), (1, 1, 1), (2, 2, 2), and (3, 3, 3).
- A 3D plot of the function z = x^2 + y^2.
- A 3D plot of the sphere with radius 3 centered at (0, 0, 0).
# Creating animations and movies
To create animations and movies in matplotlib, you can use the `animation` module. For example, to create an animation of a circle with radius 3 centered at (0, 0) that rotates around the z-axis, you can use the following code:
```python
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
def update(frame):
t = frame * 2 * np.pi / 100
x = 3 * np.cos(t)
y = 3 * np.sin(t)
z = 0
ax.clear()
ax.plot([x], [y], [z], marker='o', markersize=10, linestyle='None')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
ax.set_title('Rotating Circle')
ax.set_xlim(-4, 4)
ax.set_ylim(-4, 4)
ax.set_zlim(-1, 1)
ax.view_init(30, frame)
ani = FuncAnimation(fig, update, frames=range(100), interval=50)
plt.show()
```
## Exercise
Create the following animations and movies using matplotlib:
- An animation of a circle with radius 3 centered at (0, 0) that rotates around the z-axis.
- An animation of a sphere with radius 3 centered at (0, 0, 0) that rotates around the z-axis.
- A movie of a plot of the function y = x^2 that zooms in and out.
# Using matplotlib for scientific visualization
To use matplotlib for scientific visualization, you can customize the appearance of your plots using various keyword arguments to the `plot` function, such as `linewidth`, `marker`, `markersize`, and `markeredgewidth`. For example, to create a publication-ready plot of the function y = 2x, you can use the following code:
```python
import matplotlib.pyplot as plt
plt.plot([0, 1, 2, 3], [0, 1, 4, 9], linewidth=2, marker='o', markersize=6, markeredgewidth=2)
plt.xlabel('x', fontsize=14, fontweight='bold')
plt.ylabel('y', fontsize=14, fontweight='bold')
plt.title('Plot: y = x^2', fontsize=16, fontweight='bold')
plt.tick_params(axis='both', which='major', labelsize=12)
plt.show()
```
## Exercise
Create the following publication-ready plots using matplotlib:
- A plot of the function y = 2x.
- A plot of the circle with radius 3 centered at (0, 0).
- A plot of the parabola y = x^2.
# Advanced topics in scientific visualization
To create custom colormaps in matplotlib, you can use the `LinearSegmentedColormap` class. For example, to create a custom colormap that transitions from blue to red, you can use the following code:
```python
import matplotlib.pyplot as plt
from matplotlib.colors import LinearSegmentedColormap
cmap = LinearSegmentedColormap.from_list('custom', ['blue', 'red'])
plt.imshow([[0, 1], [1, 0]], cmap=cmap)
plt.colorbar()
plt.show()
```
To create contour plots in matplotlib, you can use the `pcolormesh` function. For example, to create a contour plot of the function y = x^2, you can use the following code:
```python
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-10, 10, 100)
y = x**2
plt.pcolormesh(x, y, cmap='viridis')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Contour Plot: y = x^2')
plt.colorbar()
plt.show()
```
## Exercise
Create the following advanced visualizations using matplotlib:
- A contour plot of the function y = x^2.
- A plot of the sphere with radius 3 centered at (0, 0, 0) with a custom colormap that transitions from blue to red.
- A plot of the parabola y = x^2 with a custom colormap that transitions from blue to red.
# Conclusion
In this textbook, you have learned how to use matplotlib for geometry and visualization. You have learned how to create various types of plots, customize their appearance, create interactive visualizations, and visualize advanced transformations and projections. You have also learned how to create 3D visualizations, animations and movies, and use matplotlib for scientific visualization.
By applying the concepts and techniques learned in this textbook, you will be able to create powerful and informative visualizations to explore and understand your data. | Textbooks |
Second-life battery systems for affordable energy access in Kenyan primary schools
Nisrine Kebir1,
Alycia Leonard1,
Michael Downey2,
Bernie Jones3,
Khaled Rabie4,
Sivapriya Mothilal Bhagavathy5 &
Stephanie A. Hirmer1
Scientific Reports volume 13, Article number: 1374 (2023) Cite this article
Climate-change adaptation
Energy grids and networks
As the world transitions to net zero, energy storage is becoming increasingly important for applications such as electric vehicles, mini-grids, and utility-scale grid stability. The growing demand for storage will constrain raw battery materials, reduce the availability of new batteries, and increase the rate of battery retirement. As retired batteries are difficult to recycle into components, to avoid huge amounts of battery waste, reuse and repurposing options are needed. In this research, we explore the feasibility of using second-life batteries (which have been retired from their first intended life) and solar photovoltaics to provide affordable energy access to primary schools in Kenya. Based on interviews with 12 East African schools, realistic system sizes were determined with varying solar photovoltaic sizes (5–10 kW in 2.5 kW increments) and lithium-ion battery capacities (5–20 kWh in 5 kWh increments). Each combination was simulated under four scenarios as a sensitivity analysis of battery transportation costs (i.e., whether they are sourced locally or imported). A techno-economic analysis is undertaken to compare new and second-life batteries in the resulting 48 system scenarios in terms of cost and performance. We find that second-life batteries decrease the levelized cost of electricity by 5.6–35.3% in 97.2% of scenarios compared to similar systems with new batteries, and by 41.9–64.5% compared to the cost of the same energy service provided by the utility grid. The systems with the smallest levelized cost of electricity (i.e., 0.11 USD/kWh) use either 7.5 kW or 10 kW of solar with 20 kWh of storage. Across all cases, the payback period is decreased by 8.2–42.9% using second-life batteries compared to new batteries; the system with the smallest payback period (i.e., 2.9 years) uses 5 kW solar and 5 kWh storage. These results show second-life batteries to be viable and cost-competitive compared to new batteries for school electrification in Kenya, providing the same benefits while reducing waste.
To meet growing global energy demands1 while combating climate change, much of the world's present energy use needs to be electrified via renewable energy2. As most renewable energy sources, including solar, wind, and tidal energy, are intermittent (i.e., not consistent over time), they must be complemented by energy storage to create systems which can consistently meet energy demands3,4.
The need for energy storage applies to both developed and developing countries. In low- and middle-income countries (LMICs), off-grid energy systems containing solar photovoltaics (PV) and batteries are remarkably common, given their plentiful solar resources5 and their need to expand energy access to rural areas distant from existing grid infrastructure6. Indeed, standalone solar-and-storage systems are increasingly the norm for expanding off-grid energy access in LMICs, and have been shown to provide social, economic, and environmental benefits7.
In off-grid energy access applications, lithium-ion batteries are often preferred over lead acid batteries due to their superior energy density and lifetime. Additionally, their ability to be continuously repaired using individual replacement cells8 makes them well suited in terms of cost, reliability, and long-term sustainability for remote applications. However, this technology comes with supply chain issues. Lithium-ion batteries are made from rare metals available in limited geographies (e.g., lithium, nickel, and cobalt)9,10 which will dwindle in supply as battery demand increases11. This is likely to raise prices, making lithium-ion batteries less viable in energy access applications where affordability is essential. It is already difficult to find a sustainable business model for micro- and mini-grids which recovers costs and can be profitable12; raising battery prices will exacerbate this situation. Therefore, to meet the demand for lithium-ion batteries at a reasonable cost, efforts must be made to recover, reuse, repurpose and recycle these rare battery materials13.
One option to address this is to give batteries a second life after they have reached 'end-of-life' in their first use. These are called 'second-life batteries' (SLBs). Using SLBs instead of new batteries (NBs) can reduce costs and waste while producing reliable renewable energy systems which recover raw materials. This is particularly important given the increasing use of batteries in electric vehicles (EVs), micro- and mini-grids, and utility-scale grid stability14,15,16,17. It is projected that the global supply of SLBs will reach 112–227 GWh by 2030; this is likely to continue escalating throughout the inevitable large-scale retirement of EVs18. It is therefore imperative to find sustainable opportunities to redeploy batteries19 as SLBs, given their considerable environmental and economic challenges at end of life20. While locally sourced SLBs are likely to be the cheapest option in many cases, there may also be an international SLB market in the near term. The increasing number of retiring batteries in developed countries is likely to raise challenges for those countries to recycle, reuse or repurpose them all internally. Exporting SLBs at low cost to LMICs could therefore become an attractive and mutually beneficial option.
In light of this, this work assesses the feasibility and potential benefits of using hybrid SLB and PV systems in energy access systems for schools in Kenya. While the idea of hybrid PV-battery energy access systems for schools has been discussed in Refs.21,22,23, these studies did not consider SLBs. The use of SLBs for rural households' electrification to improve general living conditions has been discussed in Refs.24,25; however, the idea of incorporating SLBs coupled with PV systems in rural schools proposed in the current study is novel. The techno-economic feasibility of such systems requires study to ensure that they can lead to increased and enhanced energy access at a competitive cost, making them a viable use case for end-of-life batteries.
We consider the use of SLBs coupled with PV in Kenyan primary schools. This was informed by semi-structured interviews with stakeholders and members of staff in 12 East African schools which elicited energy needs, aspirations, and technical and financial challenges. The energy characteristics of four of these schools in Kenya are studied in depth. We take one as a representative case study for techno-economic analysis, evaluating the costs and reliability of providing energy using (1) systems containing SLBs, (2) systems containing NBs, and (3) grid-connected energy. An optimal system size and type to implement is proposed using realistic assumptions from the wider literature. Considering these results, we draw conclusions on the feasibility of SLBs in this application to reduce battery waste while creating cross-cutting benefits. Note that all assumptions related to lithium-ion batteries in this paper are specifically for lithium nickel manganese cobalt oxides.
Second-life battery uses
When discussing SLBs, it is important to distinguish between reuse, repurposing, and recycling. Whereas reuse involves directly using a material again for the same purpose, repurposing gives a material a new purpose than for what it was originally intended, and recycling involves breaking down the product to a more fundamental level for reuse of constituent components. When considering SLBs, battery reuse or repurposing often maximises value compared to recycling, as these options consume less energy26. Most batteries are currently assembled without a recycling focus, meaning that certain parts—particularly the cells that contain valuable metals—are permanently connected by welding and/or glue, making them difficult to extract. As such, applications that can reuse or repurpose batteries without disassembly are likely to be the most cost effective for current SLB stock.
Lithium-ion batteries from electric vehicles (EVs) are particularly high-value SLBs. At their 'end-of-life', they still typically have 70–80% of their capacity remaining27. This equates to thousands of charge/discharge cycles and hours of usable energy storage. While these batteries may no longer be appropriate for use in EVs, there are many other applications which can make valuable use of this remaining capacity, as reviewed by a number of other authors20,28,29. Based on the literature and author field experience, SLB applications of interest in LMIC contexts include light mobility, single-building power supplies, and micro- and mini-grids; the suitability of SLBs in these cases is elaborated in Table 1. One promising opportunity for SLB redeployment is to enable affordable electricity access in school settings. This falls into the single-building power supply category listed in Table 1, which is the focus of this research, as we investigate this repurposing potential for primary schools in Kenya.
Table 1 Applications of interest for lithium-ion second-life batteries from electric vehicles in low- and middle-income countries.
Opportunities for second-life batteries in school energy access
There are approximately 32,437 primary schools in Kenya. According to a government spokesperson, in December 2017, 76% of these schools had access to electricity30. However, even where schools are connected to the grid, power is unreliable31. This means that pupils are often forced to limit their learning to daylight hours, or to when power is available. Electricity outages also limit access to modern educational technologies (e.g., computers, projectors) which, while not an educational panacea, play a complementary role and fill instructional gaps32. Not being able to use these technologies can disrupt lesson plans and limit consequent learning.
Given the resource constraints in Kenya's educational system, the high costs of using grid electricity compete with other critical school expenses. A 2019 study of 300 boarding schools in Kenya showed an average monthly spend of $4000 on electricity alone33; this is equivalent to the average cost of employing two teachers in the Nairobi area34. As such, with limited budgets available, schools may be forced to cut costs by employing fewer teachers or minimising electricity use, each of which negatively impacts learning.
Alternative and complementary off-grid energy sources for schools, such as solar PV systems with batteries, can offer a more cost effective and reliable energy alternative. Indeed, the use of hybrid systems for electrification has shown benefits in East Africa, and for schools in particular which rely on lighting and information technologies to provide quality education21,35,36. However, particularly in poorer rural communities, there is often still a lower willingness and ability to pay for such systems37; as such, costs need to be minimised wherever possible. The reuse of lithium-ion batteries could have a significant impact on recovering some of the costs involved in building micro-grids or off-grid systems25, passing affordability benefits onto the consumer.
There are also numerous waste management challenges associated with standalone systems, as previously discussed38. To illustrate, around 700 tonnes of solar e-waste were discarded in Kenya alone in 201639. When discarded improperly, batteries can cause major health concerns to nearby communities40; as such, reducing their wastage where possible is critical. By instead repurposing and re-using these batteries as SLBs in off-grid systems, we hypothesise that affordable systems can be designed for school electrification which also reduce waste, avoid health harms, and provide social benefit.
Energy needs and challenges assessment
An energy needs assessment was undertaken in 12 schools (eight in Kenya and two each in Uganda and Tanzania) via semi-structured interviews. These were selected on the basis of existing contacts and data access. Interviews covered: current energy use (including devices and hours of use), sources, costs, and satisfaction; energy use aspirations (i.e., what devices they would connect if energy were more abundant, reliable and/or affordable); whether they would consider an alternative source of energy, especially from a solar/battery system; and demographics.
The principal uses of electricity identified in the interviews were classroom lighting, security lighting, information technology (e.g., computers, printers/copiers, projectors, etc.), and phone charging. These all use relatively little power, which speaks well to the potential suitability of hybrid battery and PV solutions to meet school energy needs. It was reported that these technologies impacted the ability of students to study (lighting), the security of the school community and premises (security lighting), the ability of teachers to prepare and deliver lessons (phone charging, laptops), preparation for examinations (printing and copying), and basic life/livelihood skills for better educational outcomes (information technology labs). Some aspirational productive uses of electricity were also identified by interviewees (e.g., submersible borehole pumps for water, refrigeration for food preservation). However, these are considerably more challenging to power from a hybrid battery and PV system and are also less core to the learning purpose of the school. As such, these high-powered devices were not carried forward into the techno-economic analysis.
Interviewees at grid-connected schools reported technical and financial challenges in using electricity. They reported that power was unreliable due to frequent and prolonged network outages. They also reported that expensive utility electricity contributed to schools' high costs. This, combined with a lack of (or late payment of) funds to cover school bills from central funding sources, meant that interviewed schools often had to seek additional financial support from pupils' parents to be able to pay their electricity bills. This indicates an urgent need to provide an affordable and reliable energy access alternative. The impact of schools having limited, expensive electricity was reported to result in reduced educational outcomes, increased financial burden for parents, poor staff retention, and ultimately reduced school attendance.
Basic characteristics of case study schools
Of the sampled schools, four schools in Kenya were characterised in more detail: two situated in Nairobi, and one each in Kagiado and Machakos counties, as listed in Table 2. Schools 1, 2, and 3 are grid connected, while School 4 is not electrified at present and uses a stand-alone biogas plant for school cooking needs. Schools 1 and 2 are urban while Schools 3 and 4 are rural. They are all state schools supported by the government.
Table 2 Characteristics of the four Kenyan primary schools selected for detailed study.
Drawing from the needs assessment, Table 2 reports the average number of students per school, their required hours of electricity use, the maximum aspirational load of each school, and their annual electric bills. Note that School 1 uses electricity throughout all hours of the day as there was a reported need to provide continuous low lighting for security reasons at night.
The maximum aspirational load of these schools shown in Table 2 reflects the distribution of electrical appliances desired by the four schools, as illustrated in Fig. 1. These appliances are roughly the same for the four schools and require an average installed power around 5 kW. Based on each school's appliance aspirations, projected loads were relatively consistent during school opening days during term time, and less (but not zero) during weekends and holiday closure. The most marked seasonal variation noted within projected demand was a greater use of printing and copying in the lead up to the examination period. This allows students to do practice tests and allows schools to provide students with revision material.
Quantity of each appliance type desired per school, and their rated power in watts (W).
Techno-economic analysis of second-life battery system feasibility
We next conducted a techno-economic analysis to assess the impact of introducing hybrid PV and battery systems in these schools. Both NB and SLB options were studied, and outcomes compared in terms of capital cost, transportation cost, energy reliability, and dependency on grid supply to supplement the system. While we primarily focus on the comparison of locally sourced SLBs with imported NBs, as a sensitivity analysis, we also assess the economic impact of using imported SLBs in the event of local unavailability. We also investigate potential opportunities stemming from using imported SLBs at a low capital expenditure (CAPEX) cost. Therefore, we examine 48 scenarios including different combinations of PV, NBs, SLBs, assuming that NBs are always imported and SLB might either be local, imported at average CAPEX, or imported at a minimum CAPEX (i.e., 0.04 USD/kWh41). Given the relative similarity of the four characterised schools, we focus on School 1 as a representative case study for this analysis.
We calculated the hourly annual load demand curve of the school based on the appliance aspirations resulting from the needs assessment (for a fuller description, see "Methods"). This resulted in an annual average energy need of 10,220 kWh. We then calculated the potential solar PV production using a set of PV sizes ranging from 5 to 10 kW with an increment of 2.5 kW. Similarly, the storage capacity was selected from lithium-ion battery sizes ranging from 5 to 20 kWh with an increment of 5 kWh. These sizes are selected as they are the most common available solar system sizes42 and battery sizes in the region. As such, they are most likely to be available locally as retired NBs, which are available to be redeployed as SLBs to help provide energy access in schools.
We studied the annual import of energy from the grid required by School 1 in kWh for each scenario. The results (see Fig. 2) show that, as expected, dependency on the grid reduces with increased energy system capacity. However, grid dependency increases when SLBs are used compared to NBs. This makes sense, as grid dependency is impacted by the technical performances of the batteries in terms of (1) state of charge, which is reduced for SLBs compared to NBs, and (2) self-discharge rate, which is higher for ageing batteries (see Table 3). Considering that grid reliability is already an issue in schools, reducing grid dependency in terms of the number of days is key, particularly as outages increase in the region35,36.
Energy imported from the grid per year in each system size and battery type scenario, considering second-life batteries (SLB) and new batteries (NB).
Table 3 Economic and technical assumptions for techno-economic analysis.
Figure 3 illustrates the number of days the grid is needed to provide energy to School 1 for each scenario. This directly correlates to the capacity of the battery in the scenario and how many hours this could cover in the daily demand peak of the year. This varies between 20 and 100% for both energy storage system types studied here. These results show that the yearly dependency on the grid could be reduced to 25 days for the NB case and 113 days for the SLB case, in the 10 kW PV and 20 kWh storage capacity scenario.
Number of grid-dependent days per year in each system size and battery type scenario, considering second-life batteries (SLB) and new batteries (NB).
To assess the impact of each hybrid scenario on the school's electricity cost, we compared the levelized cost of electricity (LCOE) in each scenario over 25 years with the LCOE calculated from the utility. This was estimated at 0.31 USD/kWh, resulting from the current electricity cost of 0.28 USD/kWh and considering an annual utility escalating factor of 1%. Results for each scenario show that the LCOE varies between 0.11 USD/kWh and 0.22 USD/kWh (see Fig. 4) and that in 97.2% of the scenarios studies, using SLBs is cheaper compared to NBs as per the LCOE obtained. This was calculated using the methodology described in the "Methods" section and the assumption on Table 3 and matches results on LCOE for NB in hybrid systems in Kenya from previous studies on a discount rate basis of 5%46. The various scenarios for LCOE calculation have included the impact of the NB and SLB replacement over 25 years (see Table 3) such as the operations and maintenance cost increase and the battery efficiency improvement following the replacement.
Levelized cost of energy (LCOE) per scenario, including options with new batteries transported (NB-T), second-life batteries transported (SLB-T), local second-life batteries (SLB-L), and second-life batteries transported at minimum CAPEX (SLB-T-MC). Across the 12 system size options, SLBs reduce LCOE compared to NBs in almost all of the cases studied.
The cost benefit of switching from grid connected electricity to one of the hybrid SLB or NB systems specified here is studied in terms of the system payback period (PBP). The results, illustrated in Fig. 5, show that the minimum PBP is 2.9 years, corresponding to the use of a hybrid system composed of 5 kW PV and 5 kWh SLB storage.
Payback period for each scenario, including options with new batteries transported (NB-T), second-life batteries transported (SLB-T), local second-life batteries (SLB-L), and second-life batteries transported at minimum CAPEX (SLB-T-MC). The system with 5 kW PV and 5 kWh SLB has the shortest payback period.
This work shows the potential benefits of using hybrid PV and battery systems containing lithium-ion SLBs in Kenyan schools. The enhanced lifespan and ease of repair of these batteries positions them well to offer cheaper, reliable, and more sustainable electricity access than NB alternatives; their economic and practical viability is proven for the case of school electrification in this work.
We have shown that the use of repurposed lithium-ion SLBs provide huge affordability benefits to the end user of energy systems compared to NBs. This agrees with other studies that used SLBs in similar applications (e.g., residential usage in Ref.42, mini-grid in Tanzania in Ref.24). In fact, locally sourced SLBs are cheaper than imported NBs in all of the scenarios studied, with a maximum reduction in LCOE (29.4%) for the (1) 7.5 kW PV and 20 kWh SLB storage, and (2) 10 kW PV and 20 kWh SLB storage systems. The latter reduction in LCOE has also been identified for the use of imported SLBs in the systems (1) and (2); however, it was found that importing SLBs over using local SLBs in the systems (3) 7.5 kW PV and 5 kWh SLB storage, (4) 7.5 kW PV and 10 kWh SLB storage, and (5) 5 kW PV and 20 kWh SLB storage is not economically encouraged.
The sensitivity analysis undertaken here shows that low-CAPEX SLB imports are an exciting opportunity in LMICs in the near term. The results for the low expense imported SLB case (SLB-T-MC) show an LCOE drop of 5.6–35.3% in all scenarios compared to NBs. This is a 16.7% drop compared to local or imported SLBs at average market cost for the same preferred systems. Therefore, while the international transportation of lithium-ion batteries can be costly (i.e., around 28.02 $/kWh54; the import duties of lithium-ion account for up to 35% of the cost of the product in Kenya55) and subject to complex regulations56 (e.g., due to their perception as dangerous based on previous incidents where they have caused fires61; furthermore, international transport regulations can class SLBs as waste, which is subject to different regulations than consumer goods (e.g., Ref.62 in the United Kingdom)), there are still opportunities in using imported batteries compared to NBs. If international transit difficulties can be overcome, using SLB originating from EVs in developed countries56 for hybrid systems in Kenya would be a low-cost opportunity to reduce waste and at the same time enhance energy access. Until the EV sector is established in Kenya, or the second-hand EV market begins to flourish, SLB import is therefore a viable option. As the stock of locally available SLBs increases (e.g., as EVs are increasingly used domestically), the local market is likely to undercut these low-CAPEX imports, as they can avoid custom costs and duties. These local SLBs will create new value from otherwise waste-bound batteries without the need to cross borders, reducing costs for end users.
Regarding the payback periods calculated, we find that schools will be able to pay back hybrid system costs in a shorter period of time when SLBs are used compared to NBs in all of the scenarios studied. The shortest system payback period was found to be 2.9 years for the 5 kW PV and 5 kWh SLB scenario. Beyond these cost benefits, we have also found that using SLB systems in schools can decrease grid dependency. This is important given the high number of network outages that occur in rural and urban areas of Kenya, which are greatly disruptive to learning.
While we have shown that SLBs can reduce the costs of school energy access in Kenya significantly while effectively mitigating the challenges of battery waste, there are still a number of issues that need to be overcome for SLBs to be effectively used on a wide scale. These are highlighted in the following opportunities for further research and development.
There are still limited end of life strategies for SLBs. SLBs of course bring benefits in terms of waste management of NBs, which are considered to be one of the key hazardous wastes to be reduced in sub-Saharan Africa57. However, SLB systems are not a permanent waste management solution. SLBs will also, eventually, reach the end of their second life. At this point, recycling their materials is likely to be the most economic and sustainable remaining waste management option. There is therefore still a need to research and develop better battery recycling strategies to fulfil this future need27.
SLBs require monitoring and control, which can be difficult in remote areas. Monitoring of SLBs is needed to schedule preventive and predictive maintenance, which is liable to be somewhat less predictable than that of NBs given their varying first-life uses. It is also needed to facilitate resource sharing amongst batteries, loads, and PV in unified off-grid systems. In rural areas, which as previously discussed are good candidates for off-grid SLB-based electrification, this monitoring needs to be done remotely to keep costs down, which is challenging. However, in a country like Kenya which places a high importance on mobile connectivity (e.g., as evidenced by its largely mobile-based financial system M-PESA58), there is an opportunity for internet of things (IoT) sensors combined with mobile connectivity to play this monitoring role. IoT can be used to collect important system parameters (e.g., state of charge, number of cycles) for more efficient utilisation of available devices (e.g., batteries, solar panels and loads). More specifically, narrowband IoT has a great potential to further improve the performance and reliability of SLB energy access systems59. This could facilitate (1) resource sharing within a community or school system, providing further benefit from SLBs at minimal infrastructure cost, (2) optimised performance and maximised energy efficiency and sustainability, and (3) in the case studied here, prolonged time spent at school and consequent educational benefit. Motivated by the above, as future work, the authors intend to practically investigate the achievable gains that can be obtained with the IoT-equipped SLB system.
Lack of awareness of the benefits of SLBs. Despite these significant potential benefits of SLBs, there are still barriers to their adoption, including technology awareness. For the benefits of lithium-ion SLBs to be realised, appropriate knowledge sharing needs to be undertaken with regards to the benefits of lithium-ion SLBs to increase the likelihood of adoption. In this work, a second round of interviews conducted with the same 12 East African schools pointed to a gap in knowledge of the benefits of SLBs. Stakeholders expressed a reluctance to buy into the idea of SLBs as opposed to NBs. To promote SLB uptake in this case, the educational institution would ideally be leveraged to raise awareness and train the community members to maintain the SLB system. In a similar vein, future work should also explore the potential of sustainable battery centres which build, service, maintain, and recycle end-of-life NBs situated near schools or facilities using the SLB systems. This could provide local operations and maintenance expertise and also be an economic opportunity to promote clean development.
This work uses semi-structured interview data obtained through partners in anonymized aggregate to determine school energy use, challenges, and aspirations from a representative sample. Data from 12 primary schools is used—eight in Kenya, and two each in Tanzania and Uganda. Interviews were carried out by partners with either the head teacher or the administrative manager of the school. These interviews were coordinated by Smart Villages Research Group Ltd. and undertaken by three partner organisations based in-country whose normal activities are not in education or school energy provision, making them neutral parties for data collection. Schools were made aware that there was no expectation that they would be provided with any off-grid energy technology or solutions to support their energy needs, with an aim to eliminate artificially optimistic or aspirational responses.
Interview results were only shared with the research team in anonymized aggregate. The interview process underwent ethical review through the Smart Villages Research Group Ltd. ethics committee, which consists of the principal staff and advisors, and received full approval. All methods were performed in accordance with the relevant guidelines and regulations, including informed consent, which was obtained for all interviews.
A first round of interviews was carried out in April and May of 2021, when many schools were just emerging from lockdowns. Face-to-face visits were therefore often not feasible, so part of the selection criteria for the schools was the availability of a senior local contact, who could be phoned and would be knowledgeable regarding school activities, economics, and practices. The questions in the first semi-structured interview included: current energy use (including devices and hours of use), sources of energy, average costs, and satisfaction; energy use aspirations (i.e., what devices they would connect if energy were more abundant, reliable and/or affordable); whether they would consider an alternative source of energy, especially from a solar/battery system; and demographics. A second round of interviews was then completed in March 2022 with the same schools to present them with some of the conclusions of the techno-economic analysis (e.g., some sample system costings, and estimated grid-savings and reduction of reliance on the grid) and to ask them their reactions to the options. They were asked whether any of those might be attractive to their school, and about their general understanding of batteries in solar power systems, lithium-Ion batteries, and SLBs in particular.
Electricity demand forecast and yearly load profile
The daily load profile, on an hourly timescale, through the year for School 1 was calculated based on the following assumptions:
The electrical appliances used and desired in the school, their rating and number of hours of use per day as provided by the school staff (see Table 2 and Fig. 1):
Fans turn-off in the rainy season (the long rainy season lasts from March to May, while the short rainy season lasts from October to November).
Printers on four hours during terms and eight hours in exams periods.
The school terms and holidays scheduled between 2021 and 2023, to build an average yearly figure (see Table 4):
School days are Monday–Saturday.
The term dates are always January to April, May to August and September to November.
Table 4 Kenya school terms and holidays periods (2021–2023).
Solar PV production modelling
The annual electricity generated by the PV systems for the various sizing on an hourly basis through year 2019 (as the year of reference), was calculated using a combination of inputs:
Expected yield factor from solar array (2019 data) are taken from the https://www.renewables.ninja/ website.
Capacity of the PV array for each scenario.
Assessing electrification costs
To compare the utility cost to the grid-connected hybrid systems proposed including PV arrays and battery energy storage systems, we have used the following LCOE (USD per kWh) formula based on the Net Present Values (NPV) calculation46:
$$LCOE= \frac{{C}_{0}+\sum_{t=1}^{N}\frac{{C}_{PV+Storage+Grid}^{t}}{{(1+r)}^{t}}}{\sum_{t=1}^{N}\frac{{E}_{PV+Storage+Grid}^{t}}{{(1+r)}^{t}}}.$$
\({C}_{0}\) represents the capital cost initially invested (including transportation costs if relevant depending on the scenario studied), \({C}_{PV+Storage+Grid}\) and \({E}_{PV+Storage+Grid}\) are the yearly costs and energy produced by the hybrid system over the lifetime of the project (\(N\)), and \(r\) is the discount rate. This formula applies for both NBs and SLBs.
To calculate the number of years the school will need to recover what has been invested, the payback period has also been calculated for each scenario by dividing the investment cost by the yearly cash flow60.
Raw interview data collected to inform the current study are not publicly available to protect the privacy and anonymity of the schools involved. The research team only accessed these data in anonymized aggregate. The aggregated results which are pertinent to the analysis presented here are shown in-text in Table 2 and Fig. 1. Further details are available from the corresponding author on reasonable request.
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This project was supported by the Royal Academy of Engineering under the Frontiers of Development scheme. Authors S.H. and A.L. are part of the Climate Compatible Growth Programme which is supported by the United Kingdom Foreign, Commonwealth, and Development Office. The authors gratefully acknowledge the Engineering and Physical Sciences Research Council's Global Challenges Research Fund, UK (Grant No: EP/R030111/1), for funding the work of N.K. The views expressed in this paper do not necessarily reflect the UK government's official policies.
Energy and Power Group, University of Oxford, Oxford, UK
Nisrine Kebir, Alycia Leonard & Stephanie A. Hirmer
Aceleron Limited, Bromsgrove, UK
Michael Downey
Smart Villages Research Group, Abingdon, UK
Bernie Jones
Department of Engineering, Manchester Metropolitan University, Manchester, UK
Khaled Rabie
Power Networks Demonstration Centre, University of Strathclyde, Glasgow, UK
Sivapriya Mothilal Bhagavathy
Nisrine Kebir
Alycia Leonard
Stephanie A. Hirmer
N.K.: Writing—original draft preparation, methodology, results, visualisation. A.L.: Writing—original draft preparation, writing—review and editing, visualisation. M.D.: Writing—review and editing. B.J.: Conceptualisation, methodology, investigation, writing—review and editing, funding acquisition. K.R.: Conceptualisation, funding acquisition, writing—review and editing, funding acquisition. S.M.B.: Conceptualisation, methodology, writing—review and editing, funding acquisition. S.H.: Methodology, writing—original draft preparation, project administration, supervision.
Correspondence to Stephanie A. Hirmer.
M.D. works for Aceleron, an energy storage company working on lithium-ion battery technology. B.J. works for the Smart Villages Research Group, an organisation focused on off-grid clean energy and technology access in developing contexts. Authors N.K., S.H., A.L., and S.M.B. declare no competing interests.
Kebir, N., Leonard, A., Downey, M. et al. Second-life battery systems for affordable energy access in Kenyan primary schools. Sci Rep 13, 1374 (2023). https://doi.org/10.1038/s41598-023-28377-7 | CommonCrawl |
Analysis of Nonlinear Partial Differential Equations
Lead Research Organisation: University of Oxford
Department Name: Mathematical Institute
Partial differential equations (PDEs) are equations that relate the partial derivatives, usually with respect to space and time coordinates, of unknown quantities. They are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of phenomena in the physical, natural and social sciences, often arising from fundamental conservation laws such as for mass, momentum and energy. Significant application areas include geophysics, the bio-sciences, engineering, materials science, physics and chemistry, economics and finance. Length-scales of natural phenomena modelled by PDEs range from sub-atomic to astronomical, and time-scales may range from nanoseconds to millennia. The behaviour of every material object can be modelled either by PDEs, usually at various different length- or time-scales, or by other equations for which similar techniques of analysis and computation apply. A striking example of such an object is Planet Earth itself.Linear PDEs are ones for which linear combinations of solutions are also solutions. For example, the linear wave equation models electromagnetic waves, which can be decomposed into sums of elementary waves of different frequencies, each of these elementary waves also being solutions. However, most of the PDEs that accurately model nature are nonlinear and, in general, there is no way of writing their solutions explicitly. Indeed, whether the equations have solutions, what their properties are, and how they may be computed numerically are difficult questions that can be approached only by methods of mathematical analysis. These involve, among other things, precisely specifying what is meant by a solution and the classes of functions in which solutions are sought, and establishing ways in which approximate solutions can be constructed which can be rigorously shown to converge to actual solutions. The analysis of nonlinear PDEs is thus a crucial ingredient in the understanding of the world about us.As recognized by the recent International Review of Mathematics, the analysis of nonlinear PDEs is an area of mathematics in which the UK, despite some notable experts, lags significantly behind our scientific competitors, both in quantity and overall quality. This has a serious detrimental effect on mathematics as a whole, on the scientific and other disciplines which depend on an understanding of PDEs, and on the knowledge-based economy, which in particular makes increasing use of simulations of PDEs instead of more costly or impractical alternatives such as laboratory testing.The proposal responds to the national need in this crucial research area through the formation of a forward-looking world-class research centre in Oxford, in order to provide a sharper focus for fundamental research in the field in the UK and raise the potential of its successful and durable impact within and outside mathematics. The centre will involve the whole UK research community having interests in nonlinear PDEs, for example through the formation of a national steering committee that will organize nationwide activities such as conferences and workshops.Oxford is an ideal location for such a research centre on account of an existing nucleus of high quality researchers in the field, and very strong research groups both in related areas of mathematics and across the range of disciplines that depend on the understanding of nonlinear PDEs. In addition, two-way knowledge transfer with industry will be achieved using the expertise and facilities of the internationally renowned mathematical modelling group based in OCIAM which, through successful Study Groups with Industry, has a track-record of forging strong links to numerous branches of science, industry, engineering and commerce. The university is committed to the formation of the centre and will provide a significant financial contribution, in particular upgrading one of the EPSRC-funded lectureships to a Chair
Funded Value:
Funded Period:
Funder:
Project Category:
Project Reference:
EP/E035027/1
John Ball
Research Subject:
Mathematical sciences (100%)
Mathematical Analysis (100%)
If populated the following is a graphic depicting where in the UK the given postcode is located.
University of Oxford, United Kingdom (Lead Research Organisation)
HEFCE, United Kingdom (Co-funder)
John Ball (Principal Investigator) http://orcid.org/0000-0003-2496-2020
Nick Woodhouse (Co-Investigator)
Keith Burnett (Researcher)
Title Publication Date Published
|< < 1 2 3 4 5 6 7 8 > >|
Chrusciel P (2010) A Uniqueness Theorem for Degenerate Kerr-Newman Black Holes in Annales Henri Poincaré
Chrusciel P (2008) On Mason's Rigidity Theorem in Communications in Mathematical Physics
Chrusciel P (2009) Topological Censorship for Kaluza-Klein Space-Times in Annales Henri Poincaré
Chrusciel P (2008) Singular Yamabe Metrics and Initial Data with Exactly Kottler-Schwarzschild-de Sitter Ends in Annales Henri Poincaré
Chrusciel P (2009) On higher dimensional black holes with Abelian isometry group in Journal of Mathematical Physics
Chrusciel P (2010) On smoothness of black saturns in Journal of High Energy Physics
Chrusciel P (2011) Ghost points in inverse scattering constructions of stationary Einstein metrics in General Relativity and Gravitation
Condette N (2010) Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth in Mathematics of Computation
Dai S (2010) Crossover in coarsening rates for the monopole approximation of the Mullins-Sekerka model with kinetic drag in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Deng X (2011) Global solutions of shock reflection by wedges for the nonlinear wave equation in Chinese Annals of Mathematics, Series B
Diening L (2013) Finite Element Approximation of Steady Flows of Incompressible Fluids with Implicit Power-Law-Like Rheology in SIAM Journal on Numerical Analysis
Ding M (2012) Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations in Zeitschrift für angewandte Mathematik und Physik
Ding M (2013) Global existence and non-relativistic global limits of entropy solutions to the 1D piston problem for the isentropic relativistic Euler equations in Journal of Mathematical Physics
Dolzmann G (2012) BMO and uniform estimates for multi-well problems in Manuscripta Mathematica
Duzaar F (2007) The existence of regular boundary points for non-linear elliptic systems in Journal für die reine und angewandte Mathematik (Crelles Journal)
Fang B (2016) The uniqueness of transonic shocks in supersonic flow past a 2-D wedge in Journal of Mathematical Analysis and Applications
Figueroa L (2012) Greedy Approximation of High-Dimensional Ornstein-Uhlenbeck Operators in Foundations of Computational Mathematics
Fuchs M (2008) Existence of global solutions for a parabolic system related to the nonlinear Stokes problem in Journal of Mathematical Sciences
Gallagher I (2012) A profile decomposition approach to the $$L^\infty _t(L^{3}_x)$$ Navier-Stokes regularity criterion in Mathematische Annalen
Giannoulis J (2008) Lagrangian and Hamiltonian two-scale reduction in Journal of Mathematical Physics
Helmers M (2012) Kinks in two-phase lipid bilayer membranes in Calculus of Variations and Partial Differential Equations
Helmers M (2014) CONVERGENCE OF AN APPROXIMATION FOR ROTATIONALLY SYMMETRIC TWO-PHASE LIPID BILAYER MEMBRANES in The Quarterly Journal of Mathematics
HELMERS M (2011) SNAPPING ELASTIC CURVES AS A ONE-DIMENSIONAL ANALOGUE OF TWO-COMPONENT LIPID BILAYERS in Mathematical Models and Methods in Applied Sciences
Herrmann M (2010) Action Minimising Fronts in General FPU-type Chains in Journal of Nonlinear Science
Herrmann M (2009) Self-similar solutions for the LSW model with encounters in Journal of Differential Equations
Further Funding
Description This was a broad grant designed to help consolidate research on nonlinear partial differential equations in the UK. In particular the Oxford Centre for Nonlinear PDE was founded as a result of the grant and is now a leading international centre. As regards specific research advances, these were in various areas of applications of PDE, for example to fluid and solid mechnaics, liquid crystals, electromagnetism, and relativity.
Exploitation Route Through publications and consultation with current and former members of OxPDE.
Sectors Aerospace, Defence and Marine,Chemicals,Construction,Electronics,Energy,Environment
URL https://www0.maths.ox.ac.uk/groups/oxpde
Description Advanced Investigator Grant
Amount € 2,006,998 (EUR)
Funding ID 291053
Organisation European Research Council (ERC)
Sector Public
Country Belgium
Start 03/2012
End 03/2017 | CommonCrawl |
\begin{document}
\title{Modelling elastic structures with strong nonlinearities with application to stick-slip friction}
\author{Robert Szalai}
\affiliation{University of Bristol, Queen's Bldg., University Walk, Bristol, BS8 1TR, UK, email: }
\email{[email protected]}
\date{5th September 2013} \begin{abstract} An exact transformation method is introduced that reduces the governing equations of a continuum structure coupled to strong nonlinearities to a low dimensional equation with memory. The method is general and well suited to problems with point discontinuities such as friction and impact at point contact. It is assumed that the structure is composed of two parts: a continuum but linear structure and finitely many discrete but strong nonlinearites acting at various contact points of the elastic structure. The localised nonlinearities include discontinuities, e.g., the Coulomb friction law. Despite the discontinuities in the model, we demonstrate that contact forces are Lipschitz continuous in time at the onset of sticking for certain classes of structures. The general formalism is illustrated for a continuum elastic body coupled to a Coulomb-like friction model. \end{abstract} \maketitle
\section{Background}
One of the greatest concerns of engineers is modelling friction and impact. These two strong nonlinearities occur in many mechanical structures, e.g., underplatform dampers of turbine blades \citep{Firrone,PetrovFEA}, tyre models \citep{PacejkaMagic}, or in general any jointed structure \citep{Quinn,Segalman}. The most common way of modelling such systems is to take a finite dimensional approximation assuming that the omitted dynamics has only a small effect on the overall result. Such models can then be analysed using the well established theory of non-smooth dynamical systems \citep{diBernardoBook,FilippovBook}. The application of this theory to engineering structures provides a great insight, even though not all phenomena could be experimentally confirmed \citep{Popp}. Recent results however indicate significant deficiencies that question the predictive power of finite dimensional non-smooth models. It was shown that when a rigid constraint becomes slightly compliant in a friction-type system small-scale instabilities develop \citep{SieberSingular}. This means that refining the model by including more degrees of freedom can lead to qualitatively disagreeing results. The solution can also become non-deterministic \citep{ColomboJeffrey,PainleveAlan} or non-unique in forward time for an otherwise well specified initial condition. Therefore a better modelling framework is necessary that either eliminates inconsistency and non-determinism or at least provides a hint about the physical mechanism that causes such behaviour.
The most apparent problem with finite dimensional non-smooth models of mechanical systems is that they use rigid body dynamics to describe the motion. This includes finite mode approximation of elastic structures, where each mode has a non-zero modal mass \citep{EwinsBook}. When two contacting elastic bodies slip and then suddenly stick their contact points will experience a jump in acceleration. In case of a finite mass at the contact point, the contact force also has a jump. In reality, however, the mass of the contact point or contact surface is zero, which implies that the contact force must be continuous. For this reason standard finite mode description of elastic bodies is qualitatively inaccurate. The continuity of contact force should be preserved by mechanical models.
In this paper we propose a formalism that helps better understand and perhaps solve the above problems. We investigate mechanical systems that consist of linear elastic structures coupled at isolated points of contact with strong nonlinearities. This class of problems include mechanisms with Coulomb-like friction models. The dynamics of impact is considered in a companion paper \citep{SzalaiMZImpact}. Our formalism accounts for the zero mass of the contact point without artificially introducing coupling springs as in \citep{Melcher} to regularise the problem. To achieve such model reduction and still provide an exact description we introduce memory terms. Within this new framework the dynamics is described by a low-dimensional delay equation. We show that in our formulation contact forces are Lipschitz or continuous for certain classes of structures when the solution transitions onto the discontinuity surface. The new formalism also leads to well defined dynamics, since small perturbations of the reduced model do not affect the qualitative features of the dynamics in general.
Time-delayed models have already been in use when modelling friction \citep{HessSoom,Putelat2011}. In these cases, however the delay parameters are fitted to experimental observations. We hope that through our theory these empirical parameters can gain physical meaning.
The paper is organised as follows. In section \ref{sec:Mechmod} we present our general mechanical model. In section \ref{sec:ModRed} we describe our model reduction technique, discuss the convergence of the method and its implications to non-smooth systems. The derivation of the memory term is illustrated through the examples of a pre-tensed string and a cantilever beam. In section \ref{sec:Bowstring} we present the example of a bowed string. We demonstrate the properties of the transformed equation of motion in particular its convergence as the number of vibration modes goes to infinity.
\section{Mechanical model\label{sec:Mechmod}}
\begin{figure}
\caption{(colour online) A linear elastic structure with contact points $\chi_{j}$. Each contact point has a three-dimensional motion, however we project that motion to vectors $\boldsymbol{\phi}_{j}$ to obtain a scalar valued resolved variable $y_{j}$. If we need to resolve more than one direction of the motion of the contact point $\chi_{j}$, we attach multiple labels $\chi_{j}=\chi_{j+1}=\chi_{j+2}$ to the same point. Then the motion is projected by the linearly independent vectors $\phi_{j},\,\phi_{j+1},\,\phi_{j+2}$ to yield the resolved variables $y_{j},\, y_{j+1},\, y_{j+2}$. By this definition we ensure a one-to-one mapping between indices of labels $\chi_{j}$ and vectors $\phi_{j}$.}
\label{fig:potato}
\end{figure} The mechanical model of our structure is divided into two parts, a linear elastic body and several discrete non-smooth nonlinearities that are coupled to the continuum structure. First we describe our assumptions on the linear but infinite dimensional part of the model and then we explain how non-smooth nonlinearities are coupled to the system. The description is sufficiently general to describe friction oscillators and impact phenomena. For simplicity, we only focus on a single elastic structure, but our framework is trivially extensible to mechanisms involving multiple linear structures coupled at (strongly) nonlinear joints.
We assume that the displacement of a material point $\chi$ of the structure at time $t$ is represented by $\boldsymbol{u}(\chi,t)$. We also assume that the motion $\boldsymbol{u}(\chi,t)$ can be expressed as a series \begin{equation} \boldsymbol{u}(\chi,t)=\sum_{k=1}^{\infty}\boldsymbol{\psi}_{k}(\chi)x_{k}(t),\label{eq:dispField} \end{equation} where $\boldsymbol{\psi}_{k}(\chi)$ are three dimensional vector valued functions depending on the spatial coordinates of the structure only. The \emph{generalised coordinates} $x_{k}$ can be arranged into a vector $\boldsymbol{x}=(x_{1},x_{2},\ldots)^{T}\in\mathbb{R}^{\infty}$ to simplify the notation. Due to linearity the governing equations can be written as \begin{equation} \ddot{\boldsymbol{x}}(t)+\boldsymbol{C}\dot{\boldsymbol{x}}(t)+\boldsymbol{K}\boldsymbol{x}(t)=\boldsymbol{f}_{e}(t),\label{eq:GenLinSystem} \end{equation} where $\boldsymbol{C}$ and $\boldsymbol{K}$ are the damping and stiffness matrices, respectively, both assumed being multiplied by the inverse mass matrix from the left. The forcing term $\boldsymbol{f}_{e}(t)$ acts as a placeholder for the non-smooth part of the system and will be replaced with with specific terms. Equation (\ref{eq:GenLinSystem}) allows for internal resonances. We assume that these resonances are restricted to arbirarily large but finite dimensional subspaces of the state space, which is necessary to guarantee basic convergence properties of the solution $ $$\boldsymbol{x}(t)$ as shown in appendix \ref{sub:C0semigroup}. We also assume that (\ref{eq:GenLinSystem}) is stable in the Lyapunov sense for the same reason.
When $\boldsymbol{C}$ and $\boldsymbol{K}$ matrices are simultaneously diagonalisable the equation of motion can be written in the form of \begin{equation} \ddot{\boldsymbol{x}}(t)+2\boldsymbol{D}\boldsymbol{\Omega}\dot{\boldsymbol{x}}(t)+\boldsymbol{\Omega}^{2}\boldsymbol{x}(t)=\boldsymbol{f}_{e}(t),\label{eq:linModeDescr} \end{equation} where $\boldsymbol{\Omega}=\mathrm{diag}(\omega_{1},\omega_{2},\ldots)$ and $\boldsymbol{D}=\mathrm{diag}(D_{1},D_{2},\ldots)$. In the unforced case ($\boldsymbol{f}_{e}=\boldsymbol{0}$), the vector components of $\boldsymbol{x}$ on the left side of equation (\ref{eq:linModeDescr}) are decoupled, which means that the homogeneous equation can be solved for each $x_{k}$ independently. Therefore $x_{k}(t)$ and $\boldsymbol{\psi}_{k}(\chi)$ are called the \emph{mode}s and \emph{mode shape}s of the system, respectively, with $\omega_{k}$ natural frequencies and $D_{k}$ damping ratios \citep{EwinsBook}. System (\ref{eq:linModeDescr}) are called \emph{modal equations} describing the motion through the \emph{modal coordinates} $\boldsymbol{x}$. Our results do not require that the equations of motion assume the form (\ref{eq:linModeDescr}), however in section \ref{sub:L1conv} some restriction on the eigenvalues of (\ref{eq:GenLinSystem}) is necessary to characterise the convergence of the reduced equations of motion.
In order to take into account the coupling of the contact points to non-smooth nonlinearities we need to characterise their motion. For simplicity we assume point contact. Let us denote the motion of the $j$-th contact point at $\chi_{j}$ along the direction of vector $\boldsymbol{\phi}_{j}$ by \begin{equation} y_{j}(t)=\boldsymbol{\phi}_{j}\cdot\boldsymbol{u}(\chi_{j},t)\label{eq:resolvDef} \end{equation} as illustrated in Fig. \ref{fig:potato}. We call the positions $y_{j}(t)$ and the velocities $\dot{y}_{j}(t)$ of the contact points \emph{resolved variables}. We assume $M$ contact points, thus we define $\boldsymbol{y}=(y_{1},\ldots,y_{M},\dot{y}_{1},\ldots,\dot{y}_{M})^{T}$. Substituting (\ref{eq:dispField}) into (\ref{eq:resolvDef}) we obtain the motion of the contact points as a function of the solution of equation (\ref{eq:GenLinSystem}) \begin{equation} y_{j}(t)=\boldsymbol{n}{}_{j}\cdot\boldsymbol{x}(t),\label{eq:resolvDotProd} \end{equation} where \begin{equation} \boldsymbol{n}{}_{j}=\left(\boldsymbol{\phi}_{j}\cdot\boldsymbol{\psi}_{1}(\chi_{j}),\boldsymbol{\phi}_{j}\cdot\boldsymbol{\psi}_{2}(\chi_{j}),\ldots\right)^{T}. \end{equation} Vectors $\boldsymbol{n}{}_{j}$ are assumed to be linearly independent, spanning an $M$ dimensional subspace of $\mathbb{R}^{\infty}$.
The nonlinearities are incorporated into the model through the forcing term $\boldsymbol{f}_{e}(t)$. We assume that the nonlinearities only depend on the resolved variables. They are also piecewise continuous with isolated discontinuities. Thus the contact forces acting at contact points $\chi_{j}$ in the direction $\boldsymbol{\phi}_{j}$ are written as $f_{j}(\boldsymbol{y}(t),t).$ Summing up all the nonlinearities completes our model description by providing the right-hand side of equation (\ref{eq:linModeDescr}) in the form of \begin{equation} \boldsymbol{f}_{e}(t)=\sum_{j=1}^{M}\boldsymbol{n}_{j}f_{j}(\boldsymbol{y}(t),t).\label{eq:interForce} \end{equation}
\section{Reduction of the mechanical model\label{sec:ModRed}}
We aim to reduce the number of dimensions of our mechanical model (\ref{eq:interForce}) to an equation that only involves the $2M$ number of resolved variables all contained in the vector $\boldsymbol{y}$. To achieve this we use the Mori-Zwanzig formalism \citep{ChorinPNAS} to arrive at a $2M$ dimensional first order delay equation. The solution of the reduced system agrees with the solution of the full system for the resolved variables, while the rest of the variables are discarded. Our method can be viewed as a way of producing a Green's function for only parts of the system. In this formalism a convolution with the resolved variables represents the effect of the eliminated variables on the dynamics of the resolved variables. The technical details of the transformation are described in appendices \ref{sec:AppTrafo} and \ref{sec:AppMemKer}.
To simplify our calculation we transform equation (\ref{eq:linModeDescr}) into a first order form of \begin{equation} \dot{\boldsymbol{z}}(t)=\boldsymbol{R}\boldsymbol{z}(t)+\boldsymbol{f}(t),\label{eq:1storderODE} \end{equation} where \begin{equation} \boldsymbol{R}=\left(\begin{array}{cc} \boldsymbol{0} & \boldsymbol{I}\\ -\boldsymbol{K} & -\boldsymbol{C} \end{array}\right)\;\mbox{and}\;\boldsymbol{f}(t)=\left(\begin{array}{c} \boldsymbol{0}\\ \boldsymbol{f}_{e}(t) \end{array}\right).\label{eq:RmatDef} \end{equation}
We already have a way of obtaining the resolved variables from the generalised coordinates $\boldsymbol{x}$ through a dot product with vectors $\boldsymbol{n}{}_{j}$ as shown in equation (\ref{eq:resolvDotProd}). In a matrix-vector notation we write the conversion as \begin{equation} \boldsymbol{y}(t)=\boldsymbol{V}\boldsymbol{z}(t),\;\mbox{with}\;\boldsymbol{V}=\left(\begin{array}{c} \boldsymbol{v}_{1}^{T}\\ \vdots\\ \boldsymbol{v}_{2M}^{T} \end{array}\right)\,\mbox{and}\,\boldsymbol{v}_{j}=\left(\begin{array}{c} \boldsymbol{n}_{j}\\ \boldsymbol{0} \end{array}\right),\;\boldsymbol{v}_{M+j}=\left(\begin{array}{c} \boldsymbol{0}\\ \boldsymbol{n}_{j} \end{array}\right). \end{equation} To obtain our reduced model we construct a projection matrix $\boldsymbol{S}$ that acts on the generalised coordinates and has a $2M$ dimensional range. In order to do that we also define a lifting operator in the form of \begin{equation} \boldsymbol{z}(t)=\boldsymbol{W}\boldsymbol{y}(t),\;\mbox{with\;}\boldsymbol{W}=\left(\begin{array}{ccc} \boldsymbol{w}_{1} & \cdots & \boldsymbol{w}_{2M}\end{array}\right). \end{equation} Note that the lifting operator with $\boldsymbol{W}$ does not reproduce the full solution from the resolved variables, it is merely used as a technical tool. Moreover, $\boldsymbol{W}$ does not depend on the physical system, it can be chosen to suite the reduction procedure. By combining matrices $\boldsymbol{V}$ and $\boldsymbol{W}$ we obtain our projections $\boldsymbol{S}=\boldsymbol{W}\boldsymbol{V}$ and $\boldsymbol{Q}=\boldsymbol{I}-\boldsymbol{S}$ on the condition that $\boldsymbol{m}_{k}$ satisfy $\boldsymbol{m}_{k}\cdot\boldsymbol{n}_{l}=0$ if $k\neq l$ and $\boldsymbol{m}_{k}\cdot\boldsymbol{n}_{k}=1$. This constraint can also be expressed as $\boldsymbol{I}=\boldsymbol{V}\boldsymbol{W}$ (note the order of the two matrices).
To guarantee that the terms in the reduced equation are bounded the choice of $\boldsymbol{W}$ needs to be further restricted. Therefore we assume that the range of $\boldsymbol{W}$ is invariant under $\boldsymbol{R}$, that is,
\begin{equation} \boldsymbol{R}\boldsymbol{w}_{j}\in\mathrm{span}(\boldsymbol{w}_{1},\boldsymbol{w}_{2},\ldots,\boldsymbol{w}_{2M}),\;\mbox{for}\; j=1,2,\ldots,2M.\label{eq:RangeCond} \end{equation} Equivalently, $\boldsymbol{w}_{j}$ can be constructed as linear combinations of $2M$ number of eigenvectors of $\boldsymbol{R}$, since eigenvectors are invariant by definition. This assumption is key to our analysis.
In case of the modal equations (\ref{eq:linModeDescr}) the columns of $\boldsymbol{W}$ can be explicitly constructed in the form of \begin{equation} \boldsymbol{w}_{j}=\left(\begin{array}{c} \boldsymbol{m}_{j}\\ \boldsymbol{0} \end{array}\right)\;\mbox{and}\;\boldsymbol{w}_{M+j}=\left(\begin{array}{c} \boldsymbol{0}\\ \boldsymbol{m}_{j} \end{array}\right).\label{eq:WmDef} \end{equation} Due to the block diagonal form of $\boldsymbol{R}$ condition (\ref{eq:RangeCond}) holds when the $\boldsymbol{m}_{j}$ vectors are chosen such that they only have $M$ number of non-zero components: \begin{equation} \left[\boldsymbol{m}_{j}\right]_{p}=0,\quad\mbox{for}\quad p<P\;\mbox{and}\; p\ge P+M.\label{eq:mConstraint} \end{equation}
As the last step before arriving at the reduced equations we define \begin{align} \boldsymbol{A} & =\boldsymbol{V}\boldsymbol{R}\boldsymbol{W}, & & \in\mathbb{R}^{2M\times2M}\label{eq:Amat-1}\\ \boldsymbol{H}(t)\boldsymbol{z}(s) & =\boldsymbol{V}\boldsymbol{R}\boldsymbol{Q}\mathrm{e}^{\boldsymbol{R}(t-s)}\boldsymbol{z}(s), & & \in\mathbb{R}^{2M}\label{eq:noise}\\ \boldsymbol{L}_{j}(\tau) & =\boldsymbol{V}\mathrm{e}^{\boldsymbol{R}\tau}\boldsymbol{v}_{M+j}-\boldsymbol{A}\boldsymbol{V}\int_{0}^{\tau}\mathrm{e}^{\boldsymbol{R}\theta}\boldsymbol{v}_{M+j}\mathrm{d}\theta, & & \in\mathbb{R}^{2M}\label{eq:forceKern} \end{align} where $\boldsymbol{z}(s)$ is the initial condition of equation (\ref{eq:1storderODE}) at time $s$ and $\mathrm{e}^{\boldsymbol{R}t}$ is the fundamental matrix of equation (\ref{eq:1storderODE}). The matrix $\boldsymbol{A}$ is bounded if condition (\ref{eq:RangeCond}) holds, while $\boldsymbol{H}(t)\boldsymbol{z}(s)$ is bounded if the initial condition $\boldsymbol{z}(s)$ is bounded, too. To obtain the memory kernel $\boldsymbol{L}_{j}(\tau)$, one needs to solve the first order system (\ref{eq:1storderODE}) for $M$ different initial conditions $\boldsymbol{v}_{M+j}$. This solution may not be bounded which we rectify by integrating it (see sections \ref{sub:LjConst} and \ref{sub:L1conv}).
Using the expression (\ref{eq:interForce}) of the forcing term our reduced equation that is equivalent to (\ref{eq:1storderODE}) in the resolved variables becomes \begin{equation} \dot{\boldsymbol{y}}(t)=\boldsymbol{A}\boldsymbol{y}(t)+\sum_{j=1}^{M}\left[\boldsymbol{L}_{j}(0)f_{j}(\boldsymbol{y}(t),t)+\int_{0}^{t-s}\mathrm{d}_{\tau}\boldsymbol{L}_{j}(\tau)f_{j}(\boldsymbol{y}(t-\tau),t-\tau)\right]+\boldsymbol{H}(t)\boldsymbol{z}(s),\label{eq:MZequation} \end{equation} where the integral is understood in the Riemann-Stieltjes sense (see section \ref{sub:Stieltjes}). The formal equivalence of (\ref{eq:1storderODE}) and (\ref{eq:MZequation}) is proved in appendix \ref{sec:AppTrafo} and the formulae of $\boldsymbol{A}$, $\boldsymbol{H}(t)\boldsymbol{z}(s)$ and $\boldsymbol{L}_{j}(\tau)$ are derived in appendix \ref{sec:AppMemKer}. It turns out that for mechanical systems the alternative form of (\ref{eq:MZequation}) given below by equation (\ref{eq:MZL1eq}) is more appropriate.
\subsection{The meaning of the Riemann-Stieltjes integral\label{sub:Stieltjes}}
To provide some intuition about the interpretation of the integral in (\ref{eq:MZequation}) we remark that if $\boldsymbol{L}_{j}(\tau)$ is differentiable, the symbol $\mathrm{d}_{\tau}\boldsymbol{L}_{j}(\tau)$ can be replaced by $\nicefrac{\mathrm{d}}{\mathrm{d}\tau}\boldsymbol{L}_{j}(\tau)\cdots\mathrm{d}\tau$ to arrive at an ordinary Riemann integral. Similar simplification can be achieved if $\boldsymbol{L}_{j}(\tau)$ is Lipschitz continuous, however $\mathrm{d}_{\tau}\boldsymbol{L}_{j}(\tau)$ is replaced by the left derivative of $\boldsymbol{L}_{j}(\tau)$. A discontinuity of $\boldsymbol{L}_{j}(\tau)$ at $\tau^{\star}$ translates into a discrete delay term $\mathfrak{L}_{j}(\tau^{\star})f_{j}(\boldsymbol{y}(t-\tau^{\star}),t-\tau^{\star})$, where $\mathfrak{L}_{j}(\tau^{\star})$ is the gap in the function at $\tau^{\star}$. Therefore if $\boldsymbol{L}_{j}(\tau)$ is piecewise differentiable (or piecewise Lipschitz), with finite number of isolated discontinuities, the integral in (\ref{eq:MZequation}) can be replaced by a sum of Riemann integrals and discrete delays. In particular, assuming that the discontinuities occur at $\tau_{p}^{\star}$, $p=1,\ldots,P$ and that $\tau_{0}^{\star}=0$ and $\tau_{P+1}^{\star}=t$ we can write \begin{equation} \int_{0}^{t-s}\mathrm{d}_{\tau}\boldsymbol{L}_{j}(\tau)f(t-\tau)\mathrm{d}\tau=\sum_{p=0}^{P}\int_{\tau_{p}^{\star}}^{\tau_{p+1}^{*}-s}\left[\frac{\mathrm{d}}{\mathrm{d}\tau}\boldsymbol{L}_{j}(\tau)\right]f(t-\tau)\mathrm{d}\tau+\sum_{p=1}^{P}\mathfrak{L}_{j}(\tau_{p}^{\star})f(t-\tau_{p}^{\star}), \end{equation} where $\mathfrak{L}_{j}(\tau^{\star})=\lim_{\tau\to\tau*+0}\boldsymbol{L}_{j}(\tau)-\lim_{\tau\to\tau*-0}\boldsymbol{L}_{j}(\tau)$.
One can interpret the memory kernels $\boldsymbol{L}_{j}(\tau)$ as an indication how waves travel within the structure from contact point $\chi_{j}$ to all contact points including them returning to $\chi_{j}$. Indeed, the tail part of $\boldsymbol{L}_{j}(\tau)$ for $\tau>0$ describes how a force applied in the past is affecting the contact points at current time. This notion is therefore analogous to having waves that depart from point $\chi_{j}$ and arrive at all contact points $\chi_{l}$, $l=1,\ldots,M$ $\tau$ time later. In particular if waves do not disperse, they arrive simultaneously at a give material point $\chi_{l}$ exactly $\tau^{\star}$ time later and that corresponds to a discrete delay represented by a discontinuity in $\boldsymbol{L}_{j}(\tau)$ at $\tau=\tau^{\star}$.
The time delay in (\ref{eq:MZequation}) tends to infinity, since the history of $\boldsymbol{y}(t)$ to be taken into account grows with time. However, if $\boldsymbol{L}_{j}(\tau)$ tends to a constant as $\tau\to\infty$, the integrals in (\ref{eq:MZequation}) may be truncated at a finite delay time to produce an approximate model. In case of a conservative equation (\ref{eq:linModeDescr}), the delay cannot be truncated at a finite time, because the motion of the free structure will never stop due to the conservation of the kinetic energy. This is illustrated in section \ref{sub:StringVib}.
\subsection{An alternative form of the reduced equations\label{sub:LjConst}}
It was mentioned before that $\boldsymbol{L}_{j}(\tau)$ may not be a bounded function. Therefore we integrate the convolution in equation (\ref{eq:MZequation}) by parts, so that the integral of $\boldsymbol{L}_{j}(\tau)$ and the derivative of the contact force $f_{j}$ appear in the rewritten equation. However, to be able to integrate $\boldsymbol{L}_{j}(\tau)$ and ensure that the integral does not grow out of bound, we need to decompose $\boldsymbol{L}_{j}(\tau)$ into a constant and an oscillatory term \begin{equation} \boldsymbol{L}_{j}(\tau)=\boldsymbol{L}_{j}^{\infty}+\boldsymbol{L}_{j}^{0}(\tau),\label{eq:L0Def} \end{equation} where \begin{equation} \boldsymbol{L}_{j}^{\infty}=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}\boldsymbol{L}_{j}(\tau)\mathrm{d}\tau.\label{eq:LinfLimit} \end{equation} Since we have a formal expression for $\boldsymbol{L}_{j}(\tau)$ in the form of equation (\ref{eq:forceKern}), we can calculate the integral \begin{equation} \int_{0}^{t}\boldsymbol{L}_{j}(\tau)\mathrm{d}\tau=\int_{0}^{t}\left(\boldsymbol{V}\mathrm{e}^{\boldsymbol{R}\tau}-\boldsymbol{A}\boldsymbol{V}\int_{0}^{\tau}\mathrm{e}^{\boldsymbol{R}\theta}\mathrm{d}\theta\right)\mathrm{d}\tau\boldsymbol{v}_{M+j}.\label{eq:LjIntegral} \end{equation} From the simple rule of differentiating an exponential we find that $\int_{0}^{\tau}\mathrm{e}^{\boldsymbol{R}\theta}\mathrm{d}\theta=\boldsymbol{R}^{-1}\left(\mathrm{e}^{\boldsymbol{R}\tau}-\boldsymbol{I}\right)$. Using this formula twice in equation (\ref{eq:LjIntegral}), we are left with \begin{equation} \int_{0}^{t}\boldsymbol{L}_{j}(\tau)\mathrm{d}\tau=\left(\boldsymbol{V}\boldsymbol{R}^{-1}\left(\mathrm{e}^{\boldsymbol{R}t}-\boldsymbol{I}\right)-\boldsymbol{A}\boldsymbol{V}\boldsymbol{R}^{-1}\left(\boldsymbol{R}^{-1}\left(\mathrm{e}^{\boldsymbol{R}t}-\boldsymbol{I}\right)-t\right)\right)\boldsymbol{v}_{M+j}.\label{eq:LjIntegralFull} \end{equation} Assuming that $\boldsymbol{V}\int_{0}^{\tau}\mathrm{e}^{\boldsymbol{R}\theta}\mathrm{d}\theta\boldsymbol{v}_{M+j}$ is bounded (see section \ref{sub:L1conv}) the limit (\ref{eq:LinfLimit}) becomes \begin{equation} \boldsymbol{L}_{j}^{\infty}=\boldsymbol{A}\boldsymbol{V}\boldsymbol{R}^{-1}\boldsymbol{v}_{M+j}.\label{eq:LinfDef} \end{equation} Note that $\boldsymbol{V}\boldsymbol{R}^{-1}\boldsymbol{v}_{M+j}$ is the static displacement of the contact points under a static unit load at contact point $\chi_{j}$. Indeed, by expanding equation $\boldsymbol{R}\left(\boldsymbol{x},\boldsymbol{y}\right)^{T}=\left(\boldsymbol{0},\boldsymbol{n}_{j}\right)^{T}$ we get $\boldsymbol{K}\boldsymbol{x}=\boldsymbol{n}_{j}$ and $\boldsymbol{y}=\boldsymbol{0}$, where $\boldsymbol{x}$ is the static displacement of the generalised coordinates, hence, $\boldsymbol{V}\boldsymbol{x}=\boldsymbol{V}\boldsymbol{R}^{-1}\boldsymbol{v}_{M+j}$ is the static displacement of the contact points.
Our definition (\ref{eq:L0Def}) of $\boldsymbol{L}_{j}^{0}(\tau)$ implies that either $\boldsymbol{L}_{j}^{0}(\tau)\to\boldsymbol{0}$ exponentially if the system is dissipative or $\boldsymbol{L}_{j}^{0}(\tau)$ oscillates with zero mean. Therefore we define the integral \begin{equation} \boldsymbol{L}_{j}^{1}(t)=\int_{0}^{t}\boldsymbol{L}_{j}^{0}(\tau)\mathrm{d}\tau.\label{eq:L1def} \end{equation} The boundedness and smoothness of (\ref{eq:L1def}) is discussed in section \ref{sub:L1conv}. By virtue of (\ref{eq:L1def}) the forcing terms in (\ref{eq:MZequation}) can be integrated by parts as follows \begin{multline} \boldsymbol{L}_{j}(0)\boldsymbol{f}(t)+\int_{0}^{t-s}\mathrm{d}_{\tau}\boldsymbol{L}_{j}(\tau)f_{j}(\boldsymbol{y}(t-\tau),t-\tau)=\\ =\boldsymbol{L}_{j}^{\infty}f_{j}(t)+\boldsymbol{L}_{j}^{0}(t-s)f_{j}(s)+\int_{0}^{t-s}\mathrm{d}_{\tau}\boldsymbol{L}_{j}^{1}(\tau)\frac{\mathrm{d}}{\mathrm{d}t}[f_{j}(\boldsymbol{y}(t-\tau),t-\tau)], \end{multline} which has only bounded terms if the derivative $\frac{\mathrm{d}}{\mathrm{d}t}[f_{j}(t)]$ exists and is bounded almost everywhere and condition (\ref{eq:ResolventCondition}) in section \ref{sub:L1conv} holds. Using this transformation the governing equation becomes \begin{equation} \dot{\boldsymbol{y}}(t)=\boldsymbol{A}\boldsymbol{y}(t)+\sum_{j=1}^{M}\left[\boldsymbol{L}_{j}^{\infty}f_{j}(t)+\int_{0}^{t-s}\mathrm{d}_{\tau}\boldsymbol{L}_{j}^{1}(\tau)\frac{\mathrm{d}}{\mathrm{d}t}[f_{j}(\boldsymbol{y}(t-\tau),t-\tau)]\right]+\boldsymbol{g}(t),\label{eq:MZL1eq} \end{equation} where $\boldsymbol{g}(t)=\boldsymbol{H}(t)\boldsymbol{z}(s)+\sum_{j=1}^{M}\boldsymbol{L}_{j}^{0}(t-s)f_{j}(s).$
It is important to note that according to the theory of delay equations \citep{hale1993introduction} if the terms in equation (\ref{eq:MZL1eq}) are slightly perturbed, the solution of (\ref{eq:MZL1eq}) changes only slightly under general conditions. This is a clear advantage over the finite dimensional description where perturbation generally leads to qualitatively disagreeing solutions \citep{SieberSingular}.
\subsection{Vibrations of a pre-tensed string\label{sub:StringVib}}
\begin{figure}
\caption{Schematic of a string (a) and a beam (b). The displacement of material points is described by $u(\xi,t)$, which represents motion in the direction of the arrows. }
\label{fig:StVenant}
\end{figure}
In this section we illustrate the calculation of $\boldsymbol{A}$ and $\boldsymbol{L}_{j}^{1}(t)$ for a pre-tensed string without bending stiffness. We keep the calculation as general as possible so that it applies to systems with a single contact point that can be written in the form of \ref{eq:linModeDescr}. The schematic of the string is shown in Fig. \ref{fig:StVenant}(a), whose motion is described by the equation \begin{equation} \frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial\xi^{2}}+\delta(\xi-\xi^{\star})F_{c}(t)+\mbox{damping},\label{eq:StringPDE} \end{equation} where $c$ is the wave speed, $u(t,\xi)$ is the deflection of the string, $t$ stands for time and $0\le\xi\le1$ is the coordinate along the string. The boundary conditions $u(0,t)=0$ and $u(1,t)=0$ express that there is no movement at the two ends of the string. Equation (\ref{eq:StringPDE}) indicates damping, which explicitly appears in the mode decomposed system (\ref{eq:linModeDescr}) with non-zero damping ratios. The string is forced at $\xi=\xi^{\star}$ by a contact force $F_{c}(t)$, which is represented by the Dirac delta function in equation (\ref{eq:StringPDE}).
The first step is to provide a mode decomposition in the form of equation (\ref{eq:linModeDescr}), so that the vibration of the string is expressed by (\ref{eq:dispField}), where the mode shapes are the scalar valued $\psi_{k}(\xi)=\sin\left(k\pi\xi\right)$. The natural frequencies of the system are $\omega_{k}=ck\pi$ and we assume uniform damping $D_{k}=\nicefrac{1}{10}$ for all modes, which gives us \begin{equation} \boldsymbol{\Omega}=\mathrm{diag}(c\pi,c2\pi,\ldots),\quad\mbox{and}\quad\boldsymbol{D}=\mathrm{diag}(\nicefrac{1}{10},\nicefrac{1}{10},\ldots). \end{equation} The vibration at the contact point can be expressed as a linear combination of all the modes $u(\xi^{\star},t)=\boldsymbol{n}\cdot\boldsymbol{x}(t)$, where $\boldsymbol{n}=(\sin\left(\pi\xi^{\star}\right),\sin\left(2\pi\xi^{\star}\right),\ldots)^{T}$. The resolved variables therefore are $y_{1}(t)=\boldsymbol{n}\cdot\boldsymbol{x}(t)$ and $y_{2}(t)=\dot{y}_{1}(t)$. We also choose $\boldsymbol{m}=(\nicefrac{1}{\sin\pi\xi^{\star}},0,0,\ldots)^{T}$ in formula (\ref{eq:WmDef}), thus \begin{equation} \boldsymbol{A}=\left(\begin{array}{cc} 0 & 1\\ -\omega_{1}^{2} & -2D_{1}\omega_{1} \end{array}\right).\label{eq:StringAmat} \end{equation}
Next we calculate the function $\mathrm{e}^{\boldsymbol{R}t}\boldsymbol{v}_{2}$ that appears in the expression (\ref{eq:forceKern}) of $\boldsymbol{L}(\tau)$. The equation whose solution we are seeking is $\dot{\boldsymbol{z}}=\boldsymbol{R}\boldsymbol{z}$, which is expanded into \begin{equation} \dot{z}_{1,k}=z_{2,k},\quad\dot{z}_{2,k}=-2D_{k}\omega_{k}z_{2,k}-\omega_{k}^{2}z_{1,k},\quad k=1,2,3,\ldots.\label{eq:StringRExpand} \end{equation} The initial conditions that correspond to $\boldsymbol{z}(0)=\boldsymbol{v}_{2}$ are $z_{1,k}(0)=0$ and $z_{2,k}(0)=\left[\boldsymbol{n}\right]_{k}$. The solution for the modes are \begin{equation} z_{1,k}(t)=\left[\boldsymbol{n}\right]_{k}\mathrm{e}^{-D_{k}\omega_{k}t}\frac{\sin\sqrt{1-D_{k}^{2}}\omega_{k}t}{\sqrt{1-D_{k}^{2}}\omega_{k}}.\label{eq:StringFullSol} \end{equation} Without assuming the form of $z_{1,k}(t)$ and evaluating formula (\ref{eq:forceKern}) we find that \begin{align} \left[\boldsymbol{L}(t)\right]_{1} & =\boldsymbol{v}_{1}\cdot\mathrm{e}^{\boldsymbol{R}t}\boldsymbol{v}_{2}-\boldsymbol{v}_{2}\cdot\int_{0}^{t}\mathrm{e}^{\boldsymbol{R}\theta}\mathrm{d}\theta\boldsymbol{v}_{2}=0\quad\mbox{and}\nonumber \\ \left[\boldsymbol{L}(\tau)\right]_{2} & =\boldsymbol{v}_{2}\cdot\mathrm{e}^{\boldsymbol{R}t}\boldsymbol{v}_{2}+2D_{1}\omega_{1}\boldsymbol{v}_{1}\cdot\mathrm{e}^{\boldsymbol{R}t}\boldsymbol{v}_{2}+\omega_{1}^{2}\boldsymbol{v}_{1}\cdot\int_{0}^{t}\mathrm{e}^{\boldsymbol{R}\theta}\mathrm{d}\theta\boldsymbol{v}_{2}\nonumber \\
& =\sum_{k=1}^{\infty}\left[\boldsymbol{n}\right]_{k}\left(\frac{d}{dt}z_{1,k}(t)+2D_{1}\omega_{1}z_{1,k}(t)+\omega_{1}^{2}\int_{0}^{t}z_{1,k}(t)d\theta\right).\label{eq:StringL2gen} \end{align} Eventually substituting (\ref{eq:StringFullSol}) into (\ref{eq:StringL2gen}) in the conservative case ($D_{k}=0$) $\left[\boldsymbol{L}(t)\right]_{2}$ of our system (\ref{eq:StringPDE}) becomes \begin{equation} \left[\boldsymbol{L}(t)\right]_{2}=\sum_{k=1}^{\infty}\sin^{2}k\pi\xi^{\star}\frac{\omega_{1}^{2}}{\omega_{k}^{2}}+\sum_{k=2}^{\infty}\sin^{2}k\pi\xi^{\star}\left(1-\frac{\omega_{1}^{2}}{\omega_{k}^{2}}\right)\cos\omega_{k}t,\label{eq:StringL2} \end{equation} which is a divergent Fourier series, therefore equation (\ref{eq:MZequation}) cannot be utilised to describe the dynamics. The constant term in equation (\ref{eq:StringL2gen}) regardless of the damping ratios assumes the form \begin{equation} \left[\boldsymbol{L}^{\infty}\right]_{2}=-\omega_{1}^{2}\boldsymbol{v}_{1}\boldsymbol{R}^{-1}\boldsymbol{v}_{2}=\sum_{k=1}^{\infty}\left[\boldsymbol{n}\right]_{k}^{2}\frac{\omega_{1}^{2}}{\omega_{k}^{2}}.\label{eq:StringL2inf} \end{equation} Note that this is $-\omega_{1}^{2}$ times the static displacement of the string under unit load at $\xi=\xi^{\star}$. Using the expressions for $z_{1,k}(t)$ and integrating $\left[\boldsymbol{L}(t)\right]_{2}-\left[\boldsymbol{L}^{\infty}\right]_{2}$ as per definition (\ref{eq:L1def}) we get \begin{multline} \left[\boldsymbol{L}^{1}(\tau)\right]_{2}=\sum_{k=1}^{\infty}\left[\boldsymbol{n}\right]_{k}^{2}\frac{e^{-tD_{k}\omega_{k}}}{\sqrt{1-D_{k}^{2}}\omega_{k}^{3}}\left\{ 2\omega_{1}^{2}D_{k}\sqrt{1-D_{k}^{2}}\cos\left(t\sqrt{1-D_{k}^{2}}\omega_{k}\right)\right.\\ \left.+\left(\omega_{1}^{2}\left(2D_{k}^{2}-1\right)+\left(2D_{1}\omega_{1}+1\right)\omega_{k}^{2}\right)\sin\left(t\sqrt{1-D_{k}^{2}}\omega_{k}\right)\right\} -\left[\boldsymbol{n}\right]_{k}^{2}\frac{2\omega_{1}^{2}D_{k}}{\omega_{k}^{3}}.\label{eq:StringL2intGen} \end{multline} Assuming that $D_{k}=D$ are constant, the right limit of $\left[\boldsymbol{L}^{1}(\tau)\right]_{2}$ becomes \begin{equation} \lim_{\tau\to0+}\left[\boldsymbol{L}^{1}(\tau)\right]_{2}=\frac{\cos^{-1}D}{2\pi c\sqrt{1-D^{2}}}\label{eq:StringL1Lim} \end{equation} The detailed calculation of (\ref{eq:StringL1Lim}) can be found in appendix \ref{sec:L1Lim}, which also indicates the boundedness of $\left[\boldsymbol{L}^{1}(\tau)\right]_{2}$. The graph of $\left[\boldsymbol{L}^{1}(\tau)\right]_{2}$ for $c=1$ is illustrated in Fig. \ref{fig:StringL1fig}(a) for both the conservative and the damped case. \begin{figure}
\caption{(colour online) Graph of $\left[\boldsymbol{L}^{1}(\tau)\right]_{2}$ for the string equation (\ref{eq:StringPDE}) with at $\xi^{\star}=0.4,$ $c=1$. The blue line represents the conservative case and the red corresponds to the damped case with $D_{k}=\nicefrac{1}{10}$. (b) Graph of the function $\left[\boldsymbol{L}^{1}(\tau)\right]_{2}$ when truncating the series (\ref{eq:StringL2intGen}) at $20,40,80,160$ terms, denoted by $\times,\triangle,\square$ and $\ocircle$, respectively.}
\label{fig:StringL1fig}
\end{figure}
Note that the convolution kernel $\boldsymbol{L}^{1}(\tau)$ for $D_{k}=0$ is periodic and therefore the delay that occurs as the effect of nonlinearities is infinite. If damping is introduced $\boldsymbol{L}^{1}(\tau)$ decays in time so that the past of the system will have a smaller effect. This is illustrated by the red line in Fig \ref{fig:StringL1fig}(a). For non-zero damping, as an approximation one can truncate the delay to a finite time-interval. Truncation is a reasonable choice for most practical purposes, but it is not quite clear what are the theoretical implications \citep{FarkasStepan}.
It is worth noting that equation (\ref{eq:StringPDE}) in the conservative case can be solved using D'Alembert's formula that also leads to a delay-differential equation \citep{StepanForgeDETC}, which is similar to (\ref{eq:MZL1eq}).
\subsection{Euler-Bernoulli cantilever beam}
We choose the Euler-Bernoulli beam as our second example to illustrate the theory. This model can support waves of infinite speed, therefore its physical validity is questionable. Nevertheless it is worth investigating how this property of the Euler-Bernoulli beam translates into the properties of the memory kernel. The non-dimensional governing equation and boundary conditions are
\begin{equation} \frac{\partial^{2}u}{\partial t^{2}}=-\frac{\partial^{4}u}{\partial\xi^{2}}+\mbox{damping},\; u(t,0)=u'(t,0)=u''(t,1)=u'''(t,1)=0.\label{eq:EBbeam} \end{equation} The natural frequencies of (\ref{eq:EBbeam}) are determined by the equation $1+\cos\sqrt{\omega_{k}}\cosh\sqrt{\omega_{k}}=0$, which can be approximated by $\cos\sqrt{\omega_{k}}\approx0$ for $\omega_{k}$ sufficiently large. Therefore $\omega_{k}\approx\left(k\pi-\nicefrac{\pi}{2}\right)^{2}$. We use this estimate as a starting point to numerically find more accurate $\omega_{k}$ values. The mode shapes at the free end of the beam assume the values given by vector $\boldsymbol{n}=(2,-2,2,-2,\ldots)^{T}.$ We choose $\boldsymbol{m}=(\nicefrac{1}{2},0,0,\ldots)^{T}$ in formula (\ref{eq:WmDef}), so that $\boldsymbol{W}$ satisfies our assumption (\ref{eq:RangeCond}).
The general formulae that were derived in section \ref{sub:StringVib} still apply to equation (\ref{eq:EBbeam}) with the appropriate $\omega_{k}$, $D_{k}$ and $\left[\boldsymbol{n}\right]_{k}$ values. In particular, we use (\ref{eq:StringL2intGen}) to plot the memory kernel in Fig. \ref{fig:EBfig}. The graph of $\left[\boldsymbol{L}^{1}(\tau)\right]_{2}$ in Fig. \ref{fig:EBfig}(a) shows that the quadratically growing natural frequencies make the function non-smooth in the conservative case. When damping is introduced the function becomes smooth for $\tau>0$. Fig. \ref{fig:EBfig}(b) shows that for $0\le\tau\ll1$ $\left[\boldsymbol{L}^{1}(\tau)\right]_{2}$ grows like a power curve $\tau^{p}$, $0<p<1$. Therefore $\left[\boldsymbol{L}^{1}(\tau)\right]_{2}$ is not differentiable at $\tau=0$.
\begin{figure}
\caption{(colour online) The graph of $\left[\boldsymbol{L}^{1}(\tau)\right]_{2}$ for the Euler-Bernoulli cantilever beam. The blue curves correspond to the conservative case and the red curves represent the damped system with $D_{k}=\nicefrac{1}{50}$. The conservative case illustrates the lack of smoothness of $\left[\boldsymbol{L}^{1}(\tau)\right]_{2}$ for $\tau>0$. Panel (b) illustrates that $\left[\boldsymbol{L}^{1}(\tau)\right]_{2}$ is not differentiable at $\tau=0$. }
\label{fig:EBfig}
\end{figure}
\subsection{The convergence of\textmd{ $\boldsymbol{L}_{j}^{1}(\tau)$\label{sub:L1conv}}}
So far we derived the reduced equation of motion (\ref{eq:MZL1eq}) without any consideration whether intermediate terms are well-defined. To make the analysis rigorous we introduce infinite dimensional vector spaces \begin{equation} \boldsymbol{X}=\{\boldsymbol{x}\in\mathbb{R}^{\infty}:\sum_{i}x_{i}^{2}<\infty\}\;\mbox{and}\;\boldsymbol{Z}=\boldsymbol{X}^{2}\label{eq:Xspace} \end{equation} that contain the solutions of the second (\ref{eq:GenLinSystem}) and first order system (\ref{eq:1storderODE}), respectively. One can check that in the previous two examples $\boldsymbol{v}_{M+j}\notin\boldsymbol{Z}$, because their norm is infinite. This also implies that $\left\Vert \boldsymbol{V}\right\Vert =\infty$. Even if we know that $\left\Vert \mathrm{e}^{\boldsymbol{R}\tau}\right\Vert \le M_{0}<\infty$, the bound of $\boldsymbol{L}_{j}^{1}(\tau)$ cannot be directly estimated in the straightforward way, because $\left\Vert \boldsymbol{V}\mathrm{e}^{\boldsymbol{R}\tau}\boldsymbol{v}_{M+j}\right\Vert \le\left\Vert \boldsymbol{V}\right\Vert \left\Vert \mathrm{e}^{\boldsymbol{R}\tau}\right\Vert \left\Vert \boldsymbol{v}_{M+j}\right\Vert =\infty$. However the two examples in the previous sections show that $\boldsymbol{L}_{j}^{1}(\tau)$ can be bounded, but not necessarily smooth. In case of the string and without damping, $\boldsymbol{L}_{j}^{1}(\tau)$ is a piecewise-smooth function, while $\boldsymbol{L}_{j}^{1}(\tau)$ appears to be continuous but non-differentiable for the undamped Euler-Bernoulli beam. If damping is introduced, $\boldsymbol{L}_{j}^{1}(\tau)$ becomes smooth for $\tau>0$ and discontinuity or non-differentiability occurs only at $\tau=0$ in both examples.
Boundedness and smoothness depends on the eigenvalues of $\boldsymbol{R}$, which are directly related to the natural frequencies and damping ratios of system (\ref{eq:GenLinSystem}). First we assume that all the eigenvalues $\sigma(\boldsymbol{R})$ are in the left half of the complex plane including the imaginary axis, that is, \begin{equation} \sigma(\boldsymbol{R})\subset\{\lambda\in\mathbb{C}:\Re\lambda\le0\}.\label{eq:NegativeRealPart} \end{equation} This guarantees that $\mathrm{e}^{\boldsymbol{R}t}$ is a strongly continuous semigroup with $\left\Vert \mathrm{e}^{\boldsymbol{R}t}\right\Vert \le C_{0}$ as shown in appendix \ref{sub:C0semigroup}. This does not guarantee continuity or boundedness of $\boldsymbol{L}_{j}^{1}(\tau)$, but it is a necessary condition. However when condition \begin{equation}
\left|\lambda\boldsymbol{n}_{l}\cdot(\boldsymbol{K}+\lambda\boldsymbol{C}+\lambda^{2})^{-1}\boldsymbol{n}_{j}\right|\le M_{j,l}\;\mbox{for}\;\lambda\in\{\mathbb{C}:\Re\lambda=\gamma>0\}\label{eq:ResolventCondition} \end{equation} is satisfied as well, appendix \ref{sub:L1bound} shows that $\boldsymbol{L}_{j}^{1}(\tau)$ is bounded.
Smoothness of $\boldsymbol{L}_{j}^{1}(\tau)$ can be guaranteed if we replace (\ref{eq:NegativeRealPart}) and (\ref{eq:ResolventCondition}) with a stronger assumptions. We assume that there exists a $D_{0}>0$ such that the eigenvalues of $\boldsymbol{R}$ satisfy \begin{equation}
\sigma(\boldsymbol{R})\subset\{\lambda\in\mathbb{C}:\Re(\lambda)\le-D_{0}\left|\Im(\lambda)\right|\},\label{eq:SectorialCond} \end{equation} in other words the eigenvalues of $\boldsymbol{R}$ are contained in a sector of the imaginary half-plane. Instead of (\ref{eq:ResolventCondition}) we assume that \begin{equation}
\left|\lambda\boldsymbol{n}_{l}\cdot(\boldsymbol{K}+\lambda\boldsymbol{C}+\lambda^{2})^{-1}\boldsymbol{n}_{j}\right|\le M_{j,l}\;\mbox{for}\;\lambda\in\{\mathbb{C}:\left|\arg\lambda\right|=\nicefrac{\pi}{2}+\delta\},\,0<\delta<\tan^{-1}D_{0}.\label{eq:SmoothCond} \end{equation} In case of the modal equations (\ref{eq:linModeDescr}), assumption (\ref{eq:SectorialCond}) holds if there is a $D_{0}$ such that for the damping ratios \begin{equation} 0<D_{0}\le D_{k}.\label{eq:dampingAssumption} \end{equation} Condition (\ref{eq:SectorialCond}) ensures that unforced vibrations dissipate faster for higher natural frequencies which is essential for the smoothness of solutions. According to the theory of semigroups \citep{Pazy83} if (\ref{eq:SectorialCond}) holds the fundamental matrix $\mathrm{e}^{\boldsymbol{R}t}$ of equation (\ref{eq:1storderODE}) is a holomorphic function of $t$, in other words, its Taylor series converges in a sector about the non-negative real axis $t\ge0$ within the complex plane. Appendix \ref{sub:L1bound} proves that if (\ref{eq:SectorialCond}) and (\ref{eq:SmoothCond}) are satisfied $\boldsymbol{L}_{j}^{1}(\tau)$ is smooth for $\tau>0$.
Condition (\ref{eq:ResolventCondition}) has a mechanical meaning. When the structure is forced at contact point $\chi_{j}$ with $f_{j}(t)=\mathrm{e}^{\gamma t}\cos\omega t$,
$\gamma>0$, the velocity response $y_{M+l}(t)$ when scaled back with the exponential growth must be bounded independent of the forcing frequency $\omega$, that is $\left|y_{M+l}(t)e^{-\gamma t}\right|\le M_{j,l}$. For smoothness of $\boldsymbol{L}_{j}^{1}(\tau)$ we require that the decaying forcing $f_{j}(t)=\mathrm{e}^{-\delta\omega t}\cos\omega t$
produces a similarly decaying velocity with $\left|y_{M+l}(t)\mathrm{e}^{\delta\omega t}\right|\le M_{j,l}$ for $0<\delta<D_{0}$ independent of $\omega$.
In general, it is not straightforward to check whether (\ref{eq:ResolventCondition}) holds. Let us consider the modal equations (\ref{eq:linModeDescr}) without damping and assume that the natural frequencies scale as $\omega_{k}=\omega_{0}k^{\nicefrac{\ell}{2}},$
where $\ell=2,3,4,\dots$. We also assume that $\sup_{k}\left|\left[\boldsymbol{n}_{j}\right]_{k}\left[\boldsymbol{n}_{l}\right]_{k}\right|\le C_{j,l}$, which implies that \begin{equation}
\left|\lambda\boldsymbol{n}_{l}\cdot(\boldsymbol{K}+\lambda^{2})^{-1}\boldsymbol{n}_{j}\right|\le C_{j,l}\sum_{k=1}^{\infty}\frac{\lambda}{\omega_{0}^{2}k^{\ell}+\lambda^{2}}=C_{j,l}\left(\sum_{r=1}^{\ell}\frac{\psi\left(-(-1)^{r/\ell}\lambda^{2/\ell}\right)}{\ell\omega_{0}^{2/\ell}(-1)^{r-r/\ell}\lambda^{1-2/\ell}}-\frac{1}{\lambda}\right),\label{eq:UndampResolv} \end{equation} where $\psi(\cdot)$ is the logarithmic derivative of the Euler Gamma function \citep{Abramowitz}. Function $\psi$ has isolated singularities on the negative real axis, otherwise it is bounded. Therefore a $\gamma>0$
of (\ref{eq:ResolventCondition}) can be chosen such that none of the arguments of $\psi$ goes through these singularities. This means that there is an $M_{j,l}$ such that $\left|\lambda\boldsymbol{n}_{l}\cdot(\boldsymbol{K}+\lambda^{2})^{-1}\boldsymbol{n}_{j}\right|\le M_{j,l}$. In particular, for the two examples of the string and the beam we have \begin{equation}
\left|\lambda\boldsymbol{n}_{l}\cdot(\boldsymbol{K}+\lambda^{2})^{-1}\boldsymbol{n}_{j}\right|\le C_{j,l}\begin{cases}
\left|\frac{\pi\coth\nicefrac{\pi\lambda}{\omega_{0}}}{2\omega_{0}}-\frac{1}{2\lambda}\right|\le\frac{\pi}{2\omega_{0}}\frac{\sinh\frac{2\pi\gamma}{\omega_{0}}+1}{\cosh\frac{2\pi\gamma}{\omega_{0}}-1}+\frac{1}{2\gamma}, & \ell=2\\
\left|\frac{\sqrt[4]{-1}\pi\left(\cot\left(\frac{\sqrt[4]{-1}\pi\sqrt{\lambda}}{\sqrt{\omega_{0}}}\right)+\coth\left(\frac{\sqrt[4]{-1}\pi\sqrt{\lambda}}{\sqrt{\omega_{0}}}\right)\right)}{4\sqrt{\lambda\omega_{0}}}-\frac{1}{2\lambda}\right|, & \ell=4 \end{cases}. \end{equation} Note that for $\ell<2$, the sum (\ref{eq:UndampResolv}) is not uniformly bounded, due to the $\lambda^{1-2/\ell}$ term in the denominator.
\section{Non-smooth dynamics\label{sub:NSDyn}}
We are now in the position to include the strongly nonlinear contact forces (\ref{eq:interForce}) into the reduced model (\ref{eq:MZL1eq}) and investigate their effect. For sake of simplicity in this section we assume a single contact force $F_{c}(\boldsymbol{y})$, so that the governing equation becomes \begin{equation} \dot{\boldsymbol{y}}(t)=\boldsymbol{A}\boldsymbol{y}(t)+\boldsymbol{L}^{\infty}F_{c}(\boldsymbol{y}(t))+\int_{0}^{t-s}\mathrm{d}_{\tau}\boldsymbol{L}^{1}(\tau)\frac{\mathrm{d}}{\mathrm{d}t}[F_{c}(\boldsymbol{y}(t-\tau))]+\boldsymbol{g}(t),\label{eq:MZNSeq} \end{equation} where $\boldsymbol{g}(t)=\boldsymbol{H}(t)\boldsymbol{z}(s)+\boldsymbol{L}^{0}(t-s)F_{c}(\boldsymbol{y}(s)).$ The properties of solutions of (\ref{eq:MZNSeq}) strongly depend on both $\boldsymbol{L}^{1}(\tau)$ and $F_{c}(\boldsymbol{y})$. We also assume that $\boldsymbol{L}^{1}(\tau)$ is smooth for $\tau>0$ as it is outlined in section \ref{sub:L1conv}.
Our definition of solution at the discontinuities of $F_{c}(\boldsymbol{y})$ is based on a mechanical analogy. If the elastic bodies stick together there is an algebraic constraint that restricts the trajectories to sticking motion and one must be able to calculate the contact force implicitly from equation (\ref{eq:MZNSeq}). If these contact forces are admissible by physical law, the bodies will stick, otherwise they will continue slipping.
To formalise this definition, we assume that $F_{c}(\boldsymbol{y})$ is discontinuous along a smooth surface defined by $h(\boldsymbol{y})=0$, which stands for the algebraic constraint of sticking. We call $\Sigma=\{\boldsymbol{y}\in\mathbb{R}^{2M}:h(\boldsymbol{y})=0\}$ the switching surface. The physical bound of the contact force can be defined as the two limits of $F_{c}(\boldsymbol{y})$ on the two sides of $\Sigma$, that is, \begin{equation} \forall\boldsymbol{y}\in\Sigma,\quad F_{c}^{+}(\boldsymbol{y})=\lim_{\bar{\boldsymbol{y}}\to\boldsymbol{y},h(\bar{\boldsymbol{y}})>0}F_{c}(\bar{\boldsymbol{y}})\;\mbox{and}\; F_{c}^{-}(\boldsymbol{y})=\lim_{\bar{\boldsymbol{y}}\to\boldsymbol{y},h(\bar{\boldsymbol{y}})<0}F_{c}(\bar{\boldsymbol{y}}).\label{eq:FcLimits} \end{equation} Without restricting generality we assume that $F_{c}^{-}(\boldsymbol{y})<F_{c}^{+}(\boldsymbol{y})$. Alternatively, $F_{c}^{-}$ and $F_{c}^{+}$ can be defined on the switching surface $\Sigma$ independently of $F_{c}$, when one wants to distinguish between static and dynamic friction.
According to our physical interpretation of the solution, when a trajectory reaches the switching surface $\Sigma$ the trajectory either crosses $\Sigma$ or becomes part of $\Sigma$, which means sticking in the physical sense. The algebraic constraint of sticking is $h(\boldsymbol{y}(t))=0$. While sticking the contact force $F_{c}^{\star}(t)$ must stay within physical bounds \begin{equation} F_{c}^{-}(\boldsymbol{y})\le F_{c}^{\star}(t)\le F_{c}^{+}(\boldsymbol{y})\label{eq:FcBound} \end{equation} and the vector field must be tangential to $\Sigma$, that is, \begin{equation} \nabla h(\boldsymbol{y}(t))\cdot\dot{\boldsymbol{y}}(t)=0,\label{eq:NsSlideCond} \end{equation} so that the solution continues on the switching surface. If such a contact force cannot be found the solution crosses the switching surface and a discontinuity develops in the contact force. To calculate the contact force $F_{c}^{\star}(t)$ that makes the solution restricted to $\Sigma$ we substitute (\ref{eq:MZNSeq}) into (\ref{eq:NsSlideCond}), which yields \begin{equation} 0=\nabla h(\boldsymbol{y}(t))\cdot\biggl\{\boldsymbol{A}\boldsymbol{y}(t)+\boldsymbol{L}^{\infty}F_{c}^{\star}(t)+\int_{0}^{t-s}\mathrm{d}_{\tau}\boldsymbol{L}^{1}(\tau)\frac{\mathrm{d}}{\mathrm{d}t}[F_{c}^{\star}(t-\tau)]+\boldsymbol{g}(t)\biggr\}.\label{eq:LambdaEq} \end{equation} Equation (\ref{eq:LambdaEq}) involves the history of the contact force, which is either $F_{c}^{\star}(t)=F_{c}(\boldsymbol{y}(t))$ if $h(\boldsymbol{y}(t))\neq0$ or it is calculated from (\ref{eq:LambdaEq}).
The question is whether the contact force $F_{c}^{\star}(t)$ is well defined during the stick phase by equation (\ref{eq:LambdaEq}), which is an integral equation for $F_{c}^{\star}(t)$. To answer this we need to consider possible singularities of $\boldsymbol{L}^{1}(\tau)$ at $\tau=0$. Since $\boldsymbol{L}^{1}(\tau)$ is bounded one can find a maximal $0\le\alpha\le1$ and a positive constant $C$ such that $\left\Vert \boldsymbol{L}^{1}(\tau)\right\Vert <C\tau^{\alpha}$. This is called the H\"older condition and $\alpha$ is the H\"older exponent. If $\alpha<1$ we can also find a constant $\boldsymbol{L}^{1+}$ and positive $C_{0}$ such that \[ \left\Vert \boldsymbol{L}^{1}(\tau)-\boldsymbol{L}^{1+}\tau^{\alpha}\right\Vert <C_{0}\tau. \] This means that $\boldsymbol{L}^{1}(\tau)$ is a sum of the singular $\boldsymbol{L}^{1+}\tau^{\alpha}$ and a differentiable function. There are three cases to consider: \begin{enumerate} \item $\alpha=1$, so that $\boldsymbol{L}^{1}(\tau)$ is differentiable. We assume that $\nabla h(\boldsymbol{y}(t))\cdot\boldsymbol{L}(0)\neq0$.\label{enu:C0} \item $\alpha=0$, so that $\boldsymbol{L}^{1}(\tau)$ is discontinuous and $\boldsymbol{L}^{1+}=\lim_{\tau\to0+}\boldsymbol{L}^{1}(\tau)$. We assume that $\nabla h(\boldsymbol{y}(t))\cdot\boldsymbol{L}^{1+}\neq0$.\label{enu:C1} \item $0<\alpha<1$, when $\boldsymbol{L}^{1}(\tau)$ is not differentiable, but continuous. Similarly, we assume that $\nabla h(\boldsymbol{y}(t))\cdot\boldsymbol{L}^{1+}\neq0$.\label{enu:C2} \end{enumerate} In case \ref{enu:C0}, when $\boldsymbol{L}^{1}(\tau)$ is continuously differentiable on $[0,\infty)$, the integral term can be expressed using $\boldsymbol{L}(\tau)$ as in equation (\ref{eq:MZequation}). This is the case when the governing equations are finite dimensional or $\boldsymbol{n}_{j}$ have finite norms. Therefore the same dynamical phenomena should occur as in finite dimensional systems, which cannot be resolved by our method. Applying (\ref{eq:NsSlideCond}) to equation (\ref{eq:MZequation}) we find that the contact force obeys the integral equation
\begin{equation} F_{c}^{\star}(t)=\frac{-\nabla h(\boldsymbol{y}(t))}{\nabla h(\boldsymbol{y}(t))\cdot\boldsymbol{L}(0)}\cdot\biggl\{\boldsymbol{A}\boldsymbol{y}(t)+\int_{0}^{t-s}\mathrm{d}_{\tau}\boldsymbol{L}(\tau)F_{c}^{\star}(t-\tau)+\boldsymbol{H}(t)\boldsymbol{z}(s)\biggr\}.\label{eq:FcL1cont} \end{equation} Due to the differentiability of $\boldsymbol{L}^{1}(\tau)$ its derivative $\boldsymbol{L}^{0}(\tau)$ and therefore $\boldsymbol{L}(0)$ must be bounded. When calculating the contact force by equation (\ref{eq:FcL1cont}) can result in a discontinuity of $F_{c}^{\star}(t)$ at the onset of the stick phase. Another cause of singularity is when $\nabla h(\boldsymbol{y}(t))\cdot\boldsymbol{L}(0)=0$, which can occur in case of the two-fold singularity \citep{ColomboJeffrey}.
The pre-tensed string model falls into case \ref{enu:C1}. Due to the discontinuity of $\boldsymbol{L}^{1}(\tau)$, equation (\ref{eq:LambdaEq}) can be rearranged as a delay differential equation with $\nicefrac{\mathrm{d}}{\mathrm{d}t}F_{c}^{\star}(t)$ on the left-hand side, that is, \begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}F_{c}^{\star}(t)=\frac{-\nabla h(\boldsymbol{y}(t))}{\nabla h(\boldsymbol{y}(t))\cdot\boldsymbol{L}^{1+}}\cdot\biggl\{\boldsymbol{A}\boldsymbol{y}(t)+\boldsymbol{L}^{\infty}F_{c}^{\star}(t)+\int_{0+}^{t-s}\mathrm{d}_{\tau}\boldsymbol{L}^{1}(\tau)\frac{\mathrm{d}}{\mathrm{d}t}[F_{c}^{\star}(t-\tau)]+\boldsymbol{g}(t)\biggr\}.\label{eq:FcL1Jump} \end{equation} At the onset of stick at $t^{\star}$ the initial condition is $F_{c}^{\star}(t^{\star})=\lim_{t\to t^{\star}-0}F_{c}(\boldsymbol{y}(t))$. Since all the terms in (\ref{eq:FcL1Jump}) are bounded $\nicefrac{\mathrm{d}}{\mathrm{d}t}F_{c}^{\star}(t^{\star})$ must also be bounded. Therefore $F_{c}^{\star}(t)$ is a Lipschitz continuous function of time when the solution gets restricted to $\Sigma$ and all throughout the stick phase. At the transition from stick to slip $F_{c}^{\star}(t)$ is continuous if $F_{c}^{\pm}(\boldsymbol{y})$ is the limit of $F_{c}(\boldsymbol{y})$ defined by (\ref{eq:FcLimits}). If in addition the slope of $F_{c}(\boldsymbol{y})$ on the relevant side of $\Sigma$ is finite, $F_{c}^{\star}(t)$ is Lipschitz continuous. It remains to be investigated what are the dynamical consequences when $\nabla h(\boldsymbol{y}(t))\cdot\boldsymbol{L}^{1+}=0$ and whether the uniqueness of solution is preserved through such a singularity.
The Euler-Bernoulli beam falls into case \ref{enu:C2}. First we note that \begin{equation} \int_{0}^{t-s}\mathrm{d}_{\tau}\tau^{\alpha}\frac{\mathrm{d}}{\mathrm{d}t}[F_{c}^{\star}(t-\tau)]=\int_{s}^{t}(t-\theta)^{\alpha-1}\frac{\mathrm{d}}{\mathrm{d}\theta}[F_{c}^{\star}(\theta)]\mathrm{d}\theta, \end{equation} which is by definition $-\Gamma(\alpha)$ times the $\alpha$ fractional integral of $\nicefrac{\mathrm{d}}{\mathrm{d}t}F_{c}^{\star}(t)$ \citep{mcbride1979fractional}. We assume that the stick phase starts at time $t^{\star}$. Using the rules of fractional integration we find that \begin{equation} \int_{t^{\star}}^{t}(t-\theta)^{-\alpha}\left(\int_{0}^{\theta-t^{\star}}\mathrm{d}_{\tau}\tau^{\alpha}\frac{\mathrm{d}}{\mathrm{d}t}[F_{c}^{\star}(\theta-\tau)]\right)\mathrm{d}\theta=\frac{\alpha\pi}{\sin\alpha\pi}\left(F_{c}^{\star}(t)-F_{c}^{\star}(t^{\star})\right).\label{eq:FracIntTest} \end{equation} By separating the singular component of equation (\ref{eq:LambdaEq}) we get \begin{multline} \int_{0}^{t-t^{\star}}\mathrm{d}_{\tau}\tau^{\alpha}\frac{\mathrm{d}}{\mathrm{d}t}[F_{c}^{\star}(t-\tau)]=\frac{-\nabla h(\boldsymbol{y}(t))}{\nabla h(\boldsymbol{y}(t))\cdot\boldsymbol{L}^{1+}}\cdot\biggl(\boldsymbol{A}\boldsymbol{y}(t)+\boldsymbol{L}^{\infty}F_{c}^{\star}(t)+\\ +\int_{0}^{t-t\star}\mathrm{d}_{\tau}\left(\boldsymbol{L}^{1}(\tau)-\boldsymbol{L}^{1+}\tau^{\alpha}\right)\frac{\mathrm{d}}{\mathrm{d}\theta}[F_{c}^{\star}(t-\tau)]+\int_{t-t^{\star}}^{t-s}\mathrm{d}_{\tau}\boldsymbol{L}^{1}(\tau)\frac{\mathrm{d}}{\mathrm{d}t}[F_{c}^{\star}(t-\tau)]+\boldsymbol{g}(t)\biggr).\label{eq:SingSep} \end{multline} Since we assumed that $\nabla h(\boldsymbol{y}(t))\cdot\boldsymbol{L}^{1+}\neq0$, it follows that all the terms on the right side of (\ref{eq:SingSep}) are bounded by $C_{1}$. Therefore we fractional integrate (\ref{eq:SingSep}) with $1-\alpha$ exponent exactly as in (\ref{eq:FracIntTest}) and get \begin{equation}
\left|\frac{\alpha\pi}{\sin\alpha\pi}\left(F_{c}^{\star}(t)-F_{c}^{\star}(t^{\star})\right)\right|\le\int_{t^{\star}}^{t}(t-\theta)^{-\alpha}C_{1}\mathrm{d}\theta=\frac{C_{1}\left(t-t^{\star}\right)^{1-\alpha}}{1-\alpha}.\label{eq:FcHolder} \end{equation} This means that if $\nabla h(\boldsymbol{y}(t))\cdot\boldsymbol{L}^{1+}\neq0$, $F_{c}^{\star}(t)$ is H\"older continuous with exponent $1-\alpha$. We note that H\"older continuity implies continuity in the traditional sense, therefore the friction force is continuos during the transition from slip to stick.
\section{Stick-slip motion of a bowed string\label{sec:Bowstring}}
\begin{figure}
\caption{(colour online) (a) Schematic of a bowed string. The bow is pulled with a constant velocity $v_{0}$, while the string exhibits a stick-slip vibration generated by the friction between the bow and the string. (b) Graph of the Coulomb-like friction force.}
\label{fig:bowedstring}
\end{figure} To see our theory applied to a mechanical system consider the example of a bowed string in Fig. \ref{fig:bowedstring}(a). We consider the same equation of motion as in section \ref{sub:StringVib}, where we derived all the necessary ingredients of the reduced model apart from the contact force. Because $\boldsymbol{L}^{1}(\tau)$ has a discontinuity at $\tau=0$ this example falls into case \ref{enu:C1}.~of section \ref{sub:NSDyn}.
To complete the model we define the contact force $F_{c}$ of equation (\ref{eq:MZNSeq}) as the friction force between the bow and the string. We assume that the string is being bowed at $\xi=\xi^{\star}$ with velocity $v_{0}$ that generates the friction force \begin{equation}
f_{c}(v_{\textrm{rel}})=\mathrm{sign}v_{\mathrm{rel}}\left(\mu-\kappa+\kappa\exp\left(-\sigma\left|v_{\mathrm{rel}}\right|\right)\right),\label{eq:fforce} \end{equation} where $v_{\textrm{rel}}$ is the relative velocity of the string and the bow. The graph of the friction force function can be seen in Fig. \ref{fig:bowedstring}(b). In this example the static friction force is within the interval $[-\mu,\mu]$. The relative velocity between the string and the bow is expressed as a function of a resolved variable $v_{\mathrm{rel}}=y_{2}(t)-v_{0}$. We also use the relative velocity to define the switching surface $\Sigma$ by $h(\boldsymbol{y}(t))=v_{\mathrm{rel}}$. Therefore the contact force of equation (\ref{eq:MZNSeq}) becomes $F_{c}(\boldsymbol{y}(t))=f_{c}(h(\boldsymbol{y}(t)))$.
\subsection{Numerical method\label{sub:NumMeth}}
We use a simple explicit Euler method to approximate the solutions of (\ref{eq:MZNSeq}) and (\ref{eq:LambdaEq}). We assume that time is quantised in $\varepsilon$ chunks, so that $\boldsymbol{y}_{q}=\boldsymbol{y}(q\varepsilon)$, $f_{c,q}=f_{c}(q\varepsilon)$, where $q=0,1,2,\ldots$. In case of slipping the only unknown is the state variable $\boldsymbol{y}_{q}$ that is calculated using the formula \begin{multline} \boldsymbol{y}_{q+1}=\boldsymbol{y}_{q}+\varepsilon\Biggl(\boldsymbol{A}\boldsymbol{y}_{q}+\boldsymbol{L}^{\infty}f_{c,q}+\sum_{j=0}^{n}\boldsymbol{L}^{0}(j\varepsilon)\left(f_{c,q-j}-f_{c,q-j-1}\right)\\ +\boldsymbol{L}^{0}(q\varepsilon)f_{c,0}+\boldsymbol{H}(q\varepsilon)\boldsymbol{z}(0)\Biggr),\label{eq:NumStateUpdate} \end{multline} where the friction force $f_{c,q}=F_{c}\left(\left[\boldsymbol{y}_{q}\right]_{2}-v_{0}\right)$ is used. The integration is approximated by the rectangle rule. For just illustrating the theory such a crude approximation is sufficient while for better accuracy and efficiency higher order methods, such as the Runge-Kutta \citep{Iserles} method could be used. In our calculations we keep the step size reasonably short at $\varepsilon=5\times10^{-4}$.
If the relative velocity $h(\boldsymbol{y}_{q})=\left[\boldsymbol{y}_{q}\right]_{2}-v_{0}$ of the string and the bow becomes zero there are two possibilities. Either the trajectory crosses the switching surface $\Sigma$ or it will stay on $\Sigma$ satisfying the equation \begin{equation} \nabla h(\boldsymbol{y}_{q})\cdot(\boldsymbol{y}_{q+1}-\boldsymbol{y}_{q})=0,\label{eq:NumTangency} \end{equation} that is, the discretised version of $h(\boldsymbol{y}(t))\cdot\dot{\boldsymbol{y}}(t)=0$. To test which case applies, we substitute (\ref{eq:NumStateUpdate}) into (\ref{eq:NumTangency}) and solve for the friction force $f_{c,q}$ that would hold the string and the bow together, which becomes \begin{multline} f_{c,q}=\frac{-\nabla h(\boldsymbol{y}_{q})}{\nabla h(\boldsymbol{y}_{q})\cdot(\boldsymbol{L}^{\infty}+\boldsymbol{L}^{0}(0))}\Biggl(-\boldsymbol{L}^{0}(0)f_{c,q-1}+\boldsymbol{A}\boldsymbol{y}_{q}+\sum_{j=1}^{q}\boldsymbol{L}^{0}(j)\left(f_{c,q-j}-f_{c,q-j-1}\right)\\ +\boldsymbol{L}^{0}(q\varepsilon)f_{c,0}+\boldsymbol{H}(q\varepsilon)\boldsymbol{z}(0)\Biggr).\label{eq:NumFcUpdate} \end{multline} If the calculated friction force satisfies $-\mu\le f_{c,q}\le\mu$, the bow and the string stick together. For the stick phase of motion we use equation (\ref{eq:NumStateUpdate}) to advance the solution together with this dynamic friction force of equation (\ref{eq:NumFcUpdate}).
\subsection{Numerical results}
To illustrate the properties of the dimension reduced equation (\ref{eq:MZNSeq}) we calculated a typical stick-slip trajectory starting at a the initial condition $\boldsymbol{z}(0)=y_{1}(0)\boldsymbol{w}_{1}+y_{2}(0)\boldsymbol{w}_{2}$ with $y_{1}(0)=-2.9224$ and $y_{2}(0)=-2.7668$. The parameters of the friction force in equation (\ref{eq:fforce}) are $\mu=4$, $\kappa=0.32$, $\sigma=1$, the speed of the bow is $v_{0}=\nicefrac{3}{2}$, the damping ratios are $D_{k}=\nicefrac{1}{10}$, $k=1,\ldots,N$ and the wave speed on the string is $c=1$. The string is bowed at $\xi^{\star}=0.4$. \footnote{The choice of these parameters was guided by the desire of producing stick-slip motion rather than physical consideration. } We solved equation (\ref{eq:linModeDescr}) using \noun{Matlab}'s \textsf{ode113} solver and (\ref{eq:MZNSeq}) using our method described in section \ref{sub:NumMeth}. The results of the the simulation for reduced and the full model shown in Figure \ref{fig:Solution}(a,b) are nearly indistinguishable, because the only approximations are within the numerical methods. The blue curves denote the solution of (\ref{eq:MZNSeq}) and the (almost invisible) red curve underneath represents the solution of (\ref{eq:linModeDescr}) using solution techniques described by \citet{PiiroinenSim}.
\begin{figure}
\caption{Solution trajectories of equations (\ref{eq:linModeDescr}) and (\ref{eq:MZNSeq}). (a) Displacement of the string and (b) velocity of the string at the contact point. (c) Friction force between the bow and string using equation (\ref{eq:MZNSeq}). The continuous lines denote slip, the dash-dotted lines represent sticking motion. (d) The discontinuity of the friction force at the onset of sticking disappears in the continuum limit. $N$ indicates the considered number of modes.}
\label{fig:Solution}
\end{figure}
Initially the solution spends short time intervals on the switching surface and then settles into a periodic stick-slip motion. The stick phases can be recognised in Fig. \ref{fig:Solution}(b) as short horizontal sections at $y_{2}=1.5$. In Fig. \ref{fig:Solution}(c) the friction force is represented by the blue lines and the green dash-dotted lines for the slipping and the sticking motion, respectively. The friction force also appears to be discontinuous. To calculate this solution we did not use the converged $\boldsymbol{L}^{1}(\tau)$, instead we used a series of mode truncations shown in Fig.~(\ref{fig:StringL1fig})(b). On a smaller scale Fig. \ref{fig:Solution}(d) shows that the gap in the friction force (dashed line) between the slipping segment (continuous line) and the sticking segment (dash dotted line) of the friction force vanishes as increasing number of modes of system (\ref{eq:linModeDescr}) are considered. As the theory dictates the gap should vanish in the infinite dimensional case.
\section{Conclusions}
In this paper we considered vibrations of structures that are composed of linear elastic bodies coupled through strongly nonlinear contact forces such as friction. The coupling was assumed to occur at point contacts. We introduced an exact transformation based on the Mori-Zwanzig formalism that reduces the infinite dimensional system of ordinary differential equations to a description with time delay involving small number of variables. We found that the model reduction technique converges and contact forces become continuous even though the governing equation is discontinuous. We illustrated this novel technique through the example of a bowed string.
Through examples we found that if natural frequencies scale linearly with the mode number, the contact forces are Lipschitz continuous during the transition from slip to stick. This is the case of the elastic string. If the natural frequencies increase faster than linear, the contact forces are only continuous. The Euler-Bernoulli beam exhibits such a behaviour, but it also allows infinite wave speed, which can be though of as not physical. In reality however, every structure will have small scale longitudinal vibration components with linearly scaled frequencies similar to the Timoshenko beam model \citep{SzalaiMZImpact}. We expect that if all the details are considered for a linear structure, the contact forces must always be Lipschitz continuous in time. This finding together with the new form of governing equations could be used in further studies to understand the source of non-deterministic motion \citep{ColomboJeffrey}.
The reduced equations are also structurally stable. Small perturbations to the memory kernel or other terms only deform solutions but do not change their qualitative behaviour as long as the qualitative features of the memory kernel are preserved. This is a clear advantage over finite dimensional approximation of non-smooth systems, where small perturbations can cause qualitatively different solutions. Therefore once the qualitative form of the memory kernel is established non-smooth mechanical systems can be approximated more successfully using our description.
How non-smooth phenomena of low dimensional systems manifest in continuum structures is an open question. For low dimensional systems many singularities can occur that lead to chaotic and resonant vibration on invariant polygons \citep{SzalaiPolygons}, the Painleve paradox \citep{PainleveAlan} and other types of discontinuity induced bifurcations. It remains to investigate how these phenomena occur in systems involving elastic structures and hence equations with memory.
Our theory is developed for linear structures coupled to strong nonlinearities. It is however possible to extend this framework to cases where the underlying structure is nonlinear. For the weakly nonlinear case the Hartman-Grobman theorem \citep{CoddingtonLevinson,KuznetsovBook} guarantees the existence of a transformation that takes any weakly nonlinear system into a linear system about an equilibrium if that system is not undergoing a stability change. This generalisation is currently being worked on by the author.
We also assumed point contacts in our derivations. This is a significant simplification since most contact problems occur along a surface. The difficulty arises when one needs to deal with contacting surfaces that slip at one part of the contact surface while stick at others. An interesting question is if it is possible to develop a similar model reduction technique of such problems to involve only finite number of variables.
\section{Model transformation\label{sec:AppTrafo}}
In this appendix we show that the infinite dimensional system (\ref{eq:1storderODE}) can be transformed into a finite dimensional delayed equation. The delay equation involves convolution integrals that can be related to Green's functions, but only for part of the system. The procedure is based on the variation-of-parameters formula.
Consider the following linear forced system
\begin{equation} \dot{\boldsymbol{z}}(t)=\boldsymbol{R}\boldsymbol{z}(t)+\boldsymbol{v}f(t),\label{eq:AppLinEq} \end{equation} where $\boldsymbol{z}(t),\boldsymbol{v}\in\mathbb{R}^{2N}$, $f(t)\in\mathbb{R}$ and $\boldsymbol{R}\in\mathbb{R}^{2N\times2N}$. Assume matrices $\boldsymbol{V}\in\mathbb{R}^{2M\times2N}$ and $\boldsymbol{W}\in\mathbb{R}^{2N\times2M}$ such that $\boldsymbol{S}=\boldsymbol{W}\boldsymbol{V}$ is a projection matrix $\boldsymbol{S}=\boldsymbol{S}^{2}$ with a $2M$ dimensional range. $\boldsymbol{S}$ is a projection if and only if $\boldsymbol{V}\boldsymbol{W}=\boldsymbol{I}_{2M}$, where $\boldsymbol{I}_{2M}$ is the $2M$ dimensional identity. Also, define the complementary projection matrix $\boldsymbol{Q}=\boldsymbol{I}-\boldsymbol{S}$ and the resolved coordinates $\boldsymbol{y}(t)=\boldsymbol{V}\boldsymbol{z}(t)$. With this notation we rewrite equation (\ref{eq:AppLinEq}) into \begin{equation} \dot{\boldsymbol{z}}(t)=\boldsymbol{R}\boldsymbol{S}\boldsymbol{z}(t)+\boldsymbol{R}\boldsymbol{Q}\boldsymbol{z}(t)+\boldsymbol{v}f(t).\label{eq:AppSepEq} \end{equation} Assume that the solution of $\dot{\boldsymbol{z}}(t)=\boldsymbol{R}\boldsymbol{Q}\boldsymbol{z}(t)$ can be computed for specific initial conditions. Therefore the solution of (\ref{eq:AppSepEq}) can formally be expressed using the variation-of-parameters or Dyson's \citep{CoddingtonLevinson} formula as \begin{equation} \boldsymbol{z}(t)=\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}(t-s)}\boldsymbol{z}(s)+\int_{0}^{t-s}\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}\tau}\left(\boldsymbol{R}\boldsymbol{S}\boldsymbol{z}(t-\tau)+\boldsymbol{v}f(t-\tau)\right)\mathrm{d}\tau. \end{equation} Substituting this result into the second term on the right side of (\ref{eq:AppSepEq}) we get \begin{multline} \dot{\boldsymbol{z}}(t)=\boldsymbol{R}\boldsymbol{S}\boldsymbol{z}(t)+\boldsymbol{v}f(t)\\ +\boldsymbol{R}\boldsymbol{Q}\left\{ \mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}(t-s)}\boldsymbol{z}(s)+\int_{0}^{t-s}\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}\tau}\left(\boldsymbol{R}\boldsymbol{S}\boldsymbol{z}(t-\tau)+\boldsymbol{v}f(t-\tau)\right)\right\} \mathrm{d}\tau.\label{eq:AppSubs} \end{multline} Note that $\boldsymbol{S}\boldsymbol{z}(t)=\boldsymbol{W}\boldsymbol{V}\boldsymbol{z}(t)=\boldsymbol{W}\boldsymbol{y}(t)$ and project (\ref{eq:AppSubs}) using $\boldsymbol{V}$, to get \begin{multline} \dot{\boldsymbol{y}}(t)=\boldsymbol{V}\boldsymbol{R}\boldsymbol{W}\boldsymbol{y}(t)+\boldsymbol{V}\boldsymbol{v}f(t)\\ +\boldsymbol{V}\boldsymbol{R}\boldsymbol{Q}\left\{ \mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}(t-s)}\boldsymbol{z}(s)+\int_{0}^{t-s}\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}\tau}\left(\boldsymbol{R}\boldsymbol{S}\boldsymbol{z}(t-\tau)+\boldsymbol{v}f(t-\tau)\right)\right\} \mathrm{d}\tau, \end{multline} which is the reduced equation for only the resolved coordinates $\boldsymbol{y}(t)$. Note that $\boldsymbol{R}\boldsymbol{Q}\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}\tau}=\nicefrac{d}{d\tau}\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}\tau}$ hence the integrals can be rewritten with Riemann-Stieltjes integrals as \begin{equation} \dot{\boldsymbol{y}}(t)=\boldsymbol{A}\boldsymbol{y}(t)+\int_{0}^{t-s}\mathrm{d}_{\tau}\boldsymbol{K}(\tau)\boldsymbol{y}(t-\tau)+\int_{0}^{t-s}\mathrm{d}_{\tau}\boldsymbol{L}(\tau)f(t-\tau)+\boldsymbol{V}\boldsymbol{v}f(t)+\boldsymbol{H}(t)\boldsymbol{z}(s),\label{eq:FullMZeq} \end{equation} where \begin{align} \boldsymbol{A} & =\boldsymbol{V}\boldsymbol{R}\boldsymbol{W}, & & \in\mathbb{R}^{2M\times2M}\label{eq:A-Aexpr}\\ \boldsymbol{H}(t)\boldsymbol{z}(s) & =\boldsymbol{V}\boldsymbol{R}\boldsymbol{Q}\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}(t-s)}\boldsymbol{z}(s), & & \in\mathbb{R}^{2M}\label{eq:A-Hexpr}\\ \boldsymbol{K}(\tau) & =\boldsymbol{V}\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}\tau}\boldsymbol{R}\boldsymbol{W}, & & \in\mathbb{R}^{2M\times2M}\label{eq:A-Kexpr}\\ \boldsymbol{L}(\tau) & =\boldsymbol{V}\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}\tau}\boldsymbol{v}. & & \in\mathbb{R}^{2M}\label{eq:A-Lexpr} \end{align} If the range of $\boldsymbol{W}$ is invariant under $\boldsymbol{R}$, then $\boldsymbol{K}(\tau)=\boldsymbol{A}$ constant. This occurs because the image of the range of $\boldsymbol{W}$ is in the kernel of $\boldsymbol{Q}$. Consequently the integral with $\boldsymbol{K}(\tau)$ vanishes.
Note that this procedure is a simplified version of the Mori-Zwanzig formalism \citep{ChorinPNAS,EvansMorriss} for linear systems. Therefore our procedure can be extended to nonlinear systems. In the nonlinear case $\boldsymbol{A}\boldsymbol{y}$, $\boldsymbol{K}(\tau)\boldsymbol{y}$ and $\boldsymbol{L}(\tau)f$ become nonlinear functions of $\boldsymbol{y}$ and $f$, respectively.
\section{The memory kernels\label{sec:AppMemKer}}
In this appendix we show that the memory kernel can be obtained from the solution of the first order system (\ref{eq:1storderODE}) if condition (\ref{eq:RangeCond}) holds. Condition (\ref{eq:RangeCond}) implies that there is a $2M\times2M$ matrix $\boldsymbol{X}$ such that $\boldsymbol{R}\boldsymbol{W}=\boldsymbol{W}\boldsymbol{X}$. If we multiply this expression by $\boldsymbol{V}$ from the left we get the identity $\boldsymbol{A}=\boldsymbol{V}\boldsymbol{R}\boldsymbol{W}=\boldsymbol{V}\boldsymbol{W}\boldsymbol{X}=\boldsymbol{X}$. As a consequence for any integer $p$ we must have \begin{equation} \boldsymbol{R}^{p}\boldsymbol{W}=\boldsymbol{W}\boldsymbol{A}^{p}.\label{eq:WApower} \end{equation}
To investigate the expression $\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}t}$ that occurs in the definition of the memory kernel (\ref{eq:A-Lexpr}) we define \begin{equation} \boldsymbol{\varPhi}(t)=\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}t}\mathrm{e}^{-\boldsymbol{R}t} \end{equation}
so that $\mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}t}=\boldsymbol{\varPhi}(t)\mathrm{e}^{\boldsymbol{R}t}$. The power series expansion of $\boldsymbol{\varPhi}(t)$ can be written as $\boldsymbol{\varPhi}(t)=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\left.\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}\boldsymbol{\varPhi}(t)\right|_{t=0}$. The derivatives are calculated as \begin{align}
\left.\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}\boldsymbol{\varPhi}(t)\right|_{t=0} & =\sum_{k=0}^{n}\binom{n}{k}\left(\boldsymbol{R}-\boldsymbol{R}\boldsymbol{S}\right)^{k}\left(-\boldsymbol{R}\right)^{n-k}\nonumber \\
& =\sum_{k=0}^{n}\binom{n}{k}\sum_{p_{i}+q_{i}=1}\boldsymbol{R}^{p_{1}}\left(-\boldsymbol{R}\boldsymbol{S}\right)^{q_{1}}\cdots\boldsymbol{R}^{p_{k}}\left(-\boldsymbol{R}\boldsymbol{S}\right)^{q_{k}}\left(-\boldsymbol{R}\right)^{n-k}.\label{eq:phiExp-A} \end{align} We can transform the products in (\ref{eq:phiExp-A}) to simpler expressions. Assume that in the second summation for a fixed $r\in\{0,\ldots,k\}$, $q_{r}=1$ and $q_{i}=0$ if $i>r$, while the rest of $q_{i}$ are arbitrary. The sets of $p_{i},q_{i}$ are disjoint for different $r$ values and their union covers all possible $p_{i},q_{i}$ values once. Using formula (\ref{eq:WApower}) and $\boldsymbol{R}\boldsymbol{S}=\boldsymbol{W}\boldsymbol{A}\boldsymbol{V}$ we find that \begin{equation} \boldsymbol{R}^{p_{1}}\left(-\boldsymbol{R}\boldsymbol{S}\right)^{q_{1}}\cdots\boldsymbol{R}^{p_{k}}\left(-\boldsymbol{R}\boldsymbol{S}\right)^{q_{k}}=(-1)^{\sum q_{i}}\boldsymbol{W}\boldsymbol{A}^{r}\boldsymbol{V}\boldsymbol{R}^{k-r}. \end{equation} The sum of all terms corresponding to each $r\neq$0 can be written as \begin{equation} \sum_{l=0}^{r-1}\binom{r-1}{l}(-1)^{l+1}\boldsymbol{W}\boldsymbol{A}^{r}\boldsymbol{V}\boldsymbol{R}^{k-r}=\begin{cases} -\boldsymbol{W}\boldsymbol{A}\boldsymbol{V}\boldsymbol{R}^{k-1} & \mbox{if}\; r=1\\ 0 & \mbox{if}\; r>1 \end{cases}. \end{equation} For $r=0$, the sum is $\boldsymbol{R}^{k}$. Therefore the $n$-th derivative for $n>0$ reads \begin{align}
\left.\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}\boldsymbol{\varPhi}(t)\right|_{t=0} & =\left(-\boldsymbol{R}\right)^{n}+\sum_{k=1}^{n}\binom{n}{k}\left(\boldsymbol{R}^{k}-\boldsymbol{W}\boldsymbol{A}\boldsymbol{V}\boldsymbol{R}^{k-1}\right)\left(-\boldsymbol{R}\right)^{n-k}\\
& =\sum_{k=0}^{n}\binom{n}{k}\boldsymbol{R}^{k}\left(-\boldsymbol{R}\right)^{n-k}-\sum_{k=1}^{n}\binom{n}{k}\boldsymbol{W}\boldsymbol{A}\boldsymbol{V}\boldsymbol{R}^{k-1}\left(-\boldsymbol{R}\right)^{n-k}\\
& =(-1)^{n}\boldsymbol{W}\boldsymbol{A}\boldsymbol{V}\boldsymbol{R}^{n-1}. \end{align} Substituting the derivatives into the power series we are left with \begin{equation} \boldsymbol{\varPhi}(t)=\boldsymbol{I}-\sum_{n=1}^{\infty}\frac{t^{n}}{n!}(-1)^{n-1}\boldsymbol{W}\boldsymbol{A}\boldsymbol{V}\boldsymbol{R}^{n-1}=\boldsymbol{I}-\boldsymbol{W}\boldsymbol{A}\boldsymbol{V}\int_{0}^{t}\mathrm{e}^{-\boldsymbol{R}\tau}\mathrm{d}\tau. \end{equation} Multiplying $\boldsymbol{\varPhi}(t)$ from the right by $\mathrm{e}^{\boldsymbol{R}t}$ we get the formula \begin{equation} \mathrm{e}^{\boldsymbol{R}\boldsymbol{Q}t}=\mathrm{e}^{\boldsymbol{R}t}-\boldsymbol{R}\boldsymbol{S}\int_{0}^{t}\mathrm{e}^{\boldsymbol{R}(t-\tau)}\mathrm{d}\tau=\mathrm{e}^{\boldsymbol{R}t}-\boldsymbol{R}\boldsymbol{S}\int_{0}^{t}\mathrm{e}^{\boldsymbol{R}\tau}\mathrm{d}\tau. \end{equation} With this result the forcing term and the memory kernels become
\begin{gather*} \boldsymbol{H}(t)\boldsymbol{z}(s)=\boldsymbol{V}\boldsymbol{R}\boldsymbol{Q}\mathrm{e}^{\boldsymbol{R}(t-s)}\boldsymbol{z}(s),\;\boldsymbol{K}(\tau)=\boldsymbol{A},\;\boldsymbol{L}(\tau)=\left(\boldsymbol{V}\mathrm{e}^{\boldsymbol{R}\tau}-\boldsymbol{A}\boldsymbol{V}\int_{0}^{\tau}\mathrm{e}^{\boldsymbol{R}\theta}\mathrm{d}\theta\right)\boldsymbol{v}. \end{gather*}
\section{Strong continuity of $\mathrm{e}^{\boldsymbol{R}t}$\label{sub:C0semigroup}}
In order to justify our analysis we need to show that $\mathrm{e}^{\boldsymbol{R}t}$ is a strongly continuous semigroup. We use the Hille-Yosida theorem \citep{Pazy83,HillePhillips57}, which states that $\mathrm{e}^{\boldsymbol{R}t}$ is a strongly continuous semigroup satisfying $\left\Vert \mathrm{e}^{\boldsymbol{R}t}\right\Vert \le M_{0}$ if and only if $\overline{\mathcal{D}(\boldsymbol{R})}=\boldsymbol{Z}$ and \begin{equation} \left\Vert (\boldsymbol{R}-\lambda\boldsymbol{I})^{-n}\right\Vert \le M_{0}\lambda^{-n}\label{eq:resolventCond-HY} \end{equation} for $\lambda>0$ and $n=1,2,3,\ldots$, where $\boldsymbol{Z}$ is defined by equation (\ref{eq:Xspace}).
First we show that if $\boldsymbol{R}$ satisfies (\ref{eq:NegativeRealPart}) and that each eigenvalue of $\boldsymbol{R}$ has finite multiplicity then condition (\ref{eq:resolventCond-HY}) is satisfied. Using a linear transformation matrix $\boldsymbol{R}$ can be brought into its block diagonal Jordan normal form. Each block in the diagonal of the Jordan normal form corresponds to an eigenvalue $\lambda_{k}$ of $\boldsymbol{R}$ and has size $l_{k}$. The form of such a block is \begin{equation} \boldsymbol{J}_{k}=\left(\begin{array}{cccc} \lambda_{k} & 1 & 0 & 0\\ 0 & \lambda_{k} & \ddots & 0\\ 0 & \ddots & \ddots & 1\\ 0 & 0 & 0 & \lambda_{k} \end{array}\right). \end{equation} After inversion the component of $(\boldsymbol{R}-\lambda\boldsymbol{I})^{-n}$ that corresponds to $J_{k}$ becomes \begin{equation} \left(\boldsymbol{J}_{k}-\lambda\boldsymbol{I}\right)^{-n}=\left(\begin{array}{cccc} \left(\lambda_{k}-\lambda\right)^{-n} & -n\left(\lambda_{k}-\lambda\right)^{-n-1} & \cdots & (-1)^{l}\binom{n+l_{k}-1}{l_{k}}\left(\lambda_{k}-\lambda\right)^{-n-l_{k}}\\ 0 & \left(\lambda_{k}-\lambda\right)^{-n} & \ddots & \vdots\\ 0 & 0 & \ddots & -n\left(\lambda_{k}-\lambda\right)^{-n-1}\\ 0 & 0 & 0 & \left(\lambda_{k}-\lambda\right)^{-n} \end{array}\right). \end{equation} The norm of this Jordan block can be estimated by \begin{equation}
\left\Vert \left(\boldsymbol{J}_{k}-\lambda\boldsymbol{I}\right)^{-n}\right\Vert \le\left|\lambda_{k}-\lambda\right|^{-n}+n\left|\lambda_{k}-\lambda\right|^{-n-1}+\cdots+\binom{n+l_{k}-1}{l_{k}}\left|\lambda_{k}-\lambda\right|^{-n-l_{k}}. \end{equation}
Note that $\left|\lambda_{k}-\lambda\right|\ge\left|\Re\lambda_{k}-\lambda\right|$. Since $\lambda>0\ge\Re\lambda_{k}$ one can find an $M_{k}$ such that \begin{equation} \left\Vert \left(\boldsymbol{J}_{k}-\lambda\boldsymbol{I}\right)^{-n}\right\Vert \le M_{k}\lambda^{-n} \end{equation} Considering this estimate for all Jordan blocks we find that \begin{equation} \left\Vert (\boldsymbol{R}-\lambda\boldsymbol{I})^{-n}\right\Vert \le\sup_{k}M_{k}\lambda^{-n}\le M_{0}\lambda^{-n}, \end{equation} where $M_{0}=\sup_{k}M_{k}$. This proves (\ref{eq:resolventCond-HY}).
To conclude the proof we show that $\overline{\mathcal{D}(\boldsymbol{R})}=\boldsymbol{Z}$. Again, we use the Jordan normal form. We partition every vector in $\boldsymbol{Z}$ such that $\boldsymbol{z}=(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\ldots)^{T}$, where $\boldsymbol{z}_{k}$ are of the size of a Jordan block. Let $\boldsymbol{z}\in\boldsymbol{Z}$ and construct $\boldsymbol{z}^{P}=(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\ldots,\boldsymbol{z}_{P},0,\ldots)^{T}$ such that it has $P$ number of non-zero components. This guarantees that for any $\boldsymbol{z}^{P}$, $\left\Vert \boldsymbol{R}\boldsymbol{z}^{P}\right\Vert <\infty$, hence $\boldsymbol{z}^{P}\in\mathcal{D}(\boldsymbol{R})$. It is also clear that $\left\Vert \lim_{P\to\infty}\boldsymbol{z}_{P}\right\Vert <\infty$ due to its construction, thus we have shown that $\overline{\mathcal{D}(\boldsymbol{R})}=\boldsymbol{X}$.
\section{Boundedness and smoothness of $\boldsymbol{L}_{j}^{1}(\tau)$\label{sub:L1bound}}
The definition (\ref{eq:L1def}) with (\ref{eq:L0Def}) of $\boldsymbol{L}_{j}^{1}(\tau)$ has two terms both including the expression $\boldsymbol{\Upsilon}_{j}(\tau)=\boldsymbol{V}\int_{0}^{\tau}\mathrm{e}^{\boldsymbol{R}\theta}\boldsymbol{v}_{M+j}\mathrm{d\theta}$. Therefore we only need to consider $\boldsymbol{\Upsilon}_{j}(\tau)$ in our analysis to show boundedness and smoothness of $\boldsymbol{L}_{j}^{1}(\tau)$.
We use the inverse Laplace transform \citep{Pazy83} to obtain \begin{equation} \mathrm{e}^{\boldsymbol{R}\tau}\boldsymbol{x}=\frac{1}{2\pi i}\int_{\Gamma_{0}}\mathrm{e}^{\lambda\tau}\left(\lambda\boldsymbol{I}-\boldsymbol{R}\right)^{-1}\boldsymbol{x}\mathrm{d}\lambda,\label{eq:OpLaplace} \end{equation} where $\Gamma_{0}=\{\lambda\in\mathbb{C}:\Re\lambda=\gamma>0\}$. Using integration rules for the Laplace transform and multiplying (\ref{eq:OpLaplace}) by vectors $\boldsymbol{v}_{M+j}$ and $\boldsymbol{v}_{\ell}$ from the left and right, respectively, we get \begin{equation} \left[\boldsymbol{\Upsilon}_{j}(\tau)\right]_{\ell}=\frac{1}{2\pi i}\int_{\Gamma_{0}}\mathrm{e}^{\lambda\tau}\boldsymbol{v}_{\ell}\cdot\left(\lambda\boldsymbol{I}-\boldsymbol{R}\right)^{-1}\boldsymbol{v}_{M+j}\frac{\mathrm{d}\lambda}{\lambda},\label{eq:projLaplace} \end{equation} The equality makes sense if the integral converges. Evaluating the inverse operator in (\ref{eq:projLaplace}) we find that \begin{equation} \left(\lambda\boldsymbol{I}-\boldsymbol{R}\right)^{-1}\left(\begin{array}{c} \boldsymbol{0}\\ \boldsymbol{n}_{j} \end{array}\right)=\left(\begin{array}{c} -(\boldsymbol{K}+\lambda\boldsymbol{C}+\lambda^{2})^{-1}\boldsymbol{n}_{j}\\ -\lambda(\boldsymbol{K}+\lambda\boldsymbol{C}+\lambda^{2})^{-1}\boldsymbol{n}_{j} \end{array}\right).\label{eq:ResolvForm} \end{equation} Note that it is sufficient to consider $\left[\boldsymbol{\Upsilon}_{j}(\tau)\right]_{M+l}$ since $\left[\boldsymbol{\Upsilon}_{j}(\tau)\right]_{l}$ is the integral of $\left[\boldsymbol{\Upsilon}_{j}(\tau)\right]_{M+l}$. Substituting (\ref{eq:ResolvForm}) into (\ref{eq:projLaplace}) we are left with \begin{align} \left[\boldsymbol{\Upsilon}_{j}(\tau)\right]_{M+l} & =\frac{1}{2\pi i}\int_{\Gamma_{0}}\mathrm{e}^{\lambda\tau}\boldsymbol{n}_{l}\cdot(\boldsymbol{K}+\lambda\boldsymbol{C}+\lambda^{2})^{-1}\boldsymbol{n}_{j}\mathrm{d}\lambda. \end{align} The integral of the inverse Laplace transform (\ref{eq:projLaplace}) converges for all $t\ge0$ if \begin{equation}
\left|\lambda\boldsymbol{n}_{l}\cdot(\boldsymbol{K}+\lambda\boldsymbol{C}+\lambda^{2})^{-1}\boldsymbol{n}_{j}\right|\le M_{j,l}\;\mbox{for}\;\lambda\in\Gamma_{0} \end{equation} and for $j,l=1,\ldots,M$. Indeed, by estimating the bound we get \begin{equation}
\left|\int_{0}^{\tau}\left[\boldsymbol{\Upsilon}_{j}(\tau)\right]_{M+l}\right|\le\frac{1}{2\pi i}M_{j,l}\left|\int_{\Gamma_{0}}\mathrm{e}^{\lambda\tau}\frac{\mathrm{d}\lambda}{\lambda}\right|=M_{j,l}. \end{equation} This implies that $\boldsymbol{L}_{j}^{1}$ is bounded.
If the stronger conditions (\ref{eq:SectorialCond}) and (\ref{eq:SmoothCond})
are satisfied, we can alter the contour of integration so that it is within the left half of the complex plane, $\Gamma_{\delta}=\{\lambda\in\mathbb{C}:\left|\arg\lambda\right|=\nicefrac{\pi}{2}+\delta\}$, where $\delta>0$ is sufficiently small so that all the eigenvalues of $\boldsymbol{R}$ are on the left of $\Gamma_{\delta}$. We parametrise the contour $\Gamma_{\delta}$ by $\lambda=\kappa\mathrm{e}^{\pm i(\nicefrac{\pi}{2}+\delta)}$, $\kappa\ge0$. Using this contour we find that the derivative \begin{equation} \left[\boldsymbol{\Upsilon}_{j}(\tau)\right]_{M+l}=\frac{1}{2\pi i}\int_{\Gamma_{\delta}}\mathrm{e}^{\lambda\tau}\lambda\boldsymbol{n}_{l}\cdot(\boldsymbol{K}+\lambda\boldsymbol{C}+\lambda^{2})^{-1}\boldsymbol{n}_{j}\mathrm{d}\lambda \end{equation} can be estimated by \begin{equation}
\left|\left[\boldsymbol{\Upsilon}_{j}(\tau)\right]_{M+l}\right|\le\frac{1}{\pi}\int_{0}^{\infty}M_{j,l}\mathrm{e}^{-\kappa\tau\cos2\delta}\mathrm{d}\kappa=\frac{1}{\tau}\left(\frac{M_{j,l}}{\pi\cos2\delta}\right). \end{equation} This means that the derivative of $\boldsymbol{L}_{j}^{1}$ is also bounded but only for $\tau>0$. Due to $\mathrm{e}^{\boldsymbol{R}\tau}$ being a strongly continuous semigroup (Theorem 2.5.2 in \citet{Pazy83}) higher derivatives are also bounded \begin{equation}
\frac{1}{n!}\frac{\mathrm{d}^{n-1}}{\mathrm{d}\tau^{n-1}}\left|\left[\boldsymbol{\Upsilon}_{j}(\tau)\right]_{M+l}\right|\le\left(\frac{C_{j,l}\mathrm{e}}{\tau}\right)^{n}, \end{equation} for $C_{j,l}<\infty$ which proves that $\boldsymbol{L}_{j}^{1}$ is smooth for $\tau>0$.
\section{Discontinuity of the memory kernel of the pre-tensed string\label{sec:L1Lim}}
Here we show that the memory kernel $\boldsymbol{L}^{1}(\tau)$ of the bowed string is discontinuous at $\tau=0$. In equation (\ref{eq:StringL2intGen}) the terms that cause discontinuity are divided by the lowest power of $\omega_{k}$. The other terms are continuous and add up to zero at $\tau=0$. Therefore, after using $D_{k}=D$ and $\omega_{k}=ck\pi$ the following identity holds: \begin{equation} \lim_{\tau\to0+}\left[\boldsymbol{L}^{1}(\tau)\right]_{2}=\lim_{\tau\to0+}\sum_{k=1}^{\infty}\kappa_{k}(\tau),\;\kappa_{k}(\tau)=\frac{e^{-k\pi cD\tau}\sin^{2}k\pi\xi^{\star}}{k\pi c\sqrt{1-D^{2}}}\sin\left(k\pi c\sqrt{1-D^{2}}\tau\right).\label{eq:C-L1lim} \end{equation} One can expand $\kappa_{k}(\tau)$ in (\ref{eq:C-L1lim}) as a sum of exponentials \begin{multline} \kappa_{k}(\tau)=\frac{1}{4k\pi c\sqrt{D^{2}-1}}\biggl(\mathrm{e}^{k\pi c\left(\sqrt{D^{2}-1}-D\right)\tau}-\mathrm{e}^{-k\pi c\left(\sqrt{D^{2}-1}+D\right)\tau}+\frac{1}{2}\mathrm{e}^{-k\pi\left(c\left(\sqrt{D^{2}-1}+D\right)\tau+2i\zeta\right)}\\ -\frac{1}{2}\mathrm{e}^{k\pi\left(c\left(\sqrt{D^{2}-1}-D\right)\tau-2i\zeta\right)}+\frac{1}{2}\mathrm{e}^{-k\pi\left(c\left(\sqrt{D^{2}-1}+D\right)\tau-2i\zeta\right)}-\frac{1}{2}\mathrm{e}^{k\pi\left(c\left(\sqrt{D^{2}-1}-D\right)\tau+2i\zeta\right)}\biggr). \end{multline} Since $\sum_{k=1}^{\infty}\frac{\mathrm{e}^{ka}}{k}=-\log(1-\mathrm{e}^{a})$, the limit (\ref{eq:C-L1lim}) can be written as \begin{equation} \lim_{\tau\to0+}\left[\boldsymbol{L}^{1}(\tau)\right]_{2}=\lim_{\tau\to0+}\frac{-1}{4\pi c\sqrt{D^{2}-1}}\left\{ \log(1-\mathrm{e}^{a_{1}})-\log(1-\mathrm{e}^{a_{2}})+\frac{1}{2}\sum_{l=3}^{6}(-1)^{l+1}\log(1-\mathrm{e}^{a_{l}})\right\} , \end{equation} where \begin{eqnarray} a_{1}=\pi c\left(\sqrt{D^{2}-1}-D\right)\tau, & a_{2}=-\pi c\left(\sqrt{D^{2}-1}+D\right)\tau,\\ a_{3,4}=\mp\pi\left(c\left(\sqrt{D^{2}-1}+D\right)\tau+2i\zeta\right), & a_{5,6}=\mp\pi\left(c\left(\sqrt{D^{2}-1}+D\right)\tau-2i\zeta\right), \end{eqnarray} Discontinuity occurs if the path of $1-\mathrm{e}^{a_{l}(\tau)}$
crosses the non-positive real axis (including zero) at $\tau=0$. This is possible for $a_{1}$ and $a_{2}$ only if $0<\xi<1$, $\xi\neq\nicefrac{1}{2}$. Since we are taking a limit, it is sufficient to use a first order approximation at $\tau=0$, that is, $1-\mathrm{e}^{a_{l}(\tau)}\approx-a_{l}(\tau)$. Also note that $\log x=\log\left|x\right|+i\arg x$ and that $\left|a_{1}\right|=\left|a_{2}\right|=\pi c\tau$. The limit therefore becomes \begin{equation} \lim_{\tau\to0+}\left[\boldsymbol{L}^{1}(\tau)\right]_{2}=\frac{-1}{4\pi c\sqrt{1-D^{2}}}\left(\arg\left(\sqrt{D^{2}-1}-D\right)-\arg\left(-\sqrt{D^{2}-1}-D\right)\right). \end{equation} Assuming that $D=\cos\phi$, $0\le\phi\le\nicefrac{\pi}{2}$, we get $\arg\left(\sqrt{D^{2}-1}-D\right)=\pi-\phi$ and $\arg\left(-\sqrt{D^{2}-1}-D\right)=\pi+\phi$, hence \begin{equation} \lim_{\tau\to0+}\left[\boldsymbol{L}^{1}(t)\right]_{2}=\frac{\cos^{-1}D}{2\pi c\sqrt{1-D^{2}}}. \end{equation}
\end{document} | arXiv |
\begin{document}
\newtheorem{pro}[thm]{Proposition} \newtheorem{lem}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \title{Small eigenvalues of large Hankel matrices} \author{Yang Chen and Nigel Lawrence\\ Department of Mathematics\\ Imperial College\\ 180 Queen's Gate\\ London, SW7 2BZ}
\maketitle
\begin{abstract} In this paper we investigate the smallest eigenvalue, denoted as $\lambda_N,$ of a $(N+1)\times (N+1)$ Hankel or moments matrix, associated with the weight, $w(x)=\exp(-x^{\beta}), \;x>0,\;\;\beta>0$, in the large $N$ limit. Using a previous result, the asymptotics for the polynomials, $P_n(z),\;z\notin[0,\infty)$, orthonormal with respect to $w,$ which are required in the determination of $\lambda_N$ are found. Adopting an argument of Szeg\"{o} the asymptotic behaviour of $\lambda_N$, for $\beta>1/2$ where the related moment problem is determinate, is derived. This generalises the result given by Szeg\"{o} for $\beta=1$. It is shown that for $\beta>1/2$ the smallest eigenvalue of the infinite Hankel matrix is zero, while for $0<\beta<1/2$ it is greater then a positive constant. This shows a phase transition in the corresponding Hermitian random matrix model as the parameter $\beta$ varies with $\beta=1/2$ identified as the critical point. The smallest eigenvalue at this point is conjectured.
\end{abstract}
\setcounter{section}{1} \setcounter{equation}{0} \setcounter{thm}{0}
{\bf 1. Introduction.}
In the theory of Hermitian random matrices, the Hankel determinant plays an important role, \begin{eqnarray} D_N=\det_{0\leq i,j\leq N}(\mu_{i+j})\;.\nonumber \end{eqnarray}
For a given weight function $w(t)$ on $J$ ($\subseteq R$,) the moments $\mu_k$ are \begin{eqnarray} \mu_k:=\int_Jw(t)t^{k}dt\;\; ;\;\;k=0,1,2,\cdots \end{eqnarray} Associated with $w(t)$ is a Hankel matrix or moment matrix of order $N+1$, $\{H_{jk}\},$ whose entries are given by \begin{eqnarray} H_{jk}:=\mu_{j+k}\;\;;\;\;0\leq j,k\leq N\;. \end{eqnarray} It is believed that correlations between eigenvalues of random matrices are universal after a suitable rescaling. In the following treatment we will show that a fundamental quantity, namely the least eigenvalues of these Hankel matrices exhibit a critical dependence on the weight function. It is this non-universal property that motivates our investigation of this problem.
If $J$ is a single interval say $[a,b]$, where $a$ and $b$ are fixed and the Szeg\"o condition, \begin{eqnarray} \int_{a}^{b}\frac{v(x)dx}{{\sqrt {(b-x)(x-a)}}}<\infty,\;\;v:=-\ln w,\nonumber \end{eqnarray} is satisfied then the asymptotic behaviour of the Hankel determinants for large $N$ was established by Szeg\"{o}, \cite{Sz3}. Let $\lambda_N$ denote the smallest eigenvalue. Szeg\"o also investigated the behaviour of $\lambda_N$ for large $N$ \cite{Sz}. He studied the cases for which $J$ can either be a finite or infinite interval with special choices for $w.$ If $w(x)=1,\;x\in(-1,1)$ and $w(x)=1,\;x\in(0,1), $ then the respective smallest eigenvalues are for large $N$ \footnote{ Throughout this paper, the relation, $a_N\simeq b_N$ means $\lim_{N\to\infty} a_N/b_N=1.$} \begin{eqnarray} \mbox{} \lambda_N&\simeq&2^{\frac{9}{4}}\pi^{\frac{3}{2}}N^{\frac{1}{2}} ({\sqrt 2}-1)^{2N+3} \nonumber\\ \mbox{} \lambda_N&\simeq&2^{\frac{15}{4}}\pi^{\frac{3}{2}}N^{\frac{1}{2}} ({\sqrt 2}-1)^{4N+4}. \nonumber \end{eqnarray} Widom and Wilf \cite{ww} generalised Szeg\"o's results to a kind of ``universal'' law. Thus if $w(x)>0,\;x\in[a,b]$ and the Szeg\"o condition is satisfied then it was found in \cite{ww} that \begin{eqnarray} \lambda_N\simeq A\;N^{\frac{1}{2}}\;B^{-N},\nonumber \end{eqnarray} where $A$ and $B$ are computable constants depending on $w$, $a$, $b$ and are independent of $N.$
In \cite{Sz}, Szeg\"o also considered the cases of infinite intervals where $w(x)={\rm exp}\left[-x^2\right],\;x\in(-\infty,+\infty)$ and $w(x)={\rm exp}\left[-x\right],\;x\in[0,+\infty),$ are the weights of the Hermite and Laguerre polynomials\footnote{There is a factor of 4 missing from the original formula for $\lambda_N$; the last equation in page 677 of \cite{Sz}.}. The respective smallest eigenvalues are \begin{eqnarray} \mbox{} \lambda_N&\simeq& 2^{\frac{13}{4}}\pi^{\frac{3}{2}}\;e\;N^{\frac{1}{4}} {\rm exp} \left[-2(2N)^{\frac{1}{2}}\right],\nonumber\\ \mbox{} \lambda_N&\simeq& 2^{\frac{7}{2}}\pi^{\frac{3}{2}}\;e\;N^{\frac{1}{4}} {\rm exp} \left[-4N^{\frac{1}{2}}\right].\nonumber \end{eqnarray} Observe that in the examples given above the smallest eigenvalues are exponentially small. Therefore it is very hard to numerically invert the Hankel matrices associated with these weights.
It is well known that $\lambda_N$ is given by the Rayleigh quotient: \begin{eqnarray} \lambda_N=\min\left\{\frac{\sum_{j,k=0}^N H_{jk}x_j\overline{x}_k}{\sum_{j=0}^N
|x_j|^2}\right\}\;. \end{eqnarray} If $\pi_N(z)$ is a polynomial of degree $N$, with coefficients $x_j,\;j=0,...,N$ \begin{eqnarray} \pi_N(z):=\sum_{j=0}^{N}x_jz^j\;, \end{eqnarray} then \begin{eqnarray}
\sum_{j,k=0}^N H_{jk}x_j\overline{x}_k=\int_J|\pi_N(t)|^2w(t)dt \end{eqnarray} and \begin{eqnarray}
\sum_{j=0}^N|x_j|^2=\int_0^{2\pi}|\pi_N(e^{i\phi})|^2\frac{d\phi}{2\pi}. \end{eqnarray} Consequently we can rephrase the extremal expression for $\lambda_N$, (1.3), as, \begin{eqnarray}
\frac{2\pi}{\lambda_N}=\max\left\{\int_0^{2\pi}|\pi_N(e^{i\phi})|^2d\phi\;:\;
\int_J|\pi_N(t)|^2w(t)dt=1\right\}\;. \end{eqnarray} Let $\{P_n(t)\}$ be the polynomials, orthonormal with respect to $w(t)$, then $\pi_N$ has the expansion, \begin{eqnarray} \pi_N(z)=\sum_{j=0}^{N}c_jP_j(z)\;. \end{eqnarray} Thus \begin{eqnarray}
\int_0^{2\pi}|\pi_N(e^{i\phi})|^2d\phi=\sum_{j,k=0}^N K_{jk}c_j\overline{c}_k\;, \end{eqnarray} where \begin{eqnarray} K_{jk}:=\int_0^{2\pi}P_j(z)\overline{P_k(z)}d\phi \;;\;z=e^{i\phi}\;. \end{eqnarray} Therefore (1.7) is equivalent to \begin{eqnarray} \frac{2\pi}{\lambda_N}=\max\left\{\sum_{j,k=0}^NK_{jk}c_j\overline{c}_k\;:\;
\sum_{j=0}^{N}|c_j|^2=1\right\}\;. \end{eqnarray} With the Schwarz inequality, which states that for all values of $j$ and $k$ \begin{eqnarray}
|K_{jk}|\leq K_{jj}^{\frac{1}{2}}K_{kk}^{\frac{1}{2}}\;,\nonumber \end{eqnarray} and Cauchy's inequality we obtain an upper bound of (1.11): \begin{eqnarray} \mbox{}\sum_{j,k=0}^NK_{jk}c_{j}\overline{c}_{k}&\leq&
\sum_{j,k=0}^N|K_{jk}||c_j||c_k|\\ \mbox{}&\leq&\sum_{j,k=0}^N K_{jj}^{\frac{1}{2}}K_{kk}^{\frac{1}{2}}
|c_{j}||c_{k}| \nonumber\\ \mbox{}&\leq&\left(\sum_{j=0}^N K_{jj}\right)
\left(\sum_{j=0}^N |c_{j}|^2\right)\nonumber\\ \mbox{}&=&\sum_{j=0}^N K_{jj}.\nonumber \end{eqnarray} Therefore a lower bound for the smallest eigenvalue $\lambda_N$ is given by \begin{eqnarray} \frac{2\pi}{\sum_{j=0}^{N}K_{jj}}\leq \lambda_N. \end{eqnarray}
This paper is organised as follows: In section 2, by adopting a previous result \cite{Ch:La}, we obtain the asymptotic formula for the polynomials orthonormal with respect to $w(t):=\exp[-t^{\beta}],\;\beta>1/2,$ which is then employed in sections 3 and 4 for the determination of the large $N$ behaviour of $\lambda_N.$ In these sections we show, following \cite{Sz}, by an appropriate choice of the vector $\{c_j\}$, that the lower bound given by (1.13) is in fact an asymptotic estimate for large $N$. By a simple application of Laplace's method, $\sum_{j=0}^{N}K_{jj}$ is estimated. Thus the asymptotic form of $\lambda_N$ follows. In order to test the accuracy of the theory, these results are checked against numerical calculations for various $\beta$ and $N$, which were obtained using the Jacobi rotation algorithm \cite{NR} to reduce the Hankel matrix to diagonal form. This is found in section 5.
\setcounter{section}{2} \setcounter{equation}{0} \setcounter{thm}{0}
{\bf 2. The weight } $w(t)=\exp[-t^{\beta}],\;t\in[0,\infty)$.
In this case, the moments are \begin{eqnarray} \mu_n=\frac{1}{\beta}\Gamma\left(\frac{n+1}{\beta}\right). \end{eqnarray} In order to find a lower bound for the smallest eigenvalue good knowledge is required of the associated orthonormal polynomials $\{P_N(z)\}$, for $N$ large and $z\notin(0,\infty).$ In \cite{Ch:La}, by applying the linear statistics formula for matrix ensembles together with the Heine's determinant representation, asymptotic forms for the polynomials with weight $w(t)={\rm exp}[-v(t)]$, where $v(t)$ is an arbitrary convex function supported on $[0,\infty)$, are derived. The zeros of these polynomials are supported on $(a,b)\subset {\bf R}$. Here $a=0$, whilst $b(N)$ follows from the condition that ensures that $P_N(t)$ has $N$ roots on $(a,b)$, one finds that \cite{Ch:La}, \begin{eqnarray} b(N;\beta)=CN^{\frac{1}{\beta}},\;\;{\rm where\;} C=C(\beta):=4\left[\frac{\Gamma^2(\beta)}{\Gamma(2\beta)}\right]^{\frac{1}{\beta}} N^{\frac{1}{\beta}}\;. \end{eqnarray} The normalised polynomials as $N\to\infty$ are found, using \cite{Ch:La}, to be \begin{eqnarray} P_N(t)\simeq\frac{(-1)^N}{\sqrt{2\pi b}}\frac{ \exp[-f(t)+(2N+1)\ln(\sqrt{{\zeta}}+\sqrt{1+{\zeta}})]}{[{\zeta}(1+{\zeta})]^{\frac{1}{4}}} ,\;\zeta:=-\frac{t}{b},\;\;t\notin[0,b]\;, \end{eqnarray} where $f$ is given by \begin{eqnarray} f(t):=\frac{\sqrt{t(t-b)}}{2\pi}\int_0^b\frac{dy}{y-t}\frac{y^{\beta}}{\sqrt{ y(b-y)}}\;,t\notin[0,b]. \end{eqnarray} From the definition and basic properties of the hypergeometric functions \cite{Gr:Ry}, \begin{eqnarray} \mbox{} f(t)&=&-\frac{N}{\beta-\frac{1}{2}}{\sqrt {{\zeta}(1+{\zeta})}} {\;_2F_1}\left(1,1-\beta;\frac{3}{2}-\beta;-{\zeta}\right) - \frac{(-t)^{\beta}}{2}\sec\pi\beta\\ \mbox{} &=&-\frac{N}{\beta}{\sqrt {\frac{{\zeta}}{1+{\zeta}}}} {\;_2F_1}\left(1,\frac{1}{2};\beta+1;\frac{1}{1+{\zeta}}\right)\;.\nonumber
\end{eqnarray} At this point note the dichotomy of the problem, the nature of the Hypergeometric function dictates that whilst the first representation is more convenient in the large $b$ limit, where $|{\zeta}|<<1$, it cannot be used when $\beta=n+\frac{1}{2},\;\;n=1,2,\dots$, necessitating the use of the second result of (2.5) in such instances.
Using the fact that \begin{eqnarray} \ln(\sqrt{{\zeta}}+\sqrt{1+{\zeta}})=\sqrt{{\zeta}}{\;_2F_1}\left(\frac{1}{2},\frac{1}{2}; \frac{3}{2};-{\zeta}\right)\;, \end{eqnarray} we find, \begin{eqnarray} (2N+1)\ln(\sqrt{{\zeta}}+\sqrt{1+{\zeta}})\simeq\frac{(-t)^{\beta}}{\sqrt{\pi}C^{\beta}} \sum_{r=0}^{E[\beta-\frac{1}{2}]}(-1)^ra_r{\zeta}^{r+\frac{1}{2}-\beta}\;, \end{eqnarray} where $E[n]$ denotes the integer part of $n$ and \begin{eqnarray} a_r:=\frac{\Gamma(r+\frac{1}{2})}{(r+\frac{1}{2})\Gamma(r+1)}\;. \end{eqnarray} So the asymptotic expression of the polynomials for $t\notin(0,\infty)$, is, \begin{eqnarray} P_N(t)\simeq\frac{(-1)^N{\zeta}^{\frac{1}{4}}}{\sqrt{-2\pi t}}\exp \left(-f(t)+\frac{(-t)^{\beta}}{C^{\beta}\sqrt{\pi}}\sum_{r=0}^{E[\beta-\frac{1}{2} ]}(-1)^ra_r{\zeta}^{r+\frac{1}{2}-\beta}\right)\;. \end{eqnarray} To make further progress we now consider separately the two possible cases, as identified above, for $\beta>1/2$.
\setcounter{section}{3} \setcounter{equation}{0} \setcounter{thm}{0}
{\bf 3.} $\beta\neq n+\frac{1}{2},\;\;n=1,2,3\cdots$
When $\beta\neq n+\frac{1}{2}$, we use the first form for $f(t)$ in equation (2.5). The series expansion for the function ${\;_2F_1}(
{\scriptstyle 1,1-\beta;\frac{3}{2}-\beta;-{\zeta}})$, valid for $|{\zeta}|<1$, is \begin{eqnarray} {\;_2F_1}\left(1,1-\beta;\frac{3}{2}-\beta;-{\zeta}\right)= \frac{\Gamma(\frac{3}{2}-\beta)}{\Gamma(1-\beta)}\sum_{r=0}^{\infty}(-1)^r \frac{\Gamma(1-\beta+r)}{\Gamma(\frac{3}{2}-\beta+r)}{\zeta}^r\;,
\end{eqnarray} whilst for $|{\zeta}|<1$, $\sqrt{1+{\zeta}}$ may be written as \begin{eqnarray} \sqrt{1+{\zeta}}=\frac{-1}{2\sqrt{\pi}}\sum_{r=0}^{\infty}(-1)^r \frac{\Gamma(r-\frac{1}{2})}{\Gamma(r+1)}{\zeta}^r\;. \end{eqnarray} With this noted, the expansion for $f(t)$ as ${\zeta}\to 0$ is \begin{eqnarray} \mbox{}f(t)&\simeq&-\frac{1}{2\sqrt{\pi}}\frac{\Gamma(\frac{1}{2}-\beta)} {\Gamma(1-\beta)}\left(\frac{-t}{C}\right)^{\beta} \sum_{r=0}^{E[\beta-\frac{1}{2}]}(-1)^rb_r{\zeta}^{r+\frac{1}{2}-\beta}\\ \mbox{}&-&\frac{(-t)^{\beta}}{2}\sec\pi\beta\;,\nonumber \end{eqnarray} where \begin{eqnarray} b_r:=\sum_{s=0}^r\frac{\Gamma(s-\frac{1}{2})\Gamma(1-\beta+r-s)} {\Gamma(s+1)\Gamma(\frac{3}{2}-\beta+r-s)}\;. \end{eqnarray} Recall that ${\zeta}=-tC^{-1}N^{-\frac{1}{\beta}}$, and by the use of equation (2.9) we have, \begin{eqnarray} \mbox{}P_N(t)&\simeq&\frac{(-1)^N}{\sqrt{2\pi}} (-tCN^{\frac{1}{\beta}})^{-\frac{1}{4}}\exp\left[\frac{(-t)^{\beta}}{2} \sec\pi\beta\right]\\ \mbox{}&\times&\exp\left[\frac{N^{1-\frac{1}{2\beta}}}{\sqrt{\pi C}}\sum_{r=0} ^{E[\beta-\frac{1}{2}]}(-1)^rA_r\frac{(-t)^{r+\frac{1}{2}}} {(CN^{\frac{1}{\beta}})^r}\right]\nonumber\;, \end{eqnarray} with \begin{eqnarray} A_r:=a_r+\frac{\Gamma(\frac{1}{2}-\beta)}{2\Gamma(1-\beta)}b_r\;. \end{eqnarray}
Note with $\beta=1$, we find $C=4$ and $A_0=4\sqrt\pi$ and consequently recover the classical result for the Laguerre polynomials due to Perron \cite{Sz2}, \begin{eqnarray} P_N(t)\simeq\frac{(-1)^N}{2\sqrt{\pi}}(-tN)^{-\frac{1}{4}} \exp\left[2\sqrt{-tN}+\frac{t}{2}\right]\;\;,\;t\notin[0,\infty),\; N\to\infty\;. \end{eqnarray}
With $P_N(t)$ having the form (3.5), where $A_0=\frac{4\sqrt{\pi}\beta}{2\beta-1}$ is positive for $\beta>1/2$, we observe that for sufficiently large $j$ and $k$ the dominant contributions to $K_{jk}$ are from the arc of the unit circle around $t=-1$. Thus by fixing an arbitrary positive number $\omega$ and confining ourselves to values of $j$ and $k$ satisfying \begin{eqnarray} N-\omega N^{\frac{1}{2\beta}}\leq j,k \leq N\;, \end{eqnarray} we have \begin{eqnarray} K_{jk}\simeq\int_{\pi-\varepsilon}^{\pi+\varepsilon}P_j \left(e^{i\phi}\right)P_k\left(e^{-i\phi}\right)d\phi\;.
\end{eqnarray} Using the substitution $\theta=\phi-\pi$ and expanding the integrand for $|\theta|<<1$ gives the following, \begin{eqnarray} \mbox{}K_{jk}&\simeq&\frac{(-1)^{j+k}}{2\pi\sqrt{C}}e^{\sec\pi\beta} N^{-\frac{1}{2\beta}}\\ \mbox{}&\times& \int_{-\varepsilon}^{\varepsilon}\exp\Biggl[ \frac{1}{\sqrt{\pi C}}\sum_{r=0}^{E[\beta-\frac{1}{2}]}(-1)^r\frac{A_r}{C^r} \Biggl[\left(1-\frac{(2r+1)^2\theta^2}{8}\right) \left(j^{1-\frac{1}{2\beta}-\frac{r}{\beta}}+ k^{1-\frac{1}{2\beta}-\frac{r}{\beta}}\right)\nonumber\\ \mbox{}&&\hspace{0.5in}
+\frac{(2r+1)i\theta}{2}\left(j^{1-\frac{1}{2\beta} -\frac{r}{\beta}}- k^{1-\frac{1}{2\beta}-\frac{r}{\beta}}\right)\Biggr] \Biggr]d\theta\;.\nonumber \end{eqnarray} Because $j^{1-\frac{1}{2\beta}+\frac{r}{\beta}}
-k^{1-\frac{1}{2\beta}-\frac{r}{\beta}}$ remains bounded in the range specified by (3.8) we can disregard the linear term in $\theta$ in the integrand. This integral can then be approximated by extending the range of integration to the real axis, which does not affect the asymptotic behaviour, as contributions from $(-\infty,-\varepsilon)$ and $(\varepsilon,\infty)$ are sub-dominant compared to those from $[-\varepsilon,\varepsilon]$ as $j,k\to\infty$. Therefore, \begin{eqnarray} \mbox{}K_{jk}&\simeq& \frac{(-1)^{j+k}}{(\pi C)^{\frac{1}{4}}}A_0^{-\frac{1}{2}}e^{\sec\pi\beta} N^{-\frac{1}{2}-\frac{1}{4\beta}}\\ \mbox{}&\times&\exp\left[\frac{1}{\sqrt{\pi C}}\sum_{r=0}^{E[\beta-\frac{1}{2}]} (-1)^r\frac{A_r}{C^r}\left(j^{1-\frac{1}{2\beta}-\frac{r}{\beta}}+ k^{1-\frac{1}{2\beta}-\frac{r}{\beta}}\right)\right]\;. \nonumber \end{eqnarray}
From (3.11), we see that when $j$ and $k$ are sufficiently large and satisfy (3.8), \begin{eqnarray} K_{jk}\simeq(-1)^{j+k}K_{jj}^{\frac{1}{2}}K_{kk}^{\frac{1}{2}}\;. \end{eqnarray} This is especially useful as it enables the determination of the large $N$ behaviour of $\lambda_N$. By choosing the vector $\{c_j\}$, as in \cite{Sz}, such that \begin{eqnarray} c_{j}=\cases{\sigma e^{i\pi j}K_{jj}^{\frac{1}{2}}\;\;&if $E[N-\omega N^{\frac{1}{2\beta}}]\leq j\leq N$ \cr 0&if $j<E[N-\omega N^{\frac{1}{2\beta}}]\;,$\cr} \end{eqnarray} where $\sigma$ is a positive number determined by the condition \begin{eqnarray}
\sum_{j=0}^N|c_j|^2=\sigma^2\sum_{j=E[N-\omega N^{\frac{1}{2\beta}}]}^N K_{jj}=1\;, \end{eqnarray} we find, using (3.12) and (3.14), that \begin{eqnarray} \mbox{}\sum_{j,k=0}^N K_{jk}c_j\overline{c}_k&=& \sum_{j,k=E[N-\omega N^{\frac{1}{2\beta}}]}^N \sigma^2e^{i\pi(j-k)}K_{jk}K_{jj}^{\frac{1}{2}}K_{kk}^{\frac{1}{2}}\\ \mbox{}&\simeq&\sigma^2\left[\sum_{j=E[N-\omega N^{\frac{1}{2\beta}}]}^N K_{jj}\right]^2\nonumber\\ \mbox{}&=&\sum_{j=E[N-\omega N^{\frac{1}{2\beta}}]}^N K_{jj}\nonumber\;. \end{eqnarray} Recalling equation (1.11), we see that since $\omega$ is arbitrarily large the asymptotic behaviour of the maximum, by virtue of the inequality (1.13), is well approximated by $\sum_{j=0}^N K_{jj}$. Therefore we have shown that \begin{eqnarray} \frac{2\pi}{\lambda_N}\simeq\sum_{j=0}^NK_{jj}\;. \end{eqnarray} The leading behaviour of this sum for large $N$ is in turn found by replacing the sum by an integral and by applying Laplace's method, which in this context may be stated as :
If for $x\in[a,b]$, the real continuous function $\phi(x)$ has as its maximum the value $\phi(b)$, then as $N\to\infty$ \begin{eqnarray} \int_a^b f(x)\exp[N\phi(x)]dx\simeq\frac{f(b)\exp[N\phi(b)]} {N\phi^{\prime}(b)}\;. \end{eqnarray}
A simple calculation gives the expression for $\lambda_N$, \begin{eqnarray} \mbox{}\frac{2\pi}{\lambda_N}&\simeq&\frac{1}{4}\pi^{-\frac{1}{4}}C^{\frac{1}{4} } A_0^{-\frac{1}{2}}e^{sec\pi\beta}N^{-\frac{1}{2}+\frac{1}{4\beta}}\\ \mbox{}&\times&\exp\left[\frac{2N^{1-\frac{1}{2\beta}}}{\sqrt{\pi C}} \sum_{r=0}^{E[\beta-\frac{1}{2}]}(-1)^r\frac{A_r}{C^r}N^{-\frac{r}{\beta}} \right]\;.\nonumber \end{eqnarray} Putting $\beta=1$, Szeg\"{o}'s classical result for the Laguerre weight is recovered: \begin{eqnarray} \frac{2\pi}{\lambda_N}\simeq 2^{-\frac{5}{2}}\pi^{-\frac{1}{2}}e^{-1} N^{-\frac{1}{4}}\exp[4\sqrt{N}]\;. \end{eqnarray} From (3.18) we see that the smallest eigenvalue is exponentially small for large $N$ and is zero for the corresponding infinite Hankel matrix.
\setcounter{section}{4} \setcounter{equation}{0} \setcounter{thm}{0}
{\bf 4. $\beta=n+\frac{1}{2}\;,\;n=1,2,\cdots $}
In this section we investigate the case where $\beta=n+\frac{1}{2}\;, \;n\geq 1$. Such cases, as was explained previously, require the second form of $f(t)$ in (2.5). To obtain the asymptotic expansion for $f(t)$, we first note the following result for the hypergeometric function :
If $\beta=n+\frac{1}{2}$ with $n=1,2,\dots$ then \begin{eqnarray} {\;_2F_1}\left(1,\frac{1}{2};\beta+1;x\right)=L_{\beta}\frac{(x-1)^{\beta-\frac{1}{2}}} {x^{\beta+\frac{1}{2}}}\left(\sqrt{x}\ln\left[\frac{1+\sqrt{x}}{1-\sqrt{x}} \right] +\sum_{r=1}^{\beta-\frac{1}{2}}\frac{1}{L_{r-\frac{1}{2}}} \left(\frac{x}{x-1}\right)^r\right)\;, \end{eqnarray} where $L_{r}$ is given by \begin{eqnarray} L_{r}:=\frac{r}{2\pi}C^{r}(r)\;. \end{eqnarray} This is easily be proved by using an inductive argument, noting the following version of Gauss' recursion relations \cite{Gr:Ry} \begin{eqnarray} \mbox{}{\;_2F_1}\left(1,\frac{1}{2};n+\frac{5}{2};z\right)&=& \frac{(n+\frac{3}{2})(z-1)}{(n+1)z}\left[ {\;_2F_1}\left(1,\frac{1}{2};n+\frac{3}{2};z\right)- {\;_2F_1}\left(1,\frac{1}{2};n+\frac{1}{2};z\right)\right]\\ \mbox{}&+&\frac{n(n+\frac{3}{2})}{(n+1)(n+\frac{1}{2})}{\;_2F_1}\left( 1,\frac{1}{2};n+\frac{3}{2};z\right)\;\nonumber \end{eqnarray} together with the fact that \begin{eqnarray} {\;_2F_1}\left(1,\frac{1}{2};\frac{5}{2};z\right)=\frac{3}{4}\frac{(z-1)} {z^{\frac{3}{2}}}\ln\left[\frac{1+\sqrt{z}}{1-\sqrt{z}}\right] +\frac{3}{2}z\;. \end{eqnarray} Therefore, \begin{eqnarray} f(t)=\frac{(-1)^{\beta+\frac{1}{2}}}{2\pi}(-t)^{\beta}\left( \ln\left[\frac{\sqrt{1+{\zeta}}+1}{\sqrt{1+{\zeta}}-1}\right] +\sqrt{1+{\zeta}}\sum_{r=1}^{\beta-\frac{1}{2}}(-1)^r \frac{{\zeta}^{-r}}{L_{r-\frac{1}{2}}} \right)\;. \end{eqnarray} Using (3.2), we find \begin{eqnarray} \mbox{}f(t)&\simeq&\frac{(-1)^{\beta+\frac{1}{2}}(-t)^{\beta}}{2\pi} \ln\left[\frac{4}{{\zeta}}\right]\\ \mbox{}&+&\frac{(-t)^{\beta}}{4\pi^{\frac{3}{2}}}\sum_{r=0}^{\beta-\frac{1}{2}}
(-1)^r\delta_{\beta-\frac{1}{2}-r}{\zeta}^{r+\frac{1}{2}-\beta}\;,\;|\zeta|<<1\nonumber \end{eqnarray} where \begin{eqnarray} \delta_r:=\sum_{s=1}^{\beta-\frac{1}{2}}\frac{\gamma_{s-r}}{L_{s-\frac{1}{2}}} \end{eqnarray} and \begin{eqnarray} \gamma_r:=\cases{\frac{\Gamma(r-1/2)}{\Gamma(r+1)}\;\;&if $r\geq0$\cr 0&if $r<0$.\cr} \end{eqnarray} Recalling ${\zeta} =-tC^{-1}N^{-\frac{1}{\beta}}$, the strong asymptotics of the polynomials for $t\notin[0,\infty)$ reads, \begin{eqnarray} \mbox{}P_N(t)&\simeq&\frac{(-1)^N}{\sqrt{2\pi}} (-tCN^{\frac{1}{\beta}})^{\frac{1}{4}}\exp\left[\frac{(-1)^{\beta-\frac{1}{2}} (-t)^{\beta}}{2\pi}\ln\left(\frac{4CN^{\frac{1}{\beta}}}{-t}\right)\right]\\ \mbox{}&\times&\exp\left[ \frac{N^{1-\frac{1}{2\beta}}}{\sqrt{\pi C}}\sum_{r=0}^{\beta-\frac{1}{2}} (-1)^rB_r\frac{(-t)^{r+\frac{1}{2}}}{(CN^{\frac{1}{\beta}})^r} \right]\;,\nonumber \end{eqnarray} where \begin{eqnarray} B_r:=a_r-\frac{L_{\beta}}{2\beta}\delta_{\beta-\frac{1}{2}-r}\;. \end{eqnarray} Note the appearance of the logarithm in exponential. Since $B_0=\frac{4\sqrt{\pi}\beta}{2\beta-1}>0$ and using an argument similar to that in the previous section, we see that in determining $K_{jk}$ the essential contribution comes from the arc in the vicinity of $t=-1$. As before restricting $j,k$ to the range given in (3.8), we have, \begin{eqnarray} K_{jk}\simeq\int_{-\varepsilon}^{\varepsilon}P_j(-e^{i\theta})P_k(-e^{-i\theta})d\theta\;.
\end{eqnarray} We expand the exponential in the integrand for $|\theta|<<1$, keeping terms up to second order and then extend the range of integration to the infinite interval. Because $j^{1-\frac{1}{2\beta}-\frac{r}{\beta}}-k^{1-\frac{1}{2\beta}-\frac{r}{\beta}}$ and $\ln(j/k)$ remain bounded in the range given by (3.8), we find \begin{eqnarray} \mbox{}K_{jk}&\simeq&\frac{(-1)^{j+k}}{(\pi C)^{\frac{1}{4}}} B_0^{-\frac{1}{2}}N^{-\frac{1}{2}-\frac{1}{4\beta}}(4CN^{\frac{1}{\beta}})^ {\frac{(-1)^{\beta-\frac{1}{2}}}{\pi}}\\ \mbox{}&\times&\exp\left[ \frac{1}{\sqrt{\pi C}}\sum_{r=0}^{\beta-\frac{1}{2}}(-1)^r\frac{B_r}{C^r} \left(j^{1-\frac{1}{2\beta}-\frac{r}{\beta}}+k^{1-\frac{1}{2\beta}-\frac{r}{\beta}} \right)\right]\;.\nonumber \end{eqnarray} Again note that for sufficiently large $j$ and $k$, satisfying (3.8), \begin{eqnarray} K_{jj}\simeq(-1)^{j+k}K_{jj}^{\frac{1}{2}}K_{kk}^{\frac{1}{2}}\;. \end{eqnarray} Repeating the argument of the previous section, it follows that \begin{eqnarray} \frac{2\pi}{\lambda_N}\simeq\int_0^N K_{jj}dj\;. \end{eqnarray} The leading term in the asymptotic expansion of this integral as $N\to\infty$ follows from an application of Laplace's method and is given by \begin{eqnarray} \mbox{}\frac{2\pi}{\lambda_N}&\simeq&\frac{1}{4}\pi^{-\frac{1}{4}} C^{\frac{1}{4}}B_0^{-\frac{1}{2}} N^{-\frac{1}{2}+\frac{1}{4\beta}}(4CN^{\frac{1}{\beta}}) ^{\frac{(-1)^{\beta-\frac{1}{2}}}{\pi}}\\ \mbox{}&\times&\exp\left[\frac{2N^{1-\frac{1}{2\beta}}}{\sqrt{\pi C}} \sum_{r=0}^{\beta-\frac{1}{2}}(-1)^r\frac{B_r}{C^r}N^{-\frac{r}{\beta}} \right]\;.\nonumber \end{eqnarray} Effectively $\exp[\sec\pi\beta]$ in (3.18) is replaced by $(4CN^{1/\beta})^{\frac{(-1)^{\beta-1/2}}{\pi}}$. Note the alternating nature of this additional factor depending on whether $\beta-1/2$ is odd or even. Again (4.15) shows that $\lim_{N\to\infty}\lambda_N=0.$ According to standard theory\cite{Ak}, the moment problem associated with $w(x),\;x\geq 0$ is indeterminate if \begin{eqnarray} \int_{0}^{\infty}\frac{v(x)}{\sqrt{x}(1+x)}dx<\infty.\nonumber \end{eqnarray} Therefore $\beta=1/2$ is special as it marks the transition point at which the moment problem becomes indeterminate. Assuming, the result given in (2.9) holds, we have \begin{eqnarray} P_N(t)\simeq\frac{(-1)^N}{2\pi}(-t)^{-\frac{1}{4}}N^{-\frac{1}{2}}\exp\left[ \frac{\sqrt{-t}}{\pi}\left(\ln\left[\frac{4\pi N}{\sqrt{-t}}\right] +1\right)\right],\;\;t\notin[0,\infty). \end{eqnarray} Again if we confine ourselves to the range where
$j$ and $k$ are sufficiently large to enable the use of the above asymptotic representation, we find that the major contributions to $K_{jk}$ are from the arc around $t=-1$. But, due to the behaviour of $P_N(t)$ with increasing $N$, it is quite clear that $|K_{jk}|$ decreases as $j,k\to\infty$, making an analysis analogous to that of the previous sections impossible.
It is however possible to obtain an approximate lower bound for the least eigenvalue, since (1.13) still holds. Applying the Christoffel-Darboux formula \cite{Sz2} and the result given in \cite{ch} for the large $N$ off diagonal recurrence coeeficients, we find, \begin{eqnarray} \mbox{}\sum_{j=0}^NK_{jj}&=&\int_{-\pi}^{\pi}\sum_{j=0}^N P_{j}(-e^{i\theta})P_{j}(-e^{-i\theta})d\theta\\ \mbox{}&\simeq&\pi^2N^2\int_{-\pi}^{\pi} \frac{P_{N}(-e^{i\theta})P_{N+1}(-e^{-i\theta})- P_{N}(-e^{-i\theta})P_{N+1}(-e^{i\theta})}{e^{i\theta}-e^{-i\theta}}d\theta\nonumber \end{eqnarray} Thus using Laplaces method, $$\int_{a}^{b}dx\;f(x){\rm exp}[N\phi(x)] \simeq f(c){\rm exp}[N\phi(c)]{\sqrt {\frac{2\pi}{-Ng^{\prime\prime}(c)}}}\;\;, \;\;{\rm as}\;\; N\to +\infty,$$ where $c\in(a,b)$ is the maximum of $\phi(x)$ for $x\in(a,b)$, gives \begin{eqnarray} \sum_{j=0}^NK_{jj}\simeq\frac{(4\pi Ne)^{2/\pi}}{4\sqrt{\ln(4\pi Ne)}}. \end{eqnarray} So at the point $\beta=1/2$ the smallest eigenvalue appears to decrease algebraically instead of exponentially.
\setcounter{section}{4} \setcounter{equation}{0} \setcounter{thm}{0}
{\bf 5. Numerical Results}
In this section we check the accuracy of our asymptotic expressions for the least eigenvalue of the the various Hankel matrices against numerical results. Due to the fact that the moment matrices in these cases are very ill conditioned becuase of the vast range in scale of the matrix elements, the Jacobi rotation algorithm \cite{NR}, proved far more stable than the more conventional techniques for numerically determining a small selection of the eigenvalues of large symmetric matrices such as the Lanczos procedure or Householder's method \cite{Go}. This appears to be an unusual phenomenon. Because of the behaviour of the matrix elements in these problems it is necessary to implement a multiple-precision package that allows floating point arithmetic of arbitrary precision. The library of sub-routines created by Brent\cite{Br} was employed to combat the effect of rounding errors in the numerical procedures.
For $0<\beta<1/2$, the corresponding moment problem becomes indeterminate \cite{Ak}, and as a consequence the sum \begin{eqnarray}
\sum_{j=0}^{\infty}|P_j(z)|^2,\nonumber \end{eqnarray} converges for every $z$ in every compact subset of the complex plane. Therefore \begin{eqnarray} \sum_{j=0}^{\infty}K_{jj}=\xi>0,\nonumber \end{eqnarray} and the smallest eigenvalue for the corresponding infinite Hankel is a positive constant bounded below by $2\pi/\xi$. Proof of the extention of the above statement to all indeterminate moment problems and other related topics can be found in \cite{bci}. The situation for $0<\beta<1/2$ is in contrast to the results for $\beta>1/2$ where (3.18) and (4.15), as confirmed by the numerics, show that the sum diverges - A fact that is also well-known from the standard theory when the moment problem is determinate \cite{Ak}. This separation of behaviour in the two regions is the phenomenon of phase transition alluded to earlier.
The comparison between the numerical values of $\lambda_n$ and those obtained from the theoretical expressions (3.18) and (4.15) is shown in table 1. and figure 1. below.
\begin{figure}
\caption{The percentage error of the theoretical values of $\lambda_N$ when compared to those obtained numerically, for various $\beta$.}
\end{figure}
\begin{table} \begin{small} \begin{center}
\begin{tabular}{|c|c|c|c|} \hline \arraycolsep=1cm
$\beta$ & $N$ & Numerical $\lambda_N$ & Theoretical $\lambda_N$ \\ \hline
$1 $
& 50 & $2.0948\times 10^{-10}$& $2.3695 \times 10^{-10}$\\ \hline
& 100 & $2.1079\times 10^{-15}$& $2.3006 \times 10^{-15}$\\ \hline
& 150 & $2.9551\times 10^{-19}$& $3.1743 \times 10^{-19}$\\ \hline
& 200 & $1.6387\times 10^{-22}$& $1.7437 \times 10^{-22}$\\ \hline
& 300 & $5.5215\times 10^{-28}$& $5.8090 \times 10^{-28}$\\ \hline $\frac{3}{2}$
& 50 & $6.4066\times 10^{-22}$& $6.8438 \times 10^{-22}$\\ \hline
& 100 & $6.2353\times 10^{-36}$& $6.5384 \times 10^{-36}$\\ \hline
& 150 & $9.9476\times 10^{-48}$& $1.0343 \times 10^{-47}$\\ \hline
& 200 & $2.8132\times 10^{-58}$& $2.9101 \times 10^{-58}$\\ \hline
& 300 & $4.6009\times 10^{-77}$& $4.7300 \times 10^{-77}$\\ \hline $\frac{7}{4}$
& 50 & $6.4483\times 10^{-27}$& $6.6844 \times 10^{-27}$\\ \hline
& 100 & $1.6976\times 10^{-45}$& $1.7424 \times 10^{-45}$\\ \hline
& 150 & $1.5193\times 10^{-61}$& $1.5525 \times 10^{-61}$\\ \hline
& 200 & $3.9265\times 10^{-76}$& $4.0009 \times 10^{-76}$\\ \hline
& 300 & $1.4844\times 10^{-102}$& $1.5074\times 10^{-102}$\\ \hline $2 $
& 50 & $2.7356\times 10^{-31}$& $2.5449 \times 10^{-31}$\\ \hline
& 100 & $3.8907\times 10^{-54}$& $3.6415 \times 10^{-54}$\\ \hline
& 150 & $2.9557\times 10^{-74}$& $2.7769 \times 10^{-74}$\\ \hline
& 200 & $8.9775\times 10^{-93}$& $8.4574 \times 10^{-93}$\\ \hline
& 300 & $9.5593\times 10^{-127}$&$9.0396 \times 10^{-127}$\\ \hline $\frac{5}{2}$
& 50 & $2.2384\times 10^{-38}$& $2.4010 \times 10^{-38}$\\ \hline
& 100 & $1.2580\times 10^{-68}$& $1.3288 \times 10^{-68}$\\ \hline
& 150 & $5.3195\times 10^{-96}$& $5.5789 \times 10^{-96}$\\ \hline
& 200 & $1.2155\times 10^{-121}$&$1.2691 \times 10^{-121}$\\ \hline
& 300 & $1.5236\times 10^{-169}$&$1.5819 \times 10^{-169}$\\ \hline \end{tabular} \end{center} \end{small} \caption{ Numerical and theoretical values of $\lambda_N$ for various $\beta$} \end{table}
\noindent e-mail: [email protected], [email protected]
\end{document} | arXiv |
\begin{definition}[Definition:Ordered Sum/General Definition]
Let $S_1, S_2, \ldots, S_n$ be tosets.
Then we define $T_n$ as the ordered sum of $S_1, S_2, \ldots, S_n$ as:
:$\forall n \in \N_{>0}: T_n = \begin {cases}
S_1 & : n = 1 \\
T_{n - 1} + S_n & : n > 1
\end {cases}$
\end{definition} | ProofWiki |
\begin{document}
\title{A mass-structured individual-based model of the chemostat:\\ convergence and simulation} \author{Fabien Campillo\thanks{INRIA\,, \texttt{[email protected]}} \and Coralie Fritsch\thanks{Montpellier 2 University and INRA/MIA\,, \texttt{[email protected]}\protect\\ Fabien Campillo and Coralie Fritsch are members of the MODEMIC joint INRA and INRIA project-team. MODEMIC Project-Team, INRA/INRIA, UMR MISTEA, 2 place Pierre Viala, 34060 Montpellier cedex 01, France.}}
\maketitle
\begin{abstract} We propose a model of chemostat where the bacterial population is individually-based, each bacterium is explicitly represented and has a mass evolving continuously over time. The substrate concentration is represented as a conventional ordinary differential equation. These two components are coupled with the bacterial consumption. Mechanisms acting on the bacteria are explicitly described (growth, division and up-take). Bacteria interact via consumption. We set the exact Monte Carlo simulation algorithm of this model and its mathematical representation as a stochastic process. We prove the convergence of this process to the solution of an integro-differential equation when the population size tends to infinity. Finally, we propose several numerical simulations. \paragraph{Keywords:} individually-based model (IBM), mass-structured chemostat model, large population asymptotics, integro-differential equation, ecological population model, Monte Carlo.
\paragraph{Mathematics Subject Classification (MSC2010):}
60J80, 60J85 (primary); 37N25, 92D25 (secondary).
\end{abstract}
\section{Introduction} \label{sec.intro}
The chemostat is a biotechnological process of continuous culture developed in the 50s \citep{monod1950a,novick1950a} and which is at the heart of several industrial applications as well as laboratory devices \citep{smith1995a}. Bioreactors operating under this mode are maintained under perfect mixing conditions and usually at large bacterial population sizes.
These features allow such processes to be modeled by ordinary (deterministic) differential systems since, in large populations and under certain conditions, demographic randomness can be neglected. Moreover, perfect mixing conditions permit us to neglect the spatial distribution and express these models in terms of mean concentration in the chemostat. In its simplest version, the chemostat model is expressed as a system of two coupled ordinary differential equations respectively for biomass and substrate concentrations \citep{smith1995a}. This approach extends to the case of several bacterial species and several substrates. The simplicity of such models makes possible the development of efficient tools for automatic control and the improvement of the associated biotechnological processes. However, it is increasingly necessary to develop models beyond the standard assumption of perfect mixing with a bacterial population possessing uniform characteristics. For this purpose, several paths are available which take into account the different sources of randomness or the structuring of the bacterial population and its discrete nature. All these aspects have been somewhat neglected in previous models.
In addition, the recent development of so-called ``omics'' approaches such as genomics and large-scale DNA sequencing technology are the basis of a renewed interest in chemostat techniques \citep{hoskisson2005a}. These bacterial cultures may also be considered as laboratory models to study selection phenomena and evolution in bacterial ecosystems.
Beyond classical models based on systems of (deterministic) ordinary differential equations (ODE) which neglect any structuring of the bacterial populations, have also appeared in the 60's and 70's bacterial growth models structured in size or mass based on integro-differential equations (IDE) \citep{fredrickson1967a,ramkrishna1979a}, see also the monograph \cite{ramkrishna2000a} on these so-called population balance equations for growth-frag\-men\-tation models.
Various research papers have been devoted to the stochastic modeling of the chemostat. \cite{crump1979a} propose a model of pure jump for the biomass growth coupled with a differential equation for the substrate evolution. \cite{stephanopoulos1979a} propose a model with randomly fluctuating dilution rate, hence the noise is rather of an environmental nature, whereas in the previous model it is rather demographic. \cite{grasman2005a} propose a chemostat model at three trophic levels where randomness appears only in the upper trophic level. \cite{imhof2005a} also propose a stochastic chemostat model but, as in previous models, the noise is simply ``added'' to the classical deterministic model. In contrast, in \cite{campillo2011chemostat} article the demographic noise emerges from a description of the dynamics at the microscopic level.
In recent years, many models for the evolution in chemostats have been proposed either using integro-differential equations \citep{diekmann-odo2005a,mirrahimi2012a,mirrahimi2012b} or individual-based models (IBM) \citep{champagnat2013a}.
There are also computer models, for example \cite{lee-min-woo2009a} propose an IBM structured in mass for a `` batch'' culture process. In this model, as in \citep{champagnat2013a} the dynamics of the substrate is described by deterministic differential equations. Indeed, the difference in scale between a bacterial cell and a substrate molecule guaranties that, at the scale of the bacterial population, the dynamics of the substrate can be correctly represented by the fluid limit model, while the dynamics of the bacterial population is discrete and random.
We focus here on an individual-based model (IBM) of the chemostat. In contrast to deterministic models with continuous variables, in IBMs all variables, or at least some of them, are stochastic and discrete. These models are generally cumbersome in terms of simulation and difficult to analyze mathematically, but they can be useful in accounting for phenomena inaccessible in earlier models. The majority of IBMs are described initially in natural languages with simple rules. From there they are described as computer models in terms of algorithms, it is this approach that is often termed IBM. Nonetheless, they can also be described mathematically using a Markov process. The advantage of this approach is to allow the mathematical analysis of the IBM. In particular, as we will see here, the convergence of the IBM to an integro-differential model can be demonstrated.
The latter approach has been developed in a series of papers: for a simple model of position \citep{fournier2004a}, for the evolution of trait structured population \citep{champagnat2006b}, which is then extended to take into account the age of individuals \citep{vietchitran2006a,vietchitran2008a}. More recently \citet{champagnat2013a} proposed a chemostat model with multiple resources where the bacterial population has a genetic trait subject to evolution.
In the context of a growth-fragmentation model, \cite{hatzis1995a} proposed an IBM, without substrate variables, and draw a parallel between this model and an integro-differential model.
In Section \ref{sec.model} we introduce the IBM where each individual in the bacterial population is explicitly represented by its mass. We describe the phenomenon which the model will take into account at a microscopic scale: individual cell growth, cell division, up-take (substrate and bacteria are constantly withdrawn from the chemostat vessel), as well as the individual consumption described as a coupling with the ordinary differential equation which models the dynamics of the substrate. Then we describe the associated exact Monte Carlo algorithm, noting that this algorithm is asynchronous in time, i.e. different events occur at random instants which are not predetermined.
In Section \ref{sec.notations} we introduce some notation, then in Section \ref{sec.processus.microscopique} we construct the stochastic process associated with the IBM as a Markov process with values in the space of finite measures over the state-space of masses.
In Section \ref{sec.convergence} we prove the convergence, in large population limit, of the IBM towards an integro-differential equation of the population-balance equation type \citep{fredrickson1967a,ramkrishna1979a,ramkrishna2000a} coupled with an equation for the dynamics of the substrate. Finally in Section \ref{sec.simulations} we present several numerical simulations.
\section{The model} \label{sec.model}
\subsection{Description of the dynamics}
We consider an \emph{individual-based model (IBM) structured in mass} where the bacterial population is represented as individuals growing in a perfectly mixed vessel of volume $V$ (l). Each individual is solely characterized by its mass $x\in\mathcal{X} \eqdef [0,m_{\textrm{\tiny\rm max}}]$, this model does not take into account spatialization. At time $t$ the system is characterized by the pair: \begin{align} \label{eq.xi}
(S_{t},\nu_{t}) \end{align} where \begin{enumerate}
\item $S_{t}$ is the \emph{substrate concentration} (mg/l) which is assumed to be uniform in the vessel;
\item $\nu_{t}$ is the \emph{bacterial population}, that is $N_{t}$ individuals and the mass of the individual number $i$ will be denoted $x^i_{t}$ (mg) for $i=1,\dots,N_{t}$. It will be convenient to represent the population $\{x^i_{t}\}_{i=1,\dots,N_{t}}$ at time $t$ as the following punctual measure: \begin{align} \label{eq.nu}
\nu_t(\rmd x)=\sum_{i=1}^{N_t}\delta_{x_t^i}(\rmd x)\,. \end{align} \end{enumerate}
The dynamics of the chemostat combines \emph{discrete evolutions}, cell division and bacterial up-take, as well as \emph{continuous evolutions}, the growth of each individual and the dynamics of the substrate. We now describe the four components of the dynamics, first the discrete ones and then the continuous ones which occur between the discrete ones.
\begin{enumerate}
\vskip0.8em \item{\textbf{Cell division}} -- \emph{Each individual of mass $x$ divides at rate $\lambda(s,x)$ into two individuals of respective masses $\alpha\,x$ and $(1-\alpha)\,x$: \begin{center}
\includegraphics[width=5cm]{fig_division1.pdf} \end{center} where $\alpha$ is distributed according to a given probability distribution $Q(\rmd\alpha)$ on $[0,1]$, and $s$ is the substrate concentration.}
For instance, the function $\lambda(s,x)$ does not depend on the substrate concentration $s$ and could be of the following form which will be used in the simulation presented in Section \ref{sec.simulations}: \begin{center}
\includegraphics[width=4cm]{fig_function_lambda.pdf} \end{center} Thus, below a certain mass $m_{\textrm{\tiny\rm div}}$ it is assumed that the cell cannot divide. There are models where the rate also depends on the concentration $s$, see for example \citep{daoutidis2002a,henson2003b}.
We suppose that the distribution $Q(\rmd\alpha)$ is symmetric with respect to $\frac{1}{2}$, i.e. $Q(\rmd\alpha)=Q(1-\rmd \alpha)$. It also may admit a density $Q(\rmd \alpha)=q(\alpha)\,\rmd \alpha$ with the same symmetry: \begin{center}
\includegraphics[width=4cm]{fig_division2.pdf} \end{center}
Thus, the division kernel of an individual of mass $x$ is $ K(x,\rmd y) = Q(\frac{1}{x}\,\rmd y)$ with support $[0,x]$. In the case of perfect mitosis, an individual of mass $x$ is divided into two individuals of masses $\frac x 2$ and then $Q(\rmd \alpha)=\delta_{1/2}(\rmd \alpha)$.
\emph{It is therefore assumed that, relative to their mass, the division kernel is the same for all individuals. This allows us to reduce the model to a single division kernel. More complex scenarios can also be investigated.}
\vskip0.8em \item{\textbf{Up-take}} -- \emph{Each individual is withdrawn from the chemostat at rate~$D$.} One places oneself in the framework of a \emph{perfect mixing} hypothesis, where individuals are uniformly distributed in the volume $V$ independently from their mass. During a time step $\delta$, a total volume of $D\,V\,\delta$ is withdrawn from the chemostat: \begin{center}
\includegraphics[width=10cm]{fig_removal_rate.pdf} \end{center} and therefore, if we assume that all individuals have the same volume considered as negligible, during this time interval $\delta$, an individual has a probability $D\,\delta$ to be withdrawn from the chemostat, $D$ is the dilution rate. This rate could possibly depend on the mass of the individual.
\end{enumerate} When the division of an individual occurs, the size of the population instantaneously jumps from $N_{t}$ to $N_{t}+1$; when an individual is withdrawn from the vessel, the size of the population jumps instantaneously from $N_{t}$ to $N_{t}-1$; between each discrete event the size $N_{t}$ remains constant and the chemostat evolves according to the following two continuous mechanisms: \begin{enumerate} \setcounter{enumi}{2}
\vskip0.8em \item {\textbf{Growth of each individual}} -- \emph{Each individual of mass $x$ growths at speed $\rho_{\textrm{\tiny\rm\!\! g}}(S_{t},x)$}: \begin{align} \label{eq.masses}
\dot x^i_{t}
=
\rho_{\textrm{\tiny\rm\!\! g}}(S_{t},x^i_{t})\,, \quad i=1,\dots,N_{t} \end{align} where $\rho_{\textrm{\tiny\rm\!\! g}}:\mathbb{R}^2_{+}\mapsto\mathbb{R}_{+}$ is given. For the simulation we will consider the following Gompertz model: \begin{align*}
\rho_{\textrm{\tiny\rm\!\! g}}(s,x) \eqdef r(s)\,\log\Big(\frac{m_{\textrm{\tiny\rm max}}}{x}\Big)\,x \end{align*} where the growth rate $r(s)$ depends on the substrate concentration according to the Monod kinetics: \begin{align*}
r(s)=r_{\textrm{\tiny\rm max}}\,\frac{s}{k_r+s} \end{align*} here $m_{\textrm{\tiny\rm max}}$ is the maximum weight that an individual can reach. In Section \ref{subsec.modeles.deterministes} we also present an example of a function $\rho_{\textrm{\tiny\rm\!\! g}}(s,x)$ linear in $x$ which will lead to the classical model of chemostat.
\vskip0.8em \item{\textbf{Dynamic of the substrate concentration}} -- \emph{The substrate concentration evolves according to the ordinary differential equation: \begin{align} \label{eq.substrat}
\dot S_t = \rho_{\textrm{\tiny\rm\!\! s}}(S_{t},\nu_{t}) \end{align}} where \begin{align*}
\rho_{\textrm{\tiny\rm\!\! s}} (s,\nu)
&\eqdef
D({\mathbf s}_{\textrm{\tiny\rm in}}-s)-k\, \mu(s,\nu)\,, \\
\mu(s,\nu)
&\eqdef
\frac{1}{V} \int_{\XX} \rho_{\textrm{\tiny\rm\!\! g}}(s,x)\,\nu(\rmd x)
=
\frac{1}{V} \sum_{i=1}^{N} \rho_{\textrm{\tiny\rm\!\! g}}(s,x^i) \end{align*} with $\nu=\sum_{i=1}^N \delta_{x^i}$; $D$ is the dilution rate (1/h), ${\mathbf s}_{\textrm{\tiny\rm in}}$ is the input concentration (mg/l), $k$ is the stoichiometric coefficient (inverse of the yield coefficient), and $V$ is the representative volume (l). Mass balance leads to Equation \eqref{eq.substrat} and the initial condition $S_{0}$ may be random. \end{enumerate}
To ensure the existence and uniqueness of solutions of the ordinary differential equations \eqref{eq.masses} and \eqref{eq.substrat}, we assume that application $\rho_{\textrm{\tiny\rm\!\! g}}(s,x)$ is Lipschitz continuous w.r.t. $s$ uniformly in $x$: \begin{align} \label{hyp.rhog.lipschitz}
\bigl|\rho_{\textrm{\tiny\rm\!\! g}}(s_1,x)-\rho_{\textrm{\tiny\rm\!\! g}}(s_2,x)\bigr|
\leq
k_g \,|s_1-s_2| \end{align} for all $s_1,\,s_2 \geq 0$ and all $x\in\XX$. It is further assumed that: \begin{align} \label{hyp.rhog.borne}
0
\leq
\rho_{\textrm{\tiny\rm\!\! g}}(s,x)
\leq
\bar g \end{align} for all $(s,x)\in\mathbb{R}_+ \times \mathcal{X}$, and that
in the absence of substrate the bacteria do not grow: \begin{align} \label{hyp.rhog.nulle.en.0}
\rho_{\textrm{\tiny\rm\!\! g}}(0,x)
=
0 \end{align} for all $x\in \mathcal{X}$. To ensure that the mass of a bacterium stays between $0$ and $m_{\textrm{\tiny\rm max}}$, it is finally assumed that: \begin{align} \label{hyp.rhog.nulle.en.x=mmax}
\rho_{\textrm{\tiny\rm\!\! g}}(s,m_{\textrm{\tiny\rm max}})
=
0 \end{align} for any $s\geq 0$.
\subsection{Algorithm}
\begin{algorithm} \begin{center} \begin{minipage}{14cm} \begin{algorithmic} \STATE $t\ot 0$ \STATE sample $(S_0,\nu_0=\sum_{i=1}^{N_{0}}\delta_{x^i_{t}})$ \WHILE {$t\leq t_{\textrm{\tiny\rm max}}$}
\STATE $N \ot \crochet{\nu_{t},1}$
\STATE $\tau \ot (\bar\lambda+D)\,N$
\STATE $\Delta t \sim {\textrm{\rm Exp}}(\tau)$
\STATE integrate the equations for the mass \eqref{eq.masses}
and the substrate \eqref{eq.substrat} over $[t,t+\Delta t]$
\STATE $t \ot t+\Delta t$
\STATE draw $x$ uniformly in $\{x^i_{t}\,;\,i=1,\dots,N_{t}\}$
\STATE $u\sim U[0,1]$
\IF {$u\leq \lambda(S_{t},x)/(\bar\lambda+D)$}
\STATE $\alpha \sim Q$
\STATE $\nu_{t} \ot \nu_{t} -\delta_{x}+\delta_{\alpha\,x}
+\delta_{(1-\alpha)\,x}$
\COMMENT{division}
\ELSIF{$u\leq (\lambda(S_{t},x)+D)/(\bar\lambda+D)$}
\STATE $\nu_{t} \ot \nu_{t} -\delta_{x}$
\COMMENT{up-take}
\ENDIF \ENDWHILE \end{algorithmic} \end{minipage} \end{center} \caption{\itshape ``Exact'' Monte Carlo simulation of the individual-based model: approximations only lie in the numerical integration of the ODEs and in the pseudo-random numbers generators.} \label{algo.ibm} \end{algorithm}
In the model described above, the division rate $\lambda(s,x)$ depends on the concentration of substrate $s$ and on the mass $x$ of each individual which continuously evolves according to the system of coupled ordinary differential equations \eqref{eq.masses} and \eqref{eq.substrat}, so to simulate the division of the cell we make use of a rejection sampling technique. It is assumed that there exists $\bar\lambda<\infty$ such that: \[
\lambda(s,x)\leq \bar\lambda \] hence an upper bound for the rate of event, division and up-take combined, at the population level is given by: \[
\tau \eqdef (\bar\lambda+D)\,N\,. \] At time $t+\Delta t$ with $\Delta t\sim {\textrm{\rm Exp}}(\tau)$, we determine if an event has occurred and what is its type by acceptance/rejection. To this end, the masses of the $N$ individuals and the substrate concentration evolve according to the coupled ODEs \eqref{eq.masses} and \eqref{eq.substrat}. Then we chose uniformly at random an individual within the population $\nu_{(t+\Delta t)^-}$, that is the population at time $t+\Delta t$ before any possible event, let $x_{(t+\Delta t)^-}$ denotes its mass, then: \begin{enumerate}
\item With probability: \[
\frac{\bar\lambda}{(\bar\lambda+D)} \] we determine if there has been division by acceptance/rejection: \begin{itemize} \item division occurs, that is: \begin{align} \label{eq.event.division}
\nu_{t+\Delta t}
=
\nu_{(t+\Delta t)^-}
-\delta_{x_{(t+\Delta t)^-}}
+\delta_{\alpha\,x_{(t+\Delta t)^-}}
+\delta_{(1-\alpha)\,x_{(t+\Delta t)^-}}
\qquad
\textrm{with }\alpha\sim Q \end{align} with probability $\lambda(S_{t},x_{(t+\Delta t)^-})/\bar\lambda$; \item no event occurs with probability $1-\lambda(S_{t},x_{(t+\Delta t)^-})/\bar\lambda$. \end{itemize} In conclusion, the event \eqref{eq.event.division} occurs with probability: \[
\frac{\lambda\bigl(S_{t},x_{(t+\Delta t)^-}\bigr)}{\bar\lambda}
\,\frac{\bar\lambda}{(\bar\lambda+D)}
=
\frac{\lambda\bigl(S_{t},x_{(t+\Delta t)^-}\bigr)}{(\bar\lambda+D)}\,. \]
\item With probability: \[
\frac{D}{(\bar\lambda+D)}
=
1-\frac{\bar\lambda}{(\bar\lambda+D)} \] the individual is withdrawn, that is: \begin{align} \label{eq.event.soutirage}
\nu_{t+\Delta t}
=
\nu_{(t+\Delta t)^-}
-\delta_{x_{(t+\Delta t)^-}} \end{align} \end{enumerate} Finally, the events and the associated probabilities are: \begin{itemize} \item division \eqref{eq.event.division} with probability $\lambda(S_{t},x_{(t+\Delta t)^-})/(\bar\lambda+D)$, \item up-take \eqref{eq.event.soutirage} with probability ${D}/{(\bar\lambda+D)}$ \end{itemize} and no event (rejection) with the remaining probability. The details are given in Algorithm~\ref{algo.ibm}.
Technically, the numbering of individuals is as follows: at the initial time individuals are numbered from $1$ to $N$, in case division the daughter cell $\alpha \,x$ keeps the index of the parent cell and the daughter cell $(1-\alpha)\,x$ takes the index $N+1$; in case of the up-take, the individual $N$ acquires the index of the withdrawn cell.
\section{Notations} \label{sec.notations}
Before proposing an explicit mathematical description of the process $(\nu_{t})_{t\geq 0}$ we introduce some notations.
\subsection{Punctual measures}
Notation \eqref{eq.nu} designating the bacterial population seems somewhat abstract but it will bridge the gap between the ``discrete'' -- counting punctual measures -- and the ``continuous'' -- continuous measures of the population densities -- in the context of the asymptotic large population analysis. Indeed for any measure $\nu(\rmd x)$ defined on $\mathbb{R}_{+}$ and any function $\varphi:\mathbb{R}_{+}\mapsto\mathbb{R}$, we define: \[
\crochet{\nu,\varphi}
\eqdef
\int_{\mathbb{R}_{+}}\varphi(x)\,\nu(\rmd x)\,. \] This notation is valid for continuous measures as well as for punctual measures $\nu_{t}(\rmd x)$ defined by \eqref{eq.nu}, in the latter case $\crochet{\nu_{t},\varphi} = \sum_{i=1}^{N_{t}} \varphi(x^i_{t})$.
Practically, this notation allows us to link to macroscopic quantities, e.g. at time $t$ the \emph{population size} is: \[
N_{t} = \crochet{\nu_{t},1} \] and the \emph{total biomass} is: \[
X_{t} \eqdef \crochet{\nu_{t},I}
= \sum_{i=1}^{N_{t}} x^i_{t} \] where $1(x)\equiv 1$ and $I(x)\equiv x$. Finally: \[
x\in\nu_{t}=\sum_{i=1}^{N_{t}}\delta_{x^i_{t}}(\rmd x) \] will denote any individual among $\{x^1_{t},\dots,x^{N_{t}}_{t}\}$.
The set of finite and positive measures on $\mathcal{X}$ is denoted $\mathcal{M}_{F}(\XX)$, and $\mathcal{M}(\XX)$ is the subset of punctual finite measures on $\mathcal{X}$: \[
\mathcal{M}(\XX)
\eqdef
\left\{
\sum_{i=1}^N \delta_{x^i} \;; \; N \in \mathbb{N}, \, x^i \in \mathcal{X}
\right\} \] where by convention $\sum_{i=1}^0\delta_{x^i}$ is the null measure.
\subsection{Growth flow}
Let: \[ \begin{array}{rrcl}
A_{t} &:
\mathbb{R}_{+}\times\MM(\XX)
&\xrightarrow[]{\mbox{}\hskip1em} &
\mathbb{R}_{+}\times\MM(\XX)
\\
&(s,\nu)&\xrightarrow[]{\mbox{}\hskip1em} &
A_{t} (s,\nu) \end{array} \] be the differential flow associated with the couple system of ODEs \eqref{eq.substrat}--\eqref{eq.masses} apart from any event (division or up-take), i.e.: \begin{align} \label{eq.flot.A}
A_{t}(s,\nu)
=
\Biggl(
A^0_{t}(s,\nu) \;,\; \sum_{i=1}^N\delta_{A^i_{t}(s,\nu)}
\Biggr)
\quad\textrm{with }\nu=\sum_{i=1}^N\delta_{x^i}
\,. \end{align} where $A^0_{t}(s,\nu)$ and $(A^i_{t}(s,\nu)\,;\,i=1,\dots,N)$ are the coupled solutions of \eqref{eq.substrat}--\eqref{eq.masses} taken at time $t$ from the initial condition $(s,\nu)$, that is: \begin{align*}
\frac{\rmd}{\rmd t}
A_{t}^0(s,\nu)
&= \rho_{\textrm{\tiny\rm\!\! s}}\Bigl(A_{t}^0(s,\nu),\sum_{i=1}^N \delta_{A_{t}^i(s,\nu)}\Bigr) \\
&= D\,({\mathbf s}_{\textrm{\tiny\rm in}}-A_{t}^0(s,\nu))
-\frac{k}{V} \sum_{i=1}^N \rho_{\textrm{\tiny\rm\!\! g}}(A_{t}^0(s,\nu),A_{t}^i(s,\nu))\,,
&
A_{0}^0(s,\nu)=s\,, \\[0.5em]
\frac{\rmd}{\rmd t}
A_{t}^i(s,\nu)
&=
\rho_{\textrm{\tiny\rm\!\! g}}(A_{t}^0(s,\nu),A_{t}^i(s,\nu))\,,
&
A_{0}^i(s,\nu)=x^i \end{align*} for $i=1,\dots,N$. Hence the flow $A_{t}(s,\nu)$ depends implicitly on the size $N=\crochet{\nu,1}$ of the population $\nu$.
The stochastic process $(\nu_{t})_{t\geq 0}$ features a jump dynamics (division and up-take) and follows the dynamics of the flow $A_{t}$ between the jumps. We can therefore generalize a well-known formula for the pure jump process: \begin{align} \label{eq.dyn.saut.flot}
\Phi(S_{t},\nu_{t})
&=
\Phi(A_{t}(S_{0},\nu_{0}))
+
\sum_{u\leq t}
\bigl[
\Phi(A_{t-u}(S_{u},\nu_{u}))-\Phi(A_{t-u}(S_{u},\nu_{\umoins}))
\bigr]
\,,\quad t\geq 0 \end{align} for any function $\Phi$ defined on $\mathbb{R}\times\MM(\XX)$.
The sum $ \sum_{u\leq t}$ contains only a finite number of terms as the process $(\nu_{t})_{t\geq 0}$ admits only a finite number of jumps over any finite time interval. Indeed, the number of jumps in the process $(\nu_{t})_{t\geq 0}$ is bounded by a linear birth and death process with \textit{per capita} birth rate $\bar\lambda$ and \textit{per capita} death rate $D$ \citep{allen2003a}.
\section{Microscopic process} \label{sec.processus.microscopique}
Let $(S_0,\nu_0)$ denote the initial condition of the process, it is a random variable with values in $\mathbb{R}_+ \times \mathcal{M}(\mathcal{X})$.
The equation \eqref{eq.dyn.saut.flot} includes information on the flow, i.e. the dynamics between the jumps, but no information on the jumps themselves. To obtain an explicit equation for $(S_t,\nu_t)_{t\geq 0}$ we introduce Poisson random measures which manage the incoming of new individuals by cell division on the one hand, and the withdrawal of individuals by up-take on the other. To this end we consider two punctual Poisson random measures $N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta)$ and $N_2(\mathrm{d} u, \mathrm{d} j)$ respectively defined on $\mathbb{R}_+ \times \mathbb{N}^* \times \mathcal{X} \times [0,1]$ and $\mathbb{R}_+ \times \mathbb{N}^*$ with respective intensity measures: \begin{align*}
n_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta)
&\eqdef
\bar\lambda\, \mathrm{d} u \,\Big(\sum_{k \geq 0}
\delta_k(\mathrm{d} j)\Big) \, Q(\mathrm{d} \alpha) \, \mathrm{d} \theta\,, \\
n_2(\mathrm{d} u, \mathrm{d} j)
&\eqdef
D\,\mathrm{d} u \,\Big(\sum_{k \geq 0} \delta_k(\mathrm{d} j)\Big)\,. \end{align*} Suppose that $N_1$, $N_{2}$, $S_{0}$ and $\nu_{0}$ are mutually independent. Let $(\mathcal{F}_t)_{t \geq 0}$ be the canonical filtration generated by $(S_0,\nu_0)$, $N_1$ and $N_2$. According to \eqref{eq.dyn.saut.flot}, for any function $\Phi$ defined on $\mathbb{R}\times\MM(\XX)$: \begin{align} \nonumber
&\Phi(S_{t},\nu_{t})
=
\Phi(A_{t}(S_{0},\nu_{0})) \\ \nonumber
&
\quad+
\iiiint\limits_{[0,t]\times\N^*\times[0,1]^2}
1_{\{j\leq N_{\umoins}\}} \,
1_{\{0\leq \theta \leq \lambda(S_{u},x_\umoins^j)/\bar\lambda\}}\,
\bigl[\Phi(A_{t-u}(S_{u},\nu_{\umoins}
-\delta_{x^j_\umoins}
+\delta_{\alpha\,x^j_\umoins}
+\delta_{(1-\alpha)\,x^j_\umoins}))
\\[-0.7em]&\qquad\qquad\qquad\qquad\qquad\qquad\qquad \nonumber
-\Phi(A_{t-u}(S_{u},\nu_{\umoins}))\bigr]\,
N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta) \\[0.2em] \label{eq.Phi.S.nu}
&\quad+
\iint\limits_{[0,t]\times\N^*}
1_{\{j\leq N_{\umoins}\}} \,
\bigl[\Phi(A_{t-u}(S_{u},\nu_{\umoins}-\delta_{x^j_\umoins}))
-\Phi(A_{t-u}(S_{u},\nu_{\umoins}))\bigr]
\, N_2(\mathrm{d} u, \mathrm{d} j)\,. \end{align} In particular, we obtain the following equation for the couple $(S_{t},\nu_{t})$: \begin{align} \nonumber
&(S_{t},\nu_{t})
=
A_{t}(S_{0},\nu_{0}) \\ \nonumber
&
\qquad+
\iiiint\limits_{[0,t]\times\N^*\times[0,1]^2}
1_{\{j\leq N_{\umoins}\}} \,
1_{\{0\leq \theta \leq \lambda(S_{u},x_\umoins^j)/\bar\lambda\}}\,
\bigl[A_{t-u}(S_{u},\nu_{\umoins}
-\delta_{x^j_\umoins}
+\delta_{\alpha\,x^j_\umoins}
+\delta_{(1-\alpha)\,x^j_\umoins}) \\[-0.7em] \nonumber &\qquad\qquad\qquad\qquad\qquad\qquad\qquad
-A_{t-u}(S_{u},\nu_{\umoins})\bigr]\,
N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta) \\[0.2em] \label{def.proc.S.nu}
&\qquad+
\iint\limits_{[0,t]\times\N^*}
1_{\{j\leq N_{\umoins}\}} \,
\big[A_{t-u}(S_{u},\nu_{\umoins}-\delta_{x^j_\umoins})
-A_{t-u}(S_{u},\nu_{\umoins})\big]
\, N_2(\mathrm{d} u, \mathrm{d} j)\,. \end{align}
From now on, we consider test functions $\Phi$ of the form: \[
\Phi(s,\nu)
=
F(s,\crochet{\nu,f}) \] with $F \in C^{1,1}(\mathbb{R}^+ \times \mathbb{R})$ and $f \in C^1(\mathcal{X})$.
\begin{lemma} \label{lem.gen.inf} For any $t>0$: \begin{align} \nonumber
&
F(S_t,\crochet{\nu_t, f})
=
F(S_0,\crochet{\nu_0, f}) \\ \nonumber
&\quad
+ \int_0^t \left[
\rho_{\textrm{\tiny\rm\!\! s}} (S_u,\nu_u) \,
\partial_s F\left(S_u, \, \left<\nu_u, f\right>\right)
+ \left< \nu_u , \, \rho_{\textrm{\tiny\rm\!\! g}}(S_u,.)\,f'\right> \,
\partial_x F\left(S_u, \, \left<\nu_u, f\right>\right)
\right] \mathrm{d} u \\ \nonumber
&\quad
+ \iiiint\limits_{[0,t]\times\N^*\times[0,1]^2}
1_{\{j\leq N_{\umoins}\}} \,
1_{\{0\leq \theta \leq \lambda(S_{u},x_\umoins^j)/
\bar\lambda\}}\,
\bigl[
F(S_u,\crochet{\nu_{\umoins}
-\delta_{x_\umoins^j}
+\delta_{\alpha \, x_\umoins^j}
+\delta_{(1-\alpha)\, x_\umoins^j}, f}
)
\\[-0.9em]
\nonumber
& \qquad\qquad\qquad\qquad\qquad\quad
\qquad\qquad\qquad\qquad\qquad
-
F(S_u, \crochet{\nu_{\umoins}, f})
\bigr]
\;
N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta) \\ \label{eq.F.f}
&\quad
+ \iint\limits_{[0,t]\times\N^*}
1_{\{j\leq N_{\umoins}\}} \,
\bigl[
F(S_u, \crochet{\nu_{\umoins}-\delta_{x_\umoins^j}, f})
-
F\left(S_u, \, \left<\nu_{\umoins}, f\right>\right)
\bigr] \, N_2(\mathrm{d} u, \mathrm{d} j). \end{align} \end{lemma}
\begin{proof} From \eqref{eq.Phi.S.nu}: \begin{align*}
&
\crochet{\nu_t, f}
=
\sum_{i=1}^{N_0} f(A_t^i(S_0,\nu_0)) \\
&\quad
+ \iiiint\limits_{[0,t]\times\N^*\times[0,1]^2}
1_{\{j\leq N_{\umoins}\}} \,
1_{\{0\leq \theta \leq \lambda(S_{u},x_{\umoins}^j)/\bar\lambda\}}\, \\[-0.5em]
& \qquad\qquad\qquad
\times\Bigl[
\sum_{i=1}^{N_{\umoins}+1}
f(A_{t-u}^i(S_u,
\nu_{\umoins}-\delta_{x_{\umoins}^j}
+\delta_{\alpha\,x_{\umoins}^j}
+\delta_{(1-\alpha)\,x_{\umoins}^j})) \\[-1em]
& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
-\sum_{i=1}^{N_{\umoins}}
f(A_{t-u}^i(S_u,\nu_{\umoins}))
\Bigl] \,
N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta) \\
&\quad
+ \iint\limits_{[0,t]\times\N^*}
1_{\{j\leq N_{\umoins}\}} \,
\Bigl[
\sum_{i=1}^{N_{\umoins}-1}
f(A_{t-u}^i(S_u,
\nu_{\umoins}-\delta_{x_{\umoins}^j}))
-
\sum_{i=1}^{N_{\umoins}} f(A_{t-u}^i(S_u,\nu_{\umoins}))
\Bigl] \, N_2(\mathrm{d} u, \mathrm{d} j). \end{align*} According to the chain rule formula, for any $\nu=\sum_{i=1}^{N}\delta_{x^i}$: \begin{align*}
f(A_{t-u}^i(s,\nu))
& =
f(x^i)
+
\int_u^t \rho_{\textrm{\tiny\rm\!\! g}}(A_{\tau-u}^0(s,\nu),A_{\tau-u}^i(s,\nu)) \, f'(A_{\tau-u}^i(s,\nu))
\,\mathrm{d} \tau \\
& =
f(x^i)
+
\int_u^t \varphi(A_{\tau-u}^0(s,\nu),A_{\tau-u}^i(s,\nu)) \,\mathrm{d} \tau \end{align*} for $i \leq N$, with: \begin{align*}
\varphi(s,x) \eqdef \rho_{\textrm{\tiny\rm\!\! g}}(s,x) \, f'(x)\,. \end{align*} Hence: \begin{align*}
&\crochet{\nu_t, f}
=
\crochet{\nu_0, f} \\
& \qquad
+ \iiiint\limits_{[0,t]\times\N^*\times[0,1]^2}
1_{\{j\leq N_\umoins\}} \,
1_{\{0\leq \theta \leq \lambda(S_{u},x_\umoins^j)/\bar\lambda\}} \\[-0.8em]
&\qquad\qquad\qquad\qquad\qquad
\times \,
\left[
f(\alpha \, x_\umoins^j)
+ f((1-\alpha) \, x_\umoins^j)
- f(x_\umoins^j)
\right] \,
N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta) \\
&\qquad
- \iint\limits_{[0,t]\times\N^*}
1_{\{j\leq N_\umoins\}} \, f(x_\umoins^j) \,
\, N_2(\mathrm{d} u, \mathrm{d} j)
+T_0+T_1+T_2 \end{align*} where: \begin{align*}
T_0
&\eqdef
\sum_{i=1}^{N_0} \int_0^t \varphi(A_\tau^0(S_0,\nu_0), A_\tau^i(S_0,\nu_0)) \, \rmd\tau \\[1em]
T_1
&\eqdef
\iiiint\limits_{[0,t]\times\N^*\times[0,1]^2}
1_{\{j\leq N_\umoins\}} \,
1_{\{0\leq \theta \leq \lambda(S_{u},x_\umoins^j)/\bar\lambda\}} \, \\[-0.7em]
&\qquad\qquad\qquad
\times\int_u^t \Bigl[
\sum_{i=1}^{N_\umoins+1}
\varphi(A_{\tau-u}^0(S_u,
\nu_\umoins-\delta_{x_\umoins^j}
+\delta_{\alpha \, x_\umoins^j}
+\delta_{(1-\alpha) \, x_\umoins^j}), \\[-1em]
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
A_{\tau-u}^i(S_u,\nu_\umoins
-\delta_{x_\umoins^j}
+\delta_{\alpha \, x_\umoins^j}
+\delta_{(1-\alpha) \, x_\umoins^j})) \\[-0.6em]
&\qquad\qquad\qquad\qquad\qquad\qquad
-\sum_{i=1}^{N_\umoins}
\varphi(A_{\tau-u}^0(S_u,\nu_\umoins),A_{\tau-u}^i(S_u,\nu_\umoins))
\Bigr]\,\mathrm{d} \tau \\
&\qquad\qquad\qquad
\times N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta) \\[1em]
T_2
&\eqdef
\iint\limits_{[0,t]\times\N^*} 1_{\{j\leq N_\umoins\}}
\; \int_u^t
\Bigl[ \sum_{i=1}^{N_\umoins-1}
\varphi(A_{\tau-u}^0(S_u,
\nu_\umoins-\delta_{x_\umoins^j}), A_{\tau-u}^i(S_u,
\nu_\umoins-\delta_{x_\umoins^j})) \\[-0.6em]
&\qquad\qquad\qquad\qquad\qquad\qquad
-\sum_{i=1}^{N_\umoins}
\varphi(A_{\tau-u}^0(S_u,\nu_\umoins),A_{\tau-u}^i(S_u,\nu_\umoins))
\Bigr] \, \mathrm{d} \tau
\; N_2(\mathrm{d} u, \mathrm{d} j). \end{align*} Fubini's theorem applied to $T_1$ and $T_2$ leads to: \begin{align*}
T_1
&=
\int_0^t \iiiint\limits_{[0,\tau]\times\N^*\times[0,1]^2}
1_{\{j\leq N_\umoins\}} \,
\times 1_{\{0\leq \theta \leq \lambda(S_{u},x_\umoins^j)/\bar\lambda\}} \, \\[-0.7em]
&\qquad\qquad\qquad
\times \Bigl[
\sum_{i=1}^{N_\umoins+1}
\varphi(A_{\tau-u}^0(S_u,
\nu_\umoins
-\delta_{x_\umoins^j}
+\delta_{\alpha \, x_\umoins^j}
+\delta_{(1-\alpha) \, x_\umoins^j}),
\\[-1em]
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
A_{\tau-u}^i(S_u,
\nu_\umoins-\delta_{x_\umoins^j}
+\delta_{\alpha \, x_\umoins^j}
+\delta_{(1-\alpha) \, x_\umoins^j})) \\[-0.7em]
&\qquad\qquad\qquad\qquad\qquad
-\sum_{i=1}^{N_\umoins}
\varphi(A_{\tau-u}^0(S_u,\nu_\umoins),A_{\tau-u}^i(S_u,\nu_\umoins))
\Bigr] \\
&\qquad\qquad\qquad
\times N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta) \;\mathrm{d} \tau \\[1em]
T_2
&\eqdef
\int_0^t \iint\limits_{[0,\tau]\times\N^*}
1_{\{j\leq N_\umoins\}} \,
\Bigl[
\sum_{i=1}^{N_\umoins-1}
\varphi(A_{\tau-u}^0(S_u,
\nu_\umoins-\delta_{x_\umoins^j}), A_{\tau-u}^i(S_u,
\nu_\umoins-\delta_{x_\umoins^j}))
\\[-1em]
& \qquad\qquad\qquad\qquad\qquad\qquad
-\sum_{i=1}^{N_\umoins}
\varphi(A_{\tau-u}^0(S_u,\nu_\umoins),A_{\tau-u}^i(S_u,\nu_\umoins))
\Bigr]
\, N_2(\mathrm{d} u, \mathrm{d} j)\; \mathrm{d} \tau \end{align*} so, according to \eqref{eq.Phi.S.nu}: \begin{align*}
T_0+T_1+T_2
&=
\int_0^t \crochet{\nu_\tau,\varphi(S_\tau,.)} \, \mathrm{d}\tau \,. \end{align*} Finally, \begin{align*}
\crochet{\nu_t,f}
&=
\crochet{\nu_0, f}
+
\int_0^t \crochet{\nu_u,\rho_{\textrm{\tiny\rm\!\! g}}(S_u,.)\,f'} \, \mathrm{d} u \\
&\quad
+ \iiiint\limits_{[0,t]\times\N^*\times[0,1]^2}
1_{\{j\leq N_\umoins\}} \,
1_{\{0\leq \theta \leq \lambda(S_{u},x_\umoins^j)/\bar\lambda\}}\,
\bigl[f(\alpha \,x_\umoins^j)+f((1-\alpha)\,x_\umoins^j)
-f(x_\umoins^j)\bigr] \\[-1em]
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\times N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta) \\
&\quad
- \iint\limits_{[0,t]\times\N^*}
1_{\{j\leq N_\umoins\}} \, f(x_\umoins^j) \,
\, N_2(\mathrm{d} u, \mathrm{d} j). \end{align*}
Since $f$ and $f'$ are continuous and bounded (bounded as defined on a compact set), we can conclude this proof by using the Itô formula for stochastic integrals with respect to Poisson random measures \citep{rudiger2006a} to develop the differential of $F(S_{t},\crochet{\nu_t,f})$ using Equation \eqref{eq.substrat} and the previous equation. \carre \end{proof}
Consider the compensated Poisson random measures associated with $N_1$ and $N_2$: \begin{align*}
\tilde N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} y, \mathrm{d} \theta)
&\eqdef
N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} y, \mathrm{d} \theta)
-
n_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} y, \mathrm{d} \theta)\,, \\
\tilde N_2(\mathrm{d} u, \mathrm{d} j)
&\eqdef
N_2(\mathrm{d} u, \mathrm{d} j)-n_2(\mathrm{d} u, \mathrm{d} j)\,. \end{align*}
As the integrands in the Poissonian integrals of \eqref{eq.F.f} are predictable, one can make use of the result of \citet[p. 62]{ikeda1981a}:
\begin{proposition} \label{prop.martingale} Let: \begin{align*}
M^1_t
&\eqdef
\iiiint\limits_{[0,t]\times\N^*\times[0,1]^2}
1_{\{j\leq N_\umoins\}} \,
1_{\{0\leq \theta \leq \lambda(S_{u},x_\umoins^j)/\bar\lambda\}} \\[-1em]
&\qquad\qquad\qquad\qquad
\times
\left[
F\left(S_u,\left<\nu_\umoins-\delta_{x_\umoins^j}
+\delta_{\alpha \, x_\umoins^j}+\delta_{(1-\alpha)\, x_\umoins^j}, f\right> \right)
-F\left(S_u, \, \left<\nu_\umoins, f\right>\right)
\right] \\
&\qquad\qquad\qquad\qquad
\times \tilde N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta)\,, \\
M^2_t
&\eqdef
\iint\limits_{[0,t]\times\N^*}
1_{\{j\leq N_\umoins\}} \,
\left[
F\left(S_u, \left<\nu_\umoins-\delta_{x_\umoins^j}, f\right>\right)
-
F\left(S_u, \, \left<\nu_\umoins, f\right>\right)
\right] \, \tilde N_2(\mathrm{d} u, \mathrm{d} j)\,. \end{align*} We have the following properties of martingales: \begin{enumerate}
\item if for any $t\geq 0$: \begin{align*}
&\mathbb{E}\Bigl(
\int_0^t \int_{\mathcal{X}} \lambda(S_{u}, x) \int_0^1
\bigl|
F(S_u,\crochet{\nu_u-\delta_x+\delta_{\alpha \, x}
+\delta_{(1-\alpha)\,x},f}) \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad
-F(S_u,\crochet{\nu_u,f})
\bigr| \, Q(\mathrm{d} \alpha)\, \nu_u(\mathrm{d} x)\,\mathrm{d} u
\Bigr) < + \infty \end{align*} then $(M^1_t)_{t\geq 0}$ is a martingale; \item if for any $t\geq 0$ \begin{align*}
\mathbb{E}\Bigl(
\int_0^t \int_{\mathcal{X}}
\bigl|
F(S_u,\crochet{\nu_u-\delta_x,f})
-
F(S_u,\crochet{\nu_u,f})
\bigr|
\, \nu_u(\mathrm{d} x)\,\mathrm{d} u
\Bigr) < +\infty \end{align*} then $(M^2_{t})_{t\geq 0}$ is a martingale;
\item if for any $t\geq 0$ \begin{align*}
&\mathbb{E}\Bigl(
\int_0^t \int_{\mathcal{X}} \lambda(S_{u}, x) \int_0^1
\bigl|
F(S_u,\crochet{
\nu_u-\delta_x+\delta_{\alpha \, x}+\delta_{(1-\alpha)\,x},f
})
\\
& \qquad\qquad\qquad\qquad\qquad\qquad\qquad
-
F(S_u,\crochet{\nu_u,f})
\bigr|^2 \,
Q(\mathrm{d} \alpha)\,\nu_u(\mathrm{d} x)\,\mathrm{d} u
\Bigr)
< + \infty \end{align*} then $(M^1_{t})_{t\geq 0}$ is a square integrable martingale and predictable quadratic variation: \begin{align*}
&\crochet{M^1}_t
\eqdef
\int_0^t \int_{\mathcal{X}} \lambda(S_{u}, x) \, \int_0^1
\bigl[
F(S_u,
\crochet{
\nu_u-\delta_x+\delta_{\alpha \, x}+\delta_{(1-\alpha)\,x}
,f})
\\
& \qquad\qquad\qquad\qquad\qquad\qquad\qquad
-
F(S_u,\crochet{\nu_u,f})
\bigr]^2 \,
Q(\mathrm{d} \alpha)\, \nu_u(\mathrm{d} x)\,\mathrm{d} u\,; \end{align*}
\item if for any $t\geq 0$ \begin{align*}
\mathbb{E}\Bigl(\int_0^t \int_{\mathcal{X}}
\left|
F(S_u,\crochet{\nu_u-\delta_x,f})
-
F(S_u,\crochet{\nu_u,f})
\right|^2 \, \nu_u(\mathrm{d} x)\,\mathrm{d} u
\Bigr) < + \infty \end{align*} then $(M^2_{t})_{t\geq 0}$ is a square integrable martingale and predictable quadratic variation: \begin{align*}
\left<M^2\right>_t
\eqdef
D \, \int_0^t \int_{\mathcal{X}}
\left[
F(S_u,\crochet{\nu_u-\delta_x,f})
-
F(S_u,\crochet{\nu_u,f})
\right]^2 \,
\nu_u(\mathrm{d} x)\,\mathrm{d} u\,. \end{align*} \end{enumerate} \end{proposition}
\begin{lemma}[Control of the population size] \label{lem.taille.pop} Let $T>0$, if there exists $p\geq 1$ such that $\mathbb{E}(\crochet{\nu_0,1}^p) <\infty$, then: \begin{align*}
\mathbb{E}\left(
\sup_{t \in [0,T]} \left< \nu_t,1 \right>^p
\right)
\leq C_{p,T} \end{align*} where $C_{p,T}<\infty $ depends only on $p$ and $T$. \end{lemma}
\begin{proof} For any $n \in \mathbb{N}$, define the following stopping time: \begin{align*}
\tau_n \eqdef \inf \{t \geq 0, \, N_t \geq n \}\,. \end{align*} Lemma \ref{lem.gen.inf} applied to $F(s,x)=x^p$ and $f(x)=1$ leads to: \begin{multline*}
\sup_{u\in [0,t \wedge \tau_n]} \crochet{\nu_u,1}^p
\leq
\crochet{\nu_0,1}^p \\
+ \iiiint\limits_{[0,t\wedge \tau_n]\times\N^*\times[0,1]^2}
1_{\{j\leq N_\umoins\}}\,
1_{\{0\leq \theta \leq \lambda(S_{u}, x_\umoins^j)/ \bar\lambda\}}\,
\bigl[
(\crochet{ \nu_\umoins,1}+1)^p-\crochet{\nu_\umoins,1}^p
\bigr] \\[-1em]
\times N_1(\mathrm{d} u, \mathrm{d} j,\mathrm{d} \alpha, \mathrm{d} \theta)\,. \end{multline*} From inequality $(1+y)^p-y^p\leq C_p\,(1+y^{p-1})$ we get: \begin{multline*}
\sup_{u\in [0,t \wedge \tau_n]} \crochet{\nu_u,1}^p
\leq
\crochet{\nu_0,1}^p \\
+ C_p\,\iiiint\limits_{[0,t\wedge \tau_n]\times\N^*\times[0,1]^2}
1_{\{j\leq N_\umoins\}}\,
1_{\{0\leq \theta \leq \lambda(S_{u}, x_\umoins^j)/ \bar\lambda\}}\,
\bigl[(1+\crochet{\nu_\umoins,1}^{p-1}\bigr] \\[-1em]
\times N_1(\mathrm{d} u, \mathrm{d} j,\mathrm{d} \alpha, \mathrm{d} \theta)\,. \end{multline*} Proposition \ref{prop.martingale}, together with the inequality $(1+y^{p-1})\,y \leq 2\,(1+y^p)$ give: \begin{align*}
\mathbb{E} \Biggl(
\sup_{u\in [0,t \wedge \tau_n]} \crochet{\nu_u,1}^p
\Biggr)
\leq
\mathbb{E}(\crochet{\nu_0,1}^p)
+ 2\,\bar\lambda\,C_p\,
\mathbb{E}\int_0^{t}
\bigl( 1+\crochet{ \nu_{u\wedge \tau_n},1}^p\bigr)\,\mathrm{d} u\,. \end{align*} Fubini's theorem and Gronwall's inequality allow us to conclude that for any $T< \infty$: \begin{align*}
\mathbb{E}\Biggl(
\sup_{t\in [0,T \wedge \tau_n]} \crochet{\nu_t,1}^p
\Biggr)
\leq
\Bigl(\mathbb{E}\bigl(\crochet{\nu_0,1}^p\bigr) + 2\,\bar\lambda\,C_p\,T \Bigr) \,
\exp(2\,\bar\lambda\,C_p\,T)
\leq
C_{p,T} \end{align*} where $C_{p,T}<\infty$ as $\mathbb{E}(\crochet{\nu_0,1}^p)<\infty$.
In addition, the sequence of stopping times $\tau_n$ tends to infinity, otherwise there would exist $T_0<\infty$ such that $\mathbb{P}(\sup_n \tau_n <T_0)=\varepsilon_{T_0}>0$ hence $\mathbb{E}(\sup_{t \in [0,T_0 \wedge \tau_n]} \crochet{\nu_t,1}^p)\geq \varepsilon_{T_0}\,n^p$ which contradicts the above inequality. Finally, Fatou's lemma gives: \begin{align*}
\mathbb{E}\Biggl( \sup_{t \in [0,T]} \crochet{\nu_t,1}^p \Biggr)
=
\mathbb{E} \Biggl(
\liminf_{n \to \infty}
\sup_{t\in [0,T \wedge \tau_n]}
\crochet{\nu_t,1}^p
\Biggr)
\leq
\liminf_{n \to \infty}
\mathbb{E} \Biggl(
\sup_{t\in [0,T \wedge \tau_n]} \crochet{\nu_t,1}^p
\Biggr)
\leq
C_{p,T}\,. \end{align*} \vskip-1em\carre \end{proof}
\begin{remark} In particular, if $\mathbb{E}\crochet{\nu_0,1}<\infty$ and if the function $ F $ is bounded, then by Lemma \ref{lem.taille.pop} and Proposition \ref{prop.martingale}, $(M^1_{t})_{t\geq 0}$ and $(M^2_{t})_{t\geq 0}$ are martingales. \end{remark}
\begin{lemma} \label{lem.integrabilite_Sigma} If $\mathbb{E}\crochet{\nu_0,1}+\mathbb{E}(S_0)<\infty$ then: \begin{align*}
\mathbb{E}\biggl(\int_0^t |\rho_{\textrm{\tiny\rm\!\! s}} (S_u,\nu_u) |\,\mathrm{d} u \biggr)
\leq
D\, t\,\mathbb{E} (S_0\vee {\mathbf s}_{\textrm{\tiny\rm in}})
+
\frac kV \,\bar{g}\;
\mathbb{E}\biggl( \int_0^t\crochet{\nu_u,1} \, \mathrm{d} u\biggr)
<\infty\,. \end{align*} \end{lemma}
\begin{proof} As $S_u\geq 0$ and $\rho_{\textrm{\tiny\rm\!\! g}}$ is a non negative function, \begin{align*}
\rho_{\textrm{\tiny\rm\!\! s}} (S_u,\nu_u)
\leq
D \, {\mathbf s}_{\textrm{\tiny\rm in}}\,. \end{align*} Furthermore, for any $(s,x)\in\mathbb{R}_+ \times \mathcal{X}$, $\rho_{\textrm{\tiny\rm\!\! g}}(s,x)\leq \bar g$, and $S_u\leq S_0\vee {\mathbf s}_{\textrm{\tiny\rm in}}$ so: \begin{align*}
\rho_{\textrm{\tiny\rm\!\! s}} (S_u,\nu_u)
\geq
-D \, (S_0\vee {\mathbf s}_{\textrm{\tiny\rm in}})-\frac kV \, \bar{g}\,\crochet{\nu_u,1}\,. \end{align*} We therefore deduce that: \begin{align*}
\int_0^t |\rho_{\textrm{\tiny\rm\!\! s}} (S_u,\nu_u) |\, \mathrm{d} u
\leq
D\,t\,(S_0\vee {\mathbf s}_{\textrm{\tiny\rm in}})
+
\frac kV \, \bar{g}\, \int_0^t \crochet{\nu_u,1}\, \mathrm{d} u\,. \end{align*} According to Lemma \ref{lem.taille.pop}, the last term is integrable which concludes the proof. \carre \end{proof}
\begin{theorem}[Infinitesimal generator] The process $(S_t,\nu_t)_{t\geq 0}$ is Markovian with values in $\mathbb{R}_{+}\times\MM(\XX)$ and its infinitesimal generator is: \begin{multline}
\mathcal{L} \Phi(s,\nu)
\eqdef
\bigl(D({\mathbf s}_{\textrm{\tiny\rm in}}-s)-k\,\mu(s,\nu) \bigr)\,
\partial_s F(s, \crochet{\nu, f})
+
\crochet{\nu ,\rho_{\textrm{\tiny\rm\!\! g}}(s,.) \, f'} \,
\partial_x F(s, \crochet{\nu, f}) \\
+
\int_{\mathcal{X}} \lambda(s,x) \,
\int_0^1
\bigl[
F(s,
\crochet{\nu-\delta_x+\delta_{\alpha\,x}
+
\delta_{(1-\alpha)\,x},f}
)
-
F(s, \crochet{\nu,f})
\bigr]
Q(\mathrm{d} \alpha) \, \nu(\mathrm{d} x) \\ \label{eq.generateur.infinitesimal}
+ D\, \int_{\mathcal{X}}
\bigl[
F(s,\crochet{\nu-\delta_x,f})
-
F(s,\left<\nu,f\right>) \bigr]\,
\, \nu(\mathrm{d} x) \end{multline} for any $\Phi(s,\nu)=F(s,\crochet{\nu,f})$ with $F \in C_b^{1,1}(\mathbb{R}^+ \times \mathbb{R})$ and $f \in C^{1}(\mathcal{X})$. Thereafter $\mathcal{L} \Phi(s,\nu)$ is denoted $\mathcal{L} F(s,\crochet{\nu,f})$. \end{theorem}
\begin{proof} Consider deterministic initial conditions $S_{0}=s\in\mathbb{R}_{+}$ and $\nu_{0}=\nu\in\MM(\XX)$. According to Lemma \ref{lem.gen.inf}: \begin{align*}
&
\mathbb{E}\bigl(F(S_t,\crochet{\nu_t, f})\bigr)
=
F(s, \, \left<\nu, f\right>)
+
\mathbb{E}\biggl(\int_0^t
\rho_{\textrm{\tiny\rm\!\! s}} (S_u,\nu_u) \,
\partial_s F(S_u, \, \crochet{\nu_u, f})\, \mathrm{d} u
\biggr) \\
&\
+\mathbb{E} \biggl(\int_0^t
\crochet{ \nu_u , \rho_{\textrm{\tiny\rm\!\! g}}(S_u,.)\,f'} \;
\partial_x F\left(S_u, \, \left<\nu_u, f\right>\right)\,
\mathrm{d} u
\biggr) \\
&\
+ \mathbb{E}\biggl(\ \
\iiiint\limits_{[0,t]\times\N^*\times[0,1]^2} \!\!\!\!
1_{\{j\leq N_\umoins\}} \,
1_{\{0\leq \theta \leq \lambda(S_{u}, x_\umoins^j)/
\bar\lambda\}}
\Bigl[
F\bigl(
S_u
,
\crochet{\nu_\umoins
-\delta_{x_\umoins^j}
+ \delta_{\alpha \, x_\umoins^j}
+ \delta_{(1-\alpha)\, x_\umoins^j}
, f}
\bigr)
\\[-1em]
&\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad
-
F\bigl(
S_u
,
\crochet{\nu_\umoins, f}\bigr) \Bigr]\,
N_1(\mathrm{d} u, \mathrm{d} j, \mathrm{d} \alpha, \mathrm{d} \theta)
\bigr) \\
&\
+ \mathbb{E}\biggl( \ \
\iint\limits_{[0,t]\times\N^*}
1_{\{j\leq N_\umoins\}} \,
\Bigl[
F(S_u, \crochet{\nu_\umoins-\delta_{x_\umoins^j}, f})
-F(S_u,\crochet{\nu_\umoins, f})
\Bigr]\,
N_2(\mathrm{d} u,\mathrm{d} j)
\biggr). \end{align*} As functions $F$, $\partial_sF$, $\partial_x F$, $f'$ and $\rho_{\textrm{\tiny\rm\!\! g}}$ are bounded, from Lemmas \ref{lem.taille.pop} and \ref{lem.integrabilite_Sigma} and Proposition \ref{prop.martingale}, the expectation the right side of the above equation are finite.
Furthermore, from Proposition \ref{prop.martingale}: \begin{align*}
&
\mathbb{E}\left(F\left(S_t, \, \left<\nu_t, f\right>\right)\right)
=
F(s,\crochet{\nu, f})
+
\mathbb{E} (\Psi(t)) \end{align*} where: \begin{align*}
& \Psi(t)
\eqdef
\int_0^t
\rho_{\textrm{\tiny\rm\!\! s}} (S_u,\nu_u) \,
\partial_s F(S_u, \crochet{\nu_u,f})\, \mathrm{d} u \\
&\quad
+
\int_0^t
\crochet{\nu_u,\rho_{\textrm{\tiny\rm\!\! g}}(S_u,.)\,f'} \;
\partial_x F(S_u, \crochet{\nu_u,f})\,
\mathrm{d} u \\
&\quad
+
\int_0^t \int_{\mathcal{X}} \int_0^1
\lambda(S_{u}, x)\,
\left[
F\left(
S_u
,
\crochet{\nu_u -\delta_{x}
+\delta_{\alpha\, x}
+\delta_{(1-\alpha)\,x}
\,,\,
f}
\right)
\right.
\\[-0.7em]
&\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad
\left.
- F(S_u,\crochet{\nu_u,f})
\right] \,
Q(\mathrm{d} \alpha)\,\nu_u(\mathrm{d} x) \, \mathrm{d} u \\
&\quad
+
D\,
\int_0^t \int_{\mathcal{X}}
\bigl[
F(S_u,\crochet{\nu_u-\delta_{x}, f}) - F(S_u,\crochet{\nu_u, f})
\bigr]\,
\nu_u(\mathrm{d} x) \, \mathrm{d} u
\,. \end{align*} Also: \begin{align*}
&
\frac{\partial}{\partial t}
\Psi(t)\Bigr|_{t=0}
=
\bigl(D({\mathbf s}_{\textrm{\tiny\rm in}}-s) - k\,\mu(s,\nu)\bigr) \,
\partial_s F(s, \crochet{\nu,f})
+ \crochet{\nu,\rho_{\textrm{\tiny\rm\!\! g}}(s,.)\,f'} \;
\partial_x F(s,\crochet{\nu,f}) \\
&\qquad
+ \int_{\mathcal{X}}\int_0^1
\lambda(s, x) \,
\Bigl[
F\left(
s
,
\crochet{\nu-\delta_x
+\delta_{\alpha \, x}+\delta_{(1-\alpha)\,x}, f
}
\right)
-
F(s,\crochet{\nu,f})
\Bigr]\,
Q(\mathrm{d}\alpha) \, \nu(\mathrm{d} x) \\
&\qquad
+ D\, \int_{\mathcal{X}}
\Bigl[
F(s,\crochet{\nu-\delta_x,f})
-
F(s,\crochet{\nu,f})
\Bigr]\,
\nu(\mathrm{d} x), \end{align*} hence: \begin{multline*}
\biggl|
\frac{\partial}{\partial t} \Psi(t)\Bigr|_{t=0}
\biggr|
\leq
D\,({\mathbf s}_{\textrm{\tiny\rm in}}+s)
+
\left(
\frac kV\,\bar g \, \norme{\partial_s F}
+ \bar g \, \norme{f'}\,\norme{\partial_x F}
+ 2\,(\bar\lambda+D)\, \norme{F}
\right) \,
\crochet{\nu,1} \, . \end{multline*} The right side of the last equation is finite. One may apply the theorem of differentiation under the integral sign, hence the application $t\mapsto \mathbb{E}(F(S_t, \crochet{\nu_t, f}))$ is differentiable at $t=0$ with derivative $\mathcal{L} F(s,\crochet{\nu, f})$ defined by \eqref{eq.generateur.infinitesimal}. \mbox{}\hfil\carre \end{proof}
\begin{remark} We define the washout time as the stopping time: \begin{align*}
\tau_{\textrm{\tiny\rm w}}
\eqdef
\inf\{t\geq 0\,;\, N_{t}=\crochet{\nu_{t},1}=0\} \end{align*} with the convention $\inf\emptyset=+\infty$. Before $\tau_{\textrm{\tiny\rm w}}$ the infinitesimal generator is given by \eqref{eq.generateur.infinitesimal}, after this time $\nu_{t}$ is the null measure, i.e. the chemostat does not contain any bacteria, and the infinitesimal generator is simply reduced to the generator associated with the ordinary differential equation $\dot S_{t}=D\,({\mathbf s}_{\textrm{\tiny\rm in}}-S_{t})$ coupled with the null measure given by $\crochet{\nu_{t},f}=0$ for all $f$. \end{remark}
\section{Convergence in distribution of the individual-based model} \label{sec.convergence}
\subsection{Renormalization}
In this section we will prove that the coupled process of the substrate concentration and the bacterial population converges in distribution to a deterministic process in the space: \[
\CC([0,T],\mathbb{R}_{+})
\times
\D([0,T],\MM_{F}(\mathcal{X})) \] equipped with the product metric: \fenumi\ the uniform norm on $\CC([0,T],\mathbb{R}_{+})$; \fenumii\ the Skorohod metric on $\D([0,T],\MM_{F}(\mathcal{X}))$ where $\MM_{F}(\mathcal{X})$ is equipped with the topology of the weak convergence of measures (see Appendix \ref{appendix.skorohod}).
Renormalization must have the effect that the density of the bacterial population must grow to infinity. To this end, we first consider a growing volume, i.e. in the previous model the volume is replaced by: \[
V_{n} = n\,V \] and $(S^n_{t},\nu^n_{t})_{t\geq 0}$ will denote the process \eqref{def.proc.S.nu} where $V$ is replaced by $V_{n}$ and
$x_t^{n,1},\dots,x_t^{n,N_t^n}$ the $N_t^n$ individuals of $\nu_t^n$; second
we introduce the rescaled process: \begin{align} \label{def.renormalisation}
\bar\nu_t^n \eqdef \frac{1}{n}\nu_t^n\,,\quad t\geq 0 \end{align} and we suppose that: \begin{align*}
\bar\nu_0^n
=
\frac{1}{n}\nu_0^n
\xrightarrow[n\to\infty]{}
\xi_0
\textrm{ in distribution in }\MM_F(\mathcal{X}) \,. \end{align*} $\xi_0$ is the limit measure after renormalization of the population density at the initial time. It may be random, but we will assume without loss of generality that it is deterministic, moreover we suppose that $\crochet{\xi_{0},1}>0$.
Therefore, this asymptotic consists in simultaneously letting the volume of chemostat and the size of the initial population tend to infinity.
As the substrate concentration is maintained at the same value, it implies that the population tends to infinity. We will show that the rescaled process $(S^n_t,\bar\nu^n_t)_{t\geq 0}$ defined by \eqref{def.renormalisation} converges in distribution to a process $(S_t,\xi_t)_{t\geq 0}$ introduced later.
The process $(S^n_t,\nu^n_t)_{t\geq 0}$ is defined by: \begin{align*}
\dot S_t^n
&=
D\,({\mathbf s}_{\textrm{\tiny\rm in}}-S_t^n)-\frac k{V_{n}} \,
\int_{X} \rho_{\textrm{\tiny\rm\!\! g}}(S_t^n,x)\,\nu_t^n(\mathrm{d} x) \\
&=
D({\mathbf s}_{\textrm{\tiny\rm in}}-S_t^n)-\frac {k}{V} \,
\int_{X} \rho_{\textrm{\tiny\rm\!\! g}}(S_t^n,x)\,\bar\nu_t^n(\mathrm{d} x)
=
\rho_{\textrm{\tiny\rm\!\! s}}(S_{t}^n,\bar\nu_t^n) \end{align*} and \begin{align*}
\bar\nu_t^n
&=
\frac 1n \, \sum_{j=1}^{n} \delta_{A_t^j(S_0^n,\nu_0^n)} \\
&\qquad
+ \frac 1n \,
\iiiint\limits_{[0,t]\times\N^*\times[0,1]^2}
1_{\{i\leq N^n_\umoins\}} \,
1_{
\{0\leq \theta \leq
\lambda(S^n_u, x_\umoins^{n,i})/\bar\lambda\}
}\,
\Bigl[
-\sum_{j=1}^{N^n_\umoins}
\delta_{A_{t-u}^j(S^n_u,\nu^n_\umoins)} \\[-1em]
&\qquad\qquad\qquad\qquad\qquad
+\sum_{j=1}^{N^n_\umoins+1}
\delta_{
A_{t-u}^j(S^n_u \, ,\,
\nu^n_\umoins-\delta_{x_\umoins^{n,i}}
+\delta_{\alpha x_\umoins^{n,i}}
+\delta_{(1-\alpha) x_\umoins^{n,i}})
}
\Bigr] \;
N_1(\mathrm{d} u, \mathrm{d} i, \mathrm{d} \alpha, \mathrm{d} \theta) \\[0.5em]
&\qquad
+ \frac 1n \,
\iint\limits_{[0,t]\times\N^*}
1_{\{i\leq N^n_\umoins\}} \,
\Bigl[
-\sum_{j=1}^{N^n_\umoins}
\delta_{A_{t-u}^j(S^n_u,\nu^n_\umoins)}
+\sum_{j=1}^{N^n_\umoins-1}
\delta_{A_{t-u}^j(S^n_u,
\nu^n_\umoins-\delta_{x_\umoins^{n,i}})}
\Bigr]
\, N_2(\mathrm{d} u, \mathrm{d} i) \end{align*}
\begin{remark} \label{remark.cv.nu.seule} Due to the structure of the previous system and specifically the above equation, it will be sufficient to prove the convergence in distribution of the component $\bar\nu_t^n$ to deduce also the convergence of the component $S_{t}^n$. \end{remark}
\subsection{Preliminary results}
\begin{lemma} \label{lem.renorm} For all $t>0$, \begin{align*}
&
F(S_t^n,\crochet{\bar\nu_t^n, f})
=
F(S_0^n,\crochet{\bar\nu_0^n, f}) \\
&\qquad
+
\int_0^t \textstyle\Bigl(
D\,({\mathbf s}_{\textrm{\tiny\rm in}}-S_u^n)
- \frac {k}{V}\,\int_X \rho_{\textrm{\tiny\rm\!\! g}}(S_u^n,x)\,\bar\nu_u^n(\mathrm{d} x)
\Bigr)
\;
\partial_s F(S_u^n, \, \crochet{\bar\nu_u^n, f})\,\mathrm{d} u \\
&\qquad
+ \int_0^t
\crochet{\bar\nu_u^n , \rho_{\textrm{\tiny\rm\!\! g}}(S_u^n,.)\,f'} \;
\partial_x F(S_u^n, \, \crochet{\bar\nu_u^n, f}) \, \mathrm{d} u \\
&\qquad
+ n\,\int_0^t \int_{\mathcal{X}} \lambda(S^n_u, x)\,
\int_0^1
\textstyle
\Bigl[
F\bigl(
S_u^n
\,,\,
\crochet{\bar\nu_u^n,f} +\frac 1n f(\alpha \, x )
+\frac 1nf((1-\alpha) \, x)-\frac 1nf(x)
\bigr) \\[-0.5em]
& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
-F\bigl(S_u^n, \crochet{\bar\nu_u^n, f}\bigr)
\Bigr]
\,Q(\mathrm{d} \alpha)\, \bar\nu_u^n(\mathrm{d} x)\,\mathrm{d} u \\
&\qquad
+ D\,n\,
\int_0^t \int_{\mathcal{X}}
\textstyle\Bigl[
F\bigl(S_u^n, \, \crochet{\bar\nu_u^n, f}-\frac 1nf(x) \bigr)
-
F\bigl(S_u^n, \, \crochet{\bar\nu_u^n, f} \bigr)
\Bigr]\, \bar\nu_u^n(\mathrm{d} x)\,\mathrm{d} u
+ Z_t^{F,f,n} \end{align*} where \begin{align} \nonumber
Z_t^{F,f,n}
&\eqdef
\iiiint\limits_{[0,t]\times\N^*\times[0,1]^2}
1_{\{i\leq N_\umoins^n\}} \,
1_{\{0\leq \theta \leq \lambda(S^n_u, x_\umoins^{n,i})/\bar\lambda\}} \\[-1em] \nonumber
&\qquad\qquad\qquad\qquad
\textstyle
\Bigl[
F\bigl(
S_u^n
\,,\,
\crochet{\bar\nu_\umoins^n, f}
+ \frac 1n f(\alpha \,x_\umoins^{n,i})
+\frac 1nf((1-\alpha)\,x_\umoins^{n,i})
-\frac 1nf(x_\umoins^{n,i})
\bigr) \, \\[-0.5em] \nonumber
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
-F\bigl(S_u^n, \crochet{\bar\nu_\umoins^n, f}\bigr)
\Bigr]
\,\tilde N_1(\mathrm{d} u, \mathrm{d} i, \mathrm{d} \alpha, \mathrm{d} \theta) \\ \label{eq.Z,F,f}
& \qquad
+ \iint\limits_{[0,t]\times\N^*}
\textstyle
1_{\{i\leq N_\umoins^n\}} \,
\Bigl[
F\bigl(S_u^n, \crochet{\bar\nu_\umoins^n, f}
-\frac 1nf(x_\umoins^{n,i}) \bigr)
-
F\bigl(S_u^n, \crochet{\bar\nu_\umoins^n, f}\bigr)
\Bigr]
\, \tilde N_2(\mathrm{d} u, \mathrm{d} i) \end{align} \end{lemma}
\begin{proof} It is sufficient to note that $ F(S_t^n,\crochet{ \bar\nu_t^n,f })
= F(S_t^n,\crochet{\nu_t^n,\frac{1}{n}f}) $ and to apply Lemma \ref{lem.gen.inf}. \carre \end{proof}
\begin{lemma} \label{lem.esp} If $\sup_{n\in\N} \mathbb{E}(\crochet{\bar\nu_0^n,1}^p)<\infty$ for some $p \geq 1$, then: \[
\sup_{n\in\N} \mathbb{E}\Biggl(\sup_{u\in [0,t]} \crochet{\bar\nu_u^n,1}^p\Biggr)
<
C_{t,p} \] where $C_{t,p}$ depends only on $t$ and $p$. \end{lemma}
\begin{proof} Define the stopping time: \[
\tau_N^n
\eqdef
\inf \{t\geq 0, \crochet{\bar\nu_t^n,1} \geq N \}\,. \] According to Lemma \ref{lem.renorm}: \begin{align*}
&
\sup_{u \in [0, t \wedge \tau_N^n]} \crochet{\bar\nu_u^n,1}^p
\leq
\crochet{\bar\nu_0^n,1}^p \\
&\qquad\qquad
+ n\,\int_0^{t \wedge \tau_N^n} \int_{\mathcal{X}}
\lambda(S^n_u, x)\,
\textstyle
\left[\left(\crochet{ \bar\nu_u^n,1} + \frac{1}{n}\right)^p
-\crochet{ \bar\nu_u^n,1}^p \right]\,
\bar\nu_u(\mathrm{d} x)\,\mathrm{d} u \\
& \qquad\qquad
+ \iiiint\limits_{[0,t\wedge \tau_N^n]\times\N^*\times[0,1]^2}
1_{\{i \leq N_\umoins^n\}}\,
1_{\{0\leq \theta \leq \lambda(S^n_u, x_\umoins^{i,n})/ \bar\lambda\}} \\[-1em]
& \qquad\qquad\qquad\qquad\qquad\qquad\qquad
\times \,
\textstyle
\left[\left(\crochet{\bar\nu_\umoins^n,1} + \frac{1}{n}\right)^p
-\crochet{\bar\nu_\umoins^n,1}^p \right]\,
\tilde N_1(\mathrm{d} u, \mathrm{d} i,\mathrm{d} \alpha, \mathrm{d} \theta)\,. \end{align*} From the inequality $(1+y)^p-y^p \leq C_p\, (1+y^{p-1})$, one can easily check that $(\frac 1n+y)^p-y^p \leq \frac{C_p}{n}\, (1+y^{p-1})$. Taking expectation in the previous inequality and applying Proposition \ref{prop.martingale} lead to: \begin{align*}
\mathbb{E} \Biggl(
\sup_{u \in [0, t \wedge \tau_N^n]} \crochet{\bar\nu_u^n,1}^p
\Biggr)
&\leq
\mathbb{E}\bigl(\crochet{ \bar\nu_0^n,1}^p \bigr)
+
\mathbb{E} \int_0^{t \wedge \tau_N^n}
C_p\,
\bigl(1+\crochet{\bar\nu_u^n,1}^{p-1}\bigr)
\int_\mathcal{X} \lambda(S^n_u, x)\,\bar\nu_u^n(\mathrm{d} x)\, \mathrm{d} u \\
&\leq
\mathbb{E}\bigl(\crochet{\bar\nu_0^n,1 }^p \bigr)
+
\bar\lambda \, C_p\,
\int_0^t \mathbb{E} \left(
\crochet{ \bar\nu_{u \wedge \tau_N^n}^n,1 }
+
\crochet{\bar\nu_{u \wedge \tau_N^n}^n,1}^p
\right) \, \mathrm{d} u\,. \end{align*} As: \[
\crochet{ \bar\nu_{u \wedge \tau_N^n}^n,1}
+
\crochet{ \bar\nu_{u \wedge \tau_N^n}^n,1}^p
\leq
2 \, \left(1+ \crochet{\bar\nu_{u \wedge \tau_N^n}^n,1}^p \right)\,, \] we get: \begin{align*}
\mathbb{E} \Biggl(
\sup_{u \in [0, t \wedge \tau_N^n]} \crochet{\bar\nu_u^n,1}^p
\Biggr)
&\leq
\mathbb{E}\left(\crochet{\bar\nu_0^n,1}^p \right)
+ 2\, \bar\lambda \, C_p\, t
+ 2\, \bar\lambda \, C_p\, \int_0^t
\mathbb{E} \left(\sup_{u \in [0, u \wedge \tau_N^n]}
\crochet{\bar\nu_u^n,1}^p \right) \, \mathrm{d} u \end{align*} and from Gronwall's inequality we obtain: \begin{align*}
\mathbb{E} \Biggl(
\sup_{u \in [0, t \wedge \tau_N^n]} \crochet{\bar\nu_u^n,1}^p
\Biggr)
\leq
\Bigl(
\mathbb{E}(\crochet{\bar\nu_0^n,1}^p ) + 2\,\bar\lambda \, C_p\, t
\Bigr)
\,
\exp(2\,\bar\lambda \, C_p\, t)\,. \end{align*} The sequence of stopping times $\tau_N^n$ tends to infinity as $N$ tends to infinity for the same reasons as those set in the proof of Lemma \ref{lem.taille.pop}. From Fatou's lemma we deduce: \begin{align*}
\mathbb{E} \Biggl(
\sup_{u \in [0, t]} \crochet{\bar\nu_u^n,1}^p
\Biggr)
& =
\mathbb{E} \Biggl(
\liminf_{N \to \infty}
\sup_{u \in [0, t \wedge \tau_N^n]} \crochet{\bar\nu_u^n,1}^p
\Biggr) \\
&\leq
\liminf_{N \to \infty}\mathbb{E}
\Biggl(
\sup_{u \in [0,t\wedge\tau_N^n]} \crochet{\bar\nu_u^n,1}^p
\Biggr) \\
&\leq
\Bigl(
\mathbb{E}\left(\crochet{\bar\nu_0^n,1}^p \right)
+
2\, \bar\lambda \, C_p\, t
\Bigr)\,
\exp(2\,\bar\lambda\,C_p\, t) \end{align*} and as $\sup_n \mathbb{E}\left(\crochet{\bar\nu_0^n,1}^p \right) <\infty$, we deduce the proof of the lemma. \carre \end{proof}
\begin{corollary} Let $f \in C^1(\mathcal{X})$, suppose that $\mathbb{E}(\crochet{\bar\nu_0^n, 1}^2)<\infty$, then for all $t>0$: \begin{align} \nonumber
\crochet{\bar\nu_t^n, f}
&=
\crochet{\bar\nu_0^n, f}
+ \int_0^t
\left< \bar\nu_u^n , \, \rho_{\textrm{\tiny\rm\!\! g}}(S_u^n,.)\,f'\right>
\, \mathrm{d} u \\ \nonumber
&\qquad
+ \int_0^t \int_{\mathcal{X}} \lambda(S^n_u,x) \,
\int_0^1
\bigl[ f(\alpha \, x)+f((1-\alpha)\,x)-f(x)\bigr] \,
Q(\mathrm{d} \alpha)\,\bar\nu_u^n(\mathrm{d} x) \, \mathrm{d} u \\ \label{renormalisation}
&\qquad
- D\, \int_0^t \int_{\mathcal{X}} f(x) \, \bar\nu_u^n(\mathrm{d} x) \, \mathrm{d} u
+ Z_t^{f,n} \end{align} where \begin{align} \nonumber
Z_t^{f,n}
&\eqdef
\frac 1n \,
\iiiint\limits_{[0,t]\times\N^*\times[0,1]^2}
1_{\{i\leq N_{\umoins}^n\}} \,
1_{\{0\leq \theta \leq \lambda(S^n_u, x_{\umoins}^{i,n})/\bar\lambda\}} \\[-1em] \nonumber
&\qquad\qquad\qquad\qquad
\times
\left[
f(\alpha\,x_{\umoins}^{i,n})
+ f((1-\alpha)\,x_{\umoins}^{i,n})
- f(x_{\umoins}^{i,n})
\right] \,
\tilde N_1(\mathrm{d} u, \mathrm{d} i, \mathrm{d} \alpha, \mathrm{d} \theta) \\[0.5em] \label{eq.Zfn}
&\qquad\qquad\qquad
- \frac 1n \, \iint\limits_{[0,t]\times\N^*}
1_{\{i\leq N_{\umoins}^n\}} \,
f(x_{\umoins}^{i,n})
\, \tilde N_2(\mathrm{d} u, \mathrm{d} i) \end{align} is a martingale with the following predictable quadratic variation: \begin{align} \nonumber
\crochet {Z^{f,n}}_t
&=
\frac 1n
\int_0^t \int_{\mathcal{X}} \lambda(S^n_u, x) \,
\int_0^1
\left[ f(\alpha \, x)+f((1-\alpha) \, x)-f(x) \right]^2 \,
Q(\mathrm{d} \alpha) \,\bar\nu_u^n(\mathrm{d} x) \, \mathrm{d} u \\ \label{var_qua}
&\qquad\qquad\qquad
+ \frac 1n \, D\, \int_0^t \int_{\mathcal{X}}
f(x)^2 \, \bar\nu_u^n(\mathrm{d} x) \, \mathrm{d} u\,. \end{align} \end{corollary}
\begin{proof} Equation \eqref{renormalisation} is obtained by applying Lemma \ref{lem.renorm} with $F(s,x)=x$, then $Z^{f,n}$ is $Z^{F,f,n}$ defined by \eqref{eq.Z,F,f}. Moreover as the random measures $\tilde{N_1}$ and $\tilde{N_2}$ are independent, we have: \begin{align*} \crochet {Z^{f,n}}_t
= &
\frac 1{n^2} \, \crochet {M^1}_t
+\frac 1{n^2} \, \crochet {M^2}_t \end{align*} where $M^1$ and $M^2$ are defined at Proposition \ref{prop.martingale}. From this latter proposition and Lemma \ref{lem.esp} we deduce the proof of the corollary. \carre \end{proof}
\begin{remark} The infinitesimal generator of the renormalized process $(S^n_t,\bar\nu^n_t)_{t\geq 0}$ is: \begin{multline*}
\mathcal{L}^n \Phi(s,\nu)
\eqdef
\bigl(D({\mathbf s}_{\textrm{\tiny\rm in}}-s)-k\,\mu(s,\nu) \bigr)\,
\partial_s F(s, \crochet{\nu, f})
+
\crochet{\nu ,\rho_{\textrm{\tiny\rm\!\! g}}(s,.) \, f'} \,
\partial_x F(s, \crochet{\nu, f}) \\
\quad+
n\,\int_{\mathcal{X}} \lambda(s,x) \,
\int_0^1
\bigl[
F\bigl(s,
\crochet{\textstyle\nu - \frac 1n \delta_x + \frac 1n \delta_{\alpha\,x}
+ \frac 1n \delta_{(1-\alpha)\,x} \, , \, f}
\bigr)
-
F(s, \crochet{\nu,f})
\bigr] \;
Q(\mathrm{d} \alpha) \, \nu(\mathrm{d} x) \\
\quad
+ n\, D\, \int_{\mathcal{X}}
\bigl[
F\bigl(s,\crochet{\textstyle\nu- \frac 1n \delta_x \, , \, f} \bigr)
-
F(s,\left<\nu,f\right>)
\bigr]\,
\, \nu(\mathrm{d} x) \end{multline*} for any $\Phi(s,\nu)=F(s,\crochet{\nu,f})$ with $F \in C_b^{1,1}(\mathbb{R}^+ \times \mathbb{R})$ and $f \in C^{1}(\mathcal{X})$. Note that this generator has the same ``substrat'' part than that of the initial generator \eqref{eq.generateur.infinitesimal} which again justifies the Remark \ref{remark.cv.nu.seule}. \end{remark}
To prove the uniqueness of the solution of the limit IDE, we have to assume that the application $\lambda(s,x)$ is Lipschitz continous w.r.t. $s$ uniformly in $x$: \begin{align} \label{hyp.lambda.lipschitz}
\bigl|\lambda(s_1,x)-\lambda(s_2,x)\bigr|\leq k_\lambda \,|s_1-s_2| \end{align} for all $s_1,\,s_2 \geq 0$ and all $x\in\XX$. This hypothesis as well as Hypothesis \ref{hyp.rhog.lipschitz} will also be used to demonstrate the convergence of IBM, see Theorem \ref{theo.cv.ibm}.
\subsection{Convergence result}
\begin{theorem}[Convergence of the IBM towards the IDE] \label{theo.cv.ibm} Under the assumptions described above, the process $(S_t^n,\bar\nu_t^n)_{t\geq 0}$ converges in distribution in the product space $\CC([0,T],\mathbb{R}_{+})
\times
\D([0,T],\MM_{F}(\mathcal{X}))$ towards the process $(S_t,\xi_t)_{t\geq 0}$ solution of: \begin{align} \label{eq.limite.substrat.faible}
S_t
&=
S_{0}
+
\int_{0}^t\biggl[
D\,({\mathbf s}_{\textrm{\tiny\rm in}}-S_u)-\frac kV \int_\mathcal{X} \rho_{\textrm{\tiny\rm\!\! g}}(S_u,x)\,\xi_u(\mathrm{d} x)
\biggr]\,\rmd u\,, \\[1em] \nonumber
\crochet{\xi_t,f}
&=
\crochet{\xi_0,f}
+
\int_{0}^t\biggl[
\int_{\mathcal{X}} \rho_{\textrm{\tiny\rm\!\! g}}(S_u,x)\,f'(x) \, \xi_u(\mathrm{d} x) \\ \nonumber
&\qquad
+ \int_{\mathcal{X}} \int_0^1\lambda(S_u, x)\,
\Bigl[f(\alpha\, x)+ f((1-\alpha)\,x)-f(x)\Bigr] \,
Q(\mathrm{d} \alpha) \,\xi_u(\mathrm{d} x) \\ \label{eq.limite.eid.faible}
&\qquad
- D\, \int_{\mathcal{X}} f(x) \, \xi_u(\mathrm{d} x)
\biggr]\,\rmd u\,, \end{align} for any $f \in C^{1}(\mathcal{X})$. \end{theorem}
The proof is in three steps\footnote{Note that our situation is simpler than that studied by \cite{roelly1986a} and \cite{meleard1993a} since in our case $\XX$ is compact: in fact in our case the weak topology -- the smallest topology which makes the applications $\nu\to\crochet{\nu,f}$ continuous for any $f$ continuous and bounded -- and the vague topology -- the smallest topology which makes the applications $\nu\to\crochet{\nu,f}$ continuous for all $f$ continuous with compact support -- are identical.}: first the uniqueness of the solution of the limit equation \eqref{eq.limite.substrat.faible}-\eqref{eq.limite.eid.faible}, second the tightness (of the sequence of distribution) of $\bar\nu^n$ and lastly the convergence in distribution of the sequence.
\subsubsection*{Step 1: uniqueness of the solution of \eqref{eq.limite.substrat.faible}-\eqref{eq.limite.eid.faible}}
Let $(S_t,\xi_t)_{t\geq 0}$ be a solution of \eqref{eq.limite.substrat.faible}-\eqref{eq.limite.eid.faible}. We first show that $(\xi_t)_t$ is of finite mass for all $t\geq 0$: \begin{align*}
\crochet{\xi_t, 1}
& =
\crochet{\xi_0, 1}
+
\int_0^t \int_{\mathcal{X}} \int_0^1 \lambda(S_u, x)\, Q(\mathrm{d} \alpha) \,
\xi_u(\mathrm{d} x)\, \mathrm{d} u
- D\, \int_0^t \int_{\mathcal{X}} \xi_u(\mathrm{d} x) \, \mathrm{d} u\\
& \leq \left<\xi_0, 1\right>+(\bar \lambda - D) \int_0^t \left<\xi_u, 1\right> \, \mathrm{d} u \end{align*} and according to Gronwall's inequality: $\crochet{\xi_t, 1} \leq \crochet{\xi_0, 1} \, e^{(\bar \lambda - D)\,t}
< \infty $.
We introduce the following norm on $\MM_{F}(\XX)$: \[
\normm{\bar\nu}
\eqdef
\sup
\Bigl\{
|\crochet{\bar\nu,f}|
\,;\,
f \in C^1(\mathcal{X}),\, \norme{f}_\infty \leq 1,\,\norme{f'}_\infty \leq 1
\Bigr\} \] and consider two solutions $(S^1_t,\xi^1_t)_{t\geq 0}$ and $(S^2_t,\xi^2_t)_{t\geq 0}$ of \eqref{eq.limite.substrat.faible}-\eqref{eq.limite.eid.faible}.
It was previously shown that $\xi^1_t$ and $\xi^2_t$ are of finite mass on $\mathbb{R}_+$, so we can define: \[
C_t
\eqdef \sup_{0\leq u \leq t} \crochet{\xi^1_u+\xi^2_u , 1}
\,. \] According to \eqref{eq.limite.eid.faible}, for any $f \in C^1(\mathcal{X})$ such that $\norme{f}_\infty \leq 1$ and $\norme{f'}_\infty \leq 1$ we have: \begin{align*}
|\crochet{\xi^1_t-\xi^2_t,f}|
&\leq
\int_0^t \biggl|\int_\mathcal{X}
f'(x) \,
\Bigl[
\rho_{\textrm{\tiny\rm\!\! g}}(S^1_u,x)\, [\xi^1_u(\mathrm{d} x)-\xi^2_u(\mathrm{d} x)] \\
&\qquad\qquad\qquad
- [\rho_{\textrm{\tiny\rm\!\! g}}(S^2_u,x)-\rho_{\textrm{\tiny\rm\!\! g}}(S^1_u,x)]\, \xi^2_u(\mathrm{d} x)
\Bigr]
\biggr|\,\mathrm{d} u \\
&\qquad + \int_0^t \biggl|
\int_\mathcal{X} \int_0^1
\left[f(\alpha\,x)+f((1-\alpha)\,x)-f(x) \right] \,
Q(\mathrm{d} \alpha) \\
& \qquad\qquad\qquad
\left[
\lambda(S^1_u, x) \, [\xi^1_u(\mathrm{d} x)-\xi^2_u(\mathrm{d} x)]
- [\lambda(S^2_u, x)-\lambda(S^1_u, x)]\,
\xi^2_u(\mathrm{d} x)
\right] \biggr| \, \mathrm{d} u \\
&\qquad
+ D \, \int_0^t \biggl|
\int_\mathcal{X} f(x) \, (\xi^1_u(\mathrm{d} x)-\xi^2_u(\mathrm{d} x))
\biggr| \, \mathrm{d} u \\
&\leq
(\bar g +3\, \bar \lambda +D)\,
\int_0^t \normm{\xi^1_u-\xi^2_u} \, \mathrm{d} u
+ C_t\,( k_g + 3 \, k_\lambda) \,
\int_0^t | S^1_u- S^2_u |\, \mathrm{d} u . \end{align*} Taking the supremum over the functions $f$, we obtain: \begin{align*}
\normm{\xi^1_t-\xi^2_t}
&\leq
(\bar g +3\, \bar \lambda +D)\,
\int_0^t \normm{\xi^1_u-\xi^2_u} \, \mathrm{d} u
+ C_t\,( k_g + 3 \, k_\lambda) \,
\int_0^t | S^1_u-S^2_u |\, \mathrm{d} u
\,. \end{align*} Moreover, from \eqref{eq.limite.substrat.faible} we get: \begin{align*}
&
|S^1_t- S^2_t|
\leq
D \, \int_0^t |S^1_u- S^2_u|\,\mathrm{d} u \\
&\quad
+ \frac kV \int_0^t \biggl|
\int_\mathcal{X}
\Bigl(
\rho_{\textrm{\tiny\rm\!\! g}}(S^1_u,x)\, [\xi^1_u(\mathrm{d} x)-\xi^2_u(\mathrm{d} x)]
-
[\rho_{\textrm{\tiny\rm\!\! g}}(S^2_u,x)-\rho_{\textrm{\tiny\rm\!\! g}}(S^1_u,x)]
\, \xi^2_u(\mathrm{d} x)
\Bigr)
\biggr| \, \mathrm{d} u \\
&\quad
\leq
\Bigl(D+\frac kV \, C_t\,k_g\Bigr) \,
\int_0^t |S^1_u- S^2_u| \, \mathrm{d} u
+ \frac kV \,\bar g \int_0^t \normm{\xi^1_u-\xi^2_u} \, \mathrm{d} u \,. \end{align*} We define: \begin{align*}
M_t
\eqdef
\max \left\{\bar g +3\, \bar \lambda +D+ \frac kV \,\bar g
\, , \, C_t\,( k_g + 3 \, k_\lambda)+D+\frac kV \, C_t\,k_g\right\} \end{align*} hence: \begin{align*}
\normm{\xi^1_t-\xi^2_t}
+|S^1_t-S^2_t|
& \leq
M_t \, \int_0^t \Bigl(
\normm{\xi^1_u-\xi^2_u}+|S^1_u-S^2_u|
\bigr) \, \mathrm{d} u \end{align*} Finally from Gronwall's inequality we get
$\normm{\xi^1_t-\xi^2_t}+|S^1_t-S^2_t|=0$ for all $t\geq 0$, hence $\xi^1_t = \xi^2_t$ and $S^1_t=S^2_t$.
\subsubsection*{Step 2: tightness of $(\bar\nu^n)_{n\geq 0}$}
The tightness of $\bar\nu^n$ is equivalent to the fact that from any subsequence one can extract a subsequence that converges in distribution in the space $\D([0,T],\MM_{F}(\mathcal{X}))$. According to \citet[Th. 2.1]{roelly1986a} this amounts to proving the tightness of $\crochet{\bar\nu^n,f}$ in $\D([0,T],\mathbb{R})$ for all $f$ in a set dense in $\mathcal{C}(\XX)$, here we will consider $f\in \mathcal{C}^1(\XX)$. To prove the latter result, it is sufficient to check the following Aldous-Rebolledo criteria \citep[Cor. 2.3.3]{joffe1986a}: \begin{enumerate} \item The sequence $(\langle \bar\nu_t^n,f \rangle)_{n\geq 0}$ is tight for any $t\geq 0$.
\item Consider the following semimartingale decomposition: \[
\crochet{\bar\nu_t^n,f}
=
\crochet{\bar\nu_0^n,f}
+
A_t^n+Z_t^n\,. \] where $A_t^n$ is of finite variation and $Z_t^n$ is a martingale. For all $t>0$, $\varepsilon>0$, $\eta>0$ there exists $n_{0}$ such that for any sequence $\tau_n$ of stopping times with $\tau_{n}\leq t$ we have: \begin{align} \label{eq.AR.2.1}
\sup_{n\geq n_0} \sup_{\theta\in[0,\delta]}
\P\Big(
\big|
A^n_{\tau_n+ \theta} - A^n_{\tau_n}
\big|
\geq \eta
\Big)
&
\leq \varepsilon\,, \\ \label{eq.AR.2.2}
\sup_{n\geq n_0} \sup_{\theta\in[0,\delta]}
\P\Big(
\big|
\crochet{Z^n}_{\tau_n+ \theta} - \crochet{Z^n}_{\tau_n}
\big|
\geq \eta
\Big)
&
\leq \varepsilon\,. \end{align} \end{enumerate}
\subsubsection*{\it Proof of \fenumi}
For any $K>0$, \begin{align*}
\P\bigl(|\crochet{\bar\nu_t^n,f}| \geq K\bigr)
&\leq
\frac{1}{K}\,\norme{f}_\infty \,
\sup_{n\in\N} \E\bigl(\crochet{\bar\nu_t^n,1}\bigr) \end{align*} and using Lemma \ref{lem.esp}, we deduce \fenumi.
\subsubsection*{\it Proof of {\rm(\textit{ii})}}
\begin{align*}
A_t^n
&=
\int_0^t \crochet{\bar\nu_u^n,\rho_{\textrm{\tiny\rm\!\! g}}(S_u^n,.)\,f'} \, \mathrm{d} u \\
&\qquad
+ \int_0^t \int_{\mathcal{X}} \int_0^1
\lambda(S_u^n, x) \,
\bigl[ f(\alpha\,x)+f((1-\alpha)\,x)-f(x) \bigr] \,
Q(\mathrm{d} \alpha) \,\bar\nu_u^n(\mathrm{d} x) \, \mathrm{d} u \\
&\qquad
- D\,\int_0^t\int_{\mathcal{X}} f(x)\,\bar\nu_u^n(\mathrm{d} x) \, \mathrm{d} u \end{align*} hence, according to Lemma \ref{lem.esp}: \begin{align*}
\mathbb{E} | A_{\tau_n+\theta}^n-A_{\tau_n}^n|
&\leq
(
\norme{f'}_\infty\, \bar{g}
+ 3\, \norme{f}_\infty\,\bar\lambda
+ D\,\norme{f}_\infty
)
\, C_{t,1}\, \theta\,. \end{align*} Using \eqref{var_qua}, we also have: \begin{align*}
\mathbb{E} | \crochet{Z^n}_{\tau_n+\theta}-\crochet{Z^n}_{\tau_n}|
\leq
\frac{1}{n}\,\left(9 \, \bar \lambda+D \right) \,
\norme{f}_\infty^2\, C_{t,1} \, \theta\,. \end{align*} Hence
$\mathbb{E}| A_{\tau_n+\theta}^n-A_{\tau_n}^n|+\mathbb{E} | \crochet{Z^n}_{\tau_n+\theta}-\crochet{Z^n}_{\tau_n} |
\leq C \, \theta$ and we obtain \fenumii\ from the Markov inequality.
In conclusion, from the Aldous-Rebolledo criteria, the sequence $(\bar\nu^n)_{n\geq 0}$ is tight.
\subsubsection*{Step 3: convergence of the sequence $(\bar\nu^n)_{n\in\N}$}
To conclude the proof of the theorem it is suffice to show that the sequence $(\bar\nu^n)_{n\in\N}$ has a unique accumulation point and that this point is equal to $\xi$ described in Step 1. In order to characterize $\xi$, the solution of \eqref{eq.limite.eid.faible}, we introduce, for any given $f \in C^{1}(\mathcal{X})$, the following function defined for all $\zeta\in\D([0,T],\mathcal{M}_F(\mathcal{X}))$: \begin{align} \nonumber
\Psi_{t}(\zeta)
&\eqdef
\crochet{\zeta_{t},f}-\crochet{\zeta_{0},f}
-\int_{0}^t \biggl[
\int_{\mathcal{X}} \rho_{\textrm{\tiny\rm\!\! g}}(S^\zeta_u,x)\,f'(x) \, \zeta_u(\mathrm{d} x) \\ \nonumber
&\qquad
+ \int_{\mathcal{X}} \int_0^1\lambda(S^\zeta_u, x)\,
\bigl[f(\alpha\, x)+ f((1-\alpha)\,x)-f(x)\bigr] \,
Q(\mathrm{d} \alpha) \,\zeta_u(\mathrm{d} x) \\ \label{eq.Psi}
&\qquad
- D\, \int_{\mathcal{X}} f(x) \, \zeta_u(\mathrm{d} x)
\biggr]\,\rmd u \end{align} where $S_t^\zeta$ is defined by: \begin{align} \label{eq.Psi.2}
S_t^\zeta
&\eqdef
S_0
+
\int_0^t \Bigl(
D\,({\mathbf s}_{\textrm{\tiny\rm in}}-S_u^\zeta)
-\frac kV \int_\mathcal{X} \rho_{\textrm{\tiny\rm\!\! g}}(S_u^\zeta,x)\,\zeta_u(\mathrm{d} x)
\Bigr) \, \mathrm{d} u\,. \end{align} Hence, if $\Psi_{t}(\zeta)=0$ for all $t\geq 0$ and all $f \in C^{1}(\mathcal{X})$ then $(S^\zeta,\zeta)=(S,\xi)$ where $(S,\xi)$ is the unique solution of \eqref{eq.limite.substrat.faible}-\eqref{eq.limite.eid.faible}.
We consider a subsequence $\bar\nu^{n'}$ of $\bar\nu^{n}$ which converges in distribution in the space $\D([0,T],\mathcal{M}_F(\mathcal{X}))$ and $\tilde \nu$ its limit.
\subsubsection*{\it Sub-step 3.1: A.s. continuity of the limit~$\tilde \nu$.}
\begin{lemma} $\tilde \nu(\omega)\in \CC([0,T],\mathcal{M}_F(\mathcal{X}))$ for all $\omega\in\Omega$ a.s. \end{lemma}
\begin{proof} For any $f\in\mathcal{C}(\mathcal{X})$ such that $\norme{f}_\infty \leq 1$: \begin{align*}
\bigl|
\crochet{\bar\nu_t^{n'},f}-\crochet{\bar\nu_{t^-}^{n'},f}
\bigr|
\leq
\frac{1}{n'}\,
\bigl|
\crochet{\nu_t^{n'},1}-\crochet{\nu_{t^-}^{n'},1}
\bigr|\,. \end{align*}
But $| \crochet{\nu_t^{n'},1}-\crochet{\nu_{t^-}^{n'},1}|$ represents the difference between the number of individuals in $\nu_t^{n'}$ and in $\nu_{t^-}^{n'}$, which is at most 1. Hence: \begin{align*}
\sup_{t \in [0,T]} \,
\normtv{\bar\nu_t^{n'}-\bar\nu_{t^-}^{n'}}
\leq \frac{1}{n'} \end{align*} which proves that the limit process $\tilde\nu$ is a.s. continuous \cite[Th. 10.2 p. 148]{ethier1986a} as the Prokhorov metric is dominated by the total variation metric. \end{proof} \carre
\subsubsection*{\it Sub-step 3.2: Continuity of $\zeta\to\Psi_{t}(\zeta)$ in any $\zeta$ continuous.}
\begin{lemma} \label{lemma.Psi.continue} For any given $t\in[0,T]$ and $f\in C^{1}(\mathcal{X})$, the function $\Psi_{t}$ defined by \eqref{eq.Psi} is continuous from $\DD([0,T],\MM_{F}(\mathcal{X}))$ with values in $\mathbb{R}$ in any point $\zeta\in\CC([0,T],\MM_{F}(\mathcal{X}))$. \end{lemma}
\proof Consider a sequence $(\zeta^{n})_{n\in\N}$ which converges towards $\zeta$ in $\DD([0,T],\MM_{F}(\mathcal{X}))$ with respect to the Skorohod topology. As the limit $\zeta$ is continuous we have that $\zeta^{n}$ converges to $\zeta$ with the uniform topology: \begin{align} \label{eq.lemma.Psi.continue.a}
\sup_{0\leq t\leq T}
d_{\textrm{\tiny\rm PR}}({\zeta_t^{n},\zeta_{t}})
\cv_{n\to\infty} 0 \end{align} where $d_{\textrm{\tiny\rm PR}}$ is the Prokhorov metric (see Appendix \ref{appendix.skorohod}).
The functions $\lambda(s,x)$ and $\rho_{\textrm{\tiny\rm\!\! g}}(s,x)$ are Lipschitz continuous functions w.r.t. $s$ uniformly in $x$ and also bounded, see \eqref{hyp.lambda.lipschitz} and \eqref{hyp.rhog.lipschitz}, so from \eqref{eq.Psi.2} we can easily check that: \begin{align*}
|S_t^{\zeta^{n}}-S_t ^\zeta|
&\leq
C\,\int_0^t \Bigl(
|S_u^{\zeta^{n}}-S_u ^\zeta|
+ \Bigl|
\int_\mathcal{X} \rho_{\textrm{\tiny\rm\!\! g}}(S_u^{\zeta^{n}},x)\,
[\zeta^{n}_u(\mathrm{d} x)-\zeta_u(\mathrm{d} x)] \\
&\qquad\qquad\qquad\qquad\qquad\qquad
-
\int_\mathcal{X} [\rho_{\textrm{\tiny\rm\!\! g}}(S_u^\zeta,x)-\rho_{\textrm{\tiny\rm\!\! g}}(S_u^{\zeta^{n}},x)]
\,\zeta_u(\mathrm{d} x)
\Bigr|
\Bigr) \, \mathrm{d} u \\
&\leq
C\,\int_0^t \Bigl(
|S_u^{\zeta^{n}}-S_u ^\zeta|
+ |\crochet{\zeta^{n}_u-\zeta_u,1}|\Bigr) \, \mathrm{d} u \end{align*} and the Gronwall's inequality leads to: \begin{align} \label {eq.lemma.Psi.continue.b}
|S_t^{\zeta^{n}}-S_t ^\zeta|
\leq
C\,\int_{0}^t |\crochet{\zeta^n_{u}-\zeta_{u},1}|\,\rmd u\,. \end{align} Here and in the rest of the proof the constant $ C $ will depend only on $ T $, $ f $ and on the parameters of the models. Hence, from \eqref{eq.Psi}: \begin{align*}
|\Psi_t(\zeta^{n})-\Psi_t(\zeta)|
&\leq
C\,
\Bigl[
|\crochet{\zeta_t^{n}- \zeta_t,1}|
+
|\crochet{\zeta_0^{n}- \zeta_0,1}| \\
&\qquad\qquad\qquad
+
\int_0^t |S_u^{\zeta^{n}}-S_u ^\zeta| \, \mathrm{d} u
+
\int_0^t |\crochet{\zeta_u^{n}-\zeta_u,1}| \, \mathrm{d} u
\Bigr] \\
&\leq
C\,\sup_{0\leq t\leq T}|\crochet{\zeta_t^{n}-\zeta_t,1}|\,. \end{align*} Let $\delta_{t}=d_{\textrm{\tiny\rm PR}}(\zeta_t^{n},\zeta_t)$, by definition of the Prokhorov metric: \begin{align*}
\zeta_t^{n}(\XX)-\zeta_t(\XX^{\delta_{t}})&\leq \delta_{t}\,,
&
\zeta_t(\XX)-\zeta_t^{n}(\XX^{\delta_{t}})&\leq \delta_{t}\,, \end{align*}
but $\XX^{\delta_{t}}=\XX$ hence $|\zeta_t^{n}(\XX)-\zeta_t(\XX)|\leq \delta_{t}$. Note finally that $|\zeta_t^{n}(\XX)-\zeta_t(\XX)| = |\crochet{\zeta_t^{n}-\zeta_t,1}|$, so we get: \[
|\Psi_t(\zeta^{n})-\Psi_t(\zeta)|
\leq C\,
\sup_{0\leq t\leq T}d_{\textrm{\tiny\rm PR}}({\zeta_t^{n},\zeta_t}) \] which tends to zero. \carre
\subsubsection*{\it Sub-step 3.3: Convergence in distribution of $\Psi_{t}(\bar\nu^{n'})$ to $\Psi_{t}(\tilde\nu)$.}
The sequence $\bar\nu^{n'}$ converges in distribution to $\tilde\nu$ and $\tilde\nu(\omega)\in\CC([0,T],\MM_{F}(\XX))$; moreover the application $\Psi_{t}$ is continuous in any point of $\CC([0,T],\MM_{F}(\XX))$, thus according to the continuous mapping theorem \cite[Th. 2.7 p. 21]{billingsley1968a} we get: \begin{align} \label{eq.cv.loi.Psi}
\Psi_{t}(\bar\nu^{n'})
\xrightarrow[n\to\infty]{\textrm{loi}}
\Psi_{t}(\tilde\nu)\,. \end{align}
\subsubsection*{\it Sub-step 3.4: $\tilde\nu=\xi$ a.s.}
From \eqref{renormalisation}, for any $n\geq 0$ we have: \begin{align*}
\Psi_t(\bar\nu^n) = Z_t^{f,n} \end{align*} where $Z_t^{f,n}$ is defined by \eqref{eq.Zfn}. Also, \eqref{var_qua} gives: \begin{align*}
\mathbb{E}(|Z_t^{f,n}|^2)
&=
\mathbb{E} \crochet{Z^{f,n}}_t
\leq \frac{1}{n}\,(9\,\bar\lambda+D) \, \norm{f}_{\infty}^2\,C_{t,1} \, t\,. \end{align*} Hence $\Psi_{t}(\bar\nu^n)$ converges to $0$ in $L^2$ but also in $L^1$. Furthermore, we easily show that: \begin{align*}
|\Psi_{t}(\zeta)|
\leq & C_{f,t} \sup_{0 \leq u \leq t} \crochet{\zeta_u,1} \end{align*} moreover, from Lemma \ref{lem.esp}, $(\Psi_{t}(\bar\nu^{n'}))_{n'}$ is uniformly integrable. The dominated convergence theorem and \eqref{eq.cv.loi.Psi} imply: \[
0
= \lim_{n' \to \infty} \mathbb{E} |\Psi_t(\bar\nu^{n'})|
= \mathbb{E} |\Psi_t(\tilde \nu)|\,. \] So $\Psi_t(\tilde \nu)=0$ a.s. and $\tilde \nu$ is a.s. equal to $\xi$ where $(S,\xi)$ is the unique solution of \eqref{eq.limite.substrat.faible}-\eqref{eq.limite.eid.faible}.
This last step concludes the proof of Theorem \ref{theo.cv.ibm}.
\subsection{Links with deterministic models} \label{subsec.modeles.deterministes}
Equation \eqref{eq.limite.eid.faible} is actually a weak version of an integro-differential equation that can be easily identified. Indeed suppose that the solution $\xi_t$ of Equation \eqref{eq.limite.eid.faible} admits a density $p_t(x)\,\rmd x=\xi_{t}(\rmd x)$, and that $Q(\rmd \alpha)=q(\alpha)\,\rmd \alpha$, then the system of equations \eqref{eq.limite.substrat.faible}-\eqref{eq.limite.eid.faible} is a weak version of the following system: \begin{align} \label{eq.limite.substrat.fort}
&
\frac{\rmd}{\rmd t} S_t =
D\,({\mathbf s}_{\textrm{\tiny\rm in}}-S_t)-\frac kV \int_\mathcal{X} \rho_{\textrm{\tiny\rm\!\! g}}(S_t,x)\,p_t(x)\,\mathrm{d} x\,, \\ \nonumber
&
\frac{\partial}{\partial t} p_t(x)
+\frac{\partial}{\partial x} \bigl( \rho_{\textrm{\tiny\rm\!\! g}}(S_t,x)\,p_t(x)\bigr)
+ \bigl(\lambda(S_t, x)+D \bigr)\,p_t(x) \\ \label{eq.limite.eid.fort}
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
=
2\,\int_\mathcal{X}
\frac{\lambda(S_t, z)}{z}\,q\left(\frac xz \right)\,p_t(z)\,\mathrm{d} z\,. \end{align} In fact, this is the population balance equation introduced by \cite{fredrickson1967a} and \cite{ramkrishna1979a} for growth-fragmentation models.
It is easy to link the model \eqref{eq.limite.substrat.fort}-\eqref{eq.limite.eid.fort} to the classic chemostat model. Indeed suppose that the growth function $x \mapsto \rho_{\textrm{\tiny\rm\!\! g}}(s,x)$ is proportional to $ x $, i.e.: \[
\rho_{\textrm{\tiny\rm\!\! g}}(s,x)=\tilde\mu(s)\,x\,. \] The results presented now are formal insofar as a linear growth function does not verify the assumptions made in this article. We introduce the bacterial concentration: \[
Y_t \eqdef \frac1V \,\int_\mathcal{X} x\, p_t(x) \, \mathrm{d} x\,. \] As $\sup_{0\leq t\leq T}\crochet{p_{t},1}<\infty$, from \eqref{eq.limite.eid.fort}: \begin{multline*}
\frac{\mathrm{d}}{\mathrm{d} t} Y_t
- \frac1V \int_\mathcal{X} x \,
\frac{\partial}{\partial x}\bigl( \rho_{\textrm{\tiny\rm\!\! g}}(S_t,x)\,p_t(x)\bigr) \, \mathrm{d} x
+ \frac1V \int_\mathcal{X} x \, \lambda(S_t, x) \, p_t(x)\,\mathrm{d} x
+ D\, Y_t \\
= \frac2V\, \int_\mathcal{X} x \,
\int_\mathcal{X}
\frac{\lambda(S_t, z)}{z}\,q(x/z)\,p_t(z)\,
\mathrm{d} z \, \mathrm{d} x\,, \end{multline*} but \begin{align*} &\int_\mathcal{X} x \,
\int_\mathcal{X}
\frac{\lambda(S_t, z)}{z}\,
q(x/z)\,p_t(z)\,\mathrm{d} z \, \mathrm{d} x
=
\int_\mathcal{X} \int_0^1
z\, \lambda(S_t,z)\, \alpha \,q\left(\alpha \right)\,p_t(z)\,\mathrm{d} \alpha \, \mathrm{d} z \\
&\qquad\qquad = \int_\mathcal{X} \int_0^1
z\, \lambda(S_t, z)\, \alpha \,q\left(1-\alpha \right)\,p_t(z)\,\mathrm{d} \alpha \, \mathrm{d} z
\tag{by symmetry of $q$} \\
&\qquad\qquad = \int_\mathcal{X} \int_0^1
z\, \lambda(S_t, z)\, (1-\alpha) \,q\left(\alpha \right)\,p_t(z)\,\mathrm{d} \alpha \, \mathrm{d} z \\
&\qquad\qquad = - \int_\mathcal{X} \int_0^1
z\, \lambda(S_t, z)\, \alpha \,q\left(\alpha \right)\,p_t(z)\,\mathrm{d} \alpha \, \mathrm{d} z
+ \int_\mathcal{X}
z\, \lambda(S_t, z)\,p_t(z)\, \mathrm{d} z \end{align*} thus: \begin{align*}
2\, \int_\mathcal{X} x \,
\int_\mathcal{X}
\frac{\lambda(S_t, z)}{z}\,q(x/z)\,p_t(z)\,\mathrm{d} z \, \mathrm{d} x
= \int_\mathcal{X}
z\, \lambda(S_t, z)\,p_t(z)\, \mathrm{d} z. \end{align*} The function $x\mapsto p_t(x)$ is the population density at time $t$. On the one hand $p_0(x)$ has compact support. On the other hand the growth of each bacterium is defined by a differential equation whose right-hand side is bounded by a linear function in $x$, uniformly in $s$. Hence for all $t\leq T$, we can uniformly bound the mass of all the bacteria and $p_{t}(x)$ has a compact support, i.e. there exists $m_{\textrm{\tiny\rm max}}$ such that the support of $p_{t}(x)$ is included in $[0,m_{\textrm{\tiny\rm max}}]$ with $p_{t}(m_{\textrm{\tiny\rm max}})=0$, so we choose $\mathcal{X}=[0,m_{\textrm{\tiny\rm max}}]$. Moreover $\rho_{\textrm{\tiny\rm\!\! g}}(S_t,0)=0$ hence: \begin{align*}
\int_\mathcal{X}
x \, \frac{\partial}{\partial x}\bigl( \rho_{\textrm{\tiny\rm\!\! g}}(S_t,x)\,p_t(x)\bigr)
\, \mathrm{d} x
=
-\int_\mathcal{X} \rho_{\textrm{\tiny\rm\!\! g}}(S_t,x)\,p_t(x) \, \mathrm{d} x. \end{align*} Finally: \begin{align*}
\frac{\mathrm{d}}{\mathrm{d} t} Y_t
& =
\frac1V \int_\mathcal{X} \rho_{\textrm{\tiny\rm\!\! g}}(S_t,x)\,p_t(x) \, \mathrm{d} x
-D\, Y_t
= \tilde\mu(S_{t})\,Y_{t} -D\, Y_t\,. \end{align*} We deduce that the concentrations $(Y_{t},S_{t})_{t\geq 0}$ of biomass and substrate are the solution of the following closed system of ordinary differential equations: \begin{equation} \label{eq.chemostat.edo}
\begin{split}
\dot Y_t & = \bigl(\tilde\mu(S_t)-D \bigr)\, Y_t\,,
\\
\dot S_t & = D\,({\mathbf s}_{\textrm{\tiny\rm in}}-S_t)-k \,\tilde\mu(S_t)\, Y_t\,.
\end{split} \end{equation} which is none other than the classic chemostat equation \citep{smith1995a}.
\section{Simulations} \label{sec.simulations}
In this section we compare the behavior of the individual-based model (IBM) and two deterministic models: the integro-differential equation (IDE) \eqref{eq.limite.substrat.fort}-\eqref{eq.limite.eid.fort} and classic chemostat model, represented by the ordinary differential equation \eqref{eq.chemostat.edo} (ODE). Simulations of the IBM were performed following Algorithm \ref{algo.ibm}. The resolution of the integro-differential equation was made following the numerical scheme given in Appendix \ref{appendix.schema.num}, with a discretization step in the mass space of $\Delta x = 2 \times 10^{-7}$ and a discretization step in time of $\Delta t = 5 \times 10^{-4}$.
\subsection{Simulation parameters}
In the simulations proposed in this section, the division rate of an individual is given by the following function: \begin{align*}
\lambda(s,x)
& =
\frac{\bar\lambda}{\log \bigl((m_{\textrm{\tiny\rm max}}-m_{\textrm{\tiny\rm div}}) \, p_\lambda +1 \bigr)}
\,
\log \bigl((x-m_{\textrm{\tiny\rm div}}) \, p_\lambda +1 \bigr) \, 1_{\{x \geq m_{\textrm{\tiny\rm div}}\}} \end{align*} which does not depend on the substrate concentration.
The division kernel $Q(\mathrm{d} \alpha)=q(\alpha)\,\mathrm{d} \alpha$ is given by a symmetric beta distribution: \begin{align*}
q(\alpha)
& =
\frac{1}{B(p_\beta)}\,
\bigl( \alpha \,(1-\alpha) \bigr)^{p_\beta-1} \end{align*} where $B(p_\beta)
= \int_0^1 \bigl( \alpha \,(1-\alpha) \bigr)^{p_\beta-1} \, \mathrm{d} \alpha$ is a normalizing constant.
Individual growth follows a Gompertz model, with a growth rate depending on the substrate concentration: \begin{align*}
g(s,x)
& =
r_{\textrm{\tiny\rm max}}\,\frac{s}{k_r+s} \,
\log\Big(\frac{m_{\textrm{\tiny\rm max}}}{x}\Big)\,x\,. \end{align*} The masses of individuals at the initial time are sampled according to the following probability density function: \begin{align} \label{eq.d} d(x)
& =
\Biggl(
\frac{x-0.0005}{0.00025}
\,\left(1-\frac{x-0.0005}{0.00025}\right)
\Biggr)^5 \,
1_{\{0.0005 < x < 0.00075\}}\,. \end{align} This initial density will show a transient phenomenon that cannot be reproduced by the classical chemostat model described in terms of ordinary differential equations \eqref{eq.chemostat.edo}, see Figure \ref{fig.edo.ibm.eid}.
The simulations were performed using the parameters in Table \ref{table.parametres}. The parameters $V$, $N_0$ and $D$ will be specified for each simulation.
\begin{table}[h] \begin{center}
\begin{tabular}{|c|c|}
\hline
Parameters & Values \\
\hline
$S_0$ & 5 mg/l \\
${\mathbf s}_{\textrm{\tiny\rm in}}$ & 10 mg/l \\
$m_{\textrm{\tiny\rm max}}$ & 0.001 mg \\
$m_{\textrm{\tiny\rm div}}$ & 0.0004 mg \\
$\bar\lambda$ & 1 h$^{-1}$\\
$p_\lambda$ & 1000 \\
$p_\beta$ & 7 \\
$r_{\textrm{\tiny\rm max}}$ & 1 h$^{-1}$\\
$k_r$ & 10 mg/l\\
$k$ & 1\\
\hline \end{tabular} \end{center} \caption{Simulation parameters.} \label{table.parametres} \end{table}
\subsection{Comparison of the IBM and the IDE}
To illustrate the convergence in large population asymptotic of the IBM to the IDE, we performed simulations at different levels of population size. To this end we vary the volume of the chemostat and the number of individuals at the initial time. We considered three cases: \begin{enumerate} \item small size: $V=0.05$ l and $N_0=100$, \item medium size: $V=0.5$ l and $N_0=1000$, \item large size: $V=5$ l and $N_0=10000$. \end{enumerate} In each of these three cases we simulate: \begin{itemize} \item 60 independent runs of the IBM; \item the numerical approximation of \eqref{eq.limite.substrat.fort}-\eqref{eq.limite.eid.fort} using the finite difference schemes detailed in Appendix \ref{appendix.schema.num} \end{itemize} with the same initial biomass concentration distribution.
The convergence of IBM to EID is clearly illustrated in Figure \ref{evol.taille.concentrations} where the evolutions of the population size, of the biomass concentration, and of the substrate concentration are represented.
In Figure \ref{fig.evol.eid} the time evolution of the normalized mass distribution is depicted, i.e. the normalized solution of the IDE \eqref{eq.limite.eid.fort}. We have represented the simulation until time $T=10$ (h) to illustrate the transient phenomenon due to the choice of the initial distribution \eqref{eq.d}: after a few time iterations this distribution is bimodal; the upper mode (large mass) grows in mass and disappears before time $T=10$ (h). The lower mode (small mass) corresponds to the mass of the bacteria resulting from the division; the upper mode corresponds to the mass of the bacteria from the initial bacteria before their division. Thus, the upper mode is set to disappear quickly by division or by up-take. The IBM realizes this phenomenon, see Figure \ref{repartition.masse}. In contrast, the classical chemostat model presented below, see Equation \eqref{eq.chemostat.edo}, cannot account for this phenomenon.
Figure \ref{repartition.masse} presents this normalized mass distribution at three different instants, $t=1,\,4,\, 80$ (h), and the simulation of the IDE is compared to 60 independent runs of the IBM, again for the three levels of population sizes described above. Depending on whether the population is large, medium or small, we needed to adapt the number of bins of the histograms so that the resulting graphics are clear. The convergence of the IBM solution to the IDE in large population limit can be observed.
In conclusion, the IBM converges in large population limit to the IDE and variability ``around'' the asymptotic model is relatively large in small or medium population size; note that there is no reason why the IDE represents the mean value of the IBM.
\begin{figure}
\caption{From top to bottom: time evolutions of the population size, the biomass concentration, the concentration substrate and the concentrations phase portrait for the three levels of population sizes (small, medium and large). The blue curves represent the trajectories of 60 independent runs of IBM. The green curve represents the mean value of these runs. The red curve represents the solution of the IDE. The rate $ D $ is 0.2 h$^{-1}$.}
\label{evol.taille.concentrations}
\end{figure}
\begin{figure}
\caption{Time evolution of the normalized mass distribution for the IDE \eqref{eq.limite.eid.fort}: we represent the simulation until time $T=10$ (h) only to illustrate the transient phenomenon due to the choice of the initial distribution \eqref{eq.d}. After a few iterations in time this distribution is bimodal, the upped mode growths in mass and disappears before $T=10$ (h).}
\label{fig.evol.eid}
\end{figure}
\begin{figure}
\caption{Mass distribution for the time $t=1$ (above), $t=4$ (middle) and $t=80$ (bottom) in small (left), medium (middle) and large (right) population size. For each graph, the blue histograms represent the empirical mass distributions of individuals for the 60 independent runs of IBM. In order to plot the histogram we have adapted the number of bins according to the population size. The red curve represents the mass distribution given by the IDE. The dilution rate $ D $ is 0.2 h$^{-1}$. Again the convergence of the IBM solution to the IDE in large population limit is observed.}
\label{repartition.masse}
\end{figure}
\subsection{Comparison of the IBM, the IDE and the ODE}
We now compare the IBM and the IDE to the classical chemostat model described by the system of ODE \eqref{eq.chemostat.edo}. The function $\tilde\mu$ is the specific growth rate. The growth model in both the IBM and the IDE is of Monod type, so for the ODE model we also consider the classical Monod kinetics: \begin{align} \label{eq.edo.monod}
\tilde\mu(S)
& =
\mu_{\max} \, \frac{S}{K_s+S}\,. \end{align} The parameters of this Monod law are not given in the initial model and we use a least squares method to determine the value of the parameters $\mu_{\max}$ and $K_s$ which minimize the quadratic distance between $(S_{t},X_{t})_{t\leq T}$ given by \eqref{eq.chemostat.edo} and $(S_{t},X_{t}=\int_{\mathcal{X}} x\,p_{t}(x)\,\rmd x)_{t\leq T}$ given by \eqref{eq.limite.substrat.fort}-\eqref{eq.limite.eid.fort}.
The numerical integration of the ODE \eqref{eq.chemostat.edo} presents no difficulties and is performed by the function \texttt{odeint} of the module \texttt{scipy.integrate} of \texttt{Python} with the default parameters.
First we consider a simulation based on the initial mass density $d(x)$ defined by \eqref{eq.d}. With this initial density both the IDE and the IBM feature a transient phenomenon described in the previous section and illustrated in Figures \eqref{fig.evol.eid} et~\eqref{repartition.masse}.
Figure \ref{fig.edo.ibm.eid} (left) shows a significant difference between the IBM and the IDE on the one hand and the ODE on the other hand, the latter model cannot account for the transient phenomenon. With the first two models, the individual bacteria are withdrawn uniformly and independently of their mass (large mass bacteria has the same probability of withdrawal as small mass bacteria) and at the beginning of the simulation there is a decrease in biomass as at initial state $d(x)$ has a substantial proportion of large bacteria mass. The ODE is naturally not able to account for this phenomenon.
\begin{figure}
\caption{Time evolution of the biomass concentration (top), the substrate concentration (middle) and the concentration trajectories in the phase space (bottom) according to the initial mass distributions \eqref{eq.d} (left) and \eqref{eq.d'} (right). In blue, the trajectories of 60 independent runs of the IBM simulated with $V=3$ l and $N_0=20000$ (left), $N_0=25000$ (right); in green, the mean of the IBM runs; in red, the solution of IDE \eqref{eq.limite.substrat.fort}-\eqref{eq.limite.eid.fort}; in black, the solution of the ODE \eqref{eq.chemostat.edo}. The latter is fitted by the least squares method on the IDE, the parameters of the Monod law \eqref{eq.edo.monod} are $\mu_{\max}=0.341$ and $K_s = 2.862$ in the first case and $\mu_{\max}=0.397$ and $K_s = 3.996$ in the second. As initial densities are different, $N_{0}$ is adapted so that the average initial biomass concentration is the same in both cases. The dilution rate $D$ is 0.2 h$^{-1}$. In the first case, the ODE gives no account for the transient phenomenon described in the previous section, see Figures \eqref{fig.evol.eid} and \eqref{repartition.masse}, while the IDE and the IBM give coherent account for it. In the second case the three models are consistent.}
\label{fig.edo.ibm.eid}
\end{figure}
This phenomenon no longer appears when uses the following density: \begin{align} \label{eq.d'} d'(x)
& =
\Biggl(
\frac{x-0.00035}{0.0003}
\,\left(1-\frac{x-0.00035}{0.0003}\right)
\Biggr)^5 \,
1_{\{0.00035 < x < 0.00065\}}\,. \end{align}
Indeed, from Figure \ref{fig.edo.ibm.eid} (right), there is no longer any biomass decay at the beginning of the simulation and the different simulations are comparable, the ODE and the IDE match substantially.
\subsection{Study of the washout}
\begin{figure}
\caption{Time evolution of the biomass concentration. In blue, 1000 independent realizations of the IBM simulated with $V=0.5$ l and $N_0=30$; in green, the mean of these runs; in red, the solution of the IDE; in black, the solution of the ODE with parameters values $\mu_{\max}=0.482$ and $K_s = 6.741$. The dilution rate $ D $ is 0.275 h$^{-1}$. Among the 1000 independent runs of the IBM, 111 lead to washout while the deterministic models converge to an equilibrium with strictly positive biomass. The mean value of the 1000 runs of the IBM gives account for the washout probability while IDE and ODE models do not account for this question.}
\label{fig.lessivage}
\end{figure}
\begin{figure}
\caption{$\blacktriangleright$ (Top) Evolution of biomass concentration between $t=20$ and $t=90$ h: blue, 1000 independent runs of the IBM; in green, the mean value of these runs; in red the solution of the IDE; in black, the solution to the ODE with parameters $\mu_{\max}=0.814$ and $K_s = 17.547$. The parameters are $V=10$ l and $N_0=10000$, the dilution rate $D$ is 0.5 h$^{-1}$. For both deterministic models, the size of the population decreases exponentially rapidly to 0 but remains strictly positive for any finite time. However, all the runs of the IBM reach washout in finite time $\blacktriangleright$ (Bottom) empirical distribution of the washout time calculated from 7000 independent runs of the IBM and plotted using a time kernel regularization.}
\label{fig.lessivage2}
\end{figure}
One of the main differences between deterministic and stochastic models lies in their way of accounting for the washout phenomenon (or extinction phenomenon in the case of an ecosystem). With a sufficiently small dilution rate $D$, the solutions of the ODE \eqref{eq.chemostat.edo} and the IDE \eqref{eq.limite.substrat.fort}-\eqref{eq.limite.eid.fort} converge to an equilibrium point with strictly positive biomass. In fact, the washout is an unstable equilibrium point and apart from the line corresponding to the null biomass, the complete phase space corresponds to a basin of attraction leading to a solution with a strictly positive biomass asymptotic point. However, from Figure \ref{fig.lessivage}, among the 1000 independent runs of the IBM, 111 of them converge to washout before time $t=1000$ h; so the probability of washout at this instant is approximately 11\%. It may be noted that the IDE and the ODE do not correspond to the average value of the IBM since only the latter may reflect the washout in a finite time horizon.
Now we consider a sufficiently large dilution rate, $D=0.5$ h$^{-1}$, corresponding to the washout conditions. Figure \ref{fig.lessivage2} (left) presents the evolution of the biomass concentration in the different models. The runs of the IBM converge to the washout in finite time whereas both deterministic ODE and IDE models converge exponentially to washout without ever reaching it in finite time. Figure \ref{fig.lessivage2} (right) shows the empirical distribution of the washout time calculated from 7000 independent runs of the IBM. This washout time features a relatively large variance.
\section*{Conclusion}
We proposed a hybrid chemostat model. On the one hand the bacterial population is modeled as an individual-based model: each bacterium is explicitly represented through its mass. On the other hand, the substrate concentration dynamics is represented as a conventional ordinary differential equation. Mechanisms acting on the bacteria are explicitly described: growth, division and up-take. Consumption of the substrate is also described and it is through this mechanism that bacteria interact.
We described an exact Monte Carlo simulation technique of the model. Our main result is the convergence of the IBM to an integro-differential equation model when the population size tends to infinity. There is a convergence in distribution for coupled stochastic processes: the first one with càdlàg trajectories takes values in the set of finite measures on the space of masses, the second one with continuous trajectories takes values in the set of substrate concentration values. The integro-differential equation model limit has been known for many years as population-balance equation and is used in particular for growth-fragmentation models. The numerical tests have allowed us to illustrate this convergence: in large population, the integro-differential equation model accurately reflects the behavior of the IBM and thus, the randomness can be neglected. In small or medium population this randomness is not negligible. We also proposed a numerical test where the classical chemostat model, in terms of two coupled ordinary differential equations, cannot account for transient behavior observed with the integro-differential model as with the IBM. Finally, in the case of washout, thus in small population size, IBM gives account for the random washout time, whereas the conventional model as the integro-differential model merely offers a more limited vision of this phenomenon as an asymptotic convergence to washout, never reached in finite time.
It would be interesting to propose extensive simulations of this model, we can consider several axes. First, our simulations are based on a division rate $\lambda(s,x)$ which does not depend on the substrate concentration~$s$. One can easily consider a rate depending on~$s$, there are indeed such examples \citep[see e.g.][]{henson2003b}. Similarly, in our model the individual growth dynamics is deterministic, it would be appropriate to consider a stochastic individual growth dynamics. To progress in this direction it would be necessary to consider specific cases studies where more explicit growth dynamics would be specified.
It would also be pertinent to develop approximation methods based on moments, but these methods are usually ad hoc and it would be worthwhile to rely on a rigorous and systematic approach.
Finally, there are several relatively immediate extensions of the proposed model. At first, one can imagine extending the model to the case of several species and several types of substrates. One can also model the effects of aggregation of bacteria in the chemostat, for example in the form of flocculation with given rates for aggregation and fragmentation of flocs. We can also consider two classes of bacteria, attached bacteria and planktonic bacteria, to account for a biofilm dynamic.
\section*{Appendices} \addcontentsline{toc}{section}{Appendices}
\section{Skorohod topology} \label{appendix.skorohod}
The space of finite measures $\MM_{F}(\mathcal{X})$ is equipped with the topology of the weak convergence, that is the smallest topology for which the applications $\zeta\to\crochet{\zeta,f}=\int_{\XX}f(x)\,\zeta(\rmd x)$ are continuous for any $f\in\mathcal{C}(\mathcal{X})$. This topology is metrized by the Prokhorov metric: \begin{align*}
&d_{\textrm{\tiny\rm PR}}(\zeta,\zeta')
\eqdef
\inf\Bigl\{
\varepsilon >0\,;\,
\zeta(F)\leq \zeta'(F^\varepsilon)+\varepsilon\,,\ \\[-0.3em]
&\qquad\qquad\qquad\qquad\qquad\,\,\,
\zeta'(F)\leq \zeta(F^\varepsilon)+\varepsilon\,,\
\textrm{ for all closed }F\subset \XX
\Bigr\} \end{align*}
where $F^\varepsilon\eqdef\{x\in\XX;\inf_{y\in F}|x-y|<\varepsilon\}$ \citep[see][Appendix A2.5]{daley2003a}. The Prokhorov distance is bounded by the distance of the total variation $d_{\textrm{\tiny\rm TV}}(\zeta,\zeta')=\normtv{\zeta-\zeta'}$ associated with the norm defined by: \begin{align*}
\normtv{\zeta}
\eqdef \sup_{A\in\BB(\XX)}|\zeta(A)+\zeta(A^c)|
= \zeta_{+}(\XX)+ \zeta_{-}(\XX)
= \sup_{\substack{f\textrm{ continuous}\\\norm{f}_{\infty}\leq 1}}
|\crochet{\zeta,f}| \end{align*} for any finite and signed measure $\zeta$ where $\zeta=\zeta_{+}-\zeta_{-}$ is the Hahn-Jordan decomposition of $\zeta$.
The space $\D([0,T],\MM_{F}(\mathcal{X}))$ is equipped with the Skorohod metric $d_{\textrm{\tiny\rm S}}$. Instead of giving the definition of this metric \citep[see][Eq. (5.2) p. 117]{ethier1986a} we recall a characterization of the convergence for this metric.
According to the first characterization \cite[Prop. 5.3, p. 119]{ethier1986a}, a sequence $(\zeta^{n})_{n\in\N}$ converges to $\zeta$ in $\DD([0,T],\MM_{F}(\mathcal{X}))$, i.e. $d_{\textrm{\tiny\rm S}}(\zeta^{n},\zeta)\to 0$, if and only if there exists a sequence $\lambda_{n}(t)$ of time change functions such that, for all $n$, $t\to \lambda_{n}(t)$ is strictly increasing and continuous with $\lambda_{n}(0)=0$, $\lambda_{n}(t)\to_{t\to\infty}\infty$, satisfying: \begin{align} \label{appendix.skorohod.cara.2}
\sup_{0\leq t\leq T}
d_{\textrm{\tiny\rm PR}}({\zeta_t^{n},\zeta_{\lambda_{n}(t)}})
\xrightarrow[n\to\infty]{} 0 \end{align} and \begin{align} \label{appendix.skorohod.cara.3}
\sup_{0\leq t\leq T}|\lambda_{n}(t)-t|\to 0\,. \end{align}
If $(\zeta^{n})_{n\in\N}$ converges to $\zeta$ in $\DD([0,T],\MM_{F}(\mathcal{X}))$ and if $\zeta\in\CC([0,T],\MM_{F}(\XX))$ then in: \[
\sup_{0\leq t\leq T}d_{\textrm{\tiny\rm PR}}(\zeta^{n}_{t},\zeta_{t})
\leq
\sup_{0\leq t\leq T}d_{\textrm{\tiny\rm PR}}(\zeta^{n}_{t},\zeta_{\lambda_n(t)})
+
\sup_{0\leq t\leq T}d_{\textrm{\tiny\rm PR}}(\zeta_{\lambda_n(t)},\zeta_{t}) \] the first term of the right-hand side tends to 0 because of \eqref{appendix.skorohod.cara.2};
the second one tends to 0 because of \eqref{appendix.skorohod.cara.3} and the uniform continuity of $\zeta$ in $[0,T]$. This proves that $\zeta^{n}$ converges to $\zeta$ in $\DD([0,T],\MM_{F}(\mathcal{X}))$ also for the uniform metrics.
\section{Numerical integration scheme for the IDE} \label{appendix.schema.num}
To numerically solve the system of integer-differential equations \eqref{eq.limite.substrat.fort}-\eqref{eq.limite.eid.fort}, that is the strong version of the limit system \eqref{eq.limite.substrat.faible}-\eqref{eq.limite.eid.faible}, we make use of finite difference schemes.
Given a time step $\Delta t$ and a mass step $\Delta x = L/I$, with $I \in \mathbb{N}^*$, we discretize the time and mass space with: \begin{align*}
t_n & = n \, \Delta t \,
&
x_i & = i \, \Delta x\,. \end{align*} We introduce the following approximations: \begin{align*}
p_{n,i} & \simeq p_{t_n}(x_i) \,,
&
s_n & \simeq S_{t_n}\,. \end{align*} We also suppose first that at the initial time step there is no individual with null mass in the vessel, i.e. $p_{0,0}=0$; and second that individual with null mass cannot be generated during the cell division step, i.e. $q$ is regular with $q(0)=0$. This assumption was not necessary in the mathematical development presented in the previous sections but is naturally required to obtain reasonable mass of individuals in the simulation.
For time integration we use an explicit Euler scheme, for space integration, an uncentered upwind difference scheme, which leads to the coupled integration scheme: \begin{align*}
\frac{p_{n+1,i}-p_{n,i}}{\Delta t}
& =
- \rho_{\textrm{\tiny\rm\!\! g}}(s_n,x_i)\, \frac{p_{n,i}-p_{n,i-1}}{\Delta x}
- \frac{\partial}{\partial x}\rho_{\textrm{\tiny\rm\!\! g}}(s_n,x_i)\,p_{n,i} \\
&\qquad\qquad
- \bigl(\lambda(s_n, x_i)+D \bigr)\,p_{n, i}
+ 2\, \Delta x \, \sum_{j=1}^{I}
\frac{\lambda(s_n, x_j)}{x_j}\,
q\left(\frac{x_i}{x_j} \right)\,
p_{n,j} \,, \\
\frac{s_{n+1}-s_n}{\Delta t}
& =
D\,({\mathbf s}_{\textrm{\tiny\rm in}} - s_n)
- \frac{k}{V} \, \Delta x \, \sum_{j=1}^{I} \rho_{\textrm{\tiny\rm\!\! g}}(s_n, x_j) \, p_{n,j} \end{align*} for $n \in \mathbb{N}$ and $i = 1, \cdots I$, with the boundary condition: \begin{align*}
p_{n+1,0} = 0 \end{align*} and given initial conditions $p_{0,i}$ and $s_{0}$.
We finally get: \begin{align*}
p_{n+1,i}
& =
p_{n,i} + \Delta t \;
\Biggl\{
- \rho_{\textrm{\tiny\rm\!\! g}}(s_n,x_i)\, \frac{p_{n,i}-p_{n,i-1}}{\Delta x}
- \frac{\partial}{\partial x}\rho_{\textrm{\tiny\rm\!\! g}}(s_n,x_i)\,p_{n,i} \\
& \qquad\qquad
- \bigl(\lambda(s_n, x_i)+D \bigr) \, p_{n, i}
+ 2\, \Delta x \,
\sum_{j=1}^{I} \frac{\lambda(s_n, x_j)}{x_j}\,
q\left(\frac{x_i}{x_j} \right)\, p_{n,j}
\Biggr\} \\
s_{n+1}
& =
s_n + \Delta t \,
\Biggl\{
D\,({\mathbf s}_{\textrm{\tiny\rm in}} - s_n)
- \frac{k}{V} \, \Delta x \, \sum_{j=1}^{I} \rho_{\textrm{\tiny\rm\!\! g}}(s_n, x_j) \, p_{n,j}
\Biggl\} \end{align*} for $n \in \mathbb{N}$ and $i = 1, \cdots I$ with boundary condition $p_{n+1,0} = 0$ and given initial conditions $p_{0,i}$ and $s_{0}$.
\end{document} | arXiv |
\begin{document}
\title{On supersaturation for oddtown and eventown}
\author{Xin~Wei, Yuhao~Zhao, Xiande~Zhang~and~Gennian~Ge \thanks{\emph{2020 Mathematics Subject Classifications}: 05D05.} \thanks{This project was supported by the National Key Research and Development Program of China under Grant 2020YFA0712100, Grant 2018YFA0704703 and Grant 2020YFA0713100, the National Natural Science Foundation of China under Grant 11971325, Grant 12171452 and Grant 12231014, and Beijing Scholars Program.}
\thanks{X. Wei ({\tt [email protected]}) and Y. Zhao ({\tt [email protected]}) are with the School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China.}
\thanks{X. Zhang ({\tt [email protected]}) is with the School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China, and Hefei National Laboratory, Hefei, 230088, China.}
\thanks{G. Ge ({\tt [email protected]}) is with the School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China. }
} \maketitle
\begin{abstract}
We study the supersaturation problems of oddtown and eventown. Given a family $\mathcal A$ of subsets of an $n$ element set, let $op(\mathcal A)$ denote the number of distinct pairs $A,B\in \mathcal A$ for which $|A \cap B|$ is odd. We show that if $\mathcal A$ consists of $n+s$ odd-sized subsets, then $op(\mathcal A)\geq s+2$, which is tight when $s\le n-4$. This disproves a conjecture by O'Neill on the supersaturation problem of oddtown. For the supersaturation problem of eventown, we show that for large enough $n$, if $\mathcal A$ consists of $2^{\lfloor n\slash 2\rfloor}+s$ even-sized subsets, then $op(\mathcal A)\ge s\cdot2^{\lfloor n\slash 2\rfloor-1} $ for any positive integer $s\le 2^{\lfloor\frac n 8\rfloor}\slash n$. This partially proves a conjecture by O'Neill on the supersaturation problem of eventown. Previously, the correctness of this conjecture was only verified for $s=1$ and $2$. We further provide a twice weaker lower bound in this conjecture for eventown, that is $op(\mathcal{A})\ge s\cdot 2^{\lfloor n/2\rfloor-2}$ for general $n$ and $s$ by using discrete Fourier analysis. Finally, some asymptotic results for the lower bounds of $op(\mathcal A)$ are given when $s$ is large for both problems.
\end{abstract}
\begin{IEEEkeywords} \boldmath oddtown, eventown, supersaturation, intersecting set families. \end{IEEEkeywords}
\section{Introduction}
In extremal set theory, given a finite family (i.e., a collection of subsets) $\mathcal F$ and a restriction on the intersection of two subsets, the {\it restricted intersection problem} asks for the maximum size of a subfamily $\mathcal A\subset\mathcal F$ such that any two different members of $\mathcal A$ satisfy the restricted intersection. Many fundamental and classical results in extremal combinatorics can be stated as a restricted intersection problem. Let $[n]:=\{1, 2, \ldots, n\}$, $2^{[n]}$ denote the collection of all subsets of $[n]$, and $\binom{[n]}{k}$ denote the collection of all $k$-subsets of $[n]$. Then the celebrated Erd\H{o}s-Ko-Rado theorem on intersecting families~\cite{erdos1961intersection} can be viewed as a solution to the restricted intersection problem with $\mathcal F=\binom{[n]}{k}$ for $n\ge 2k$ when restricting empty pairwise intersections. As another example, Sperner's theorem on antichains~\cite{sperner1928satz} states that the maximum size of a subfamily $\mathcal A$ of $\mathcal F=2^{[n]}$ with restricted intersection $A\backslash B {=A\cap B^{c}} \neq \emptyset$ for any $A, B\in\mathcal A$ is $\binom n {\lfloor\frac n 2\rfloor}$. There are several other well-studied restricted intersection problems, such as $L$-intersecting families and bounded symmetric differences. For more information, one can refer to~\cite{fisher1940174, frankl2017stability, gerbner2018extremal, snevily1995generalization, kleitman1966combinatorial, gao2022stability}.
In this paper, we focus on the \emph{oddtown} and \emph{eventown} problems, which are also restricted intersection problems. Both of them share the same restriction that intersections of every two different members have even size. The difference is that oddtown requires the family $\mathcal F$ to consist of all odd-sized subsets of ${[n]}$, while eventown requires all even-sized subsets.
Formally, let $\mathcal A=\{A_1, A_2, \ldots, A_m\}$ be a family of subsets of $[n]$. We say $\mathcal A$ is an \emph{oddtown} (resp. \emph{eventown}) \emph{family} if all its sets have odd (resp. even) size and $$|A_i\cap A_j|\text{ is even for }1\le i<j\le m.$$ The maximum size of an oddtown family is $n$, and the maximum size of an eventown family is $2^{\lfloor\frac n 2\rfloor}$, which were determined by Berlekamp~\cite{berlekamp1969subsets} and Graver~\cite{graver1975boolean} independently. Their methods highlighted the linear algebra method~\cite{babai1988linear} in extremal combinatorics. Numerous extensions and variants of the oddtown and eventown problems can be found in the literature \cite{deza1983functions, o2022note, vu1999extremal, szabo2005exact,FRANKL1983215, vu1997extremal, sudakov2018two}, such as extending modulo $2$ to modulo general $\ell$, which is known as $\ell$-even\slash oddtown problem~\cite{babai1988linear, FRANKL1983215}, and extending pairwise restricted intersections to $k$-wise restricted intersections \cite{vu1997extremal, sudakov2018two}.
Recently in \cite{o2021towards}, O'Neill initiated the study of supersaturation problem for oddtown and eventown: if $\mathcal A\subset 2^{[n]}$ is a family of more than $n$ odd-sized subsets, or a family of more than $2^{\lfloor\frac n 2\rfloor}$ even-sized subsets, how many pairs of members in $\mathcal A$ must violate the intersecting restriction, that is, have an odd number of elements in common?
Supersaturation versions of other foundational problems in extremal set theory have also attracted a lot of attention recently. For example, works like~\cite{balogh2018kleitman, balogh2018structure, leader2003set, das2015sperner, das2016minimum} gave the supersaturation versions for Erd\H{o}s-Ko-Rado theorem and Sperner's theorem.
For a given set family $\mathcal A$, the {\it odd pair number}, denoted by $op(\mathcal A)$, is the number of pairs of distinct members $A, B\in \mathcal A$ such that $|A\cap B|$ is odd. In \cite{o2021towards}, O'Neill constructed a family $\mathcal A=\{\{i\}\}_{i\in [n]}\cup \mathcal C_s$, where $\mathcal C_s$ consists of exactly $s$ members from the extremal oddtown family: a collection of vertex disjoint $K_4^{(3)}$ - i.e., all triples on four vertices. It is easy to check that $\mathcal A$ is a family of $n+s$ odd-sized subsets, and $op(\mathcal A)=3s$. O'Neill proved that this is the best possible result for $s=1$: \begin{theorem}[\hspace{-0.01em}\cite{o2021towards}]\label{thm_old_oddtown}
Let $n\ge 1$ and $\mathcal A\subset 2^{[n]}$ consists of odd-sized subsets with $|\mathcal A|\ge n+1$. Then $op(\mathcal A)\ge 3$. \end{theorem}
O'Neill further conjectured that: \begin{conjecture}[\hspace{-0.01em}\cite{o2021towards}]\label{conj_odd_town}
Let $n\ge 1$ and fix $1\le s\le n$. If $\mathcal A\subset 2^{[n]}$ is a family of odd-sized subsets with $|\mathcal A|\ge n+s$, then $op(\mathcal A)\ge 3s$. \end{conjecture}
Our first main result is to show that Conjecture~\ref{conj_odd_town} is not true. In fact, when $n\ge s+4$, we can construct an odd-sized family of size $n+s$ but with odd pair number $s+2$, which is much smaller than the lower bound in Conjecture~\ref{conj_odd_town}. We further show that $s+2$ is best possible for any $s$ and $n\geq s+4$. The statement is summarized below. For brevity, we use the term ``odd-sized (resp. even-sized) family'' to present a family consisting of odd-sized (resp. even-sized) subsets. \begin{theorem}\label{thm_for_oddtown}
Let $n\ge 1$ and fix $1\le s\le n-4$. Any odd-sized family $\mathcal A\subset 2^{[n]}$ with $|\mathcal A|\ge n+s$ must satisfy $op(\mathcal A)\ge s+2$, and the lower bound is tight. \end{theorem}
Theorem~\ref{thm_for_oddtown} focuses on the supersaturation problem of oddtown when $|\mathcal A|$ exceeds the corresponding extremal number $n$ by some $s$ smaller than $n$. One can also ask the same question when $s$ is larger than $n$. Although we do not compute the exact value for the smallest odd pair number, we have the following asymptotic result when $n$ goes to infinity. The standard asymptotic notations like $o$, $O$ and $\Theta$ are used in this paper to compare two functions when $n$ goes to infinity, and all logarithms are under base $2$ by default.
\begin{theorem}\label{theorem_oddtown_approximation}
Given some positive integer valued function $s=s(n)$, denote $M_o(s, n)$ as the minimum number of $op(\mathcal A)$ among all odd-sized subfamily $\mathcal A\subset 2^{[n]}$ with size $|\mathcal A|=n+s$.
If $s=cn+ o(n)$ for some constant $c>0$, \begin{equation}\label{eq-mo2}
M_o(s, n)=\big(\binom{\lfloor c\rfloor+1}2+(\lfloor c\rfloor+1)(c- \lfloor c\rfloor)\big)n+ o(n).
\end{equation}
\end{theorem}
For the eventown case, O'Neill~\cite{o2021towards} constructed a family of $2^{ n\slash2}+s$ even-sized subsets whose odd pair number is $s\cdot 2^{ n\slash2-1}$ when $n$ is doubly even. His construction is as follows.
Suppose $n=2k=4\ell$ and let $X_1, X_2, \ldots, X_\ell\subset[n]$ be pairwise disjoint subsets with $X_i=\{4i-3, 4i-2, 4i-1, 4i\}$. For each $X_i$, define four subsets $A_{2i-1}=\{4i-3, 4i-2\}$, $A_{2i}=\{4i-1, 4i\}$, $B_{2i-1}=\{4i-3, 4i\}$, and $B_{2i}=\{4i-2, 4i-1\}$. Then define two collections, \begin{equation} \mathcal A=\{\cup_{j\in J}A_j: J\subset [k]\}\text{ and }\mathcal B=\{\cup_{j\in J}B_j: J\subset [k]\}. \end{equation}
Observe that both $\mathcal A$ and $\mathcal B$ are extremal eventown families. Moreover for each $B\in\mathcal B\backslash\mathcal A$, $op(\mathcal A\cup \{B\})=2^{k-1}$. Note that $|\mathcal B\backslash\mathcal A|=2^k-2^\ell$ by linear algebra. For any $s\in [2^k-2^\ell]$, consider $\mathcal A'$ formed by $\mathcal A$ and $s$ distinct members from $\mathcal B\backslash\mathcal A$. Then $|\mathcal A'|=2^k+s$ and $op(\mathcal A')=s\cdot 2^{k-1}$. O'Neill~\cite{o2021towards} proved that this is best possible for $s=1,2$ and further conjectured this is true for a large range of $s$.
\begin{conjecture}\label{conj_even_town}
Let $n\ge 1$ and fix $1\le s\le 2^{\lfloor n\slash2\rfloor}- 2^{\lfloor n\slash4\rfloor}$. If $\mathcal A\subset 2^{[n]}$ consists of even-sized subsets with $|\mathcal A|\ge 2^{\lfloor n\slash2\rfloor}+s$, then $op(\mathcal A)\ge s\cdot 2^{\lfloor n\slash2\rfloor-1}$. \end{conjecture}
Recently progress on Conjecture~\ref{conj_even_town} was made in \cite{antipov2022lovasz}, where half of the lower bound for even $n$ (but much weaker bound for odd $n$) and general $s$ was proved by spectral analysis.
\begin{theorem}[\hspace{-0.01em}\cite{antipov2022lovasz}]\label{thm_old_eventown}
Let $n, s$ be positive integers. If an even-sized family $\mathcal A\subset 2^{[n]}$ satisfies $|\mathcal A|\ge 2^{ n\slash2}+s$, then $op(\mathcal A)\ge s\cdot 2^{\lfloor n\slash2\rfloor-2}$. \end{theorem}
Our next contribution is to show that Conjecture~\ref{conj_even_town} is true for a wide range of $s$ and for sufficiently large $n$ by using extremal graph theory. We state it below. \begin{theorem}\label{thm_for_eventown}
Let $n$ be a large enough integer and fix $s\in[2^{\lfloor\frac n 8\rfloor}\slash n]$. Any even-sized family $\mathcal A\subset 2^{[n]}$ with $|\mathcal A|\ge 2^{\lfloor n\slash2\rfloor}+s$ satisfies $op(\mathcal A)\ge s\cdot 2^{\lfloor n\slash2\rfloor-1}$. \end{theorem}
Similar to the supersaturation problem of oddtown, one can consider the case when $s$ is larger than $2^{\lfloor n\slash2\rfloor}$. When $| \mathcal{A}| \ge 2^{(1-\epsilon)n}$ for some $\epsilon\in(0,1/2)$, O'Neill \cite{o2021towards} proposed the following problem. \begin{problem}
Let $\epsilon\in\left( 0,\frac{1}{2}\right) $ and $n$ be sufficiently large. Determine the maximum value $f_n(\epsilon)$ so that if $\mathcal{A}\subseteq 2^{[n]}$ is an even-sized family with $\left| \mathcal{A}\right| \ge 2^{(1-\epsilon)n}$, then $op(\mathcal{A})\ge f_n(\epsilon)\binom{|\mathcal{A}|}{2}$. \end{problem}
We show that when $| \mathcal{A}| \ge 2^{(1-\epsilon)n}$ for any given $\epsilon\in(0,1/2)$, the density of $op(\mathcal A)$ always approaches $\frac12$. The formal statement is as follows.
\begin{theorem}\label{thm_eventownlar} Let $\epsilon\in\left( 0,\frac{1}{2}\right) $ and $n\ge {1\slash\epsilon}$. We have $$
\frac{1}{2}\left( 1-2^{\left( \epsilon-\frac{1}{2}\right) n} \right) \le f_n(\epsilon) \le \begin{cases} \frac{1}{2}, & n\text{ odd;}\\ \frac12(1-\frac1{2^{n-1}-1}), & n\text{ even.} \end{cases} $$ Hence for fixed $\epsilon\in \left( 0,\frac{1}{2} \right)$, we have $\lim\limits_{n\rightarrow \infty} f_n(\epsilon) = \frac{1}{2}$.
\end{theorem}
By improving an intermediate result in the proof of Theorem~\ref{thm_eventownlar}, we show that half of the lower bound in Conjecture~\ref{conj_even_town} is true for general $n$ and $s$. Thus it completes the result in Theorem~\ref{thm_old_eventown} from \cite{antipov2022lovasz}.
\begin{theorem}\label{thm_complete_old_eventown}
For any positive integers $n$ and $s$, let $\mathcal A\subset 2^{[n]}$ be an even-sized family with $|\mathcal A|\ge 2^{\lfloor n\slash2\rfloor+s},$ then $op(\mathcal A)> s\cdot2^{\lfloor n\slash2\rfloor-2}$. \end{theorem}
The proofs of Theorem~\ref{thm_eventownlar} and Theorem~\ref{thm_complete_old_eventown} apply Fourier analysis on finite abelian groups~\cite{stein2011fourier}.
The rest of this paper is organized as follows. In Section~\ref{sec_preliminary}, we introduce some necessary notations and basic results in Fourier analysis, and then give a glance at the supersaturation problem of Tur\'an theorem. We prove Theorems~\ref{thm_for_oddtown}-\ref{theorem_oddtown_approximation} in Section~\ref{sec_supersatuation_oddtown}, and Theorems~\ref{thm_for_eventown}-\ref{thm_complete_old_eventown} in Section~\ref{sec_supersaturation_eventown}. Finally, a conclusion and some remarks are listed in Section~\ref{sec-conc}.
\section{Preliminary}\label{sec_preliminary}
We begin with some useful notations used throughout this paper. For two integers $a\le b$, we always use $[a, b]$ to denote the set of consecutive integers from $a$ to $b$, i.e., $\{a, a+1, \ldots, b\}$. For any set $S$ and positive integer $k$, $\binom S k$ stands for the collection of all $k$-sized subsets of $S$, and $2^S$ stands for the collection of all subsets of $S$. For a subset $A\subseteq[n]$, let $\bm v_A\in \mathbb F_2^n$ be the characteristic vector of $A$, that is, for any $j\in [n]$, $\bm v_A(j)=1$ if and only if $j\in A$. For a vector subset $W$ of $\mathbb F_2^n$, let $W^\perp=\{\bm v\in \mathbb F_2^n: (\bm v, \bm u)=0, ~ \forall\bm u \in W\}$, where $(\cdot , \cdot)$ means the natural inner product in $\mathbb F_2^n$. If $W=\{\bm u\}$ for some $\bm u$, we simply write $\bm u^\perp$ instead of $\{\bm u\}^\perp$.
Given a graph $H=(V, E)$, let $v(H)$ and $e(H)$ denote the numbers of vertices and edges in $H$, respectively. For a vertex set $U\subset V$, the induced subgraph of $H$ on set $U$ is denoted by $H[U]$, which has vertex set $U$ and edge set $E\cap\binom U 2$. Similarly, for two disjoint $U_1, U_2\subset V$, let $H[U_1, U_2]$ denote the induced bipartite subgraph of $H$ with two parts $U_1$ and $U_2$ and let $e(U_1, U_2)$ denote the number of edges in $H[U_1, U_2]$. The complement of $H$ is denoted as $\overline H$. For any vertex subset $U\subset V$, the neighborhood of $U$ on $H$, denoted by $N_H(U)$, is the set of vertices in $V\setminus U$ which is adjacent to at least one vertex in $U$. If $U=\{v\}$, we simply write $N_H(v)$. The degree of $v$ is denoted by $d_H(v)$. The subscript $H$ will be omitted if $H$ is clear.
For any set family $\mathcal A\subseteq 2^{[n]}$, the \emph{odd pair graph} of $\mathcal A$, denoted by $H(\mathcal A)$, is constructed in the following way: the vertices are members in $\mathcal A$, and two different members $A, B\in \mathcal A$ are adjacent if and only if $|A\cap B|$ is odd. Consequently, the number of edges in $H(\mathcal A)$ is exactly $op(\mathcal A)$. There is another way to construct the odd pair graph: let $V\subseteq \mathbb F_2^n$, and define graph $H(V)$ as vertex set $V$ and edge set $\{\{\bm u, \bm v\}: \bm u\ne \bm v\in V; (\bm u, \bm v)=1\}$. It is easy to check that under bijection: $A\mapsto \bm v_A$ from $2^{[n]}$ to $\mathbb F_2^n$, these two constructions are equivalent when $V$ is the image of $\mathcal A$. Thus when considering the odd pair graph, a vertex, its corresponding subset $A\subseteq [n]$, and its characteristic vector $\bm v_\mathcal A \in \mathbb F_2^n$ are seen as the same object, and sometimes they share the same notation. Denote the odd pair graph of the largest collection $2^{[n]}$ as $H_0=H(2^{[n]})=H(\mathbb F_2^n)$. Under these notations, for any subfamilies $\mathcal A_1\subseteq \mathcal A_2\subseteq 2^{[n]}$, $H(\mathcal A_1)$ is the induced subgraph of $ H(\mathcal A_2)$ on $\mathcal A_1$, and specially, $H(\mathcal A_1)=H_0[\mathcal A_1]$.
\subsection{Fourier analysis}
Given a finite abelian group $G$ with the additive notation, a {\it character} on $G$ is defined as a complex valued function $\chi: G\to S^1$, where $S^1$ means the unit circle in the complex plane, such that for all $a, b\in G$, $\chi(a+b)=\chi(a)\chi(b)$. Denote the set of all characters of $G$ as $\widehat G$. Then $\widehat G=Hom(G, S^1)$ is a multiplicative abelian group with multiplication defined by $(\chi_1\cdot\chi_2)(a)=\chi_1(a)\chi_2(a)$ for all $a\in G$. Since group $G$ is finite abelian, one can use the fundamental theorem for finite abelian groups~\cite{kurzweil2004theory} to show that $\widehat{G}\cong G$, where $\cong$ means the group isomorphism. In particular, $|\widehat G|=|G|$.
Denote the vector space of complex-valued functions on $G$ as $L(G)$. Define a Hermitian inner product on $L(G)$ by
$$(f, g)=\frac 1{|G|}\sum_{a\in G}f(a)\overline{g(a)}, \text{ for any }f, g\in L(G).$$
It can be proved that the characters of $G$ form an orthonormal basis of $L(G)$ with the inner product defined above. Given a function $f\in L(G)$, define the {\it Fourier transform} of $f$ as a function $\hat f: \widehat G\to \mathbb C$, such that for any $\chi\in\widehat G$, $$\hat f(\chi)=(f, \chi)=\frac 1{|G|}\sum_{a\in G}f(a)\overline{\chi(a)},$$
which leads to the following {\it Fourier inversion formula}, $$f=\sum_{\chi\in\widehat G}(f, \chi)\chi= \sum_{\chi\in\widehat G}\hat f(\chi)\chi.$$ Define the norm of $f\in L(G)$ from the inner product as $\Vert f\Vert=(f, f)^{1\slash 2}$. Then the following is the corresponding {\it Plancherel formula,}
$$\Vert f\Vert^2=\sum_{\chi\in\widehat G}|\hat f(\chi)|^2.$$
\subsection{Supersaturation of Tur\'an theorem} Next we introduce some results on the supersaturation problem of Tur\'an theorem~\cite{erdHos1983supersaturated}. Denote $ex(n, K_t)$ as the maximum possible number of edges in a graph on $n$ vertices which does not contain $K_t$ as a subgraph. Tur\'an \cite{turan1941extremalaufgabe} proved that for any positive integers $n$ and $r$, $ex(n, K_{r+1})=e(T_{n, r})$, where $T_{n, r}$ is the unique $n$ vertex $r$-partite complete graph with each part of size $\lfloor\frac n r\rfloor$ or $\lceil\frac n r\rceil$. An approximate form of Tur\'an theorem shows that: \begin{theorem}\label{coro_turan} For positive integers $n$ and $r$, $$ex(n, K_{r+1})\le\big(\frac{r-1}r \big)\frac{n^2}2.$$ \end{theorem} If the edge density of some graph $G$ on $n$ vertices exceeds the density of Tur\'an graph $T_{n, r}$, which is $\frac{r-1}r$, then a large amount of forbidden structures, i.e., $K_{r+1}$ as subgraphs of $G$, will emerge. \begin{theorem}[\hspace{-0.01em}\cite{erdHos1983supersaturated}]\label{thm_graph_supersaturation} For every $\epsilon>0$, there exist some $\delta=\delta(\epsilon)>0$ and integer $n_0=n_0(\epsilon)$ such that every graph on $n\ge n_0$ vertices with at least $(\frac{r-1}r+\epsilon)\binom n 2$ edges contains at least $\delta n^{r+1}$ copies of $K_{r+1}$ as a subgraph. \end{theorem}
\section{Supersaturation problem of oddtown}\label{sec_supersatuation_oddtown}
\subsection{Proof of Theorem~\ref{thm_for_oddtown}} We first prove the lower bound.
\begin{lemma}\label{lemma_oddtown_lower_bound}
Let $n\ge s\ge 1$. Any odd-sized family $\mathcal A\subset 2^{[n]}$ with $|\mathcal A|\ge n+s$ must satisfy $op(\mathcal A)\ge s+2$. \end{lemma} \begin{proof} We prove it by induction on the value of $s$. For the base case when $s=1$, the statement is true from Theorem~\ref{thm_old_oddtown}. Assume that the statement is true for all $s\le k$ for some $k>0$. Consider the case when $s=k+1$.
Suppose on the contrary, there exists some odd-sized family $\mathcal A$ on $[n]$ for some $n\ge k+1$ with $|\mathcal A|\ge n+k+1$ and $op(\mathcal A)\le k+2$. Since $op(\mathcal A)\ge 1$, we can choose a member $A\in\mathcal A$ such that $A$ is not an isolated vertex in $H(\mathcal A)$. Denote $\mathcal A'=\mathcal A\backslash\{A\}$, which is also an odd-sized collection on $[n]$, but with $|\mathcal A'|\ge n+k$ and $op(\mathcal A')\le op(\mathcal A)-1\le k+1$. This leads to a contradiction to the statement when $s=k$. \end{proof}
It is left to give a construction of odd-sized family $\mathcal A$ with $|\mathcal A|=n+s$ and $op(\mathcal A)=s+2$ when $n\geq s+4$. We first give a construction when $n=s+4$.
\begin{construction}\label{cons_A_s} For any integer $s\ge 1$, let $n=s+4$. Construct a family $\mathcal A_s$ of $2s+4$ odd-sized subsets of $[n]$ as follows. \begin{itemize}
\item[(1)] There are six special subsets $\{1, 2, 3\}$, $\{1, 2, 4\}$, $\{1, 2, 5\}$, $\{1, 3, 4\}$, $\{1, 3, 5\}$, $\{1, 4, 5\}$. We call the collection of those six subsets the center of $\mathcal A_s$, denoted by $\mathcal{C}$, which is irrelevant to the value of $s$. \item[(2)] The remaining $2s-2$ subsets form $s-1$ pairs: $\{i+5\}$ and $\{ 2, 3, 4, 5, i+5\}$, $i\in [s-1]$. \end{itemize} \end{construction}
\begin{example} When $s=3$, $n=7$, the family $\mathcal A_3$ consists of ten odd-sized subsets in $[7]$: six subsets in the center, i.e., $\{1, 2, 3\}$, $\{1, 2, 4\}$, $\{1, 2, 5\}$, $\{1, 3, 4\}$, $\{1, 3, 5\}$ and $\{1, 4, 5\}$, and four additional subsets forming two pairs, i.e., $\{6\}$, $\{2, 3, 4, 5, 6\}$ and $\{7\}$, $\{2, 3, 4, 5, 7\}$. \end{example}
Next, we show that the odd pair number of $\mathcal A_s$ is $s+2$.
\begin{lemma}\label{lemma_A_s_is_good} For any integer $s\ge 1$, $op(\mathcal A_s)=s+2$. \end{lemma} \begin{proof}
It is easy to check that $op(\mathcal{C})=3$, and the two subsets in each of the $s-1$ pairs out of the center have odd-sized intersection. We claim that no other two members in $\mathcal A_s$ can have odd-sized intersections, and hence $op(\mathcal A_s)=op(\mathcal{C})+(s-1)=s+2$. There are only two possible cases need to be checked: (1) exactly one member is in the center; (2) the two members are from different pairs.
For the first case, without loss of generality, suppose $A_1\in \mathcal{C}$ and $A_2$ is from the $i$th pair for some $i\in [s-1]$. Then $A_1=\{1, x, y\}$ for some $x\ne y\in [2, 5]$. If $A_2={i+5}$, then $A_1\cap A_2=\emptyset$. If $A_2=\{2, 3, 4, 5, i+5\}$, then $A_1\cap A_2=\{x, y\}$. Both situations lead to even-sized intersections.
For the second case, there exist some $i\ne j\in [s-1]$ such that $A_1$ and $A_2$ are in the $i$th and the $j$th pair, respectively. Then $A_1$ is one of $\{i+5\}$ and $\{2, 3, 4, 5, i+5\}$, while $A_2$ is one of $\{j+5\}$ and $\{2, 3, 4, 5, j+5\}$. All of the four situations lead to even-sized intersections.
\end{proof}
From the proof of Lemma~\ref{lemma_A_s_is_good}, we see that the odd pair graph $H(\mathcal A_s)$ is a perfect matching on $2s+4$ vertices, which does not contain an extremal oddtown family of size $s+4$. Now we extend the result in Lemma~\ref{lemma_A_s_is_good} to any $n\geq s+4$. \begin{lemma}\label{lemma_s+2_is_achivable}
For any fixed integers $s\ge 1$ and $n\ge s+4$, there exists an odd-sized family $\mathcal A\subset 2^{[n]}$ with $|\mathcal A|=n+s$ and $op(\mathcal A)=s+2$. \end{lemma} \begin{proof} Consider $\mathcal A$ consisting of the family $\mathcal A_s$ together with the following subsets: $\{i\}$, $i\in [s+5,n]$. Then $\mathcal A$ is an odd-sized family on $[n]$ with cardinality $(2s+4)+(n-s-4)=n+s$. It is easy to see those new-added subsets with single elements do not add new odd-sized intersections. So by Lemma~\ref{lemma_A_s_is_good}, $op(\mathcal A)=op(\mathcal A_s)=s+2$. \end{proof}
Combining Lemma~\ref{lemma_oddtown_lower_bound} and Lemma~\ref{lemma_s+2_is_achivable}, Theorem~\ref{thm_for_oddtown} is proved.
We remark that the construction in the proof of Lemma~\ref{lemma_s+2_is_achivable} is not the only extremal construction under the equivalence of permutation. There is another method to construct good odd-sized families with large size and small odd pair number based on the existence of eventown families. \begin{construction}~\label{cons_A_m_I} For given integers $n> m\ge 1$, divide $[n]$ into two parts $[m]$ and $[m+1, n]$. Let $\mathcal E$ be an eventown family on $[m]$. Define $\mathcal A(\mathcal E, m,n)$ as the product family of $\mathcal E$ and $\{\{i\}: i\in[m+1, n]\}$. That is, $$\mathcal A(\mathcal E, m,n)=\{E\cup \{i\}: E\in\mathcal E, i\in [m+1, n]\}.$$ \end{construction}
Then $\mathcal A(\mathcal E, m,n)$ is a family on $[n]$ of $|\mathcal E|(n-m)$ odd-sized subsets. For any two different sets $A_1= E_1\cup \{i_1\}$ and $A_2= E_2\cup \{i_2\}$ from $\mathcal A(\mathcal E, m,n)$, where $E_1, E_2\in \mathcal E$ and $i_1, i_2\in[m+1, n]$, the size $|A_1 \cap A_2|$ is odd if and only if $i_1= i_2$. Thus, $op(\mathcal A(\mathcal E, m,n))=\binom {|\mathcal E|} 2 (n-m)$.
For any $n\ge 3$, if we set $\mathcal E=\{\emptyset, \{1, 2\}\}$, $\mathcal A(\mathcal E, 2, n)$ is with size $2n-4$ and $op(\mathcal A(\mathcal E, 2, n))=n-2$, which meets Lemma~\ref{lemma_oddtown_lower_bound} when $s=n-4$. Moreover, for any $1\le s\le n-4$, $\mathcal A(\mathcal E, 2, n)\backslash\{\{1, 2, i\}: i> s+4\}$ contains $n+s$ subsets and its odd pair number is $s+2$. So it is another example achieving the lower bound in Lemma~\ref{lemma_oddtown_lower_bound}. This example is not equivalent to the constructions from Lemma~\ref{lemma_s+2_is_achivable}, because of the different size distributions of the two families. Construction~\ref{cons_A_m_I} will be used to prove Theorem~\ref{theorem_oddtown_approximation}.
\subsection{Asymptotic result of oddtown} The lower bound of Theorem~\ref{theorem_oddtown_approximation} comes from the following lemma. \begin{lemma}\label{lemma_odd_approx_lower}
For any odd-sized family $\mathcal A\subset 2^{[n]}$ with $|\mathcal A|=n+s$, we have \begin{equation}\label{eq-app} op(\mathcal A)\ge \binom{\lfloor s\slash n\rfloor+1}2 n+(\lfloor s\slash n\rfloor+1)(s-n\lfloor s\slash n\rfloor).
\end{equation} \end{lemma} \begin{proof}
We define a sequence of disjoint subcollections of $\mathcal A$: $\mathcal A_1, \mathcal A_2, \ldots$ recursively. First let $\mathcal A_1$ be a maximum oddtown subfamily of $\mathcal A$, i.e., the largest subfamily of $\mathcal A$ which satisfies the oddtown property. Let $\mathcal A_2$ be the maximum oddtown subfamily in $\mathcal A\setminus \mathcal A_1$. Then for any $i\ge 2$, as long as $\cup_{j\in [i]}\mathcal A_j\not=\mathcal A$, define $\mathcal A_{i+1}$ as the maximum oddtown subfamily in $\mathcal A\backslash \cup_{j\in [i]}\mathcal A_j$. This process will terminate when some $\mathcal A_r\neq\emptyset$ {satisfies} $\cup_{j\in [r]}\mathcal A_j=\mathcal A$. Trivially $r\le|\mathcal A|$. Since each $\mathcal A_i$ is an oddtown subfamily, $|\mathcal A_i|\le n$ and $r\ge {\lceil s\slash n\rceil}+1$.
Because of the maximality of each $\mathcal A_i$, for any $j\in [r]$ with $j>i$, there exists at least one edge from $A$ to $\mathcal A_i$ in $H(\mathcal A)$ for any $A\in\mathcal A_j$. This means in $H(\mathcal A)$, $e(\mathcal A_i, \mathcal A_j)\ge |\mathcal A_j|$. As a consequence, \begin{equation*} \begin{aligned} op(\mathcal A)&\ge e(\mathcal A_1, \mathcal A\backslash\mathcal A_1)+ e(\mathcal A_2, \mathcal A\backslash(\mathcal A_1\cup \mathcal A_2))+\cdots +e(\mathcal A_{r-1}, \mathcal A\backslash \cup_{j\le r-1}\mathcal A_j)\\
&\ge |\mathcal A\backslash\mathcal A_1|+|\mathcal A\backslash(\mathcal A_1\cup \mathcal A_2)|+\cdots+ |\mathcal A\backslash \cup_{j\le r-1}\mathcal A_j|\\ &\ge s+ (s-n)+\cdots +(s-n\lfloor s\slash n\rfloor)\\ &=\binom{\lfloor s\slash n\rfloor+1}2 n+(\lfloor s\slash n\rfloor+1)(s-n\lfloor s\slash n\rfloor). \end{aligned} \end{equation*} \end{proof} \begin{proof}[Proof of Theorem~\ref{theorem_oddtown_approximation}]
When $s=cn+o(n)$ for some constant $c$,
$$\binom{\lfloor s\slash n\rfloor+1}2 n+(\lfloor s\slash n\rfloor+1)(s-n\lfloor s\slash n\rfloor)=\binom{\lfloor c\rfloor+1}2 n+(\lfloor c\rfloor+1)(c-\lfloor c\rfloor)n+ o(n).$$
For the tightness, choose $m=2\lceil\log{(\lfloor c\rfloor+3)}\rceil$, which is a constant integer satisfying $2^{\lfloor m\slash2\rfloor}\ge\lfloor c\rfloor+3.$ By the extremal size of an eventown family, there exists an eventown subfamily $\mathcal E_1\subset 2^{[m]}$ with size $\lfloor c\rfloor+3.$ Consider the family $\mathcal A(\mathcal E_1, m, n)$ from Construction~\ref{cons_A_m_I}. By the definition of $m$, $ |\mathcal A(\mathcal E_1, m, n)|=(\lfloor c\rfloor+3)(n-m)\geq s+n$. Choose our family $\mathcal A$ as a subfamily of $\mathcal A(\mathcal E_1, m, n)$ with size $s+n$, such that
each element in $[m+1,n]$ appears almost equally often in $\mathcal A$. Note that $|\mathcal A(\mathcal E_1, m, n)|-|\mathcal A|= n+ (1+\lfloor c\rfloor-c)n+o(n)$ and $m$ is a small constant. This means when $c>\lfloor c\rfloor$, there are in total $(c-\lfloor c\rfloor)n-o(n)$ elements in $[m+1, n]$ each appearing $\lfloor c\rfloor+2$ times in $\mathcal A$ and $(1+\lfloor c\rfloor-c)n+o(n)$ elements in $ [m+1, n]$ each appearing $\lfloor c\rfloor+1$ times in $\mathcal A$.
{When $c=\lfloor c\rfloor$, there are $n-o(n)$ elements each appearing $\lfloor c\rfloor+1$ times, and $o(n)$ elements each appearing $\lfloor c\rfloor+2$ times if $|\mathcal A(\mathcal E_1, m, n)|-|\mathcal A|<2(n-m)$, or $\lfloor c\rfloor$ times if $|\mathcal A(\mathcal E_1, m, n)|-|\mathcal A|>2(n-m)$.} Since subsets in $\mathcal A$ only have odd intersections with the subsets in $\mathcal A$ sharing the same element in $[m+1, n]$, for both cases, \begin{equation*} \begin{aligned} op(\mathcal A)&=\binom{\lfloor c\rfloor+2}2(c-\lfloor c\rfloor)n+ \binom{\lfloor c\rfloor+1}2(1+\lfloor c\rfloor-c)n+o(n)\\ &=\binom{\lfloor c\rfloor+1}2 n+(\lfloor c\rfloor+1)(c-\lfloor c\rfloor)n+ o(n). \end{aligned} \end{equation*} \end{proof} By using the same analysis as in the proof of Theorem~\ref{theorem_oddtown_approximation}, we can give more asymptotic results for $M_o(s, n)$ for much larger $s$ by choosing some proper $m$. We list some results here and omit the proofs for brevity. \begin{itemize} \item As long as $s=o(n^2\slash\log n)$ and $n=o(s)$, $M_o(s, n)=\frac12s^2\slash n+\frac12 s+o(s)$. \item As long as $\log s=o(n)$ and $n=o(s)$, $M_o(s, n)=\frac12s^2\slash n+o(s^2\slash n)$. \end{itemize} As one can see, more restrictions on $s$ lead to a more accurate result.
\section{Supersaturation problem of eventown}\label{sec_supersaturation_eventown}
The first part of this section devotes to prove Theorem~\ref{thm_for_eventown}. Motivated by the proof strategy in \cite{o2021towards}, for a given even-sized family $\mathcal A$ on $[n]$, we pay special attention to the maximum eventown subfamily $\mathcal A'$ of $\mathcal A$, i.e., the largest subfamily of $\mathcal A$ which satisfies eventown property. Then $\mathcal A'$ is an independent set of $H(\mathcal A)$ with maximum size. Further, any independent set of $H(\mathcal A)$ is an eventown subfamily of $\mathcal A$, and vise versa. We first show that if $|\mathcal A'|$ is either too large or too small, $\mathcal A$ must have large odd pair number. For convenience, let $N:=2^{\lfloor n\slash 2\rfloor}+s$. \begin{lemma}\label{lemma_size_o_max_subeventown}
Let $n$ be a positive integer and $s\in [2^{\lfloor n\slash2\rfloor}- 2^{\lfloor n\slash4\rfloor}]$ satisfying $\lfloor\frac n 2\rfloor> 2\log(s+1)$. Let $\mathcal A$ be an even-sized family on $[n]$ with $|\mathcal A|=N$ and $\mathcal A'$ be its maximum eventown subfamily. If $|\mathcal A'|\ge 2^{\lfloor n\slash 2\rfloor-1}+s$ or $|\mathcal A'|\le \lceil\frac N {s+2}\rceil$, then $op(\mathcal A)\ge s\cdot 2^{\lfloor n\slash 2\rfloor-1}$. \end{lemma}
\begin{proof} Let $t=|\mathcal A'|$, and without loss of generality assume that $\mathcal A'=\{A_1, A_2, \ldots A_t\}$.
First we consider the case $t\ge 2^{\lfloor n\slash 2\rfloor-1}+s$. For each $A_i$, write $\bm v_i=\bm v_{A_i}$ for short, that is the characteristic vector of $A_i$ in $\mathbb F_2^n$. Consider $W=span(\bm v_1, \bm v_2, \ldots, \bm v_t)$, which is a subspace of $\mathbb F_2^n$. Since $\mathcal A'$ is an eventown family on $[n]$, the inner product of any two vectors in $W$ is zero, and hence $W\subset W^\perp$. By the fact that $\dim(W)+\dim(W^\perp)=\dim(\mathbb F_2^n)=n$, we have $\dim(W)\le \lfloor n\slash 2\rfloor$. However, the subspace $W$ has at least $t> 2^{\lfloor n\slash 2\rfloor-1}$ different vectors, so $\dim(W)= \lfloor n\slash 2\rfloor$.
Note that $\emptyset\not\in\mathcal A\backslash \mathcal A'$, otherwise $\mathcal A'\cup \{\emptyset\}$ is a larger eventown subfamily of $\mathcal A$. So for any $A\in \mathcal A\backslash \mathcal A'$, its characteristic vector $\bm v_A$ is not zero. For any $A_i\in \mathcal A'$, $|A\cap A_i|$ is odd if and only if $(\bm v_A, \bm v_i)=1$. Since $\mathcal A'\cup\{A\}$ is no longer an eventown family, $W\not\subset \bm v_A^\perp$ and hence $\dim(W\cap\bm v_A^\perp)=\dim(W)-1$. Let $N(A)$ denote the number of elements in $\mathcal A$ which has odd-sized intersection with $A$. Then $|N(A)|\ge |\mathcal A'|-|W\cap \bm v_A^\perp| {\ge} t-2^{\lfloor n\slash 2\rfloor-1}$, and \begin{equation}
op(\mathcal A)\ge \sum_{A\in\mathcal A\backslash \mathcal A'}|N(A)|\geq(N-t)(t-2^{\lfloor n\slash 2\rfloor-1}). \end{equation} Since $t\ge 2^{\lfloor n\slash 2\rfloor-1}+s$ and $t\le 2^{\lfloor n\slash 2\rfloor}$ (the latter is from that $\mathcal A'$ is an eventown family), $op(\mathcal A)\ge s\cdot 2^{\lfloor n\slash 2\rfloor-1}$.
Next we consider the case $t\le \lceil\frac N {s+2}\rceil:=\alpha$. Since $\lfloor\frac n 2\rfloor> 2\log(s+1)$, we have $(s+1)\alpha<N \leq (s+2)\alpha$. Then $\lceil\frac N \alpha\rceil= s+2$.
Similar to the process in the proof of Lemma~\ref{lemma_odd_approx_lower}, we define a sequence of disjoint subcollections of $\mathcal A$: $\mathcal A_1, \mathcal A_2, \ldots$ recursively. First let $\mathcal A_1=\mathcal A'$, and let $\mathcal A_2$ be the maximum eventown subfamily in $\mathcal A\setminus \mathcal A'$. Then for any $i\ge 2$, as long as $\cup_{j\in [i]}\mathcal A_j\not=\mathcal A$, define $\mathcal A_{i+1}$ as the maximum eventown subfamily in $\mathcal A\backslash \cup_{j\in [i]}\mathcal A_j$. This process will terminate when some $\mathcal A_r\neq\emptyset$ {satisfies} $\cup_{j\in [r]}\mathcal A_j=\mathcal A$. Trivially $r\le|\mathcal A|$. Since each $\mathcal A_i, i\ge 1$ is also an eventown subfamily in $\mathcal A$, then $|\mathcal A_i|\le |\mathcal A'|\le \alpha$ for any $i\ge 1$. So $r\ge \lceil\frac N \alpha\rceil=s+2$.
Consider the odd pair graph $H(\mathcal A)$. Since $\mathcal A_1$ is a maximal independent set of $H(\mathcal A)$, each vertex in $\mathcal A\backslash\mathcal A_1$ has at least one neighbour in $\mathcal A_1$. So $e(\mathcal A_1, \mathcal A\backslash\mathcal A_1)\ge |\mathcal A\backslash\mathcal A_1|\ge N-\alpha$. Similarly, for any $i\in [s+1]$, $\mathcal A_i$ is a maximal independent set of $\mathcal A\backslash \cup_{j\le i-1}\mathcal A_j$. Then each vertex in $\mathcal A\backslash \cup_{j\le i}\mathcal A_j$ has at least one neighbour in $\mathcal A_i$. So $e(\mathcal A_i, \mathcal A\backslash \cup_{j\le i}\mathcal A_j)\ge |\mathcal A\backslash \cup_{j\le i}\mathcal A_j|\ge N-i\cdot \alpha$. As a consequence, the number of edges in $H(\mathcal A)$ is \begin{equation*} \begin{aligned} op(\mathcal A)&\ge e(\mathcal A_1, \mathcal A\backslash\mathcal A_1)+ e(\mathcal A_2, \mathcal A\backslash(\mathcal A_1\cup \mathcal A_2))+\cdots +e(\mathcal A_{r-1}, \mathcal A\backslash \cup_{j\le r-1}\mathcal A_j)\\
&\ge |\mathcal A\backslash\mathcal A_1|+|\mathcal A\backslash(\mathcal A_1\cup \mathcal A_2)|+\cdots+ |\mathcal A\backslash \cup_{j\le r-1}\mathcal A_j|\\ &\ge (N-\alpha)+ (N-2\alpha)+\cdots +(N-(s+1)\alpha)\\ &=(N-\frac{(s+2)\alpha}2)(s+1). \end{aligned} \end{equation*} Since $(s+2)\alpha=\lceil\frac N\alpha\rceil\alpha\le N+\alpha$, we have $op(\mathcal A)\ge (N-\frac{N+\alpha}2)(s+1)=\frac{s+1}2(N-\alpha)$. If $\frac{s+1}2(N-\alpha)\ge s\cdot 2^{\lfloor\frac n 2\rfloor-1}$, then our proof is finished. In fact, by doubling both sides and computing their difference, we have \begin{equation*} \begin{aligned} (s+1)(N-\alpha)-s\cdot2^{\lfloor\frac n 2\rfloor}&=s(2^{\lfloor\frac n 2\rfloor}+s-2^{\lfloor\frac n 2\rfloor}-\alpha)+N-\alpha\\ &=s^2+N-(s+1)\alpha, \end{aligned} \end{equation*} which is positive since $(s+1)\alpha<N$.
\end{proof}
To complete the proof of Theorem~\ref{thm_for_eventown}, we are left to check the case when $|\mathcal A'|$ is in the range $ [\lceil\frac N{s+2}\rceil+1, 2^{\lfloor n\slash 2\rfloor -1}+s-1]$, for which we have the following lemma.
\begin{lemma}\label{lemma_middlesize}
Let $n$ be a sufficiently large integer and $s\in [2^{\lfloor\frac n 8\rfloor}\slash n]$. Let $\mathcal A$ be an even-sized family on $[n]$ with $|\mathcal A|=N$ and $\mathcal A'$ be its maximum eventown subfamily. If $|\mathcal A'|\in [\lceil\frac N{s+2}\rceil+1, 2^{\lfloor n\slash 2\rfloor -1}+s-1]$ and $op(\mathcal A)< s\cdot 2^{\lfloor n\slash 2\rfloor -1}$, then for any $A\in\mathcal A\backslash \mathcal A'$, $op(\mathcal A'\cup\{A\})\ge s+1.$
\end{lemma}
Lemma~\ref{lemma_middlesize} trivially leads to a contradiction by \begin{equation*}
op(\mathcal A)> |\mathcal A\backslash\mathcal A'|\cdot s\ge (2^{\lfloor\frac n 2\rfloor} +s-(2^{\lfloor n\slash 2\rfloor -1}+s-1))\cdot s> s\cdot 2^{\lfloor n\slash 2\rfloor -1}, \end{equation*}and thus completes the proof of Theorem~\ref{thm_for_eventown}. However the proof of Lemma~\ref{lemma_middlesize} is more involved and far from trivial, and we defer it to the next subsection.
\subsection{Proof of Lemma~\ref{lemma_middlesize}}
In this subsection, we always assume that the conditions in Lemma~\ref{lemma_middlesize} are all satisfied. Remember that we use $H_0$ to denote the odd pair graph of $2^{[n]}$. Consider the induced bipartite graph $H_0[\mathcal A', X]$ with $X\triangleq \mathcal A\backslash\mathcal A'$. It is easy to check that $H_0[\mathcal A', X]$ is a subgraph of $H(\mathcal A)$.
Note that any vertex $x$ in $X$ has at least one neighbor in $\mathcal A'$, otherwise $\mathcal A'\cup\{x\}$ is a larger eventown subfamily. Under these notations, Lemma~\ref{lemma_middlesize} is equivalent to saying that $d_{H_0[\mathcal A', X]}(x)\ge s+1$ for any $x\in X$. Suppose on the contrary that there exists a vertex $x\in X$ with degree at most $s$ in $H_0[\mathcal A', X]$. Denote its neighborhood in $H_0[\mathcal A', X]$ as $N(x)\subset \mathcal A'$. Further denote $Y$ as the set of vertices in $X$ whose neighborhood in $H_0[\mathcal A', X]$ is contained in $N(x)$, i.e., $Y=\{y\in X: N(y)\subseteq N(x)\}$. As $x\in Y$, $Y$ is not empty. We claim that the size of $Y$ is very small. \begin{claim}\label{claim_Y_is_small}
The size $|Y|<s\cdot (2^{n\slash 4}+1)$. \end{claim}
\begin{proof} Denote $\ell=|Y|$. We first claim that $Y$ does not contain an independent set of ${H_0[X]}$ of size $s+1$. Otherwise, say $I\subset Y$ is an independent set with $|I|=s+1$. Then ${\mathcal B}=\big(\mathcal A'\backslash N(x)\big)\cup I$ is also an independent set, i.e., an eventown subfamily of ${\mathcal A}$, since $N(I)\subseteq N(Y)\subseteq N(x)$. Since $|N(x)|\le s$, ${\mathcal B}$ is of size $|\mathcal A'|-|N(x)|+|I|\ge |\mathcal A'|+1$, which contradicts the maximality of $|\mathcal A'|$.
Consider the induced subgraph $H(Y)=H(\mathcal A)[Y]=H_0[Y]$, which does not contain an independent set of size $s+1$ by the above analysis. Equivalently, the complement of $H(Y)$, that is $\overline {H}(Y)$, does not contain any copy of $K_{s+1}$. By Theorem~\ref{coro_turan}, \begin{equation*} e(\overline {H}(Y))\le \frac{\ell^2}2\cdot\frac{s-1}s. \end{equation*} So $e(H(Y))\ge \binom{\ell}2-{\ell^2(s-1)}\slash 2s={\ell^2}\slash{2s}- {\ell}\slash2$. Since $e(H(Y))\le e(H(\mathcal A))=op(\mathcal A)< s\cdot 2^{\lfloor\frac n 2\rfloor-1}$, then ${\ell^2}\slash{2s}- {\ell}\slash2< s\cdot 2^{\lfloor\frac n 2\rfloor-1}$, which leads to our desired result. \end{proof} \begin{remark}
Note that in our analysis $|\mathcal A'|<2^{\lfloor\frac n 2\rfloor-1}+s$, so $|X|=N-|\mathcal A'|>2^{\lfloor\frac n 2\rfloor-1}$, and hence $|Y|=o(|X|)$. This means $|X\setminus Y|$ has the same order as $|X|$. \end{remark}
Since $N(x)\neq \emptyset \neq Y$, we consider the induced subgraph $H_0[\mathcal A'\backslash N(x), X\backslash Y]$. Note that any vertex in $X\backslash Y$ has at least one neighbor in $\mathcal A'\backslash N(x)$ by the definition of $Y$. The following claim shows that after deleting $N(x)$ and $Y$, the part $X\backslash Y$ still cannot reach the minimum degree $s+1$ in $H_0[\mathcal A'\backslash N(x), X\backslash Y]$. \begin{claim}\label{claim_smallest_degree_is_small_after_del} There exists a vertex $v\in X\backslash Y$ with degree at most $ s$ on $H_0[\mathcal A'\backslash N(x), X\backslash Y]$. \end{claim} \begin{proof}
We prove a stronger claim that there exists a vertex $v\in X\backslash Y$ with degree at most $ s$ on $H_0[\mathcal A', X]$. Suppose on the contrary, every vertex in $X\backslash Y$ has degree at least $s+1$. Then $op(\mathcal A)\ge e(\mathcal A', X\backslash Y)\ge (|X|-|Y|)(s+1)$. Since $|X|>2^{\lfloor\frac n 2\rfloor-1}$ and $|Y|<s\cdot (2^{n\slash 4}+1)$, then $$ \begin{aligned} op(\mathcal A)-s\cdot 2^{\lfloor\frac n 2\rfloor-1}&\ge (2^{\lfloor\frac n 2\rfloor-1}+1-s(2^{\frac n 4}+1))(s+1)- s\cdot 2^{\lfloor\frac n 2\rfloor-1}\\ &{=2^{\lfloor\frac n 2\rfloor-1}-s(s+1)(2^{\frac n 4}+1)+s+1.} \end{aligned} $$ Since $s\le 2^{\lfloor\frac n 8\rfloor}\slash n$, ${s(s+1)(2^{\frac n 4}+1)}=O(2^{\frac n 2}\slash n^2)=o(2^{\lfloor\frac n 2\rfloor-1})$. So $op(\mathcal A)>s\cdot 2^{\lfloor\frac n 2\rfloor-1}$, which contradicts the assumption in Lemma~\ref{lemma_middlesize}.
\end{proof}
By Claim~\ref{claim_smallest_degree_is_small_after_del}, we can see the graph $H_0[\mathcal A'\backslash N(x), X\backslash Y]$ has the same property as $H_0[\mathcal A', X]$, i.e., the part $X\setminus Y$ has no isolated vertex but has a small degree ($\leq s$) vertex. Motivated by this, we can set $\mathcal A'\backslash N(x)$ and $X\backslash Y$ as new $\mathcal A'$ and $X$, and do the same analysis as Claims~\ref{claim_Y_is_small} and~\ref{claim_smallest_degree_is_small_after_del} iteratively. The detail of the induction is as follows, where we denote $W_{\mathcal B}$ the subspace $span(\bm v_B:B\in \mathcal B)\subset \mathbb F_2^n$ for any $\mathcal B\subset \mathcal A$.
For a general index $i\geq 1$, we define the following conditions and notations for a pair $(\mathcal A_i, X_i)$: \begin{itemize}
\item[ (C1)]$\mathcal A_i\subset \mathcal A'$ and $|\mathcal A_i|\geq |\mathcal A'|-(i-1)s$;
\item[ (C2)]$\dim(W_{\mathcal A_i})= \dim(W_{\mathcal A'})+1-i$;
\item[ (C3)]$X_i\subset X$ and $|X_i|\ge |X|- \binom{i}{2} s(2^{\frac n 4}+1)$; and
\item[ (C4)]in $H_i:=H_0[\mathcal A_i,X_i]$, there is a vertex $x_i\in X_i$ such that $1\leq d_{H_i}(x_i)\leq s$.
\end{itemize} Then based on the chosen $x_i$, define $N_i:=N_{H_i}(x_i) \subset \mathcal A_i$, which satisfies $1\leq |N_i|\leq s$ by (C4), and define $Y_i:=\{y\in X_i: N_{H_i}(y)\subset N_i\}\subset X_i$, which is nonempty since $x_i\in Y_i$. For the index $i+1$, set
$\mathcal A_{i+1}:=\mathcal A_i\setminus N_i$, $X_{i+1}:=X_i\setminus Y_i$ and $H_{i+1}:=H_0[\mathcal A_{i+1},X_{i+1}]$.
Notice that for large $i$, the existence of a pair $(\mathcal A_i, X_i)$ satisfying (C1)-(C4) itself can lead to a contradiction. To see this, we mention that the lower bound of $|\mathcal A_i|$ and the upper bound of $\dim(W_{\mathcal A_i})$ are given in (C1) and (C2), respectively, but $\log |\mathcal A_i|$ can never exceed $\dim(W_{\mathcal A_i})$. As a consequence, our strategy is to prove the existence of a pair of $(\mathcal A_i, X_i)$ iteratively from $i=1$ to some $i$ large enough to trigger such a contradiction.
As the base case, we set $\mathcal A_1:=\mathcal A'$, $X_1:=X$ and $H_1:=H_0[\mathcal A_1,X_1]$. There exists a vertex $x_1:=x\in X_1$ with degree less than $s+1$ in $H_1$ by the original assumption, and $ d_{H_1}(x_1)\geq 1$ by the maximality of $\mathcal A'$. So conditions (C1)-(C4) trivially hold for $i=1$.
Now we check the conditions for $i=2$. Define $N_1:=N(x)=N_{H_1}(x_1)$, which is not empty and has size at most $s$, and define $Y_1:=Y=\{y\in X_1: N_{H_1}(y)\subset N_1\}$, which has size at most $s\cdot (2^{n\slash 4}+1)$ by Claim~\ref{claim_Y_is_small}. Then (C1) and (C3) hold for $i=2$. Since $ d_{H_1}(x_1)\geq 1$, that is, $\mathcal A' \not\subset x_1^\perp$, we have $\dim (W_{\mathcal A'} \cap x_1^\perp)=\dim(W_{\mathcal A'})-1$. Note that $\mathcal A_2=\mathcal A'\cap x_1^\perp$, so $\dim(W_{\mathcal A_2})=\dim(W_{\mathcal A'})-1$ and (C2) holds for $i=2$. Finally, (C4) holds for $i=2$ by Claim~\ref{claim_smallest_degree_is_small_after_del}, and the nonzero degree is from the definition of $Y_1$.
In general, we assume that (C1)-(C4) hold for $i$. We will show that as long as both $\mathcal A_{i+1}$ and $X_{i+1}$ are nonempty and $i\leq n\slash 2$, all conditions (C1)-(C4) still hold for $i+1$.
First, since $|N_i|=d_{H_i}(x_i)\leq s$ by (C4) and $|\mathcal A_i|\geq |\mathcal A'|-(i-1)s$ by (C1), we have $|\mathcal A_{i+1}|= |\mathcal A_{i}|-|N_i|\geq |\mathcal A'|-(i-1)s -s=|\mathcal A'|-is$, i.e., (C1) holds for $i+1$. Second, since $d_{H_i}(x_i)\geq 1$ by (C4), that is $\mathcal A_i\not\subset x_i^\perp$, then $\dim(W_{\mathcal A_{i+1}})=\dim (W_{\mathcal A_i} \cap x_i^\perp)=\dim(W_{\mathcal A_i})-1= \dim (W_{\mathcal A'})-i$ by (C2). So we have proved (C2) for $i+1$.
Third, to prove (C3) for $i+1$, we need to show the following result as in Claim~\ref{claim_Y_is_small}.
\begin{claim}\label{claim_Y_i_is_small}
The size $|Y_i|<i s\cdot (2^{n\slash 4}+1)$.
\end{claim} \begin{proof}The proof is similar to that of Claim~\ref{claim_Y_is_small}.
Denote $\ell_i=|Y_i|$. We first claim that $H_0[Y_i]$ does not have an independent set of size $is+1$. Otherwise, say $I_i\subset Y_i$ is an independent set with $|I_i|=is+1$. Then ${\mathcal B_i}=\big(\mathcal A_i\backslash N_i\big)\cup I_i$ is also an independent set in $H(\mathcal A)$, i.e., an eventown subfamily of ${\mathcal A}$, since $N_{H_1}(I_i)\subset N_{H_1}(Y_i) \subset N_i \cup ({\mathcal A}\setminus \mathcal A_i)$. Since $|N_i|\le s$, ${\mathcal B_i}$ is of size $|\mathcal A_i|-|N_i|+|I_i|\ge |\mathcal A'|-(i-1)s-s+is+1\geq |\mathcal A'|+1$ by (C1), which contradicts the maximality of $|\mathcal A'|$.
Then the induced subgraph $H_0[Y_i]$ does not contain an independent set of size $is+1$ by the above analysis. That is, the complement $\overline {H_0}[Y_i]$ does not contain any copy of $K_{is+1}$. By Corollary~\ref{coro_turan}, \begin{equation*} e(\overline {H_0}[Y_i])\le \frac{\ell_i^2}2\cdot\frac{is-1}{is}. \end{equation*} So $e(H_0[Y_i])\ge \binom{\ell_i}2-{\ell_i^2(is-1)}\slash 2is={\ell_i^2}\slash{2is}- {\ell_i}\slash2$. Since $e(H_0[Y_i])\le e(H(\mathcal A))=op(\mathcal A)< s\cdot 2^{\lfloor\frac n 2\rfloor-1}$, then ${\ell_i^2}\slash{2is}- {\ell_i}\slash2< s\cdot 2^{\lfloor\frac n 2\rfloor-1}$, which leads to $\ell_i<is(2^{\frac n 4}+1)$.
\end{proof}
By Claim~\ref{claim_Y_i_is_small}, $|X_{i+1}|=|X_i|-|Y_i|\geq |X|-\binom{i}{2} s(2^{\frac n 4}+1)-i s\cdot (2^{n\slash 4}+1)=|X|-\binom{i+1}{2} s(2^{\frac n 4}+1)$. So (C3) holds for $i+1$. It is left to prove (C4) for $i+1$. The nonzero degree of each vertex in $X_{i+1}$ in $H_{i+1}$ is from the definition of $Y_i$, so we only need to prove a result similar to Claim~\ref{claim_smallest_degree_is_small_after_del}.
\begin{claim}\label{claim_small_degree_for_all_s} If $i\leq n\slash 2$, then there exists a vertex $v\in X_{i+1}$ with degree at most $ s$ on $H_{i+1}$. \end{claim}
\begin{proof} As in the proof of Claim~\ref{claim_smallest_degree_is_small_after_del}, we assume that each vertex $v\in X_{i+1}$ has degree at least $s+1$ in the bigger graph $H_i$. Then $op(\mathcal A)\ge e(\mathcal A_i, X_{i+1})\ge |X_{i+1}|(s+1)$. Since $|X_{i+1}|\ge |X|-\binom{i+1}{2}s(2^{\frac n 4}+1)$ by (C3) for $i+1$, $$ \begin{aligned} & op(\mathcal A)-s\cdot2^{\lfloor\frac n 2\rfloor-1}\\
\geq & \left(|X|-\binom{i+1}{2}s(2^{\frac n 4}+1)\right)(s+1)-s\cdot2^{\lfloor\frac n 2\rfloor-1}\\ \ge & \left(2^{\lfloor\frac n 2\rfloor-1}+1-\frac{i(i+1)}2(2^{\frac n 4}+1)s\right)(s+1)-s\cdot2^{\lfloor\frac n 2\rfloor-1}\\ =& s-\frac{i(i+1)}2 s^2(2^{\frac n 4}+1) + 2^{\lfloor\frac n 2\rfloor-1}+1-\frac{i(i+1)}2(2^{\frac n 4}+1)s\\ =& 2^{\lfloor\frac n 2\rfloor-1}-\frac{i(i+1)} 2 s(s+1)(2^{\frac n 4}+1)+s+1. \end{aligned} $$ Since $i\le n\slash 2$ and $s\le 2^{\lfloor\frac n 8\rfloor}\slash n$, $$ \begin{aligned} & \frac{i(i+1)} 2 s(s+1)(2^{\frac n 4}+1)\\ \le& \frac{n(n+2)} 8 s(s+1)(2^{\frac n 4}+1)\\ =&\frac 1 8 (2^{\lfloor\frac n 8\rfloor})(2^{\lfloor\frac n 8\rfloor}+o(2^{\frac n 8}))(2^{\frac n 4}+1)\\ \le & 2^{\frac n 2-3}+o(2^\frac n 2)< 2^{\lfloor\frac n 2\rfloor-1}. \end{aligned} $$ So $op(\mathcal A)>s\cdot2^{\lfloor\frac n 2\rfloor-1}$, contradicting to the hypothesis of Lemma~\ref{lemma_middlesize} again. \end{proof}
By Claim~\ref{claim_small_degree_for_all_s}, (C4) holds for $i+1$. Thus we have proved that all conditions (C1)-(C4) hold for $i+1$ if $i\leq n/2$.
So we can continue this induction until some $i<n\slash 2$ such that either $\mathcal A_i$ or $X_i$ is empty, or, until $i=\lfloor n\slash2\rfloor$. Finally, we claim that both $\mathcal A_{\lfloor n\slash2\rfloor}$ and $X_{\lfloor n\slash2\rfloor}$ are of large sizes (larger than any given constant), but $\dim(W_{\mathcal A_{\lfloor n\slash2\rfloor}})=\dim(W_{\mathcal A'})+1-\lfloor n\slash2\rfloor\leq \lfloor n\slash2\rfloor+1-\lfloor n\slash2\rfloor=1$ by (C2), which is a contradiction. This means the original assumption ``there exists a vertex $x\in X$ with degree at most $s$ in $H_0[\mathcal A', X]$'' should not happen at the beginning, so Lemma~\ref{lemma_middlesize} is proved.
Now we show that $|\mathcal A_{\lfloor n\slash2\rfloor}|$ and $|X_{\lfloor n\slash2\rfloor}|$ are both large.
Remember that $N=2^{\lfloor n\slash 2\rfloor}+s$, $s\le 2^{\lfloor\frac n 8\rfloor}\slash n$, and $\frac N {s+2}\le \lceil\frac N {s+2}\rceil <|\mathcal A'|< 2^{\lfloor\frac n 2\rfloor-1}+s$. By (C1), $|\mathcal A_{\lfloor n\slash2\rfloor}|\ge |\mathcal A'|-ns\slash 2> \frac N {s+2}-ns\slash 2$. This leads to $|\mathcal A_{\lfloor n\slash2\rfloor}|\ge \frac N {s+2}(1-o(1))>4$, since $$\frac N {s+2}\ge\frac{2^{\lfloor\frac n 2\rfloor}+s}{2^{\lfloor\frac n 8\rfloor}\slash n+2}=\Theta(n\cdot 2^{\frac 3 8 n}), \text{ but } ns=O(2^\frac n 8).$$
By (C3) we have $|X_{\lfloor n\slash2\rfloor}|\ge |X|-n^2s(2^{\frac n 4}+1)$. Since $|X|= |\mathcal A|- |\mathcal A'| > 2^{\lfloor\frac n 2\rfloor-1}$, $|X_{\lfloor n\slash2\rfloor}|\ge 2^{\lfloor\frac n 2\rfloor-1} -O(n\cdot 2^{\frac 3 8 n})=2^{\lfloor\frac n 2\rfloor-1}(1-o(1))$. So both $|\mathcal A_{\lfloor n\slash2\rfloor}|$ and $|X_{\lfloor n\slash2\rfloor}|$ are large when $n$ is large.
\subsection{Asymptotic result}
This subsection is devoted to prove Theorem~\ref{thm_eventownlar} and Theorem~\ref{thm_complete_old_eventown} by using Fourier analysis. Consider the characters on additive abelian group $\mathbb F_2^n$. For any $\bm m\in \mathbb F_2^n$, define $\chi_{\bm m}: \mathbb F_2^n\to\mathbb C^*$ as $\chi_{\bm m}(a)=e^{\pi i(\bm m, \bm a)}=(-1)^{(\bm m, \bm a)}$ for any $\bm a\in \mathbb F_2^n$. It is easy to check that for all $\bm a, \bm b\in \mathbb F_2^n$, $\chi_{\bm m}(\bm a+\bm b)=\chi_{\bm m}(\bm a)\chi_{\bm m}(\bm b)$, so $\chi_{\bm m}\in\widehat{\mathbb F_2^n}$. Moreover, for any $\bm m\ne \bm m'\in \mathbb F_2^n$, we have $\chi_{\bm m}\ne\chi_{\bm m'}$. The reason is that there always exists some $\bm a\in \mathbb F_2^n$ such that $(\bm a, \bm m-\bm m')=1$, which leads to $\chi_{\bm m}(\bm a)=(-1)^{(\bm m, \bm a)}\ne (-1)^{(\bm m', \bm a)}= \chi_{\bm m'}(\bm a).$ From $\{\chi_{\bm m}: \bm m\in \mathbb F_2^n\}\subseteq\widehat {\mathbb F_2^n}$ and $|\{\chi_{\bm m}: \bm m\in \mathbb F_2^n\}|=|\mathbb F_2^n|=|\widehat {\mathbb F_2^n}|$, we have $\widehat {\mathbb F_2^n}= \{\chi_{\bm m}: \bm m\in \mathbb F_2^n\}$.
For any subfamily $\mathcal A\subset 2^{[n]}$, consider its odd pair graph $H=H(\mathcal A)$, which is an induced subgraph of $H_0$. Denote $V(H)$ as the vertex set of $H$. We have the following concentration result for the edge number of $H$.
\begin{lemma}\label{lemma_concentration_result} Let $H=H(\mathcal A)$ for some subfamily $\mathcal A\subset 2^{[n]}$. Let $v_o(H)$ be the number of vertices in $H$ which are odd-sized elements in $\mathcal A$. Then we have
$$|v(H)^2-4e(H)-2v_o(H)|\le 2^{n\slash 2}v(H).$$ In particular, when $\mathcal A$ is even-sized,
$$|v(H)^2-4e(H)|\le 2^{n\slash 2}v(H).$$ \end{lemma} \begin{proof} Define $f: \mathbb F_2^n\to\mathbb C$ as the indicator function of $V(H)$. In other words, $f(\bm a)=1$ if $\bm a$ is a vertex of $H$, and $f(\bm a)=0$ otherwise. Then for any $\bm m\in V(H)$, $$ \begin{aligned}
\hat f(\chi_{\bm m})&=\frac 1{|\mathbb F_2^n|}\sum_{\bm a\in \mathbb F_2^n}f(\bm a)\overline{\chi_{\bm m}(\bm a)}\\ &=\frac 1 {2^n}\sum_{\bm a\in\mathbb F_2^n} f(\bm a)(-1)^{(\bm m, \bm a)}\\ &=\frac 1 {2^n}\sum_{\bm a\in V(H)} (-1)^{(\bm m, \bm a)}.
\end{aligned} $$ For $\bm a\ne \bm m$, $(\bm a, \bm m)=1$ if and only if $\{\bm a, \bm m\}$ forms an edge in $H$. For $\bm a=\bm m$, $(\bm m, \bm m)=1$ if and only if $(\bm 1, \bm m)=1$, where $\bm 1$ means the all-one vector. As a result, if $\bm m$ is odd-sized, $\hat f(\chi_{\bm m})= \frac 1 {2^n}[(-1)(d_{H}(\bm m)+1)+ 1\cdot(v(H)-d_{H}(\bm m)-1)]= \frac 1 {2^n}(v(H)-2d_{H}(\bm m)-2)$; if $\bm m$ is even-sized, $\hat f(\chi_{\bm m})= \frac 1 {2^n}[(-1)d_{H}(\bm m)+ 1\cdot(v(H)-d_{H}(\bm m))]= \frac 1 {2^n}(v(H)-2d_{H}(\bm m))$. To sum up, for any $\bm m\in v(H)$, \[\hat f(\chi_{\bm m})= \frac 1 {2^n}(v(H)-2d_{H}(\bm m)-2(\bm 1, \bm m)).\]
By Plancherel's formula,
$$\frac 1 {2^n}v(H)=\Vert f\Vert^2=\sum_{\bm m\in\mathbb F_2^n}|\hat f(\chi_{\bm m})|^2\ge \sum_{\bm m\in V(H)}|\hat f(\chi_{\bm m})|^2.$$ By Cauchy-Schwarz inequality, \begin{equation*}
\begin{split}
\sum_{\bm m\in V(H)} \left| \hat{f}(\chi_{\bm m})\right|^2 &= \sum_{\bm m\in V(H)} \left( \frac{1}{2^n} \big( v(H)-2d_{H}(\bm m)-2(\bm 1, \bm m)\big) \right)^2 \\
&= \frac{1}{2^{2n}} \sum_{\bm m\in V(H)} \big( v(H)-2d_{H}(\bm m)-2(\bm 1, \bm m)\big)^2\\
&\ge \frac{1}{2^{2n}} \cdot \frac{1}{v(H)} \left( \sum_{\bm m\in V(H)} \big( v(H)-2d_{H}(\bm m)-2(\bm 1, \bm m)\big) \right)^2\\
&= \frac{1}{2^{2n}} \cdot \frac{1}{v(H)} \left( v(H)^2-4e(H)-2v_o(H) \right)^2.
\end{split}
\end{equation*}
Combining both inequalities above, $$\frac1{2^n}v(H)\ge \frac1{2^{2n}}\frac1{v(H)}(v(H)^2-4e(H)-2v_o(H))^2,$$ and hence $|v(H)^2-4e(H)-2v_o(H)|\le 2^\frac n 2v(H)$. \end{proof}
$~$
\begin{proof}[Proof of Theorem~\ref{thm_eventownlar}]
Since $n\ge 1\slash\epsilon$, $2^{(1-\epsilon)n}\le 2^{n-1}$. There exists some even-sized $\mathcal A\subset 2^{[n]}$ such that $|\mathcal A|\ge 2^{(1-\epsilon)n}$. Let $H=H(\mathcal A)$ be the odd pair graph of any such $\mathcal A$. From Lemma~\ref{lemma_concentration_result}, we have $${e(H)}\slash{\binom {v(H)} 2}\ge \frac{v(H)^2-2^{n\slash 2}v(H)}{2v(H)(v(H)-1)}\ge \frac12(1-2^{n\slash2}v(H)^{-1})\ge \frac 1 2(1-2^{(\epsilon-1\slash2)n}).$$
Hence, $f_n(\epsilon)=\min\{\frac{e(H(\mathcal A))}{\binom{|\mathcal A|} 2}: \text{ even-sized }\mathcal A\subset2^{[n]}; |\mathcal A|\ge 2^{(1-\epsilon)n}\}\ge \frac 1 2(1-2^{(\epsilon-1\slash2)n}).$
For the upper bound, let $\mathcal A$ be the collection of all even-sized sets from $2^{[n]}$. Then $|\mathcal A|=2^{n-1}\ge 2^{(1-\epsilon)n}$, and $V(H)$ forms a subspace of $\mathbb F_2^n$. For any $\bm v\in V(H)$, it is clear that $d_H(\bm v)=0$ if and only if one of the following two cases happens: \begin{itemize} \item[(1)] $\bm v=\bm 0$; \item[(2)] $\bm v=\bm 1$ and $n$ is even. \end{itemize}
Otherwise, $d_H(\bm v)=v(H)-|V(H)\cap {\bm v}^\perp|= 2^{n-1}-2^{n-2}=2^{n-2}$.
Thus, when $n$ is odd, $e(H)=\frac12\sum_{\bm v\in V(H)\backslash\{\bm 0\}}2^{n-2}=\frac12\binom{v(H)}2$; when $n$ is even, $e(H)=\frac12\sum_{\bm v\in V(H)\backslash\{\bm 0, \bm 1\}}2^{n-2}=v(H)(v(H)-2)\slash4$.
\end{proof}
$~$
When $n$ is odd, we can further improve Lemma~\ref{lemma_concentration_result} by considering the Fourier analysis on finite additive abelian subgroup $\bm 1^\perp\subset \mathbb F_2^n$. Note that $\bm 1^\perp$ is a subspace of $\mathbb F_2^n$ with dimension $n-1$ consisting of all vectors with even numbers of $1$, and hence the additive abelian subgroup structure follows naturally from the subspace structure. For any $\bm m\in\bm 1^\perp$, consider $\chi_{\bm m}'$ as the restriction of $\chi_{\bm m}$ under $\bm 1^\perp$, i.e., it maps $\bm a$ to $\chi_{\bm m}(\bm a)=(-1)^{(\bm m, \bm a)}$ for any $\bm a\in \bm 1^\perp$. From $\chi_{\bm m}\in\widehat{\mathbb F_2^n}$, $\chi_{\bm m}'\in\widehat{\bm 1^\perp}$. Moreover, we claim that $\chi_{\bm m_1}'= \chi_{\bm m_2}'$ if and only if $\bm m_1=\bm m_2$ with $\bm m_1, \bm m_2\in \bm 1^\perp$. In fact, the equality $\chi_{\bm m_1}'= \chi_{\bm m_2}'$ means $(\bm m_1-\bm m_2, \bm a)=(\bm m_1, \bm a)-(\bm m_2, \bm a)=0$ for any $\bm a\in\bm 1^\perp$, which means $\bm m_1-\bm m_2\in(\bm 1^\perp)^\perp=span\{\bm 1\}$. If $\bm m_1\ne\bm m_2$, the only choice is $\bm m_1-\bm m_2=\bm 1$, which contradicts to both $\bm m_1, \bm m_2\in\bm 1^\perp$ for $n$ odd. Hence from the same analysis as in $ \widehat{\mathbb F_2^n}$, $\widehat {\bm 1^\perp}= \{\chi_{\bm m}': \bm m\in \bm 1^\perp\}$.
\begin{lemma}\label{lemma_concentration_result_odd} Let $H=H(\mathcal A)$ for some subfamily $\mathcal A\subset 2^{[n]}$. When $n$ is odd and $\mathcal A$ is even-sized, we have
$$|v(H)^2-4e(H)|\le 2^{(n-1)\slash 2}v(H).$$ \end{lemma} \begin{proof} Define $g: \bm 1^\perp\to\mathbb C$ as the indicator function of $V(H)$. Since $\mathcal A$ is even-sized, $V(H)\subseteq\bm 1^\perp$. For any $\bm m\in V(H)$, $$ \begin{aligned}
\hat g(\chi_{\bm m}')&=\frac 1{|\bm 1^\perp|}\sum_{\bm a\in \bm 1^\perp}g(\bm a)\overline{\chi_{\bm m}'(\bm a)}\\ &=\frac 1 {2^{n-1}}\sum_{\bm a\in V(H)} (-1)^{(\bm m, \bm a)}\\ &=\frac 1 {2^{n-1}}[(-1)d_{H}(\bm m)+ 1\cdot(v(H)-d_{H}(\bm m))]\\ &=\frac 1 {2^{n-1}}(v(H)-2d_{H}(\bm m)). \end{aligned} $$ Plancherel's formula gives us that
$$\frac 1 {2^{n-1}}v(H)=\Vert g\Vert^2=\sum_{\bm m\in\bm 1^\perp}|\hat g(\chi_{\bm m}')|^2\ge \sum_{\bm m\in V(H)}|\hat g(\chi_{\bm m}')|^2.$$ By Cauchy-Schwarz inequality, \begin{equation*}
\begin{split}
\sum_{\bm m\in V(H)} \left| \hat{g}(\chi_{\bm m}')\right|^2 &= \sum_{\bm m\in V(H)} \left( \frac{1}{2^{n-1}} \big( v(H)-2d_{H}(\bm m)\big) \right)^2 \\
&= \frac{1}{2^{2n-2}} \sum_{\bm m\in V(H)} \big( v(H)-2d_{H}(\bm m)\big)^2\\
&\ge \frac{1}{2^{2n-2}} \cdot \frac{1}{v(H)} \left( \sum_{\bm m\in V(H)} \big( v(H)-2d_{H}(\bm m)\big) \right)^2\\
&= \frac{1}{2^{2n-2}} \cdot \frac{1}{v(H)} \left( v(H)^2-4e(H) \right)^2.
\end{split}
\end{equation*}
Combining above two inequalities, we get $$\frac1{2^{n-1}}v(H)\ge \frac1{2^{2n-2}}\frac1{v(H)}(v(H)^2-4e(H))^2,$$ and hence $|v(H)^2-4e(H)|\le 2^\frac {n-1} 2v(H)$. \end{proof}
$~$
\begin{proof}[Proof of Theorem~\ref{thm_complete_old_eventown}]
Let $n, s$ be positive integers. If $n$ is even, for any even-sized subfamily $\mathcal A\subset 2^{[n]}$ with $|\mathcal A|\ge 2^{\lfloor n\slash2\rfloor}+s= 2^{ n\slash2}+s$, let $H=H(\mathcal A)$ and by Lemma \ref{lemma_concentration_result} we have $$4e(H)\ge v(H)^2-2^{n\slash2}v(H)\ge s(2^\frac n 2+s)>s\cdot2^{\lfloor n\slash2\rfloor},$$ and hence $op(\mathcal A)=e(H)>s\cdot2^{\lfloor n\slash2\rfloor-2}$.
If $n$ is odd, for any even-sized subfamily $\mathcal A\subset 2^{[n]}$ with $|\mathcal A|\ge 2^{\lfloor n\slash2\rfloor}+s= 2^{(n-1)\slash2}+s$, let $H=H(\mathcal A)$ and by Lemma \ref{lemma_concentration_result_odd}, $$4e(H)\ge v(H)^2-2^{{(n-1)}\slash2}v(H)\ge s(2^\frac {n-1} 2+s)>s\cdot2^{ (n-1)\slash2}.$$ Hence $op(\mathcal A)=e(H)>s\cdot2^{\lfloor n\slash2\rfloor-2}$. \end{proof}
\section{Conclusion}\label{sec-conc} We studied the supersaturation problems of oddtown and eventown. It is well known that the maximum size of an oddtown (resp. eventown) family $\mathcal A$ over an $n$ element set is at most $n$ (resp. $2^{\lfloor\frac n 2\rfloor}$). The supersaturation problem counts the number of pairs of subsets with odd-sized intersection in $\mathcal A$ if the size of $\mathcal A$ exceeds the corresponding extremal value. O'Neill~\cite{o2021towards} initiated the study of this problem and gave two conjectures on the odd pair numbers for oddtown and eventown respectively, and a problem on the asymptotic supersaturation result for eventown. We disproved the conjecture for oddtown, and proved the conjecture for eventown partially when $n$ is large enough.
Asymptotic supersaturation results for the oddtown and eventown subfamilies are given, resulting in different formulas for the minimum odd pair numbers of $\mathcal A$ for different exceeding numbers $s$. {We also completed a result for eventown reaching half of the conjectured lower bound for general $n$ and $s$ proposed by Antipov et al. \cite{antipov2022lovasz}.} Methods like Fourier analysis and extremal graph theory are included. Here we list some open problems. \begin{itemize} \item For the supersaturation problem of oddtown family, when $s\le n-4$, the families reaching the tight bounds are not unique under the equivalence of permutation. It is interesting to determine all extremal structures under the equivalence of permutation. \item In the supersaturation problem of oddtown family, no result about the exact value of the minimum odd pair number is known for $s>n-4$. We believe that the constraint $s\le n-4$ is best possible for the tightness of the bound $op(\mathcal A)\ge s+2$. It is interesting to find the exact values of minimum odd pair numbers for more $s$ systematically. \item Our eventown supersaturation result Theorem~\ref{thm_for_eventown} only works for sufficiently large $n$, and we do not take efforts to determine the explicit lower bound. It is interesting to give a good explicit lower bound of $n$ such that Theorem~\ref{thm_for_eventown} is satisfied. \item For Conjecture~\ref{conj_even_town}, we proved that it is true when $s\leq 2^{\lfloor\frac n 8\rfloor}\slash n$, and left a large gap in the conjectured range $[2^{\lfloor\frac n 2\rfloor}-2^{\lfloor\frac n 4\rfloor}]$ of $s$. We are interested in how to further shrink this gap. \item For the asymptotic supersaturation results, we have studied the following cases.
\begin{itemize}
\item[(1)] Oddtown family, and $\lim_{n\to\infty}s\slash n= c$ for some constant $c$.
\item[(2)] Oddtown family, with $n=o(s)$ but $\log s= o(n)$.
\item[(3)] Eventown family, and $s\ge 2^{(1-\epsilon)n}$ for some $\epsilon\in(0, \frac12)$.
\end{itemize}
Moreover, remind that Lemma~\ref{lemma_concentration_result} also works for odd-sized family. Consequently, by using the same analysis as in Case (3), one can prove that an odd-sized subfamily shares the same performance on the minimum odd pair number density when the exceeding number $s\ge 2^{(1-\epsilon)n}$ for $\epsilon\in(0, \frac12)$, i.e., $$\lim_{n\to\infty}\min\{\frac{e(H(\mathcal A))}{\binom{|\mathcal A|}2}:\text{ odd-sized }\mathcal A\subset 2^{[n]}; |\mathcal A|\ge 2^{(1-\epsilon)n}\}=\frac12.$$
Note that for Cases (1) and (2), this minimum density limit is zero for odd-sized family. If $\lim_{n\to\infty}\frac{\log s}n=c$ for some constant $c<\frac12$, by Construction~\ref{cons_A_m_I} with some suitable $m$, we can also determine that the minimum density limit is zero.
So it is interesting to determine the minimum density limit when $\lim_{n\to\infty}\frac{\log s}n=\frac12$ for odd-sized family.
\item Theorem~\ref{thm_eventownlar} and Theorem~\ref{thm_complete_old_eventown} are proved by using Fourier analysis, which shows the priority of this method. We look forward to more new results in this area derived from Fourier analysis. \end{itemize}
\vskip 10pt
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August 2013, 7(3): 1051-1074. doi: 10.3934/ipi.2013.7.1051
The Gaussian beam method for the wigner equation with discontinuous potentials
Dongsheng Yin 1, , Min Tang 2, and Shi Jin 3,
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084
Department of Mathematics, Institute of Nature Science, and Ministry of Education Key Laboratory in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240, China
Department of Mathematics, Institute of Natural Sciences, and MOE Key Lab in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240
Received June 2012 Revised January 2013 Published September 2013
For the Wigner equation with discontinuous potentials, a phase space Gaussian beam (PSGB) summation method is proposed in this paper. We first derive the equations satisfied by the parameters for PSGBs and establish the relations for parameters of the Gaussian beams between the physical space (GBs) and the phase space, which motivates an efficient initial data preparation thus a reduced computational cost than previous method in the literature. The method consists of three steps: 1) Decompose the initial value of the wave function into the sum of GBs and use the parameter relations to prepare the initial values of PSGBs; 2) Solve the evolution equations for each PSGB; 3) Sum all the PSGBs to construct the approximate solution of the Wigner equation. Additionally, in order to connect PSGBs at the discontinuous points of the potential, we provide interface conditions for a single phase space Gaussian beam. Numerical examples are given to verify the validity and accuracy of method.
Keywords: Wigner equation, discontinuous potential., Gaussian beam.
Mathematics Subject Classification: Primary: 65M75, 81Q20, 35Q41; Secondary: 65Z05, 34E0.
Citation: Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems & Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051
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Dongsheng Yin Min Tang Shi Jin | CommonCrawl |
Lester S. Hill
Lester S. Hill (1891–1961) was an American mathematician and educator who was interested in applications of mathematics to communications. He received a bachelor's degree from Columbia College (1911) and a Ph.D. from Yale University (1926). He taught at the University of Montana, Princeton University, the University of Maine, Yale University, and Hunter College. Among his notable contributions was the Hill cipher. He also developed methods for detecting errors in telegraphed code numbers and wrote two books.
Dr.
Lester S. Hill
Lester S. Hill on May 16, 1956
Born
Lester Sanders Hil[1]
(1891-01-18)January 18, 1891
New York City
DiedJanuary 9, 1961(1961-01-09) (aged 69)[2][3]
Bronxville, New York[2]
NationalityAmerican
Occupation(s)mathematician and cryptographer
Known forthe Hill cipher (1929)
Notable workCryptography in an Algebraic Alphabet (1929)[4]
References
1. Christensen, Chris (2014). "Lester Hill Revisited". Cryptologia. 38 (4): 300–301. doi:10.1080/01611194.2014.915260. S2CID 45603857. Retrieved November 9, 2018 – via ResearchGate.
2. "Dr. Lester S. Hill". Chicago Tribune. January 10, 1961. p. 10. Dr. Lester S. Hill, 70, mathematician and cryptographer, died today in Lawrence hospital after a long illness. Hill was commended for application, of higher mathematics to the construction of secret codes.
3. "LESTER HILL DIES; A MATHEMATICIAN; Ex-Hunter Professor Was 70 -- Cryptographer Cited for Service to U. S." The New York Times. January 10, 1961. Retrieved November 9, 2018.(subscription required)
4. Dooley, John (January 1, 2018). "10.1 The Shoulders of Giants: Friedman, Hill, and Shannon". History of Cryptography and Cryptanalysis: Codes, Ciphers, and Their Algorithms. Springer. p. 167. ISBN 978-3-319-90442-9 – via Google Books.
• Rosen, Kenneth (2005). Elementary Number Theory and its Applications, fifth edition, Addison-Wesley, p. 292.
Authority control: Academics
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Artin–Schreier theory
In mathematics, Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic p. Artin and Schreier (1927) introduced Artin–Schreier theory for extensions of prime degree p, and Witt (1936) generalized it to extensions of prime power degree pn.
For the result about real-closed fields, see Artin–Schreier theorem.
If K is a field of characteristic p, a prime number, any polynomial of the form
$X^{p}-X-\alpha ,\,$
for $\alpha $ in K, is called an Artin–Schreier polynomial. When $\alpha \neq \beta ^{p}-\beta $ for all $\beta \in K$, this polynomial is irreducible in K[X], and its splitting field over K is a cyclic extension of K of degree p. This follows since for any root β, the numbers β + i, for $1\leq i\leq p$, form all the roots—by Fermat's little theorem—so the splitting field is $K(\beta )$.
Conversely, any Galois extension of K of degree p equal to the characteristic of K is the splitting field of an Artin–Schreier polynomial. This can be proved using additive counterparts of the methods involved in Kummer theory, such as Hilbert's theorem 90 and additive Galois cohomology. These extensions are called Artin–Schreier extensions.
Artin–Schreier extensions play a role in the theory of solvability by radicals, in characteristic p, representing one of the possible classes of extensions in a solvable chain.
They also play a part in the theory of abelian varieties and their isogenies. In characteristic p, an isogeny of degree p of abelian varieties must, for their function fields, give either an Artin–Schreier extension or a purely inseparable extension.
Artin–Schreier–Witt extensions
There is an analogue of Artin–Schreier theory which describes cyclic extensions in characteristic p of p-power degree (not just degree p itself), using Witt vectors, developed by Witt (1936).
References
• Artin, Emil; Schreier, Otto (1927), "Eine Kennzeichnung der reell abgeschlossenen Körper", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Springer Berlin / Heidelberg, 5: 225–231, doi:10.1007/BF02952522, ISSN 0025-5858
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001 Section VI.6
• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001 Section VI.1
• Witt, Ernst (1936), "Zyklische Körper und Algebren der Characteristik p vom Grad pn. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik pn", Journal für die reine und angewandte Mathematik (in German), 176: 126–140, doi:10.1515/crll.1937.176.126
| Wikipedia |
Math For ML
Linear Algebra Basics 3: Linear Transformations and Matrix Multiplication
What is Linear Transformations? What is Vector Space? How to do Vector Multiplication (Matrix Multiplication)? Conceptualizing a Linear Transformation is also key to understanding a transformation, so ..
Casper Hansen
MSc AI Student @ DTU. This is my Machine Learning journey 'From Scratch'. Conveying what I learned, in an easy-to-understand fashion is my priority.
More posts by Casper Hansen.
What is Linear Transformations?
What is Vector Space?
How to do Matrix Multiplication?
This post continues on from the last post, which is a prerequisite to reading this one. In the last post, I covered basis vectors, span and linear combinations.
Conceptualizing Linear Transformations
A Linear Transformation is just a function, a function $f(x)$. It takes an input, a number x, and gives us an ouput for that number. In Linear Algebra though, we use the letter T for transformation.
$$T(input_x) = output_x$$
Or with vector coordinates as input and the corresponding vector coordinates output
$$ \begin{bmatrix} x_{input}\\ y_{input} \end{bmatrix} = \begin{bmatrix} x_{output}\\ y_{output} \end{bmatrix} $$
We might think of a transformation in terms of transforming a vector, where we essentially transform vector coordinates, or even the basis vectors, for some vector. The idea is, that we give some vector coordinates as input, and then we get an output from those vector coordinates.
Any time we do that, we can visualize the transformation by imagining a vector moving from one position to another — every vector in the space moves with a transformation.
The nice thing about transformations is the fact, that we just need
Any vector coordinates in our space, and
The basis vectors
Then if we do a transformation, we would transform all vectors in our space, along with the basis vectors. That means we just need to find the transformed basis vectors to calculate any transformed vector in our space.
How to do a Linear Transformation?
We can write a general equation like this for a vector $\vec{v}$ with vector coordinates $\begin{bmatrix}x\\y\end{bmatrix}$ and basis vectors $\hat{i} = \begin{bmatrix}i_1\\i_2\end{bmatrix}$ and $\hat{j} = \begin{bmatrix}j_1\\j_2\end{bmatrix}$
$$ T\left(\begin{bmatrix}x\\y\end{bmatrix}\right)= x\begin{bmatrix} i_1\\ i_2 \end{bmatrix} + y\begin{bmatrix} j_1\\ j_2 \end{bmatrix} $$
So that means we would just have to replace $\hat{i}$ and $\hat{j}$ after the transformation, and we could just do the multiplication as learned in Linear Algebra Basics 1.
An alternative way to represent the above, and perhaps a more intuitive way of understanding a transformation numerically would be something like this:
$$ transformed_{\vec{v}} = x(transformed_{\hat{i}}) + y(transformed_{\hat{j}}) $$
To get the transformed vector (output), we take the input vector coordinates x and y, and scale with the transformed basis vectors $\hat{i}$ and $\hat{j}$. Then what x and y end up as, is the transformed vector coordinates for the vector $\vec{v}$.
What happens numerically
— e.g. what happens inbetween input and output?
All that happens numerically, is that we define a rule, which dictates how we transform any vector — that is how you do a transformation numerically. Such a rule can be wrong, but hang with me; So what could a rule be? Often what happens is, that we define which dimension our current vector space should transform to.
Vector space is the set of all vectors in our space. We can do operations on these vectors, e.g. vector addition or scaling.
So what happens is, that all of vector space transforms, as we do a linear transformation, therefore we want to define which dimension our current vector space should transform from and to.
T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}
The above reads: our transformation T maps from two-dimensional space to 2-dimensional space. The $\mathbb{R}^{2}$ means all real numbers in the 2-dimensional space.
Before we head further, we have not touched upon multiplying vectors. So how do we go about multiplying matrices? Let me expand upon what you learned in Linear Algebra Basics 2 — scaling a vector by a number.
Say we have a matrix A defined as such, where a, b, c and d are real numbers
$$ A = \begin{bmatrix} a & b\\ c & d \end{bmatrix} $$
And a vector $\vec{v}$ defined by
$$ \vec{v} = \begin{bmatrix} x\\ y \end{bmatrix} $$
Then we could multiply them together, exactly like this, using vector multiplication:
$$ \begin{bmatrix} a & b\\ c & d \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = x \begin{bmatrix} a\\ c \end{bmatrix} + y \begin{bmatrix} b\\ d \end{bmatrix} = \begin{bmatrix} ax + by\\ cx + dy \end{bmatrix} $$
Note that we could define the vector as a matrix, so we could also call this matrix multiplication.
Matrix Multiplication: We multiply rows by coloumns. This means you take the first number in the first row of the second matrix and scale (multiply) it with the first coloumn in the first matrix. You do this with each number in the row and coloumn, then move to the next row and coloumn and do the same.
Now we can define the linear transformation. We can start by giving the matrix A numbers and then letting vector $\vec{v}$ be any possible vector in our vector space
$$ A = \begin{bmatrix} -2 & 6\\ 3 & 1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} $$
Then we can choose and say that we define our linear transformation by $T(\vec{v}) = A \vec{v}$. That means, for every vector coordinate in our vector $\vec{v}$, we have to multiply that by the matrix A.
$$ T(\vec{v}) = A \vec{v} \rightarrow \begin{bmatrix} -2 & 6\\ 3 & 1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} $$
As I just showed you above, where we defined the matrix A by a, b, c and d, we can do multiplication as follows
$$ T(\vec{v}) = A \vec{v} \rightarrow \begin{bmatrix} -2 & 6\\ 3 & 1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = x \begin{bmatrix} -2\\ 3 \end{bmatrix} + y \begin{bmatrix} 6\\ 1 \end{bmatrix} = \begin{bmatrix} -2x + 6y\\ 3x + 1y \end{bmatrix} $$
Calculation of a transformed vector?
To calculate any vector after a transformation, all we simply need to do, as described further up in this post, is record the basis vectors. If we know where the basis vectors are after a transformation, the calculation of any transformed vector is almost infuriatingly simple.
I have been emphasizing this matrix A throughout this whole post, for a very specific reason. Imagine the first coloumn of the matrix being $\hat{i}$ and the second coloumn being $\hat{j}$
$$ A = \begin{bmatrix} \hat{i}_x & \hat{j}_x\\ \hat{i}_y & \hat{j}_y \end{bmatrix} $$
Now it becomes evidently clear that we can, given any vector, calculate any vector in the transformed vector space. So, if we have $\hat{i}$ and $\hat{j}$ transformed, we can just pass any vector to this formular along with the transformed basis vectors, and it would give us the transformed vector (where T stands for transformed):
$$ \begin{bmatrix} T_{\hat{i}_x} & T_{\hat{j}_x}\\ T_{\hat{i}_y} & T_{\hat{j}_y} \end{bmatrix} \begin{bmatrix} \vec{v}_x \\ \vec{v}_y \end{bmatrix} $$
What about 2x2 matrix multiplication?
This case is a simple matter too. Suppose we have a matrix A (left) and B (right). We would simply multiply the rows in the first matrix by the coloumns in the second matrix:
$$ \begin{bmatrix} a & b\\ c & d \end{bmatrix} \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix} $$
BUT! That is not easy to remember, so here is the intuitive way, as I showed earlier. We split the process into 2:
Multiply matrix A by the first coloumn in matrix B
Multiply matrix A by the second coloumn in matrix B
This is a known scenario, I showed you this earlier:
$$ \begin{bmatrix} a & b\\ c & d \end{bmatrix} \begin{bmatrix} e \\ g \end{bmatrix} = e \begin{bmatrix} a \\ c \end{bmatrix} + g \begin{bmatrix} b \\ d \end{bmatrix} = \begin{bmatrix} ae + bg \\ ce + dg \end{bmatrix} $$
$$ \begin{bmatrix} a & b\\ c & d \end{bmatrix} \begin{bmatrix} f \\ h \end{bmatrix} = f \begin{bmatrix} a \\ c \end{bmatrix} + h \begin{bmatrix} b \\ d \end{bmatrix} = \begin{bmatrix} af + bh \\ cf + dh \end{bmatrix} $$
Adding the two results together, we get the equation further up:
This was clearly a lot to learn. But we could also go further and ask the impending question of, what can be a wrong linear transformation? Because we just defined some linear transformation and assumed that it is right. For now, I will not go deeper into the subject, but as WolframAlpha suggests, we need 2 conditions to be true, before we can call it a linear transformation.
Summary (of the questions at the top):
Linear transformations are a function $T(x)$, where we get some input and transform that input by some definition of a rule. An example is $T(\vec{v})=A \vec{v}$, where for every vector coordinate in our vector $\vec{v}$, we have to multiply that by the matrix A.
Vector space is the set of all vectors in our space, which we define in dimensions. We can do operations on these vectors, e.g. vector addition or scaling.
With a matrix $A = \begin{bmatrix}a & b\\c & d \end{bmatrix}$, where a, b, c and d are real numbers.
Just remember this: We multiply rows by coloumns. This means you take the first number in the first row of the second matrix and scale (multiply) it with the first coloumn in the first matrix. You do this with each number in the row and coloumn, then move to the next row and coloumn and do the same.
It could be two vectors, where a and c are one vector and b and d are another vector. Secondly, with a vector $\vec{v}=\begin{bmatrix}x\\y \end{bmatrix}$ ($\vec{v}$ implied being a matrix), we would multiply them like this.
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What are basis vectors? Why are they so important? What is Span and a Linear Combination? Come read, quick recap at the bottom.
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Arden's rule
In theoretical computer science, Arden's rule, also known as Arden's lemma, is a mathematical statement about a certain form of language equations.
Background
A (formal) language is simply a set of strings. Such sets can be specified by means of some language equation, which in turn is based on operations on languages. Language equations are mathematical statements that resemble numerical equations, but the variables assume values of formal languages rather than numbers. Among the most common operations on two languages A and B are the set union A ∪ B, and their concatenation A⋅B. Finally, as an operation taking a single operand, the set A* denotes the Kleene star of the language A.
Statement of Arden's rule
Arden's rule states that the set A*⋅B is the smallest language that is a solution for X in the linear equation X = A⋅X ∪ B where X, A, B are sets of strings. Moreover, if the set A does not contain the empty word, then this solution is unique.[1][2]
Equivalently, the set B⋅A* is the smallest language that is a solution for X in X = X⋅A ∪ B.
Application
Arden's rule can be used to help convert some finite automatons to regular expressions, as in Kleene's algorithm.
See also
• Regular expression
• Nondeterministic finite automaton
Notes
1. Daintith, John (2004). "Arden's Rule". Archived from the original on 13 February 2010. Retrieved 10 March 2010.
2. Sutner, Klaus. "Algebra of Regular Languages" (PDF). Archived from the original (PDF) on 2011-07-08. Retrieved 15 Feb 2011.
References
• Arden, D. N. (1960). Delayed logic and finite state machines, Theory of Computing Machine Design, pp. 1-35, University of Michigan Press, Ann Arbor, Michigan, USA.
• Dean N. Arden (Oct 1961). "Delayed Logic and Finite State Machines". Proc. 2nd Ann. Symp. on Switching Circuit Theory and Logical Design (SWCT), Detroit/MI. (open-access abstract)
• John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley Publishing, Reading Massachusetts, 1979. ISBN 0-201-02988-X. Chapter 2: Finite Automata and Regular Expressions, p.54.
• Arden, D.N. An Introduction to the Theory of Finite State Machines, Monograph No. 12, Discrete System Concepts Project, 28 June 1965.
| Wikipedia |
\begin{document}
\draft \title{Quantum Information Processing with Microfabricated Optical Elements}
\author{F.B.J. Buchkremer, R. Dumke, M. Volk, T. M\"uther, G. Birkl, and W. Ertmer}
\address{Institut f\"ur Quantenoptik, Universit\"at Hannover, Welfengarten 1, 30167 Hannover}
\date{\today} \maketitle
\begin{abstract}
We discuss a new direction in the field of quantum information processing with neutral atoms. It is based on the use of microfabricated optical elements. With these elements versatile and integrated atom optical devices can be created in a compact fashion. This approach opens the possibility to scale, parallelize, and miniaturize atom optics for new investigations in fundamental research and applications towards quantum computing with neutral atoms. The exploitation of the unique features of the quantum mechanical behavior of matter waves and the capabilities of powerful state-of-the-art micro- and nanofabrication techniques lend this approach a special attraction.
\end{abstract}
\pacs{}
\narrowtext
\section {Introduction}
Following the spectacular theoretical results in the field of quantum information processing of recent years \cite{Bouwmeester}, there is now also a growing number of experimental groups working in this area. Among the many currently investigated approaches, which range from schemes in quantum optics to superconducting electronics \cite{Fortschritte}, the field of atomic physics seems to be particularly promising due to the experimentally achieved remarkable control of single qubit systems and the understanding of the relevant coherent and incoherent processes.
While there have been successful implementions of quantum logic with charged atomic particles in ion traps \cite{Monroe}, quantum information schemes based on neutral atoms \cite{Hagley,Jaksch99,Brennen,Jaksch00} are an attractive alternative due to the weak coupling of neutral atoms to their environment. A further attraction of neutral atoms lies in the fact that many of the requirements for the implementation of quantum computation \cite{DiVincenzo} are potentially met by the newly emerging miniaturized and integrated atom optical setups.
These miniaturized setups can be obtained by using different types of microfabricated structures:
The trapping and guiding of neutral atoms in microfabricated {\bf charged and current carrying} structures has been pursued by a number of groups in recent years \cite{Weinstein,Hindsreview,Schmiedmayer,Haensch,Cornell,Prentiss,Engels,Hinds}.
A new approach to generate miniaturized and integrated atom optical systems has been recently introduced by our group \cite{Birkl}: We proposed the application of microfabricated {\bf optical} elements (microoptical elements) for the manipulation of atoms and atomic matter waves with laser light. This enables one to exploit the vast industrial and research interest in the field of applied optics directed towards the development of micro-optical elements, which has already lead to a wide range of state-of-the-art optical system applications \cite{Herzig,Sinzinger} in this field. Applying these elements to the field of quantum information processing, however, constitutes a novel approach. Together with systems based on miniaturized and microfabricated mechanical as well as electrostatic and magnetic devices, the application of microoptical systems will launch a new field in atom optics which we call {\bf ATOMICS} for {\bf AT}om {\bf O}ptics with {\bf MIC}ro-{\bf S}tructures. This field will combine the unique features of devices based on the quantum mechanical behavior of atomic matter waves with the tremendous potential of micro- and nanofabrication technology and will lead to setups that are also very attractive for quantum information processing.
\section {Microoptical Elements for Quantum Information Processing}
A special attraction of using microoptical elements lies in the fact, that most of the currently used techniques in atom manipulation are based on the optical interaction with the atoms. The use of microfabricated optical elements is therefore in many ways the canonical extension of the conventional optical methods into the micro-regime, so that much of the knowledge and experience that has been acquired in atom optics can be applied to this new regime in a very straightforward way. There are however, as we will show in the following, a number of additional inherent advantages in using microoptics which significantly enhance the applicability of atom optics and will lead to a range of new developments that were not achievable until now: The use of state-of-the-art lithographic manufacturing techniques adapted from semiconductor processing enables the optical engineer to fabricate structures with dimensions in the micrometer range and submicrometer features with a large amount of flexibility and in a large variety of materials (glass, quartz, semiconductor materials, plastics, etc.). The flexibility of the manufacturing process allows the realization of complex optical elements which create light fields not achievable with standard optical components. Another advantage lies in the fact, that microoptics is often produced with many identical elements fabricated in parallel on the same substrate, so that multiple realizations of a single conventional setup can be created in a straightforward way. A further attraction of the flexibility in the design and manufacturing process of microoptical components results from the huge potential for integration of different elements on a single substrate, or, by using bonding techniques, for the integration of differently manufactured parts into one system. No additional restrictions arise from the small size of microoptical components since for most applications in atom optics, the defining parameter of an optical system is its numerical aperture, which for microoptical components can easily be as high as NA=0.5, due to the small focal lengths achievable.
Among the plethora of microoptical elements that can be used for quantum information processing applications are refractive or diffractive microoptics, computer generated holograms, microprisms and micromirrors, integrated waveguide optics, near-field optics, and integrated techniques such as planar optics or micro-opto-electro-mechanical systems (MOEMS). Excellent overviews of microoptics can be found in \cite{Herzig,Sinzinger}. To our knowledge, of all these elements only computer generated holograms and phase gratings have been used in atom optics so far for guiding \cite{Schiffer} and trapping \cite{Michaud,Fournier,Ozeri} of atoms, while a new type of atom trap based on the near field of laser radiation has been proposed in \cite{Klimov}.
In this paper we give an overview of the novel possibilities that arise for quantum information processing with neutral atoms if one employs microfabricated optical elements. We show how crucial components for miniaturized systems for quantum information processing with neutral atoms can be realized with microoptical elements.
\section {Experimental Setup}
The key elements in quantum information processing with neutral atoms are atom traps which act as storage devices for the logical qubits inscribed in the internal states of the atoms. The experimental setup that we employ for our studies of the applicability of microoptical elements to generate novel trapping structures is depicted in Fig. \ref{setup}. The focal plane of a microoptical component illuminated by a red detuned laser beam is transferred into the vacuum chamber with the help of two achromats (f = 300 mm, D = 50 mm) so that we can load the atoms into the trapping structures by overlapping them with a magnetooptical trap (MOT). This setup has the advantage, that the performance of the MOT is not disturbed by the microoptical component and that we can experiment with a variety of microoptical elements without the need to open the vacuum chamber.
A particularly simple atom trap is based on the dipole potential of a single focused red detuned laser beam. This trap has been first realized by Chu {\it et al.} \cite{Chu_dipole} and has remained an important element ever since \cite{Grimm}. We have achieved this trapping structure by using a single lens as the microoptical component in Fig. \ref{setup} and have managed to trap approximately $10^5$ $^{85}Rb$ atoms with a lifetime of 166 ms (Fig. \ref{Single-trap}) in a dipole trap of potential depth $U_0/k_B = -1.9 mK$ (laser power P = 50 mW, detunig of 2 nm, Gaussian waist of focus $w_0$ = 15 $\mu$m).
\section {Multiple Atom Traps}
A new approach arises from the application of one- or two-dimensional arrays of spherical microlenses for atom trapping (Fig. \ref{Microlenses}). Microlenses have typical diameters of ten to several hundreds of $\mu$m. Due to their short focal lengths of typically 100$\mu$m to 1mm, their numerical aperture can be easily as high as 0.5, resulting in foci whose focal size q (defined as the radius of the first minimum of the Bessel function which results from the illumination of an individual microlens with a plane wave) can be as low as q=1$\mu$m for visible laser light.
By focusing a single red-detuned laser beam extending over multiple microlenses with a spherical microlens array, we obtain one- or two-dimensional arrays of a large number of dipole traps (Fig. \ref{Klassiker}), in which we store multiple-atom samples \cite{Dumke}.
For frequently used atomic species and commonly used laser sources one can easily obtain a large number of atom traps of considerable depth with rather moderate laser power. For typical laser parameters and 100 atom traps, the trap depth is significantly larger than the kinetic energy of the atoms achievable with Doppler cooling (0.141 mK $\times k_B$ for rubidium) \cite{Birkl}. The low rates of spontaneous scattering that are achievable with sufficiently far-detuned trapping light ensure long storage and coherence times as required for successful quantum information processing, while a strong localization of the atoms strongly suppresses heating of the atoms and makes it possible to cool the atoms to the ground state of the dipole potential via sideband cooling in all dimensions. For the parameters given in \cite{Birkl} the size of the atomic wavefunction reaches values that are significantly smaller than 100 nm, even approaching 10 nm in many cases,
thus making microlens arrays well suited for the generation of strongly confined and well localized atom samples.
The lateral distances between the individual traps (typically 100 $\mu$m) make it easy to selectively detect and address the atom samples in each dipole trap. While the natural way of addressing an individual trap consists in sending the addressing laser beams through the corresponding microlens, there are also more sophisticated methods possible, e.g. with a two-photon Raman-excitation technique as depicted in Fig. \ref{Klassiker}. This technique has been applied frequently to create superposition states in alkali atoms \cite{Kasevich_Raman} and can be used to implement single qubit rotations for quantum information processing. It relies on the simultaneous interaction of the atoms with two mutually coherent laser fields. For a sufficiently large detuning from the single photon resonance, only the atoms in the trap that is addressed by both laser beams are affected by them.
These factors open the possibility to prepare and modify quantum states in a controlled way ("quantum engineering") in each trap, which is a necessary ingredient for quantum computing. As a first and easily achievable implementation, single qubits associated with long-lived internal states can be prepared and rotated in each individual trap of a two-dimensional dipole trap array (Fig. \ref{Microlenses}), stored and later read out again. Thus, this device can serve as a quantum state register.
The manipulation of atoms with microlens arrays is extremely flexible: It is easily possible to temporarily modify the distances between individual traps if smaller or adjustable distances between traps are required. This can be accomplished either by using two independent microlens arrays which are laterally shifted with respect to each other or by illuminating a microlens array with two beams (possibly of different wavelength) under slightly different angles, thereby generating two distinct sets of dipole trap arrays. Their mutual distance can be controlled by changing the angle between the two beams. With a fast beam deflector, this can be done in real-time during the experiment.
Due to this flexibility, setups based on microlens arrays are also well suited for the implementation of two-qubit gates. Considering, for example, quantum phase gates based on dipole-dipole interactions between atoms \cite{Brennen} all requirements are fulfilled in the configuration depicted in Fig. \ref{Klassiker}. Atoms localized in neighboring traps can be first initialized and then be brought close to each other with a definable separation in the single-micron range and for a predefined duration, in order to inscribe the required phase shift. Especially well suited is this configuration also for quantum gates based on the dipole-dipole interaction of low-lying Rydberg states in constant electric fields, as proposed in \cite{Jaksch00}.
\section{Integration}
The huge potential for integration of microoptical components can be used for a large variety of further atom optical purposes. Due to their large numerical aperture, microoptical components can also be used for efficient spatially resolved read-out of quantum information (Fig. \ref{readout}). In most cases the state of a qubit is recorded by exciting the atom state-selectively with resonant light and collecting the fluorescence light. Microoptical components can be used for the collection optics. Furthermore, the optical detection of quantum states with microoptical components is not restricted to optical trapping structures (Fig. \ref{readout} (a)): Since the same techniques are applied for the fabrication of microoptical components and microstructured wires on surfaces, microoptical components can be easily combined with the magnetic and electric structures of \cite{Weinstein,Hindsreview,Schmiedmayer,Haensch,Cornell,Prentiss,Engels,Hinds} (Fig. \ref{readout} (b)). In addition, microfabricated atom-optical components can be integrated with optical fibres and waveguides so that quantum information after being read-out by detection of the scattered light can be further processed by optical means (Fig. \ref{readout} (c)).
Another canonical extension is given by the integration of microoptical components with optoelectronic devices such as semiconductor laser sources and photodiode detectors. In this case, the communication with the outside world can take place fully electronically, with the required laser light created in situ and the optical signals converted back to electrical signals on the same integrated structure. Fig. \ref{VCSEL} illustrates a simple but powerful configuration based on this approach. The depicted setup utilizes vertical cavity surface emitting lasers (VCSELs) \cite{Sinzinger,Iga,Jewell} which are directly mounted onto a two-dimensional microlens array. VCSELs emit circular symmetrical non-astigmatic beams while their structure is optimized for the lithographic fabrication of densely packed 2D arrays. Since VCSELs generally emit a single longitudinal mode at a wavelength which can be tuned by changing the current, the light coming from a two-dimensional array of VCSELs can be directly focused by the mirolens array to create a two-dimensional array of stable dipole traps (Fig. \ref{VCSEL}). With VCSELs it becomes now possible to selectively switch on and off individual traps or to selectively change their potential depth resulting in further flexibility in the atom manipulation.
\section {Conclusion}
In this paper we have discussed the new research direction of using microfabricated optical elements for quantum information processing with neutral atoms. This application hugely benefits from the many inherent advantages of microoptical components. Specifically, an approach based on microoptical systems addresses two of the most important requirements for the technological implementation of quantum information processing: parallelization and scalability. In addition, the possibility to selectively address individual qubits is essential for most schemes proposed for quantum computing with neutral atoms.
Thus, all steps required for quantum information processing with neutral atoms - i.e. the preparation, manipulation and storage of qubits, entanglement and gate operations as well as the efficient read-out of quantum information - can be performed using microfabricated optical elements.
\section {Acknowledgements}
This work is supported by the program ACQUIRE (IST-1999-11055) of the European Commission as well as the SFB 407 and the {\it Schwerpunktprogramm Quanten-Informationsverarbeitung} of the {\it Deutsche Forschungsgemeinschaft}.
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\begin{figure}
\caption{Experimental setup. The light of an incoming red-detuned laser beam is
shaped by a microoptical component and the focal plane is imaged into the vacuum chamber.
The atoms are first collected in a MOT and then transfered into the
optical microstructure.}
\label{setup}
\end{figure}
\begin{figure}
\caption{Inset: Density distribution of atoms in single dipole trap obtained by using a
single microlens in the setup depicted in Fig. \ref{setup}. Main picture: Number of atoms
remaining in dipole trap as a function of time. An exponential fit to the data yields a
storage time of 166 ms.}
\label{Single-trap}
\end{figure}
\begin{figure}
\caption{Refractive (a) and diffractive (b) array of spherical microlenses.}
\label{Microlenses}
\end{figure}
\begin{figure}
\caption{Two-dimensional array of dipole traps created by focusing a red-detuned laser
beam with an array of microlenses. Due to their large separation (typically 100$\mu$m)
individual traps can be addressed selectively, e.g. by two-photon Raman-excitation,
as depicted.}
\label{Klassiker}
\end{figure}
\begin{figure}
\caption{Spatially resolved readout of the internal and external states
of atoms (e.g. the state of a qubit) using microlens arrays: (a) Integration
of two spherical microlens arrays creates a combined system of dipole
traps and efficient detection optics.
(b) Integration of a microlens array (for readout) with microfabricated
magnetic or electrostatic trapping
structures. (c) Optical waveguides and fibres can also be integrated on the substrate.}
\label{readout}
\end{figure}
\begin{figure}
\caption{Integration of microoptical components and laser sources. An array of
vertical cavity surface emitting lasers (VCSELs)
illuminates a microlens array with matched lens separation. Each trap of
the resulting two-dimensional
array of dipole traps can be individually
switched on and off because the individual VCSELs are selectively addressable.}
\label{VCSEL}
\end{figure}
\end{document} | arXiv |
Seven mathematicians met up one week.
The first mathematician shook hands with all the others.
The second one shook hands with all the others apart from the first one (since they had already shaken hands).
The third one shook hands with all the others apart from the first and the second mathematicians, and so on, until everyone had shaken hands with everyone else.
How many handshakes were there altogether?
The next week, eight mathematicians met. How many handshakes took place this time?
Sam is trying to work out how many handshakes there would be if 20 mathematicians met. He says that since each mathematician shakes hands 19 times, there must be $20 \times 19$ handshakes altogether. Helen disagrees; she worked out $19 + 18 + 17 + ... + 2 + 1$ and got a different answer. What is wrong with Sam's reasoning? How should he modify his method?
One day, 161 mathematicians met. How many handshakes took place this time?
Can you describe a quick way of working out the number of handshakes for any size of mathematical gathering?
Could there be exactly 4851 handshakes at a gathering where everyone shakes hands? How many mathematicians would there be?
What about the following numbers of handshakes?
You may wish to try the problems Picturing Triangle Numbers and Mystic Rose. Can you see why we chose to publish these three problems together?
You may also be interested in reading the article Clever Carl, the story of a young mathematician who came up with an efficient method for adding lots of consecutive numbers.
Curious. Visualising. Interactivities. Games. Mathematical reasoning & proof. Triangle numbers. Generalising. Creating and manipulating expressions and formulae. Patterned numbers. Factors and multiples. | CommonCrawl |
\begin{document}
\title{Some Remarks on Schauder Bases in Lipschitz Free Spaces}
\author{Mat\v ej Novotn\'y} \address{Department of Mathematics\\Faculty of Electrical Engineering\\ Czech Technical University in Prague\\ Jugosl\'{a}vsk\'{y}ch Partyz\'an\r{u} 1580/3, 160 00, Prague}
\address{Department of Industrial Informatics\\Czech Institute of Informatics, Robotics, and Cybernetics\\ Czech Technical University in Prague\\ Jugosl\'{a}vsk\'{y}ch Partyz\'an\r{u} 1580/3, 160 00, Prague} \email{[email protected]}
\subjclass[2000]{46B03, 46B10.} \keywords{Lipschitz-free space, Schauder basis, extension operator, unconditionality} \thanks{The work was supported in part by GA\v CR 16-073785, RVO: 67985840 in part by grant SGS18/064/OHK4/1T/13 of CTU in Prague and in part by Ministry of Education, Youth and Sport of the Czech Republic within the project Cluster 4.0 number CZ.02.1.01/0.0/0.0/16\_ 026/0008432.} \subjclass[2010]{46B20, 46T20}
\begin{abstract} We show that the basis constant of every retractional Schauder basis on the Free space of a graph circle increases with the radius. As a consequence, there exists a uniformly discrete subset $M\subseteq\R^2$ such that $\mathcal F(M)$ does not have a retractional Schauder basis. Furthermore, we show that for any net $ N\subseteq\R^n$, $n\geq 2$, there is no retractional unconditional basis on the Free space $\mathcal F(N)$. \end{abstract}
\maketitle
\section{Introduction}
Let $(M,d)$ be a metric space with a distinguished point $0\in M$. Denote $\operatorname{Lip}_0(M)$ the space of all Lipschitz functions $f:M\to\R$ with the property $f(0)=0$. Such a space can be equipped with the Lipschitz norm $\| f\|=\operatorname{sup}_{x\neq y}\frac{|f(x)-f(y)|}{d(x,y)}$, which turns it into a Banach space. We see that each point in $M$ can be naturally embedded into $\operatorname{Lip}_0(M)^*$ via the Dirac mapping $\delta$: $\delta_x(f)=f(x)$, $f\in\operatorname{Lip}_0(M)$, $x\in M$. The norm-closure of the subspace generated by functionals $\delta_x$, $x\in M$, i.e.
$$\overline{\operatorname{span}}^{\operatorname{Lip}_0(M)^*}\{\delta_x|x\in M\right\rbrace$$ is the Lipschitz Free space over $M$, denoted $\mathcal F(M)$. Lipschitz Free spaces were introduced already by Arens and Eells in \cite{AE}, although the authors did not use the name Lipschitz Free spaces. Free spaces are called Arens-Eells spaces in \cite{W}, where a lot of results regarding the topic is presented.
Lipschitz Free spaces gained a lot of interest in last decades, connecting nonlinear theory with the linear one. Given two pointed metric spaces $M,N$, every Lipschitz mapping $\varphi:M\to N$ which fixes the point $0$ extends to a bounded linear map $F:\mathcal F(M)\to\mathcal F(N)$, making the following diagram commute: $$\begin{CD} \mathcal F(M) @>{F}>> \mathcal F(N)\\ @A{\delta_M}AA @AA{\delta_N}A\\ M @>{\varphi}>> N \end{CD}$$ We focus on structural properties of Lipschitz Free spaces. It is well-known that $\operatorname{Lip}_0(\R)= L_\infty$, which yields $\mathcal F(\R)= L_1$ isometrically and similarly $\mathcal F(\N)=\ell_1$. In \cite{CDW}, the authors prove that $\mathcal F(M)$ contains a complemented copy of $\ell_1(\N)$ if $M$ is infinite (has at least cardinality $\aleph_0$), which was further extended from $\N$ to all cardinalities in \cite{HN}. However, $\mathcal F(\R^2)$ cannot be embedded in $\mathcal F(\R)=L_1$ (see \cite{NS}).
Certain results were obtained concerning approximation properties in Free spaces, including \cite{PS},\cite{LP},\cite{HLP},\cite{K},\cite{Godefroy},\cite{GO} and of course \cite{GK}. However, not much is known yet about Schauder bases in Free spaces. H\'{a}jek and Perneck\'{a} \cite{HP} constructed a Schauder basis for the Free spaces $\mathcal F(\ell_1)$ and $\mathcal F(\R^n)$. From \cite{Kaufmann} we have $\mathcal F(M)$ is isomorphic to $\mathcal F(\R^n)$ for every $M$ with non-empty interior, which gives existence of Schauder basis on such $\mathcal F(M)$.
This article follows up the article \cite{HN}, where the authors proved existence (and in the case of $c_0$ constructively) of a Schauder basis on $\mathcal F(N)$, for any net $N$ in spaces $C(K)$ for $K$ metrizable compact (hence for $c_0$ and $\R^n$). In section \ref{nonexistence} we show that the same construction as in \cite{HN} cannot be used for constructing bases in $\mathcal F(N)$ for arbitrary uniformly discrete subset $N$. In section \ref{unconditionality} we prove that bases constructed in \cite{HN} are not unconditional and that for nets in $\R^n$, no Schauder basis on $\mathcal F(N)$ arising from the technique using retractions can be unconditional. \section{Preliminaries} As we mentioned, we are interested in constructing a Schauder basis on Lipschitz Free space. However, constructing such basis directly on the Free space is rather complicated, wherefore we prefer to work with its adjoint space and transfer the results to the Free space. The next theorem shows a way to construct a Schauder basis through operators on $\operatorname{Lip}_0(M)$.
\begin{theorem} \label{operator} Let $M$ be a pointed metric space. Suppose there exists a sequence of linear operators $E_n:\operatorname{Lip}_0(M)\to \operatorname{Lip}_0(M)$, which satisfies the following conditions: \begin{enumerate} \item $\dim E_n \left(\operatorname{Lip}_0(M)\right)=n$ for every $n\in\N$, \item There exists $K>0$ such that $E_n$ is $K$-bounded for every $n\in\N$, \item $E_m E_n=E_n E_m=E_n$ for every $m,n\in\N$, $n\leq m$, \item \label{weak}For every $n$, the operator $E_n$ is continuous with respect to topology of pointwise convergence on $\operatorname{Lip}_0(M)$, \item \label{continuity} For every $f\in\operatorname{Lip}_0(M)$ the function sequence $E_n f$ converges pointwise to $f$. \end{enumerate} Then the space $\mathcal F(M)$ has a Schauder basis with the basis constant at most $K$. \end{theorem} \begin{proof} Note first that the topology of pointwise convergence coincides with the $w^*$-topology on bounded subsets of $\operatorname{Lip}_0(M)$. Therefore, from the condition (\ref{weak}), the operators $E_n$ are $w^*$ to $w^*$ continuous on bounded subsets of $\operatorname{Lip}_0(M)$ and hence there exist linear operators $P_n:\mathcal F(M)\to\mathcal F(M)$ such that $P^*_n=E_n$ for every $n\in\N$. It is now clear that $\| P_n\|\leq K$, $\dim P_n \left(\mathcal F(M)\right)=n$ and that $P_m P_n=P_n P_m=P_n$ for every $m,n\in\N$, $n\leq m$. Furthermore (\ref{continuity}) together with the fact that the topology of pointwise convergence coincides with the $w^*$-topology on bounded subsets of $\operatorname{Lip}_0(M)$ means, that for every $f\in \operatorname{Lip}_0(M)$ the sequence $E_n f$ converges $w^*$ to $f$, and that for every $\mu\in\mathcal F(M)$ the sequence $P_n\mu$ converges weakly to $\mu$. But that means $\| P_n\mu-\mu\|\to 0$ for every $\mu\in\mathcal F(M)$. Indeed, if there were $\mu\in\mathcal F(M)$, $c>0$ and a subsequence $P_{n_k}$, such that $\| P_{n_k}\mu-\mu\|>c$ for all $k\in\N$, then for every $n\geq n_1$, there exists a $k\in\N$ such that $n\leq n_k$, which yields
$$c<\| P_{n_k}\mu-\mu\|\leq\| P_{n_k}\mu-P_n\mu\| + \| P_{n}\mu-\mu\|\leq (K+1)\| P_n\mu-\mu\|.$$
From $P_1(\mathcal F(M))\subseteq P_2(\mathcal F(M))\subseteq P_3(\mathcal F(M))\subseteq...$ we get $E=\bigcup_{n=1}^{\infty}P_n(\mathcal F(M))$ is a convex set and as all $P_n$ are commuting projections, we have that $\mu\notin\overline E$. Indeed, if $\mu \in \overline E$, then there is a sequence $\{x_k\right\rbrace_{k=1}^\infty\subseteq E$, such that $x_k\to \mu$. If we choose an increasing sequence of numbers $l_k\in\N$, $l_k>n_1$, which satisfy $P_{l_k}x_k=x_k$, we get that $$\| P_{l_k}x_k-\mu\|\geq \| P_{l_k}\mu-\mu\|-\| P_{l_k}\mu-P_{l_k}x_k\|\geq \frac{c}{K+1}-K\| \mu-x_k\|.$$
Limiting $k\to\infty$ yields $0\geq \frac{c}{K+1}$, which is a contradiction. Therefore $\mu\notin\overline E$. Hence Hahn-Banach theorem gives us the existence of a linear functional $f\in\operatorname{Lip}_0(M)$, $\| f\|=1$ with $f|_E=0$ and $f(\mu)>0$. But that is a contradiction as $P_n\mu\overset{w}\to\mu$. Therefore $P_n \mu\to \mu$. \end{proof}
The following corollary appears already in \cite{HN}, p.12. It gives us a way to construct the Schauder basis on $\mathcal F(M)$ only by using the metric space $M$.
\begin{corollary}\label{one} Let $M$ be a metric space with a distinguished point $0$. Suppose there exists a sequence of distinct points $\{\mu_n\right\rbrace_{n=0}^{\infty}\subseteq M$, $\mu_0=0$, together with a sequence of retractions $\{\varphi_n\right\rbrace_{n=0}^\infty$, $\varphi_n:M\to M$, $n\in\N_0$ which satisfy the following conditions: \begin{enumerate}[(i)] \item $\varphi_n(M)=\{\mu_j\right\rbrace_{j=0}^{n}$ for every $n\in\N_0$,\label{bed1} \item $\overline{\bigcup_{j=0}^{\infty}\{\mu_j\right\rbrace}=M$, \label{bed2} \item There exists $K>0$ such that $\varphi_n$ is $K$-Lipschitz for every $n\in\N_0$,\label{bed3} \item $\varphi_m\varphi_n=\varphi_n\varphi_m=\varphi_n$ for every $m,n\in\N_0$, $n\leq m$. \label{bed4} \end{enumerate} Then the space $\mathcal F(M)$ has a Schauder basis with the basis constant at most $K$. \end{corollary} \begin{proof} It is not difficult to see that for each $n\in\N$ the formula $E_nf=f\circ\varphi_n$, $f\in\operatorname{Lip}_0(M)$ defines a linear operator $E_n:\operatorname{Lip}_0(M)\to\operatorname{Lip}_0(M)$, such that the sequence $E_n$ satisfies the assumptions of Theorem \ref{operator}. \end{proof}
The last two theorems lead us to the following definition.
\begin{definition} Let $M$ be an infinite metric space such that $\mathcal F(M)$ has a Schauder basis $E$ with projections $P_n$, $n\in\N$. We say $E$ is an extensional Schauder basis if there exist finite sets $\{0\right\rbrace=M_0\subseteq M_1\subseteq M_2\subseteq...$ such that $\bigcup_{n=1}^\infty M_n$ is dense in $M$ and we have that for every $n\in\N$ the adjoint $P_n^*$ is a linear extension operator $P_n^*:\operatorname{Lip}_0(M_n)\to\operatorname{Lip}_0(M)$ with $P_n^*f|_{M_n}=f$ (or equivalently $P_n$ is a projection onto $\mathcal F(M_n)$). We say $E$ is a retractional Schauder basis, if there exist retractions $\{\varphi_n\right\rbrace_{n=0}^\infty$, $\varphi_n:M\to M$ which satisfy the conditions of Corollary \ref{one} and such that they give rise to the basis $E$, i.e. the adjoints $P_n^*$ satisfy $P_n^*f=f\circ\varphi_n$, $f\in\operatorname{Lip}_0(M)$. \end{definition}
It is clear that in the definition we actually have $|M_n\setminus M_{n-1}|=1$ for every $n\in\N$. Note also that every retractional Schauder basis is a special case of an extensional Schauder basis. The next lemma shows in more detail what form the basis vectors take. \begin{lemma}\label{char} Let $M$ be a metric space such that there is a sequence of distinct points $0=\mu_0,\mu_1,\mu_2,...\in M$ such that $\bigcup_{n=1}^\infty \{\mu_0,\mu_1,...,\mu_n\right\rbrace$ is dense in $M$. For every $n\in\N_0$ denote $M_n=\{\mu_0,...,\mu_n\right\rbrace$. Suppose $\mathcal F(M)$ has a Schauder basis $B=\{e_n\right\rbrace_{n=1}^\infty$. Then the following are equivalent: \begin{enumerate} \item\label{extb} $B$ is an extensional Schauder basis with extension operators $E_n:\operatorname{Lip}_0(M_n)\to\operatorname{Lip}_0(M)$. \item\label{basf} For every $n\in\N$, there are constants $0\neq c_n,a^n_i\in\R$, $i\in\{1,...,n-1\right\rbrace$ such that we have $c_n e_n=\delta_{\mu_n}-\sum_{i=1}^{n-1}a_i^{n}\delta_{\mu_i}$. \end{enumerate} \end{lemma} \begin{proof} $(\ref{basf})\Rightarrow(\ref{extb})$. Note first that for every $n\in\N$, we have $e_n\in \operatorname{Im} P_n\cap\ker P_{n-1}$. From that it follows inductively for every $n\in\N$ that $\operatorname{Im} P_n=\operatorname{span}\{\delta_{\mu_1},...,\delta_{\mu_n}\right\rbrace$.
$(\ref{extb})\Rightarrow(\ref{basf})$ The fact that $E_n=P^*_n$ is a bounded linear extension from $M_n$ to $M$ implies that each $P_n$ maps $\mathcal F(M)$ onto $\mathcal F(M_n)$, which means each basis vector $e_n$ is a linear combination of Dirac functionals at the points of $M_n$, such that the coefficients at $\delta_{\mu_n}$ do not vanish. \end{proof} Keeping the notation from previous lemma, we see that for each $n\in\N$ we may define a finite dimensional operator $R_{n}:\operatorname{Lip}_0(M_{n-1})\to\operatorname{Lip}_0(M_{n})$ via $$R_nf(\mu_{j})=\begin{cases} f(\mu_j) & j\in\{0,...,n-1\right\rbrace\,,\\ \sum_{i=1}^{n-1}a_i^{n}f(\mu_{i}) & j=n\,.\\ \end{cases}$$ The operator $E_n=P^*_n$ can be then reconstructed through a $w^*$-limit of operator composition $\lim_k R_kR_{k-1}...R_{n+1}$. The constants $c_n$ were in the lemma only for scaling of the basis vectors $e_n$.
In case of a retractional basis, the basis vectors take form of two-point molecules: For every $n\in\N$ and $i\in\{1,...,n-1\right\rbrace$ exactly one of the coefficients $a_i^n$ is non-zero, namely has the value $1$. If for example $a_j^n=1$, then $\varphi_j(\mu_n)=\mu_j$, which means $e_n=\delta_{\mu_n}-\delta_{\mu_{j}}$.
Throughout this article, given a metric space $M$, $d$ will denote its metric. If $M$ is a countable (even finite) uniformly discrete metric space with $\mathcal F(M)$ having a retractional Schauder basis, by symbols $\mu_0,\mu_1,\mu_2...$, resp. $\varphi_0,\varphi_1,\varphi_2,...$ we will always mean points $\mu_i\in M$, resp. retractions $\varphi_i:M\to M$ which satisfy Corollary \ref{one}. Obviously the finite analogues of Corollary \ref{one} and Theorem \ref{operator} also hold. We are going to look in more detail on some properties of retractional Schauder basis.
Following the notation of Corollary \ref{one} (or the proof of Lemma 14 in \cite{HN}) we find useful to denote the set-valued functions $F_i=\varphi_i^{-1}:M\to 2^M$, $F_i(x)=\{y|\ \varphi_i(y)=x\right\rbrace$, $i\in\N_0$. Clearly $F_0(0)=M$. From the commutativity of the $\varphi_i$'s further follows that for any $i<j$ we have $$F_i(\mu_i)\cap F_j(\mu_j)\in\{\emptyset,F_j(\mu_j)\right\rbrace.$$
\begin{definition} Let $M$ and $\mu_i$, $\varphi_i$, $i\in\N_0$ satisfy the assumptions from Corollary \ref{one} and $M=\{\mu_i\right\rbrace_{i=0}^\infty$. A finite or infinite sequence of points $(\mu_{k_1},\mu_{k_2},\mu_{k_3},...)$ is called a chain whenever $k_1<k_2<k_3<...$ and $\varphi_{k_i-1}(\mu_{k_i})=\mu_{k_{i-1}}$ for every $i\in\{2,3,...\right\rbrace$.
\end{definition} Note that for every chain $(\mu_{k_1},\mu_{k_2},\mu_{k_3}...)$ we have $F_{k_1}(\mu_{k_1})\supseteq F_{k_2}(\mu_{k_2})\supseteq F_{k_3}(\mu_{k_3})\supseteq ...$. We can also introduce partial order on $M$ by $\mu_i\prec\mu_j$ if and only if there exist $n\in\N_0$ points $\mu_{k_1},...,\mu_{k_n}\in M$ such that $(\mu_i,\mu_{k_1},...,\mu_{k_n},\mu_j)$ is a chain. Note also that for two chains $S,T$ the difference $S\setminus T$ and intersection $S\cap T$ are also chains, if nonempty. For a finite chain $(x_1,x_2,...,x_n)$ we call the point $x_1$ its initial point and $x_n$ its final point.
Every chain can be viewed as a path or its segment from $0\in M$ to a given point $x\in M$. Indeed, for every $x\in M$ there exists $n\in\N$ such that for every $i\geq n$ one has $\varphi_i(x)=x$. Assuming $n$ is the least number with that property we can define the set $T^{x}_0=\bigcup_{i=0}^{n}\{\varphi_i(x)\right\rbrace$ which contains exactly the points of the chain with initial point $0$ and final point $x$. Regarding $T_0^x$ as an ordered set (the order $\prec$ is linear on $T_0^x$), it is clear that given $x\in M$, there exists exactly one chain $T_0^x$ from $0$ to $x$.
Note also that for every chain $(\mu_{n_1},...,\mu_{n_k})$, $k\geq 2$ there exist constants $c_{n_i}$ (the constants from Lemma \ref{char}) such that for basis vectors $e_{n_1},...,e_{n_k}$ we have $$\sum_{i=2}^kc_{n_i}e_{n_i}=\delta_{\mu_{n_k}}-\delta_{\mu_{n_1}}.$$ The following lemma says that, if the space is not too "porous", basis vectors can be made only of two-point molecules in points which are not too far from each other.
\begin{lemma}[Step lemma]\label{step} Let $M$ be a countable metric space, $\alpha>0, K\geq 1$ and $\varphi_n:M\to M$ a system of retractions from Corollary \ref{one}. If $(\mu_{i_1},...,\mu_{i_j})$, $j>1$ is a chain and there exist distinct points $x_1,...,x_k\in M$ with $d(x_l,x_{l+1})\leq\alpha$, $l\in\{1,...,k-1\right\rbrace$, $x_1=\mu_{i_j}$, $x_k=\mu_{i_1}$ and $\operatorname{sup}_{i_1\leq n\leq i_j}\operatorname{Lip}\varphi_n\leq K$, then $d(\mu_{i_{m-1}},\mu_{i_{m}})\leq 2K\alpha$ for all $m\in\{2,...,j\right\rbrace$. \end{lemma} \begin{proof} Suppose $d(\mu_{i_{m-1}},\mu_{i_{m}})> 2K\alpha$ for some $m\in\{2,...,j\right\rbrace$. We know $\varphi_{i_m}(\mu_{i_j})=\mu_{i_m}$ and $\varphi_{i_m-1}(\mu_{i_j})=\mu_{i_{m-1}}$. We prove by induction for all $l\in\{1,...,k\right\rbrace$ that $\varphi_{i_m}(x_l)=\mu_{i_m}$ and $\varphi_{i_m-1}(x_l)=\mu_{i_{m-1}}$, which is a contradiction as $x_k=\mu_{i_1}$ and $\varphi_{i_m}(\mu_{i_1})=\mu_{i_1}\neq \mu_{i_m}$. For $l=1$ we have $x_l=\mu_{i_j}$ and the statement clearly holds. Suppose it holds for all $l=1,...,s-1<k$. From $d(x_{s-1},x_{s})\leq\alpha$ it follows that $d(\mu_{i_m},\varphi_{i_m}(x_{s}))\leq K\alpha$ and $d(\mu_{i_{m-1}},\varphi_{i_{m}-1}(x_{s}))\leq K\alpha$. From commutativity of all $\varphi_n$'s follows that either $\varphi_{i_{m}-1}(x_s)=\varphi_{i_m}(x_s)\notin\{\mu_{i_m}\right\rbrace$ holds or $\varphi_{i_m}(x_s)=\mu_{i_m}$ and $\varphi_{i_m-1}(x_s)=\mu_{i_{m-1}}$ is true. Since $B_{K\alpha}(\mu_{i_m})\cap B_{K\alpha}(\mu_{i_{m-1}})=\emptyset$ we conclude the latter is true, which completes the induction step and the contradiction is obtained. \end{proof}
In the following section, we are going to prove that there are spaces $\mathcal F(M)$ with no retractional Schauder basis yet having Schauder basis, moreover extensional.
\section{Nonexistence of retractional Schauder bases}\label{nonexistence}
\begin{definition} Let $x_0,x_1,...,x_n$, $n\in\N$ be distinct points. The set $C^0_n=\{x_0,x_1,x_2...,x_n\right\rbrace$ with the (standard graph) metric $d(x_k,x_0)=n$, $k\neq 0$, $d(x_k,x_l)=\min\{|k-l|,n-|k-l|\right\rbrace$, $k,l>0$ we call a circle or a circle of radius $n$ with centre $x_0$. \end{definition} In the following, we regard the centre $x_0$ as the base point in the pointed metric space $(C^0_n,d,x_0)$ and denote it $0$.
We are also going to use an uncentered circle, i.e. a subgraph $C_n=\{x_1,x_2,...,x_n\right\rbrace$ with the induced metric. On $C_n$, we define orientation: We say point $x_l$ lies to the left of the point $x_k$, $k,l\in\{1,...,n\right\rbrace$, if one of these situations happens: \begin{enumerate} \item $k>\frac{n-1}{2}$ and $l\in\{k,k-1,...,k-\lfloor\frac{n+1}{2}\rfloor+1\right\rbrace$, \item $k\leq \frac{n-1}{2}$ and $l\in\{k,k-1,...,1\right\rbrace\cup\{n,n-1,...,n-\lfloor\frac{n+1}{2}\rfloor+k+1\right\rbrace$. \end{enumerate} Analogously, we say $x_l$ lies to the right of $x_k$ if one of the following conditions is satisfied: \begin{enumerate} \item $k\leq\frac{n-1}{2}$ and $l\in\{k,k+1,...,k+\lfloor\frac{n+1}{2}\rfloor\right\rbrace$, \item $k>\frac{n-1}{2}$ and $l\in\{k,k+1,...,n\right\rbrace\cup\{1,2,...,\lfloor\frac{n+1}{2}\rfloor-(n-k+1)\right\rbrace$. \end{enumerate} We show that every retractional Schauder basis on $\mathcal F(C_n^0)$ has a basis constant which is increasing with $n$.
\begin{theorem}\label{circle} Let $n\in\N$, $n\geq 10$ and let $\{\varphi_i\right\rbrace_{i=0}^n$ be a system of retractions on a circle $C^0_n$ satisfying the conditions of Corollary \ref{one} . Then there is an $s\in\{1,...,n\right\rbrace$ such that $$\operatorname{Lip}\varphi_s\geq\frac{\sqrt{8n+1}-1}{8}.$$ \end{theorem} \begin{proof} Let us fix $n\geq 10$ and denote $K=\frac{\sqrt{8n+1}-1}{8}$. We have $\mu_0=0$ and $\mu_1\in C_n$ with $\varphi_1(x)=\mu_1$ for all $x\in C_n$ and $\varphi_1(0)=0$. Indeed, if $\varphi_1(y)=0$ for some $y\in C_n$, then the sets $F_1(0)$ and $F_1(\mu_1)$ have distance $1$. Since they are finite, there exist $w\in F_1(0)$, $z\in F_1(\mu_1)$ such that $d(w,z)=1$ and clearly $d(\varphi_1(w),\varphi_1(z))=n$, which trivially yields the result, as $n>K$. We prove the theorem by contradiction and assume therefore, $\operatorname{Lip}\varphi_i< K$ for all $i\in\{1,...,n\right\rbrace$.
For every point $x\in C_n$ there exists a $k\in\{1,...,n\right\rbrace$ such that $\{x\right\rbrace=\{\mu_k\right\rbrace=\varphi_k(C_n)\setminus\varphi_{k-1}(C_n)$ and therefore there exists exactly one chain $S_x=(\mu_{k_1},\mu_{k_2},...,\mu_{k_l})$, such that $\mu_{k_1}=\mu_1$ and $\mu_{k_l}=x$ (equivalently $k_1=1$ and $k_l=k$).
Let us introduce sets
$$A=\{y|\ y\in C_n\setminus\{\mu_1\right\rbrace,d(y,\mu_1)\leq 3K,\text{$y$ lies to the left of $\mu_1$}\right\rbrace$$
$$B=\{y|\ y\in C_n\setminus\{\mu_1\right\rbrace,d(y,\mu_1)\leq 3K,\text{$y$ lies to the right of $\mu_1$}\right\rbrace$$ and a mapping $f:C_n\setminus\{\mu_1\right\rbrace\to\{A,B\right\rbrace$, $$f(w)=\begin{cases} A & \text{there is a $z\in S_w\cap A$ such that for every $y\in S_w$, $z\prec y$, we have $y\notin A\cup B$}\,, \\ B & \text{there is a $z\in S_w\cap B$ such that for every $y\in S_w$, $z\prec y$, we have $y\notin A\cup B$}\,. \\ \end{cases}$$ Note that the definitions of $A,B$ make perfect sense, as $3K<\frac{n}{2}$. Also, the mapping $f$ is well-defined, as for every $w\in C_n\setminus\{\mu_1\right\rbrace$ the intersection $S_w\cap(A\cup B)$ is nonempty. Indeed, according to Step lemma \ref{step} applied on the $C_n$, the distance between any two adjacent points in a chain is smaller than $2K$ and therefore for the second element $z\in S_w$ (meaning $S_w=(\mu_1,z,...,w)$) we have $d(\mu_1,z)\leq 2K$ and thus $z\in A$ or $z\in B$.
Observe that $f(w)=A$ for every $w\in A$ and $f(w)=B$ for every $w\in B$. We prove there exist two points $a,b\in C_n\setminus\left(\{\mu_1\right\rbrace\cup A\cup B\right)$ such that $d(a,b)=1$, $f(a)=A$ and $f(b)=B$.
Let us assume for contradiction that $f(w)=A$ for all points $w\in C_n\setminus\left(\{\mu_1\right\rbrace\cup A\cup B\right)$. Denote $z$ the closest point to the right of the set $B$, i.e. the only point with $3K <d(z,\mu_1)\leq 3K+1$ and $d(z,B)=1$. We have $f(z)=A$, which means the chain $S_z=(\mu_1,\mu_{k_2},...,z)$ leaves the set $A$ and goes to the left around (meaning omitting the set $A\cup B$) the circle to the point $z$, with steps smaller than $2K$. Therefore there exists a point $\mu_l\in S_z$ such that $d(\mu_1,\mu_l)\geq\frac{n-2K}{2}$. But then we have $\varphi_{l}(\mu_1)=\mu_1$, $\varphi_{l}(z)=\mu_l$, which yields $$\operatorname{Lip}\varphi_l\geq \frac{d(\mu_1,\mu_l)}{d(\mu_1,z)}\geq\frac{n-2K}{2(3K+1)}\geq K,$$ as $n\geq 10$, which contradicts our assumption.
Therefore, let there exist two points $a,b\in C_n\setminus\left(\{\mu_1\right\rbrace\cup A\cup B\right)$ such that $d(a,b)=1$, $f(a)=A$ and $f(b)=B$. Consider the two chains $S_a=(\mu_1,\mu_{k_1},\mu_{k_2}...,a)$ and $S_b=(\mu_1,\mu_{l_1},\mu_{l_2}...,b)$ and let $i$ and $j$ be such that $\mu_{k_i}\in A$, $\mu_{l_j}\in B$ and we have $\mu,\nu\notin A\cup B\cup\{\mu_1\right\rbrace$ for every $\mu\in S_a$, $\mu_{k_i}\prec \mu$ and every $\nu\in S_b$, $\mu_{l_j}\prec \nu$. Note that $d(\mu_1,\mu_{l_j})\geq K+1$ and $d(\mu_1,\mu_{k_i})\geq K+1$.
Without loss of generality suppose $k_i<l_j$. Then $\varphi_{l_j}(b)=\mu_{l_j}$ and $d(\varphi_{l_j}(a),\mu_{l_j})\leq K$. This implies $u_a:=\varphi_{l_j}(a)$ has distance at most $K$ from the set $B$, which yields $d(\mu_1,u_a)\leq 4K$ and the chain $S=(\mu_{k_i},...,u_a)$ must go from the set $A$ to the left around the circle closer to the set $B$. Thus there must exist a point $v=\mu_s\in S$ such that $d(v,\mu_1)\geq\frac{n-2K}{2}$. It follows that $$\operatorname{Lip} \varphi_s\geq \frac{d(\varphi_s(u_a),\varphi_s(\mu_1))}{d(u_a,\mu_1)}=\frac{d(v,\mu_1)}{d(u_a,\mu_1)}\geq\frac{n-2K}{8K}=K,$$ which is again a contradiction. We conclude there exists an $s\in\{1,...,n\right\rbrace$ such that $\operatorname{Lip}\varphi_s\geq K=\frac{\sqrt{8n+1}-1}{8}$. \end{proof} \begin{corollary}\label{no basis} There exists a uniformly discrete set $N\subseteq\R^2$ such that the Free space $\mathcal F(N)$ has no retractional Schauder basis. \end{corollary} \begin{proof} Let $N=\bigcup_{n=1}^\infty C^0_{4^n}$ be a union of circles with the same centre $0$ and with radii $4^n$, $n\in\N$. Suppose $d_n$ is the metric on $C^0_{4^n}$. Let us define a metric on $N$ in the following way: $$d(x,y)=\begin{cases} d_n(x,y) & \text{ if } x,y\in C^0_{4^n}\,,\\ \max\{4^i,4^j\right\rbrace & \text{ if } x\in C_{4^i},y\in C_{4^j},i\neq j\,.\\ \end{cases}$$ It is clear that $d$ is indeed a metric on $N$ and one has no difficulties to embed $N$ into $\R^2$ in a bilipschitz way, actually with distortion not worse than $2\pi$. We show that every sequence of retractions $\varphi_i:N\to N$ satisfying conditions $(i)$ and $(iv)$ from Corollary \ref{one} cannot satisfy the condition $(iii)$ of that corollary.
Let therefore $\varphi_i:N\to N$ be a commuting sequence of retractions such that $\varphi_0(0)=0$ and $|\varphi_i(N)|=i+1$. We show that for every $k\in\N$, $k\geq 4$, there exists an $n=n_k\in\N$ such that $\operatorname{Lip}\varphi_{n_k}\geq k.$ Pick therefore $k\in\N$, $k\geq 4$, and find the smallest $n$ such that $\mu_n\in C_{4^k}$. Then $\varphi_i(\mu_n)=\mu_n$ for every $i\geq n$. If there exist $j\geq n$ and $x\in C_{4^k}$ such that $\varphi_j(x)\notin C_{4^k}$, we have $\operatorname{Lip}\varphi_j\geq 4^k\geq k$ and the proof is finished. Indeed, if we take $x\in C_{4^k}$ such that $\varphi_j(x)\notin C_{4^k}$ and without loss of generality we assume $x$ is such that $d(x,\mu_n)$ is minimal among all $x\in C_{4^k}$ with $\varphi_j(x)\notin C_{4^k}$, we have $d\left(\varphi_j(y),\varphi_j(x)\right)\geq 4^k\geq k$ for one of $x$'s neighbours $y$ (i.e. $d(x,y)=1$). This means $\operatorname{Lip}\varphi_j\geq k$. If, on the contrary, we have $\varphi_i(x)\in C_{4^k}$ for all $i\geq n$ and all $x\in C_{4^k}$, we find ourselves in the case of Theorem \ref{circle}. Indeed, if we view the circle $C^0_{4^k}$ as a set $C^0_{4^k}=\{0,\mu_{s_1},\mu_{s_2},...,\mu_{s_{4^k}}\right\rbrace$ (for some eligible $s_1,s_2,...,s_{4^k}\in\N$) and look only at retractions $\varphi_0,\varphi_{s_1},\varphi_{s_2},...,\varphi_{s_{4^k}}$ restricted to the circle $C^0_{4^k}$, we apply \ref{circle} and conclude $\max\{\operatorname{Lip}\varphi_{s_1},\operatorname{Lip}\varphi_{s_2},...,\operatorname{Lip}\varphi_{s_{4^k}}\right\rbrace\geq\frac{\sqrt{8\cdot 4^k+1}-1}{8}\geq k$. \end{proof} We see it is impossible to build a retractional Schauder basis on $\mathcal F(N)$. However, the space $\mathcal F(N)$ has an extensional Schauder basis as we are going to show in the next proposition: \begin{proposition} Let $N=\bigcup_{n=1}^\infty C^0_{4^n}$ be the metric space from Corollary \ref{no basis}. Then $\mathcal F(N)$ has an extensional monotone Schauder basis. \end{proposition} \begin{proof} First, note that we have orientation of every $C_{4^n}$, $n\in\N$. For every $i\in\N$, define $k=k(i)$ as the unique integer such that $\frac{4^{k}-1}{3}\leq i< \frac{4^{k+1}-1}{3}$. Let $N=\{0,x_1,x_2,x_3,...\right\rbrace$ be enumerated in such way that for every $i\in\N$ we have $x_i\in C_{4^{k}}$ and that the enumeration respects the orientation on every circle $C_{4^k}$. Namely, if $x_i,x_{i+1}\in C_{4^k}$, we have that $d(x_i,x_{i+1})=1$ and $x_{i+1}$ lies to the right of $x_i$. Denote $D_i=\{0,x_1,x_2,...x_{i}\right\rbrace$.
We are going to define a sequence of extension operators $P_i:\operatorname{Lip}_0(D_i)\to\operatorname{Lip}_0(N)$ and prove they satisfy the assumptions of Theorem \ref{operator}. In order to do that, let us define some preparatory notions. Define the left and the right "$D_i$-neighbour" functions $\nu^l_i,\nu^r_i:\bigcup_{n=1}^{k(i)} C_{4^n}^0\to D_i$ as follows: For each $n\in\{1,2,...,k(i)\right\rbrace$ and $x\in C_{4^{n}}$, let $\nu^l_i(x)\in D_i$ be the closest point to the left of $x$ and let $\nu^r_i(x)\in D_i$ be the closest point to the right of $x$. We set $\nu^l_i(0)=\nu^r_i(0)=0$. Note that $\nu^l_i(x)=\nu^{r}_i(x)=x$ if and only if $x\in D_i$. Further we need to define "right-" and "left-" metric function (not proper metrics) on every circle $C_{4^n}$. For points $x,y\in C_{4^n}$ we set the value $d^l(x,y)$ as the length of the path (in the graph $C_{4^n}$) going from $x$ to the left up to $y$. Analogously, we set $d^r(x,y)$ as the length of the path going from $x$ to the right up to $y$. It is clear that for $x,y\in C_{4^n}$ we have $d^l(x,y)=d^r(y,x)$ and $d(x,y)=\min\{d^r(x,y),d^l(x,y)\right\rbrace$.
Further we define for every $i\in\N$ the $i$-th interpolation function $I_i:\operatorname{Lip}_0(D_i)\times \bigcup_{n=1}^{k(i)} C_{4^n}^0\to\R$ via $$I_i(f,x)=\frac{d^r(x,\nu^r_i(x))f(\nu^l_i(x))+d^l(x,\nu^l_i(x))f(\nu^r_i(x))}{d^l(x,\nu^l_i(x))+d^r(x,\nu^r_i(x))}\ \ \ \text{if } x\neq \nu^l_i(x) \text{ or } x\neq \nu^r_i(x)$$ and $I_i(f,x)=f(x)$ for $x=\nu^l_i(x)=\nu^r_i(x)$. Clearly, $I_i(f,x)$ is the value of linear interpolation of the function $f$ between closest points of $x$ to the left and to the right from the set $D_i$, given we take $x$ itself to be the closest point to $x$ in any direction if $x\in D_i$. Let now $f\in\operatorname{Lip}_0(D_i)$. Then we define our (extension) operators $P_i$, $i\in\N$ simply as $$P_if(x)=\begin{cases} I_i(f,x) & x\in\bigcup_{n=1}^{k(i)} C_{4^n}^0\,,\\ 0 & x\in\bigcup_{n=k(i)+1}^{\infty} C_{4^n} \end{cases}$$ and of course, $P_0=0$. Clearly $P_i$ is a linear operator for every $i\in\N$ and the function $P_if$ is Lipschitz with the same constant as $f$. Indeed, if we take $x\in C_{4^{n}}$ and $y\in C_{4^{m}}$ with $m< n$, we see from the definition of $I_i$ that $\min_{z\in C_{4^{n}}}f(z)\leq P_if(x)\leq\max_{z\in C_{4^{n}}}f(z)$ and $\min_{w\in C_{4^{m}}}f(w)\leq P_if(y)\leq\max_{w\in C_{4^{m}}}f(w)$. From that and from the fact that $d(z,w)=d(x,y)=4^n$ holds for all $z\in C_{4^n}$ and $w\in C_{4^m}$, we get
$$|P_if(x)-P_if(y)|\leq \max_{\substack{
z\in C_{4^{n}}\\
w\in C_{4^{m}}}}
|f(z)-f(w)|\leq 4^n \| f\|=d(x,y)\| f\|.$$
For $x\in C_{4^{n}}$ and $0$ we have clearly $|P_if(x)-P_if(0)|\leq\max_{z\in C_{4^{n}}}|f(z)|\leq d(x,0)\| f\|$.
The only nontrivial case to prove is the case $x,y\in C_{4^{k(i)}}$. Let therefore $x,y\in C_{4^{k(i)}}$. There are three cases. If $x,y\in D_i$, then $P_if(x)=f(x)$ and $P_if(y)=f(y)$, which is trivial. Let $x,y\notin D_i$ and $\nu^r_i(x)=\nu^r_i(y)=a$, $\nu_i^l(x)=\nu_i^l(y)=b$. We can assume $d^r(b,x)\leq d^r(b,y)$, for the roles of $x$ and $y$ are symetrical. From that we have $d^r(y,a)\leq d^r(x,a)$. If $d(x,y)=d^r(x,y)$, we have \begin{align*}
|P_if(x)-P_if(y)|&=\left|\frac{f(b)d^r(x,a)+f(a)d^r(b,x)}{d^r(b,a)}-\frac{f(b)d^r(y,a)+f(a)d^r(b,y)}{d^r(b,a)}\right|\\
&=\left|\frac{f(b)d^r(x,y)-f(a)d^r(x,y)}{d^r(b,a)}\right|\\ &\leq \frac{\| f\| d(a,b)}{d^r(b,a)}d^r(x,y)\leq \| f\| d(x,y). \end{align*} If $d(x,y)=d^l(x,y)$, then $d(x,y)=d^r(y,a)+d^r(a,b)+d^r(b,x)$ and then from $d^r(b,a)-d^r(x,y)=d^r(b,x)+d^r(y,a)$ we have by triangle inequality \begin{align*}
|P_if(x)-P_if(y)|&=\left|\frac{f(b)d^r(x,y)-f(a)d^r(x,y)}{d^r(b,a)}\right|\\
&=\left|\frac{f(b)\left(d^r(x,y)-d^r(b,a)\right)+(f(b)-f(a))d^r(b,a)+f(a)\left(d^r(b,a)-d^r(x,y)\right)}{d^r(b,a)}\right|\\
&=\left|\frac{\left(d^r(b,x)+d^r(y,a)\right)\left(f(a)-f(b)\right)+(f(b)-f(a))d^r(b,a)}{d^r(b,a)}\right|\\ &\leq \| f\|\left(\frac{d(a,b)}{d^r(a,b)}\left(d^r(b,x)+d^r(y,a)\right)+d(a,b)\right)\leq\| f\| d(x,y)\\ \end{align*} The case $x\in D_i$, $y\notin D_i$ is proved in a similar way.
We see that the functions $f_j=P_i\left(\chi_{\{x_j\right\rbrace}\right)$, $1\leq j\leq i$ create a basis of each space $P_i(\operatorname{Lip}_0(N))$, hence $\dim P_i(\operatorname{Lip}_0(N))=i$ for every $i\in\N$.
To prove the commutativity it suffices to prove $P_{i+1}P_i=P_iP_{i+1}=P_i$ for every $i\in\N$. While $P_{i}P_{i+1}=P_i$ is clear, we prove for every $f\in\operatorname{Lip}_0(N)$ we have $P_{i+1}P_if=P_if$. Fix $f\in\operatorname{Lip}_0(N)$. If $k(i+1)>k(i)$, then $D_i=\bigcup_{n=1}^{k(i)} C_{4^n}^0$ and $P_{i+1}P_i f(x)=f(x)=P_i f(x)$ for all $x\in \bigcup_{n=1}^{k(i)} C_{4^n}^0$ and $P_{i+1}P_i f(x)=0=P_i f(x)$ for all $x\notin \bigcup_{n=1}^{k(i)} C_{4^n}^0$. Let therefore $k(i+1)=k(i)$.
Denote $a=x_i=\nu^l_{i}(x_{i+1})$ and $b=\nu^r_{i}(x_{i+1})$. All we need to check is $P_{i+1}P_i f(y)=P_i f(y)$ holds for all $y\in C_{4^{k(i)}}\setminus D_i$. Indeed, for all other points $x$ we have $P_if(x)=P_{i+1}f(x)$. Take therefore a point $y\neq x_{i+1}$ (otherwise it is trivial). Note that $\nu^l_{i}(y)=a$, $\nu^r_{i}(y)=b$ and that $d^r(a,x_{i+1})=1$. Then we have \begin{align*} P_{i+1}(P_i f)(y)&=\frac{d^r(x_{i+1},y)P_if(b)+d^r(y,b)P_if(x_{i+1})}{d^r(x_{i+1},b)}\\ &=\frac{d^r(x_{i+1},y)f(b)+d^r(y,b)\cdot\frac{f(b)+d^r(x_{i+1},b)f(a)}{d^r(a,b)}}{d^r(x_{i+1},b)}\\ &=\frac{d^r(x_{i+1},y)d^r(a,b)+d^r(y,b)}{d^r(x_{i+1},b)d^r(a,b)}\cdot f(b)+\frac{d^r(y,b)}{d^r(a,b)}\cdot f(a)\\ &=\frac{d^r(x_{i+1},y)d^r(x_{i+1},b)+d^r(x_{i+1},y)+d^r(y,b)}{d^r(x_{i+1},b)d^r(a,b)}\cdot f(b)+\frac{d^r(y,b)}{d^r(a,b)}\cdot f(a)\\ &=\frac{d^r(x_{i+1},b)\left(1+d^r(x_{i+1},y)\right)}{d^r(x_{i+1},b)d^r(a,b)}\cdot f(b)+\frac{d^r(a,y)}{d^r(a,b)}\cdot f(a)\\ &=\frac{d^r(a,y)}{d^r(a,b)}\cdot f(b)+\frac{d^r(a,y)}{d^r(a,b)}\cdot f(a)\\ &=P_i f(y)\\ \end{align*} and the commutativity is proved.
Let $i\in\N$. If $f_{\alpha}\to f$ pointwise, then for every $x\in D_i$ we have $P_if_{\alpha}(x)=f_{\alpha}(x)\to f(x)=P_if(x)$ and for every $x\in\bigcup_{l=k(i)+1}^\infty C_{4^{l}}$ we have $P_if_{\alpha}(x)=0=P_if(x)$. Finally, for every $x\in C_{4^{k(i)}}\setminus D_i$ we have $P_if_{\alpha}(x)=\gamma_x f_{\alpha}(a_x)+(1-\gamma_x)f_{\alpha}(b_x)$, for some eligible $\gamma_x\in [0,1]$, $a_x,b_x\in D_i$ and the choice of these points depends only on $x$ (and $i$ of course). Therefore $P_if_{\alpha}\to P_if$ pointwise, which means that every operator $P_i$ is continuous with respect to topology of pointwise convergence.
Finally the sequence $P_if$ converges pointwise to $f$. Indeed, for every $y\in N$ there exists $i\in\N$ such that $y\in D_i\subseteq D_{i+1}\subseteq D_{i+2}\dots$, which yields $P_i f(y)=P_{j}f(y)=f(y)$ for all $j\geq i$. Hence $P_i f\to f$ pointwise.
Since the operators $P_i$ meet all assumptions from Theorem \ref{operator}, we get that there is a sequence of operators $T_i:\mathcal F(N)\to\mathcal F(N)$, $i\in\N_0$ with $T_i^*=P_i$ which build a monotone Schauder basis for $\mathcal F(N)$. \end{proof} \begin{remark*} It was not necessary for the construction of $P_i$'s to enumerate the set $N$ with respect to orientation on every circle $C_{4^k}$. Actually any enumeration which satisfies $x_i\in C_{4^{k}}$ for every $i\in\N$ works. Our choice only slightly simplifies the proof. \end{remark*} \section{Unconditionality of retractional Schauder bases}\label{unconditionality} As we construct a Schauder basis on $\mathcal F(M)$ via sequence of retractions, as described in Corollary \ref{one}, properties of such a basis depend also on properties of the metric space $M$. Naturally it leads us to the question: What can $M$ be like such that there is an unconditional retractional Schauder basis on $\mathcal F(M)$? The next lemma sets a condition on the chains under which the acquired basis is conditional. It is further used in Theorem \ref{main}, which shows that retractional bases on Free spaces of nets in finite-dimensional spaces are conditional.
\begin{lemma}\label{alligned chains} Let $\alpha,\beta>0$ and let $N$ be an $\alpha$-separated metric space, such that there exist retractions $\varphi_i:N\to N$ satisfying the conditions from Corollary \ref{one}. Suppose there exists $n_0\in\N$ such that for every $n\in\N$, $n\geq n_0$ there exist chains $S=(\mu_{0},\mu_{k_1},...,\mu_{k_s})$ and $T=(\mu_0,\mu_{l_1},...,\mu_{l_m})$, $s,m\in\N$ with $d(\mu_{k_s},\mu_{l_m})\leq\beta$ and $|S\setminus T|\geq n$. Then the retractional Schauder basis on $\mathcal F(N)$ corresponding to the retractions $\varphi_i$ is conditional. \end{lemma} \begin{proof} Let now $P_i$ be the associated Schauder projection to the mapping $\varphi_i$ for each $i\in\N_0$, i.e. the projection to the subspace $\operatorname{span}\{\delta_{\mu_0},\delta_{\mu_1},...,\delta_{\mu_i}\right\rbrace$. Instead of working directly with $P_0,P_1,P_2,...$ we will use their adjoints $P_0^*,P_1^*,P_2^*,...$ and for every $n\in\N$, $n\geq n_0$ we construct a function $f_n\in\operatorname{Lip}_0(N)$ with $\| f_n\|\leq 1$ and find a sequence of signs $ \varepsilon_0, \varepsilon_1,..., \varepsilon_{k_s}$ for some $s\geq n$ such that the following inequality holds $$\left\Vert\sum_{i=0}^{k_s} \varepsilon_i(P_{i+1}^*-P_{i}^*)f_n\right\Vert\geq \frac{\alpha (n-1)}{\beta}.$$
Fix $n\in\N$ and chains $S=(\mu_0,\mu_{k_1},...,\mu_{k_s})$, $T=(\mu_0,\mu_{l_1},...,\mu_{l_m})$ for which we have $d(\mu_{k_s},\mu_{l_m})<\beta$ and $|S\setminus T|\geq n$. Suppose now $t\in\{0,1,2,...,s-n\right\rbrace$ is such that $\mu_{k_t}\in T$ and $\mu_{k_{t+1}}\notin T$ (we set $\mu_{k_0}=\mu_0$). We define the function $f_n$ on $N$ via the formula
$$f_n(x)=\begin{cases} \frac{\alpha}{2} & x=\mu_{k_j}\text{ for } j \text{ odd},j>t\,,\\ \frac{-\alpha}{2} & x=\mu_{k_j}\text{ for } j \text{ even},j> t\,,\\ 0 & \text{else}\,. \end{cases}$$ Clearly, $f_n(\mu_0)=0$ and $\| f_n\|\leq 1$. For the following choice of sings $ \varepsilon_0=1$, $$ \varepsilon_i=\begin{cases} - \varepsilon_{i-1} & i=k_j \text{ for some }j\in\N\,,\\ \varepsilon_{i-1} & \text{else}\,, \end{cases}$$ we have $$\sum_{i=0}^{k_s} \varepsilon_i(P_{i+1}^*-P_{i}^*)=-P_0^{*}+2\sum_{j=1}^s(-1)^{j+1}P^*_{k_j}+(-1)^sP^{*}_{k_s+1}=:P$$ and then \begin{align*} \left\Vert P\right\Vert&\geq \left\Vert P f_n\right\Vert\geq \left\Vert \frac{Pf_n(\mu_{k_s})-Pf_n(\mu_{l_m})}{d(\mu_{k_s},\mu_{l_m})}\right\Vert\geq\frac{1}{\beta}\left\Vert Pf_n(\mu_{k_s})-Pf_n(\mu_{l_m})\right\Vert=\\ &=\frac{1}{\beta}\left\vert -f_n(0)+2\sum_{j=1}^s(-1)^{j+1}f_n(\mu_{k_j})+(-1)^sf_n(\mu_{k_s})+0\right\vert\\ &=\frac{1}{\beta}\left\vert 2\sum_{j=t+1}^s\frac{\alpha}{2}-\frac{\alpha}{2}\right\vert\geq\frac{\alpha(s-t-1)}{\beta}\geq\frac{\alpha (n-1)}{\beta}.\\ \end{align*} \end{proof} Recall that a subset $S$ of a metric space $M$ is called an $\alpha,\beta$-net whenever $S$ is $\alpha$-separated and $\beta$-dense in $M$, i.e. $\inf_{x\neq y}d(x,y)\geq \alpha$, $x,y\in S$ and $\operatorname{sup}_{x\in M} d(x,S)\leq \beta$.
In \cite{HN}, the authors constructed a system of retractions on the integer lattice in $c_0$ which satisfies the conditions of Corollary \ref{one}. Through suitable homomorphisms they further showed the existence of a basis on any Free space of a net in a separable $C(K)$ space or a net in $c_0^+$, the positive cone in $c_0$.
\begin{corollary} Let $N$ be a net in any of the following metric spaces: $C(K)$, $K$ metrizable compact, or $c_0^+$ (the subset of $c_0$ consisting of elements with non-negative coordinates). The basis on $\mathcal F(N)$ constructed in \cite{HN} is conditional. \end{corollary} \begin{proof} First we consider the case $N=\mathbb{Z}^{<\omega}\subseteq c_0$, the integer lattice in $c_0$. Following the proof of Lemma $14$ in \cite{HN} we see, there are chains which go parallelly along the first coordinate axis (or any other coordinate axis). Every such two chains hence satisfy the conditions of the previous lemma, which yields that a basis arising from these retractions cannot be unconditional. As the existence of bases in other cases than $N$ being the integer lattice in $c_0$ was proven only by isomorphisms, we conclude that none of them are unconditional. \end{proof} \begin{theorem}\label{main} Let $N$ be an $\alpha,\beta$-net in a finite-dimensional normed space $X$ with $\dim X\geq 2$. Let $E=\{e_i\right\rbrace_{i=1}^\infty$ be a retractional Schauder basis on $\mathcal F(N)$. Then $E$ is conditional. \end{theorem} In the following, $B_{ \varepsilon}(x)$ denotes closed ball of radius $ \varepsilon>0$ and centre $x\in X$, $B_{ \varepsilon}^{\circ}(x)$ denotes its interior. In the same way $B_{ \varepsilon}:=B_{ \varepsilon}(0)$ and $S_{ \varepsilon}$ denotes sphere of radius $ \varepsilon$ and centre $0$. \begin{proof}
Let $\varphi_i:N\to N$ be the corresponding retractions to the basis $E$. We prove the theorem by showing that the assumptions of Lemma \ref{alligned chains} are met. Denote $\operatorname{sup}_{i\in\N}\operatorname{Lip}\varphi_i=K<\infty$. Pick $n\in\N$, such that $n>8K$. Define annulus with radii $r$ and $w$, $w<r$ as $A(r,w)=B_{r+w}(0)\setminus B^{\circ}_{r-w}(0)$. Our aim is to prove there exist chains $T,Z$ with final points $t,z\in A(3K\beta n+\beta,\beta)\cap N$ with $d(t,z)\leq 2\beta$ such that $x\in B_{K\beta n}$ holds for $x=x_{t,z}$, the final point of the chain $T\cap Z$. Then we have $d(t,x)\geq 3K\beta n-K\beta n=2K\beta n$ and Step lemma \ref{step} yields $|T\setminus Z|\geq n$, which by Lemma \ref{alligned chains} concludes the proof.
For the following, for every two points $x,y\in N$ with $x\prec y$ denote $T_x^y$ the chain with initial point $x$ and final point $y$. Assume now for contradiction, for every pair of points $t,z\in A(3K\beta n+\beta,\beta)\cap N=:A$ with $d(t,z)\leq 2\beta$ the final point $x_{t,z}$ of $T_0^t\cap T_0^z$ lies outside the ball $B_{K\beta n}$. That means there exists a point $\mu_m\in N$ for some $m\in\N$ with $d(0,\mu_m)>K\beta n$ such that $\varphi_m(t)=\mu_m$ for every $t\in A$. To prove this, note that $0\in\bigcap_{t\in A}T_0^t$ and as $\bigcap_{t\in A}T_0^t$ is a chain, it has a final point which we denote $\mu_m$ and prove that $d(0,\mu_m)>K\beta n$. We show that for every two points $t,z\in A$ the final point $x_{t,z}$ of the chain $T_0^t\cap T_0^z$ is of greater norm than $K\beta n$. Clearly, if $d(t,z)\leq 2\beta$, the statement holds as assumed. If $d(t,z)> 2\beta$, we can find a finite sequence of points $y_1,...,y_l\in A$, $l\in\N$ such that $d(y_i,y_{i+1})\leq 2\beta$ for every $i\in\{1,...,l-1\right\rbrace$ and that $y_1=t$ and $y_l=z$. Then $x_{t,z}\in\{x_{y_i,y_{i+1}}|\ i\in\{1,...,l-1\right\rbrace\right\rbrace$, which means $\| x_{t,z}\|> K\beta n$. Note that for any three points $s,t,z\in A$ the final point $x_{s,t,z}$ of the chain $T_0^s\cap T_0^t\cap T_0^z$ is equal to one of the points $x_{s,t},x_{t,z},x_{s,z}$. Indeed, as $x_{s,t},x_{t,z}\in T_0^t$, we have that either $x_{s,t}\prec x_{t,z}$ or $x_{s,t}\succ x_{t,z}$. If $x_{s,t}\prec x_{t,z}$, then $x_{s,z}=x_{s,t}=x_{s,t,z}$ and the other case follows symmetrically. But from that we get inductively that for any finite number of points $t_1,...,t_v$, there are indices $i,j\in\{1,...,v\right\rbrace$, such that the final point $x_{t_1,...,t_v}$ of the chain $\bigcap_{l=1}T_0^{t_l}$ equals $x_{t_i,t_j}$. Because for each two $t,z\in A$ we have $\| x_{t,z}\|>K\beta n$ and $A$ is finite, we have $\| \mu_m\|> K\beta n$.
Observe further, that $T_{\mu_m}^t\cap B_{\beta n}=\emptyset$ holds for every chain $T_{\mu_m}^t$ with initial point $\mu_m$ and final point $t\in A$. Indeed, if $\mu_p\in T_{\mu_m}^t$, $p\in\N$ is such that $\|\mu_p\|\leq \beta n$, we have $\operatorname{Lip}\varphi_m\geq \frac{\|\varphi_m(0)-\varphi_m(\mu_p)\|}{\| \mu_p\|}=\frac{\|\mu_m\|}{\|\mu_p\|}>\frac{K\beta n}{\beta n}=K$, which is not possible.
Let us denote $S=\bigcup_{t\in A}T_{\mu_m}^{t}$ the set of all chains from $\mu_m$ to points of $A$. Let $S=\{\mu_{k_1},...,\mu_{k_q}\right\rbrace$ for some $k_1<k_2<...<k_q$, $q\in\N$. Note that $\mu_{k_1}=\mu_m$. For every chain $T=(t_1,...,t_l)$, $l\in\N$ define a trajectory of the chain $\operatorname{Tr}(T)$ as the union of the line segments $\bigcup_{i=1}^{l-1} [t_i,t_{i+1}]$. Denote $D=\bigcup_{t\in A}\operatorname{Tr}\left(T_{\mu_m}^{t}\right)$. Define now a function $F:[1,q]\times A\to D$ via $$F(t,x)=(t-i)\varphi_{k_{i+1}}(x)+(1-(t-i))\varphi_{k_{i}}(x), x\in A, t\in [i,i+1), i\in\{1,...,q-1\right\rbrace$$ and $F(q,x)=x$, $x\in A$. We see that for each $t\in [1,q]$, the function $F(t,\cdot)$ is $K$-Lipschitz and that we have $F(1,x)=\mu_m$ for all $x\in A$.
Let $ \varepsilon=3K\beta n+\beta$ and consider $B=\{B^{\mathrm{o}}_{2\beta}(x)\cap S_{ \varepsilon}\right\rbrace_{x\in A}$ as an open cover of $S_{ \varepsilon}$ and find a partition of unity $\{\psi_a\right\rbrace_{a\in A}$ subordinated to the cover $B$. Define a function $R:[1,q]\times S_{ \varepsilon}\to X$ by
$$R(t,x)=\sum_{a\in A}\psi_a(x)F(t,a),\ \ t\in [1,q],\ x\in S_{ \varepsilon}.$$
We see, that $R(1,x)=\mu_m$ for all $x\in S_{ \varepsilon}$ and that $$\operatorname{sup}_{x\in S_{ \varepsilon}}|R(q,x)-x|\leq 2\beta.$$ Of course, $R$ is continuous on $[1,q]\times S_{ \varepsilon}$. Our goal is to prove there is a continuous deformation of $S_{ \varepsilon}$ into one point $\mu_m$ avoiding the origin, which is a contradiction. For that we define a straight-line homotopy between identity and $R(q,\cdot)$ by $W:[0,1]\times S_{ \varepsilon}\to A( \varepsilon,2\beta)$, $W(t,x)=tR(q,x)+(1-t)x$. Joining mappings $W$ and $R$ we get a mapping $Z:[0,q]\times S_{ \varepsilon}\to X$ precisely defined by $$Z(t,x)=\begin{cases} W(t,x) & t\in[0,1), x\in S_{ \varepsilon}\,,\\ R\left(\frac{q}{t},x\right) & t\in[1,q], x\in S_{ \varepsilon}\,.\\ \end{cases}$$ All there is left to prove is that $R([1,q]\times S_{ \varepsilon})\cap \{0\right\rbrace=\emptyset$. To see that, note that the value $R(t,x)$ is a convex combination of values $F(t,a)$, where $a\in A$ are such that $d(x,a)<2\beta$. Fix therefore $x\in S_{ \varepsilon}$ and let $\mu_{l_1},\mu_{l_2},...,\mu_{l_p}\in A$ be such that $d(x,\mu_{l_i})<2\beta$ for all $i$. From the preceding paragraphs it follows that the trajectory $\operatorname{Tr}(T_{\mu_m}^{\mu_{l_i}})$ of each chain from $\mu_m$ to $\mu_{l_i}$ has no intersection with $B_{\beta \frac{n}{2}}$. Indeed, as the chain $T_{\mu_{m}}^{\mu_{l_i}}$ avoids the ball $B_{\beta n}$ and the distance between two consecutive points in a chain is bounded by $2\beta K$ and $n>2K$, we get the result. From the fact that $d(\mu_{l_j},\mu_{l_i})\leq 4\beta$ for all $i,j$, we have that $\| F(t,\mu_{l_i})-F(t,\mu_{l_j})\|\leq 4K\beta$ for all $t\in [1,q]$. But as $n>8K$ we get $R(t,x)\neq 0$ for any $t\in [0,1]$. Altogether we obtain $Z([0,q]\times S_{ \varepsilon})\cap \{0\right\rbrace=\emptyset$, which was to prove. \end{proof} One could ask in general what are the metric spaces $M$ such that $\mathcal F(M)$ has an unconditional Schauder basis. It is clear that if $M$ contains a line segment, then $ L_1$ is contained in $\mathcal F(M)$ and therefore $\mathcal F(M)$ cannot have an unconditional Schauder basis. The only interesting cases are then topologically discrete spaces $M$. Our guess is that if $\mathcal F(M)$ has an unconditional Schauder basis, it is isomorphic to $\ell_1$. \\ \\\textbf{Open problem 1} \textit{Suppose $\mathcal F(M)$ has an unconditional Schauder basis. Is it isomorphic to $\ell_1$?}
In \cite{Gd}, one sees that $\mathcal F(M)$ is a complemented subspace of $L_1$ if and only if $M$ can be bi-Lipschitzly embedded into an $\R$-tree. A complemented subspace of $ L_1$ with unconditional basis is isomorphic to the space $\ell_1$ due to \cite{LiPe}. One can therefore restate the conjecture above into: Suppose $\mathcal F(M)$ has a Schauder basis $B$. If $M$ cannot be embedded into an $\R$-tree, is it true that $B$ is conditional? \\ \\\textbf{Open problem 2} \textit{Is it true that for every uniformly discrete set $N\subseteq\R^2$ the space $\mathcal F(N)$ has a Schauder basis?}
It follows from Corollary \ref{no basis} the answer is no if we restrict ourselves only to retractional Schauder bases. However, we don't know if, supposed the answer is yes, we can find for every uniformly discrete set $N\subseteq\R^2$ an extensional Schauder basis on $\mathcal F(N)$.
\end{document} | arXiv |
The expression with n terms is (1/2)(2/3)(3/4)...)((n-1)/n)=1/n. The limit is obviously 0.
(n-1)*(n-2)*(n-3)!/n*(n-1)*(n-2)*(n-3)!. Now I understand the result.
Last edited by skipjack; October 4th, 2018 at 11:28 PM.
Is it okay how I solve it? I wanted to know where the result 1/n appeared from.
The result comes intuitively comes from pattern recognition and formally from mathematical induction.
Last edited by skipjack; October 4th, 2018 at 11:29 PM.
which tends to 0 as $n \to \infty$.
You did not understand what I wrote. (n-1)/n is the least term of the product. When it is concatenated, the product is (n-1)!/n!. | CommonCrawl |
Sampling fraction
In sampling theory, the sampling fraction is the ratio of sample size to population size or, in the context of stratified sampling, the ratio of the sample size to the size of the stratum.[1] The formula for the sampling fraction is
$f={\frac {n}{N}},$
where n is the sample size and N is the population size. A sampling fraction value close to 1 will occur if the sample size is relatively close to the population size. When sampling from a finite population without replacement, this may cause dependence between individual samples. To correct for this dependence when calculating the sample variance, a finite population correction (or finite population multiplier) of (N-n)/(N-1) may be used. If the sampling fraction is small, less than 0.05, then the sample variance is not appreciably affected by dependence, and the finite population correction may be ignored. [2][3]
References
1. Dodge, Yadolah (2003). The Oxford Dictionary of Statistical Terms. Oxford: Oxford University Press. ISBN 0-19-920613-9.
2. Bain, Lee J.; Engelhardt, Max (1992). Introduction to probability and mathematical statistics (2nd ed.). Boston: PWS-KENT Pub. ISBN 0534929303. OCLC 24142279.
3. Scheaffer, Richard L.; Mendenhall, William; Ott, Lyman (2006). Elementary survey sampling (6th ed.). Southbank, Vic.: Thomson Brooks/Cole. ISBN 0495018627. OCLC 58425200.
| Wikipedia |
\begin{document}
\title{Characters of some simple supercuspidal representations on split tori}
\author{Moshe Adrian}
\maketitle
\section{Introduction}
Let $F$ be a nonarchimedean local field of characteristic zero, $\mathfrak{o}$ its ring of integers, $\mathfrak{p}$ the maximal ideal, $p$ the residual characteristic, $q$ the order of the residue field, and $\varpi$ a fixed uniformizer of $F$. Let $G$ be a connected reductive group defined over $F$. If $\pi$ is an irreducible admissible representation of $G$, we denote by $\theta_{\pi}$ its distribution character, which is a linear functional on $C_c^{\infty}(G)$, the locally constant, compactly supported functions on $G$. Harish-Chandra showed that $\theta_{\pi}$ can be represented by a locally constant function on the regular semisimple set of $G$, which we will also denote $\theta_{\pi}$.
Suppose that $\pi$ is a supercuspidal representation. Much is known about $\theta_{\pi}$. The first supercuspidal characters were computed by Sally and Shalika in \cite{sallyshalika}, where they investigated the supercuspidal representations of $SL(2,F)$ when $p \neq 2$. Shimizu calculated the supercuspidal characters of $GL(2,F)$ in \cite{shimizu}, for $p \neq 2$. Kutzko began a study of the supercuspidal characters of $GL(\ell,F)$, $\ell$ a prime (see \cite{kutzko}), when $\ell \neq p$, and DeBacker computed these characters on elliptic tori (see \cite{debacker}). Later, Spice calculated the supercuspidal characters of $SL(\ell,F)$, $\ell$ an odd prime (see \cite{spice}), for $\ell \neq p$ and together with Adler, they computed a large class of supercuspidal characters for very general connected reductive groups (see \cite{adlerspice}).
Many times one wants to determine character values of a representation on a particular torus, as this can carry much of the information of the representation. For example, discrete series representations of real groups are determined by their character values on the compact (mod center) torus. On the $p$-adic side, it is known (see \cite{adrian}) that the supercuspidal representations of $GL(n,F)$, $n$ prime, are determined by their character values on a specific elliptic torus, for $p > 2n$.
In this paper, we compute the character values of the simple supercuspidal representations (recently discovered by Gross and Reeder) of $SL(2,F)$ and $SL(3,F)$ on the maximal split torus of each group when $p$ is arbitrary. The character values for $SL(2,F)$ are especially elegant, being a fixed constant times a sum of $q^{\ell(w)}$ over appropriate affine Weyl group elements $w$, where $\ell(w)$ is the length of $w$. We had hoped that the character values of simple supercuspidals for $SL(3,F)$ and more general reductive groups would be as elegant, but this is unfortunately not the case. As one might expect, various Gauss-type sums appear for $SL(3,F)$.
Let $T$ denote the maximal split torus of $SL(2)$, $Z$ the center, and $W^a$ the affine Weyl group. Let $val$ denote valuation. Using the Frobenius character formula for supercuspidal representations (see \cite{sally}), we will first prove the following theorem.
\begin{theorem}\label{maintheorem} Let $g = \mat{a}{0}{0}{a^{-1}} \in SL(2,F)$ where $a \in 1 + \mathfrak{p}$, and set $r := val(a - a^{-1})$. Let $\pi$ be a simple supercuspidal representation of $SL(2,F)$. Then $$\theta_{\pi}(g) = c_q \left( \displaystyle\sum_{w \in W^{a} : \ell(w) < r} q^{\ell(w)} \right)$$ where \begin{equation*} c_q := \left\{ \begin{array}{rl} \frac{q-1}{2} & \text{if } p \neq 2 \\ q-1 & \text{if } p = 2 \end{array} \right. \end{equation*} \end{theorem}
It will be clear from our calculations in later sections that $\theta_{\pi}$ vanishes on $T(F) \setminus Z(F) T(1 + \mathfrak{p})$, so up to the central character, we have computed $\theta_{\pi}$ on all of the split torus. Moreover, since the central character is given by the data forming the simple supercuspidal representation, we have therefore computed all of $\theta_{\pi}$ on the split torus.
We will also compute $\theta_{\pi}(g)$ for simple supercuspidal representations $\pi$ of $SL(3,F)$, where $g$ is in the maximal split torus of $SL(3,F)$. Specifically, we will compute the character values on $T(1 + \mathfrak{p})$, where here $T$ is the maximal split torus of $SL(3)$. Then by the same reasoning as for $SL(2,F)$, this will be enough to give the character values on the entire maximal split torus of $SL(3,F)$. However, since the theorem for $SL(3,F)$ is much more complicated, we will defer its statement to a later section. Moreover, as there are nontrivial Gauss sums in this formula, we will compute them in complete generality in the last section of this paper.
We note that the term $a - a^{-1}$ in Theorem \ref{maintheorem} is, up to a sign, a canonical square root of the Weyl denominator. In particular, if $D(g)$ is the standard Weyl denominator, then $-D(g) = (a-a^{-1})^2$.
We wish to make another note. In \cite{sallyshalika}, Sally/Shalika have character values on the split torus of $SL(2,F)$ for an arbitrary supercuspidal representation of $SL(2,F)$ when $p \neq 2$, which we briefly recall. For any quadratic extension $V = F(\sqrt{\theta})$ of $F$, let $C_{\theta}$ denote the kernel of the norm $N_{V/F}$, and $\mathfrak{p}_{\theta}$ the prime ideal in $V$. If $V$ is ramified, set $C_{\theta}^{(h)} = (1 + \mathfrak{p}_{\theta}^{2h+1}) \cap C_{\theta}, h \geq 0$. If $\psi \in \hat{C}_{\theta}$, denote the conductor of $\psi$ by cond $\psi$ (this is the largest subgroup in the filtration $\{ C_{\theta}^{(h)} \}$ on which $\psi$ is trivial). The ramified discrete series are indexed by a nontrivial additive character $\eta$ of $F$ and a nontrivial character $\psi \in \hat{C}_{\theta}$, where $V$ is ramified. The corresponding representation is denoted $\Pi(\eta, \psi, V)$. If $g = \mat{a}{0}{0}{a^{-1}} \in SL(2,F)$, $a \in 1 + \mathfrak{p}$, and $\Pi(\eta, \psi, V)$ is a ramified discrete series, then $$\theta_{\Pi(\eta, \psi, V)}(g) = \frac{1}{|a-a^{-1}|} - \frac{1}{2} q^h \left( \frac{q+1}{q} \right)$$ where $V = F(\sqrt{\theta})$ and cond $\psi = C_{\theta}^{(h)}, h \geq 1$.
Since these character values are in particular valid for the simple supercuspidal representations when $p \neq 2$ (since simple supercuspidal representations are, in particular, ramified discrete series), we may compare their character values to ours. After some calculation, one can see that their character values agree with ours in the case of simple supercuspidal representations on the split torus.
We now briefly present an outline of the paper. In section \ref{background}, we define simple supercuspidal representation and we present the relevant background theory that we use to compute the character values. In section \ref{sl2section}, we compute the character formula for $SL(2,F)$. In section \ref{sl3section}, we compute the character formula for $SL(3,F)$. In particular, the formula contains various Gauss-type sums. In section \ref{morecalculations}, we compute these Gauss sums.
Acknowledgements: This paper has benefited from conversations with Gordan Savin, Aaron Wood, Chris Kocs, and Loren Spice.
\section{Background}\label{background}
Let us recall some basic definitions. Let $G$ be a split, simply connected, almost simple, connected reductive group, and $T$ a maximal $F$-split torus in $G$. Associated to $T$ we have the set of roots $\Phi$ of $T$ in $G$, an apartment, together with a set of affine roots $\Psi$, and an affine Weyl group $W^a$. We also have a canonical length function $\ell(w)$ on $W^a$. Fix a Chevalley basis in the Lie algebra of $G$. To each $\psi \in \Psi$ we have an associated affine root group $U_{\psi}$. Fix an alcove $C$ in the apartment with corresponding simple and positive affine roots $\Pi \subset \Psi^+$. Let $T(\mathfrak{o})$ be the maximal compact subgroup of $T(F)$. Let $T(1+\mathfrak{p}) := < t \in T(\mathfrak{o}) : \lambda(t) \in 1 + \mathfrak{p} \ \forall \lambda \in X^*(T)>$, where $X^*(T)$ is the character lattice of $T$. Let $I = <T(\mathfrak{o}), U_{\psi} : \psi \in \Psi^+>$ denote the corresponding Iwahori subgroup, and $I_+ = <T(1 + \mathfrak{p}), U_{\psi} : \psi \in \Psi^+>$ its pro-unipotent radical. Set $I_{++} := <T(1 + \mathfrak{p}), U_{\psi} : \psi \in \Psi^+ \setminus \Pi>$. We set $H := Z(F) I_+$, where $Z$ is the center of $G$. Let $N$ denote the normalizer of $T(F)$ in $G(F)$.
\begin{lemma}(see \cite{grossreeder}) The subgroup $I_{++}$ is normal in $I_+$, with quotient $$I_+ / I_{++} \cong \displaystyle\bigoplus_{\psi \in \Pi} U_{\psi} / U_{\psi+1}$$ as $T(\mathfrak{o})$-modules. \end{lemma}
\begin{definition}(see \cite{grossreeder}) A character $\chi : H \rightarrow \mathbb{C}^*$ is called \emph{affine generic} if
(i) $\chi$ is trivial on $I_{++}$ and
(ii) $\chi$ is nontrivial on $U_{\psi}$ for every $\psi \in \Pi$. \end{definition}
\begin{theorem} (see \cite{grossreeder}) Let $\chi : H \rightarrow \mathbb{C}^*$ be an affine generic character. Then $cInd_{H}^{G(F)} \chi$ is an irreducible supercuspidal representation, called a \emph{simple supercuspidal representation}, where $cInd$ denotes compact induction. \end{theorem}
Now suppose that $\pi$ is an irreducible smooth supercuspidal representation of $G(F)$. Let $K$ be an open, compact subgroup of $G(F)$, and suppose that $\sigma$ is an irreducible representation of $K$ such that $$\pi = cInd_K^{G(F)} \sigma.$$ Let $\chi_{\sigma}$ denote the distribution character of $\sigma$. The following is the Frobenius formula for the induced character $\theta_{\pi}$.
\begin{theorem}\label{sally} (see \cite{sally}) Let $g$ be a regular element of $G(F)$. Then $$\theta_{\pi}(g) = \displaystyle\sum_{x \in K \backslash G(F) / K} \displaystyle\sum_{y \in K \backslash K x K} \dot{\chi}_{\sigma}(ygy^{-1})$$ where
\begin{equation*} \dot{\chi}_{\sigma}(k) = \left\{ \begin{array}{rl} \chi_{\sigma}(k) & \text{if } k \in K \\ 0 & \text{if } k \in G(F) \setminus K \end{array} \right. \end{equation*} \end{theorem}
There is an integral version of this formula as well (see \cite{sally}), and these formulas are a main tool in computing characters of supercuspidal representations. We will use this theorem to compute the character values of the simple supercuspidal representations of $SL(2,F)$ and $SL(3,F)$ on their maximal split tori.
We first show that the formula in Theorem \ref{sally} simplifies considerably in our situation. Let us first recall the following basic theory about double coset decompositions. If $G$ is a connected reductive group and $K$ is a compact open subgroup of $G$, let us choose a set of representatives $\{t_{\alpha} \}$ for the double cosets of $K \backslash G / K$. Then $K t_{\alpha} K$ is the disjoint union of the cosets $K t_{\alpha} s_1, K t_{\alpha} s_2, ..., K t_{\alpha} s_m$, where $s_1, s_2, ..., s_m$ is a set of representatives of $K / (K \cap t_{\alpha}^{-1} K t_{\alpha})$. We will use this fact repeatedly in this paper. We can now state our reduction formula.
\begin{proposition}\label{reduction}
Let $G$ be simply connected, and $\chi$ an affine generic character of $H$. Set $\pi := cInd_{H}^{G(F)} \chi$. Then $$\theta_{\pi}(g) = |T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p})| \displaystyle\sum_{x \in W^a} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}_{\sigma}(ygy^{-1}),$$ where the outer sum is meant to be taken over any set of representatives $x$ in $W^a$. \end{proposition}
\proof Recall the affine Bruhat decomposition $I \backslash G / I \leftrightarrow W^a \cong N / T(\mathfrak{o})$. As $I / I_+ \cong T(\mathfrak{o}) / T(1 + \mathfrak{p})$, the affine Bruhat decomposition descends to $H \backslash G / H \leftrightarrow N / Z(F) T(1 + \mathfrak{p})$ (we are using here that $G$ is simply connected, so that $Z(F) = Z(\mathfrak{o})$ and therefore $Z(\mathfrak{o}) I_+ = H \subset I$ and $I / H = T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p})$), and we have the short exact sequence $$1 \rightarrow T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p}) \rightarrow N / Z(F) T(1 + \mathfrak{p}) \rightarrow W^a = N / T(\mathfrak{o}) \rightarrow 1$$ Therefore, $$\theta_{\pi}(g) = \displaystyle\sum_{x \in N / Z(F) T(1 + \mathfrak{p})} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}_{\sigma}(ygy^{-1})$$ Write $$\sigma(x) := \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}_{\sigma}(ygy^{-1})$$ for $x \in N$. We claim that $\sigma(x) = \sigma(xt) \ \forall t \in T(\mathfrak{o})$. Write $HxH = \displaystyle\cup_{i=1}^m Hx x_i$. Recall that $x_i$ are representatives of $H / (H \cap x^{-1} Hx)$. As $x \in N$, $x^{-1} H x = Z(F) I'_+$, where $I'_+$ is the pro-unipotent radical of another Iwahori subgroup $I'$. It is then easy to see that $H / (H \cap x^{-1} Hx)$ is a direct sum of spaces of the form $U_{\gamma} / U_{\gamma + n}$, where $\gamma \in \Phi^+$ or $\gamma = \gamma' + 1$ where $\gamma' \in \Phi^-$, and where $n$ is a non-negative integer. Now, $$\sigma(xt) = \displaystyle\sum_{y \in H \backslash H xt H} \dot{\chi}_{\sigma}(ygy^{-1}).$$ Write $HxtH = \displaystyle\cup_{j} Hxt y_j$. $y_j$ are representatives of $H / (H \cap (xt)^{-1} H xt)$. Since $t \in T(\mathfrak{o})$, we have that $t^{-1} x^{-1} Hxt = x^{-1} Hx$. Therefore, in particular, $HxtH = \displaystyle\cup_{i=1}^m Hxt x_i$. Then $$\sigma(xt) = \displaystyle\sum_{i=1}^m \dot{\chi}_{\sigma}(xtx_i gx_i^{-1} t^{-1} x^{-1}) = $$ $$\displaystyle\sum_{i = 1}^m \dot{\chi}_{\sigma}(xtx_i t^{-1} t g t^{-1} tx_i^{-1} t^{-1} x^{-1}) = \displaystyle\sum_{i=1}^m \dot{\chi}_{\sigma}(xtx_i t^{-1} g tx_i^{-1} t^{-1} x^{-1}).$$ Since this sum is over spaces of the form $U_{\gamma} / U_{\gamma+n}$ as noted above, and since conjugation by an element $t \in T(\mathfrak{o})$ preserves $U_{\gamma} / U_{\gamma+n}$, we get $$\displaystyle\sum_{i=1}^m \dot{\chi}_{\sigma}(xtx_i t^{-1} g tx_i^{-1} t^{-1} x^{-1}) = \displaystyle\sum_{i=1}^m \dot{\chi}_{\sigma}(xx_i g x_i^{-1} x^{-1}) = \sigma(x).$$
Therefore, since $\sigma$ is constant along fibers of the above exact sequence, we have that
$$\theta_{\pi}(g) = |T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p})| \displaystyle\sum_{x \in W^a} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}_{\sigma}(ygy^{-1})$$ \qed
Therefore, we only need to compute $\sigma(x)$ as $x$ varies over a set of representatives of $W^a$. Moreover, since we are only computing $\theta_{\pi}$ on $T(1+\mathfrak{p})$, we only need the data of the affine generic character $\chi$ on $I_+$ (and not on all of $ZI_+$), since if $g \in T(1+\mathfrak{p})$, then the terms $ygy^{-1}$ in $\theta_{\pi}(g)$ will always live in $I_+$.
\section{The character formula for $SL(2,F)$}\label{sl2section}
In this section we prove Theorem \ref{maintheorem}. We prove this theorem with a case by case investigation. We compute the inner sum $\sigma(x)$ in Proposition \ref{reduction} for any set of representatives of $W^a$ by decomposing $HxH$ into a union of left cosets, as in the paragraph that immediately precedes proposition \ref{reduction}. Afterwards, we sum everything up to get $\theta_{\pi}$.
We fix a Haar measure on $F$ such that $\mathfrak{o}$ has volume $1$, and we use the abbreviation $vol$ to denote volume. Fix an element $g = \mat{a}{0}{0}{a^{-1}} \in T(1+\mathfrak{p})$, and set $r := val(a - a^{-1})$
\begin{proposition}\label{innersum1} Let $x = \mat{b}{0}{0}{b^{-1}}$. Then
\begin{equation*} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \left\{ \begin{array}{rl} vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{-2n+r}) & \text{if } val(b) = n \geq 0 \ \mathrm{and} \ 2n < r \\ vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{2n+r}) & \text{if } val(b) = n < 0 \ \mathrm{and} \ -2n < r \\ 0 & \text{otherwise} \end{array} \right. \end{equation*} \end{proposition}
\proof We first rewrite the double coset $H x H$ as a finite union of single right cosets. Suppose $val(b) = n > 0.$ Since $$H \cap x^{-1} H x = Z(F) \mat{1 + \mathfrak{p}}{\mathfrak{o}}{\mathfrak{p}^{2n+1}}{1 + \mathfrak{p}},$$ we can obtain an explicit disjoint union $$H = \displaystyle\bigcup_{z \in \mathfrak{p} / \mathfrak{p}^{2n+1}} Z(F) \mat{1 + \mathfrak{p}}{\mathfrak{o}}{\mathfrak{p}^{2n+1}}{1 + \mathfrak{p}} \mat{1}{0}{z}{1},$$ where $$\mat{1 + \mathfrak{p}}{\mathfrak{o}}{\mathfrak{p}^{2n+1}}{1 + \mathfrak{p}} := \left\{ \mat{x_1}{x_2}{x_3}{x_4} \in SL(2,F): x_1,x_4 \in 1 + \mathfrak{p}, x_2 \in \mathfrak{o}, x_3 \in \mathfrak{p}^{2n+1} \right\} $$ (we will use this last type of notation throughout the paper). Therefore, we have a disjoint union $$H x H = \displaystyle\bigcup_{z \in \mathfrak{p} / \mathfrak{p}^{2n+1}} H x \mat{1}{0}{z}{1}$$
Now suppose that $val(b) = n < 0.$ Then similarly, we get $$H \cap x^{-1} H x = Z \mat{1 + \mathfrak{p}}{\mathfrak{p}^{-2n}}{\mathfrak{p}}{1 + \mathfrak{p}},$$ $$H = \displaystyle\bigcup_{z \in \mathfrak{o} / \mathfrak{p}^{-2n}} Z \mat{1 + \mathfrak{p}}{\mathfrak{p}^{-2n}}{\mathfrak{p}}{1 + \mathfrak{p}} \mat{1}{z}{0}{1}.$$ Therefore, we have a disjoint union $$H x H = \displaystyle\bigcup_{z \in \mathfrak{o} / \mathfrak{p}^{-2n}} H x \mat{1}{z}{0}{1}$$
Finally, if $val(b) = 0$, then we get $H \cap x H x^{-1} = H$. Therefore, $H x H = H x$.
Let us return to the case $val(b) = n > 0$. Suppose $y \in H x H$. We need to check when $y g y^{-1} \in H$ since $\dot{\chi}$ vanishes outside $H$. Using our above double coset decomposition, write $y = i x \tilde{z}$, where $\tilde{z}$ is of the form $\mat{1}{0}{z}{1}$ for some $z \in \mathfrak{p} / \mathfrak{p}^{2n+1}$ and for some $i \in H$. Then $y g y^{-1} \in H \Leftrightarrow x \tilde{z} g \tilde{z}^{-1} x^{-1} \in H$. Moreover, $$x \tilde{z} g \tilde{z}^{-1} x^{-1} = \mat{a}{0}{b^{-2} z(a-a^{-1})}{a^{-1}}.$$ Notice that $a \in \pm (1 + \mathfrak{p})$ is forced upon us here in order to have $x \tilde{z} g \tilde{z}^{-1} x^{-1} \in H$. (We note that the condition $a \in \pm (1 + \mathfrak{p})$ continues to be forced upon us, for the same reason, when you compute the terms $ygy^{-1}$ that appear in $\theta_{\pi}(g)$ for any other representative $x$ of any element of the affine Weyl group,as simple computations will show. This shows, therefore, that $\theta_{\pi}$ vanishes on $T(F) \setminus Z(F) T(1 + \mathfrak{p})$).
Write $a - a^{-1} = \varpi^r u$ for some unit $u$. Absorbing all units into the $z$ term, we may write $b^{-2} z(a - a^{-1}) = \varpi^{-2n} \varpi^r z'$ for some $z' \in \mathfrak{p} / \mathfrak{p}^{2n+1}$.
Recall that we are only interested in $\chi|_{I_+}$. We will abuse notation and write $\chi$ for $\chi|_{I_+}$. Now, write $\chi$ on $I_+$ as $$\chi : I_+ \rightarrow \mathbb{C}^*$$ $$\mat{d_{11}}{d_{12}}{d_{21}}{d_{22}} \mapsto \chi_1(d_{12}) \chi_2(d_{21}) $$ where $\chi_1$ is a level $1$ character of $\mathfrak{o}$ and where $\chi_2(d_{21}) = \chi_2'(\frac{1}{\varpi} d_{21})$, where $\chi_2'$ is a level $1$ character of $\mathfrak{o}$ (a character of $\mathfrak{o}$ is said to be level $1$ if it is trivial on $\mathfrak{p}$, but nontrivial on $\mathfrak{o}$). Set $\dot{\chi}_1(z) := \chi_1(z) \ \forall z \in \mathfrak{o}$ and $\dot{\chi}_1(z) = 0 \ \forall z \in F \setminus \mathfrak{o}$. Moreover, set $\dot{\chi}_2(z) := \chi_2'(z) \ \forall z \in \mathfrak{o}$ and $\dot{\chi}_2(z) = 0 \ \forall z \in F \setminus \mathfrak{o}$.
Therefore, $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum_{z' \in \mathfrak{p} / \mathfrak{p}^{2n+1}} \dot{\chi}_2(\varpi^{-2n+r-1} z')$$ Making a change of variables, we get $$\displaystyle\sum_{z' \in \mathfrak{p} / \mathfrak{p}^{2n+1}} \dot{\chi}_2(\varpi^{-2n+r-1} z') = \displaystyle\sum_{z'' \in \mathfrak{p}^{-2n+r} / \mathfrak{p}^{r}} \dot{\chi}_2(z'') = vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{-2n+r} \cap \mathfrak{o}} \dot{\chi}_2(z'') d(z'')$$ since $\dot{\chi}$ vanishes outside $H$. If $\mathfrak{p}^{-2n+r} \supseteq \mathfrak{o}$, then this integral vanishes since the integral of a nontrivial character over a group vanishes. However, if $\mathfrak{p}^{-2n+r} \varsubsetneq \mathfrak{o}$, which is precisely the condition that $2n < r$, then $$vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{-2n+r} \cap \mathfrak{o}} \dot{\chi}_2(z'') d(z'') = vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{-2n+r}} d(z'') = vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{-2n+r})$$ since $\dot{\chi}_2$ is trivial on $\mathfrak{p}$.
Now consider the case $val(b) = n < 0$. Suppose $y \in H x H$. By our above double coset decomposition, write $y = i x \tilde{z}$, where $\tilde{z}$ is of the form $\mat{1}{z}{0}{1}$ for some $z \in \mathfrak{o} / \mathfrak{p}^{-2n}$ and for some $i \in H$. Moreover, $x \tilde{z} g \tilde{z}^{-1} x^{-1} = \mat{a}{b^2 z(a^{-1} - a)}{0}{a^{-1}}.$ Therefore, $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum_{z \in \mathfrak{o} / \mathfrak{p}^{-2n}} \dot{\chi}_1( b^2 z (a^{-1} - a))$$ We rewrite $b^{2} z(a^{-1} - a) = \varpi^{2n} p^r z'$, where $z' \in \mathfrak{o} / \mathfrak{p}^{-2n}$. Again, after a change of variables, we get $$\displaystyle\sum_{z \in \mathfrak{o} / \mathfrak{p}^{-2n}} \dot{\chi}_1(b^2 z (a^{-1} - a)) = \displaystyle\sum_{z' \in \mathfrak{o} / \mathfrak{p}^{-2n}} \dot{\chi}_1(\varpi^{2n} p^r z' ) = $$ $$ \displaystyle\sum_{z'' \in \mathfrak{p}^{2n+r} / \mathfrak{p}^{r}} \dot{\chi}_1(z'') = vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{2n+r} \cap \mathfrak{o}} \dot{\chi}_1(z'') d(z'')$$ If $\mathfrak{p}^{2n+r} \supseteq \mathfrak{o}$, then again this integral vanishes. However, if $\mathfrak{p}^{2n+r} \varsubsetneq \mathfrak{o}$, which is precisely the condition that $-2n < r$ then $$vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{2n+r} \cap \mathfrak{o}} \dot{\chi}_1(z'') d(z'') = vol(\mathfrak{p}^r)^{-1} \int_{\mathfrak{p}^{2n+r}} d(z'') = vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{2n+r})$$ since $\dot{\chi}_1$ is trivial on $\mathfrak{p}$.
Now consider the case $val(b) = 0$. Suppose $y \in H x H$. Recall that in this case, $H x H = H x$. Moreover, $x g x^{-1} = g.$ Therefore, $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \dot{\chi}(g) = 1$$ \qed
\begin{proposition} Let $x = \mat{0}{c}{-c^{-1}}{0}$. Then
\begin{equation*} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \left\{ \begin{array}{rl} vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{-2n+r-1}) & \text{if } val(c) = n \geq 0 \ \mathrm{and} \ 2n +1 < r \\ vol(\mathfrak{p}^r)^{-1} vol(\mathfrak{p}^{2n+r+1}) & \text{if } val(c) = n < 0 \ \mathrm{and} \ -2n -1 < r \\ 0 & \text{otherwise} \end{array} \right. \end{equation*} \end{proposition}
\proof The proof is completely analogous to that of Proposition \ref{innersum1}. \qed
Now note that
\begin{equation*}
|T(\mathfrak{o}) / Z(F)T(1+\mathfrak{p})| = |\mathfrak{o}^* / (\pm (1 + \mathfrak{p}))| = \left\{ \begin{array}{rl} \frac{q-1}{2} & \text{if } p \neq 2 \\ q-1 & \text{if } p = 2 \end{array} \right. \end{equation*}
(note that if $p = 2$, $1 + \mathfrak{p} = -1 + \mathfrak{p}$). Thus, our character formula is
\[ \theta_{\pi_{\chi}}(g) = c_q vol(\mathfrak{p}^r)^{-1} \bigg[ \displaystyle\sum_{n \in \mathbb{N}, 0 \leq 2n < r} vol(\mathfrak{p}^{-2n+r}) + \displaystyle\sum_{n \in \mathbb{N}, 0 < -2n < r} vol(\mathfrak{p}^{2n+r}) \] \[ + \displaystyle\sum_{n \in \mathbb{N}, 0 \leq 2n+1 < r} \left( vol(\mathfrak{p}^{-2n+r-1}) \right) + \displaystyle\sum_{n \in \mathbb{N}, 0 < -2n-1 < r} \left( vol(\mathfrak{p}^{2n+r+1}) \right) \bigg] \]
It is a straightforward calculation to show that if $x = \mat{b}{0}{0}{b^{-1}}$ and $val(b) = k$, then $\ell(x) = |2k|$. Moreover, if $x = \mat{0}{c}{-c^{-1}}{0}$ and $val(c) = k$, then $\ell(x) = |2k+1|$.
Making the relevant substitutions, and noting that $vol(\mathfrak{p}^d) = q^{-d}$ for $d > 0$ by our choice of measure, one can see that we have proven Theorem \ref{maintheorem}.
\section{The character formula for $SL(3,F)$}\label{sl3section}
In this section we compute the character formula for $SL(3,F)$ on the split maximal torus. As in the case of $SL(2,F)$, we compute the formula via a case by case investigation. We compute the inner sum $\sigma(x)$ in Proposition \ref{reduction} for any set of representatives of $W^a$. Afterwards, we sum everything up to get $\theta_{\pi}$.
Let $g = \matt{\alpha}{0}{0}{0}{\beta}{0}{0}{0}{\gamma} \in T(1+\mathfrak{p})$. Suppose $\alpha - \beta = \varpi^r u$, $\beta - \gamma = \varpi^s u'$, $\alpha - \gamma = \varpi^t u''$ for some units $u, u', u''$. Again, we fix a Haar measure on $F$ such that $\mathfrak{o}$ has volume $1$.
Before we state the main theorem, we need to make a few simplifications. First, notice that $val(\alpha - \gamma) = val( (\alpha - \beta) + (\beta - \gamma) ) \geq inf \{val(\alpha - \beta), val(\beta - \gamma) \}$. Similarly, $val(\alpha - \beta) \geq inf \{val(\alpha - \gamma), val(\beta - \gamma) \}$ and $val(\beta - \gamma) \geq inf \{val(\alpha - \gamma), val(\alpha - \beta) \}$. One can conclude therefore that either $t \geq r = s$, $s \geq r = t$, or $r \geq s = t$. Since everything in sight is symmetric, we will assume without loss of generality that $t \geq r = s$, and we will state the main theorem in this case. One can easily state the analogous results in the two other cases.
So assume $t \geq r = s$. A complicated impediment is the character values are different in the cases $t = r = s$, $t = r+ 1 = s + 1$, and $t > r + 1 = s + 1$. We therefore have to state the character formula separately for these three cases.
We need some notation first. Let $\mathcal{A}_{n_i}$ denote the set of affine Weyl group elements who representatives in $N_G(T)$ are of the form $$\matt{a}{0}{0}{0}{b}{0}{0}{0}{c}, \matt{0}{a}{0}{0}{0}{b}{c}{0}{0}, \matt{0}{0}{a}{b}{0}{0}{0}{c}{0}, $$ $$\matt{a}{0}{0}{0}{0}{b}{0}{c}{0}, \matt{0}{0}{a}{0}{b}{0}{c}{0}{0}, \ \mathrm{and} \ \matt{0}{a}{0}{b}{0}{0}{0}{0}{c},$$ respectively. For any of the above types of matrices, let $n_{12} = val(a/b), n_{21} = val(b/a), n_{23} = val(b/c), n_{32} = val(c/b), n_{13} = val(a/c), n_{31} = val(c/a)$. We now define a long list of notation that we need in order to state the main theorem of this section.
Let $\mathcal{B}_1, \mathcal{B}_2, \mathcal{B}_3, \mathcal{B}_4, \mathcal{B}_5, \mathcal{B}_6$ be the inequality conditions $\{n_{21} < r, -n_{31} < t, n_{32} < s \}, \{n_{13} < r-1, n_{21} < s, -n_{23} < t + 1 \}, \{-n_{12} < t + 1, n_{13} < s - 1, n_{32} < r \}, \{n_{21} < t, -n_{31} < r, -n_{23} < s + 1 \}, \{n_{13} < t - 1, -n_{12} < s + 1, -n_{23} < r + 1 \},$ and $\{-n_{12} < r + 1, n_{32} < t, -n_{31} < s \}$, respectively. For example, if $x = \matt{a}{0}{0}{0}{b}{0}{0}{0}{c}$ and $val(b/a) < r , -val(c/a) < t, val(c/b) < s$, then we say that ``$x$ satisfies $\mathcal{B}_1$''.
Let $\mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3, \mathcal{C}_4, \mathcal{C}_5, \mathcal{C}_6$ be the inequality conditions $\{n_{31} \leq t, -n_{32} \leq s, -n_{21} \leq r \}, \{n_{23} \leq t - 1, -n_{13} \leq r + 1, -n_{21} \leq s \}, \{-n_{32} \leq r, -n_{13} \leq s + 1, n_{12} \leq t - 1 \}, \{n_{23} \leq s - 1, -n_{21} \leq t, n_{31} \leq r \}, \{n_{23} \leq r - 1, n_{12} \leq s - 1, -n_{13} \leq t + 1 \},$ and $\{n_{31} \leq s, n_{12} \leq r -1, -n_{32} \leq t \}$, respectively.
Let $\mathcal{D}_1, \mathcal{D}_2, \mathcal{D}_3, \mathcal{D}_4, \mathcal{D}_5, \mathcal{D}_6$ be the exact same inequality conditions as $\mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3, \mathcal{C}_4, \mathcal{C}_5, \mathcal{C}_6$, respectively, except that we replace every $\leq$ sign with a $>$ sign, and moreover, within each $\mathcal{C}_i$, replace every comma by the word ``or''. For example, if $x = \matt{0}{a}{0}{0}{0}{b}{c}{0}{0}$ satisfies at least one of the inequalities $val(b/c) > t - 1, -val(a/c) > r + 1$, or $-val(b/a) > s$, then we say that ``$x$ satisfies $\mathcal{D}_2$''.
Let $\mathcal{E}_1^0, \mathcal{E}_2^0, \mathcal{E}_3^0, \mathcal{E}_4^0, \mathcal{E}_5^0, \mathcal{E}_6^0$ be the inequality conditions $\{n_{21} \geq 0, n_{31} \geq 0, n_{32} \geq 0 \} \cup \{n_{21} \geq 0, n_{31} < 0, n_{32} < 0 \} \cup \{n_{21} < 0, n_{31} < 0, n_{32} \geq 0 \}, \{n_{13} \geq 0, n_{23} \geq 0, n_{21} \geq 0 \} \cup \{n_{13} < 0, n_{23} < 0, n_{21} \geq 0 \} \cup \{n_{13} \geq 0, n_{23} < 0, n_{21} < 0 \}, \{n_{32} < 0, n_{12} < 0, n_{13} \geq 0 \} \cup \{n_{32} \geq 0, n_{12} < 0, n_{13} < 0 \} \cup \{n_{32} \geq 0, n_{12} \geq 0, n_{13} \geq 0 \}, \{n_{31} < 0, n_{21} \geq 0, n_{23} \geq 0 \} \cup \{n_{31} < 0, n_{21} < 0, n_{23} < 0 \} \cup \{n_{31} \geq 0, n_{21} \geq 0, n_{23} < 0 \}, \{n_{23} \geq 0, n_{13} \geq 0, n_{12} < 0 \} \cup \{n_{23} < 0, n_{13} \geq 0, n_{12} \geq 0 \} \cup \{n_{23} < 0, n_{13} < 0, n_{12} < 0 \},$ and $\{n_{12} < 0, n_{32} \geq 0, n_{31} \geq 0 \} \cup \{n_{12} \geq 0, n_{32} \geq 0, n_{31} < 0 \} \cup \{n_{12} < 0, n_{32} < 0, n_{31} < 0 \}$, respectively. For example, if $x = \matt{a}{0}{0}{0}{b}{0}{0}{0}{c}$ satisfies $val(b/a) \geq 0, val(c/a) \geq 0$, and $val(c/b) \geq 0$, then we say that ``$x$ satisfies $\mathcal{E}_1^0$. If $x = \matt{a}{0}{0}{0}{b}{0}{0}{0}{c}$ satisfies $val(b/a) \geq 0, val(c/a) < 0$, and $val(c/b) < 0$, then we say that ``$x$ satisfies $\mathcal{E}_1^0$. If $x = \matt{a}{0}{0}{0}{b}{0}{0}{0}{c}$ satisfies $val(b/a) < 0, val(c/a) < 0$, and $val(c/b) \geq 0$, then we say that ``$x$ satisfies $\mathcal{E}_1^0$.
Let $\mathcal{E}_1^2, \mathcal{E}_2^2, \mathcal{E}_3^2, \mathcal{E}_4^2, \mathcal{E}_5^2, \mathcal{E}_6^2$ be the inequality conditions $\mathcal{E}_1^0 \setminus \{n_{21} \geq 0, n_{31} \geq 0, n_{32} \geq 0 \}$, $\mathcal{E}_2^0 \setminus \{n_{13} \geq 0, n_{23} \geq 0, n_{21} \geq 0 \}$, $\mathcal{E}_3^0 \setminus \{n_{32} \geq 0, n_{12} \geq 0, n_{13} \geq 0 \}$, $\mathcal{E}_4^0 \setminus \{n_{31} < 0, n_{21} < 0, n_{23} < 0 \}$, $\mathcal{E}_5^0 \setminus \{n_{23} < 0, n_{13} < 0, n_{12} < 0 \}$, and $\mathcal{E}_6^0 \setminus \{n_{12} < 0, n_{32} < 0, n_{31} < 0 \}$, respectively. For example, if $x = \matt{a}{0}{0}{0}{b}{0}{0}{0}{c}$ satisfies $val(b/a) \geq 0, val(c/a) \geq 0$, and $val(c/b) \geq 0$, then $x$ does not satisfy $\mathcal{E}_1^2$. If $x = \matt{a}{0}{0}{0}{b}{0}{0}{0}{c}$ satisfies $val(b/a) \geq 0, val(c/a) < 0$, and $val(c/b) < 0$, then ``$x$ satisfies $\mathcal{E}_1^2$. If $x = \matt{a}{0}{0}{0}{b}{0}{0}{0}{c}$ satisfies $val(b/a) < 0, val(c/a) < 0$, and $val(c/b) \geq 0$, then ``$x$ satisfies $\mathcal{E}_1^2$.
Set $\mathcal{E}_1^1 = \mathcal{E}_1^2, \mathcal{E}_2^1 = \mathcal{E}_2^2, \mathcal{E}_3^1 = \mathcal{E}_3^2, \mathcal{E}_4^1 = \mathcal{E}_4^0, \mathcal{E}_5^1 = \mathcal{E}_5^0, \mathcal{E}_6^1 = \mathcal{E}_6^0$. For example, if $x = \matt{a}{0}{0}{0}{b}{0}{0}{0}{c}$ satisfies $val(b/a) \geq 0, val(c/a) \geq 0$, and $val(c/b) \geq 0$, then $x$ does not satisfy $\mathcal{E}_1^1$, but it does satisfy $\mathcal{E}_1^0$.
Finally, let $\mathcal{F}_i^j$ be the condition that if $x \in \mathcal{A}_{n_i}$, then $x$ satisfies $\mathcal{B}_i, \mathcal{C}_i,$ and $\mathcal{E}_i^j$. Moreover, let $\mathcal{G}_i^j$ be the condition that if $x \in \mathcal{A}_{n_i}$, then $x$ satisfies $\mathcal{B}_i, \mathcal{D}_i,$ and $\mathcal{E}_i^j$. For example, if $x = \matt{a}{0}{0}{0}{0}{b}{0}{c}{0}$, and if $x$ satisfies $val(b/a) < t, -val(c/a) < r, -val(b/c) < s + 1$, and if $x$ satisfies $val(b/c) > s - 1$, and if $x$ satisfies $val(c/a) < 0, val(b/a) \geq 0, val(b/c) \geq 0$, then $x$ satisfies $\mathcal{G}_4^j$, for $j = 0,1,$ and $2$.
We need a few more notations. As in the case of $SL(2,F)$, the character values will be sums of powers of $q$. Some of these powers will be lengths of certain affine Weyl group elements, as before, but some will not. Some powers will be ``truncated lengths'' of affine Weyl group elements. We will not define ``truncated length''. However, we will do an example in the next section, and the computation is analogous in every other case. If an affine Weyl group element $x$ is of type $\mathcal{G}_i^0, \mathcal{G}_i^1$, or $\mathcal{G}_i^2$, then $x$ will contribute the term $q^{\ell' (x)}$ to $\theta_{\pi}(g)$, where $\ell'(x)$ denotes the ``truncated length'' of $x$. There will also be two types of Gauss sums that appear in the character formula. If an affine Weyl group element $x$ is of a certain type to be discussed later, then a Gauss sum corresponding to this element will contribute to the character, and we will denote this Gauss sum by either $\Gamma(x)$ or $\Xi(x)$. We will do examples of how these Gauss sums arise in the next section.
Let $\mathcal{H}_i$ be the condition that if $x \in \mathcal{A}_{n_i}$, then $x$ does not satisfy $\mathcal{E}_i^0$. Let $\mathcal{E}_i^3$ be the inequality condition $\mathcal{E}_i^0 \setminus \mathcal{E}_i^2$. Let $\mathcal{J}_i$ be the condition that if $x \in \mathcal{A}_{n_i}$, then $x$ satisfies $\mathcal{E}_i^3$. Write $\Upsilon := |T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p})|$. Finally, in the statement of the main theorem, when we write a summation over $x \in W^a$, we mean that we are summing over any set of representatives of the elements in $W^a$. Our main theorem for $SL(3,F)$ is the following.
\begin{theorem} \begin{equation*} \frac{\theta_{\pi}(g)}{\Upsilon} = \left\{ \begin{array}{ll}
\displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{F}_i^0 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{G}_i^0 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell'(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that} \\ x \ \mathrm{satisfies} \ \mathcal{H}_i \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} \Gamma(x) & \text{if } t = r \\
\displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{F}_i^1 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{G}_i^1 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell'(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that} \\ x \ \mathrm{satisfies} \ \mathcal{H}_i \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} \Gamma(x) + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{J}_i \\ \mathrm{for \ any} \ 1 \leq i \leq 3}} \Xi(x) & \text{if } t = r + 1 \\
\displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{F}_i^2 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{G}_i^2 \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} q^{\ell'(x)} + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that} \\ x \ \mathrm{satisfies} \ \mathcal{H}_i \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} \Gamma(x) + \displaystyle\sum_{\substack{x \in W^a \\ \mathrm{such \ that}\\x \ \mathrm{satisfies} \ \mathcal{J}_i \\ \mathrm{for \ any} \ 1 \leq i \leq 6}} \Xi(x) & \text{if } t > r + 1 \end{array} \right. \end{equation*} \end{theorem}
We note that the value of term $\Upsilon = |T(\mathfrak{o}) / Z(F) T(1 + \mathfrak{p})|$ depends on whether or not there are cube roots of unity in $F$, which is why we leave this term as is in the above theorem.
\subsection{The case of $x \in \mathcal{A}_{n_1}$}
In this section we compute $\sigma(x)$ when $x \in \mathcal{A}_{n_1}$. When $x \in \mathcal{A}_{n_2}$ or $x \in \mathcal{A}_{n_3}$, the calculations are similar. There is a very slight difference, however, when $x \in \mathcal{A}_{n_4}$, $x \in \mathcal{A}_{n_5}$, and $\mathcal{A}_{n_6}$, although these three latter cases are almost completely analogous. We will address them in the next section.
The main result of this section is the following proposition. Some of the notation in the proposition will be explained in the proof.
\begin{proposition}\label{firsttheorem} Let $x = \matt{a}{0}{0}{0}{b}{0}{0}{0}{c}$. Suppose $n_{21} \geq 0, n_{31} \geq 0, n_{32} \geq 0$. If $r \geq t$, we have
\begin{equation*} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \left\{ \begin{array}{rl} vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) q^{n_{21} + n_{32}} & \text{if } -n_{21} + r > 0 \ \mathrm{and} \ -n_{32} + s > 0 \ \\ 0 & \text{otherwise} \end{array} \right. \end{equation*}
If $r < t$, then \begin{equation*} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \Xi(x) \end{equation*}
\end{proposition}
\proof We first rewrite the double coset $H x H$ as a finite union of single right cosets, as in the case of $SL(2,F)$. Since $$H \cap x^{-1} H x = Z(F) \matt{1 + \mathfrak{p}}{\mathfrak{p}^{n_{21}}}{\mathfrak{p}^{n_{31}}}{\mathfrak{p}}{1 + \mathfrak{p}}{\mathfrak{p}^{n_{32}}}{\mathfrak{p}}{\mathfrak{p}}{1 + \mathfrak{p}},$$ we have a disjoint union $$H x H = \displaystyle\bigcup_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}} \\ z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ z_{31} \in \mathfrak{o} / \mathfrak{p}^{n_{31}}}} H x \matt{1}{z_{21}}{z_{31}}{0}{1}{z_{32}}{0}{0}{1}$$
Now, if $z = \matt{1}{z_{21}}{z_{31}}{0}{1}{z_{32}}{0}{0}{1}$, then $$xzgz^{-1}x^{-1} = \matt{\alpha}{\frac{a}{b} z_{21} (\beta - \alpha)}{\frac{a}{c}[z_{21} z_{32}(\alpha - \beta) + z_{31}(\gamma - \alpha)]}{0}{\beta}{\frac{b}{c} z_{32} (\gamma - \beta)}{0}{0}{\gamma}$$ Notice that $g \in Z(F) T(1+\mathfrak{p})$ is forced upon us here in order to have $x z g z^{-1} x^{-1} \in H$ (so that $\dot{\chi}$ doesn't vanish on $x z g z^{-1} x^{-1}$). We note that the condition $g \in Z(F) T(1+\mathfrak{p})$ continues to be forced upon us, for the same reason, when you compute the terms $ygy^{-1}$ that appear in $\theta_{\pi}(g)$ for any other representative $x$ of any element of the affine Weyl group, but we won't include these calculations. This shows, therefore, that $\theta_{\pi}$ vanishes on $T(F) \setminus Z(F) T(1+\mathfrak{p})$.
Now, write $\chi$ on $I_+$ as $$\chi : I_+ \rightarrow \mathbb{C}^*$$ $$\matt{d_{11}}{d_{12}}{d_{13}}{d_{21}}{d_{22}}{d_{23}}{d_{31}}{d_{32}}{d_{33}} \mapsto \chi_1(d_{12}) \chi_2(d_{23}) \chi_3(d_{31})$$ where $\chi_1, \chi_2$ are level $1$ characters of $\mathfrak{o}$ and where $\chi_3(d_{31}) = \chi_3'(\frac{1}{\varpi} d_{31})$, where $\chi_3'$ is a level $1$ character of $\mathfrak{o}$.
We would like to say that we therefore have $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}} \\ z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ z_{31} \in \mathfrak{o} / \mathfrak{p}^{n_{31}}}} \dot{\chi}_1\left(\frac{a}{b} z_{21} (\beta - \alpha) \right) \dot{\chi}_2 \left(\frac{b}{c} z_{32} (\gamma - \beta) \right)$$ where $\dot{\chi_i}(z) = \chi_i(z) \ \forall z \in \mathfrak{o}$ and $\dot{\chi_i}(z) = 0 \ \forall z \in F \setminus \mathfrak{o}$, for $i = 1,2$, and where $\dot{\chi}_3(z) = \chi_3'(z) \ \forall z \in \mathfrak{o}$ and $\dot{\chi}_3(z) = 0 \ \forall z \in F \setminus \mathfrak{o}$.
However, we need to take into account the fact that we have the term $\frac{a}{c}[z_{21} z_{32}(\alpha - \beta) + z_{31}(\gamma - \alpha)]$. $\dot{\chi}$ is zero outside of $Z(F) I_+$, and so we have to take into account the condition that $\frac{a}{c}[z_{21} z_{32}(\alpha - \beta) + z_{31}(\gamma - \alpha)] \in \mathfrak{o}$.
Absorbing all units into the $z_{21}, z_{32}, z_{31}$ terms, and recalling that $\alpha - \beta = \varpi^r u$, $\beta - \gamma = \varpi^s u'$, $\alpha - \gamma = \varpi^t u''$, we therefore wish to understand the condition $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31} \in \mathfrak{o}$. We separate this into two cases.
Case 1) Suppose $r \geq t$. Notice that there might be negative powers of $\varpi$ in $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31}$ because of the $-n_{31}$ terms. Fix any $z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}}$ and any $z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}}$. Then, $\varpi^{-n_{31}+r} z_{21} z_{32}$ can certainly contain negative powers of $\varpi$. However, we can use $z_{31}$ to cancel out these negative powers of $\varpi$, since $r \geq t$. In order to force $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31} \in \mathfrak{o}$, it is evident that all of the negative $\varpi$ power terms in $\varpi^{-n_{31} + t} z_{31}$ are uniquely determined by the negative $\varpi$ power terms in $\varpi^{-n_{31} + r} z_{21} z_{32}$. Moreover, since $r \geq t$, no matter what $z_{21}$ or $z_{32}$ are, we can always find a $z_{31}$ to force $\varpi^{-n_{31} + r} z_{21} z_{32} + \varpi^{-n_{31}+t} z_{31} \in \mathfrak{o}$. Once we determine the negative $\varpi$ power terms of $\varpi^{-n_{31} + t} z_{31}$ so that the total sum $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31}$ is in $\mathfrak{o}$, we have complete leeway in the non-negative $\varpi$ power terms in $\varpi^{-n_{31} + t} z_{31}$. Therefore, it appears that we have obtained
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}} \\ z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ z_{31} \in \mathfrak{p}^{n_{31}-t} / \mathfrak{p}^{n_{31}}}} \dot{\chi}_1\left(\frac{a}{b} z_{21} (\beta - \alpha) \right) \dot{\chi}_2 \left(\frac{b}{c} z_{32} (\gamma - \beta) \right)$$
However, the last thing we need to notice is that $z_{31}$ still has to be in $\mathfrak{o}$, by a condition from earlier. Therefore, the condition $z_{31} \in \mathfrak{p}^{n_{31}-t} / \mathfrak{p}^{n_{31}}$ is not quite correct since it could be the case that $n_{31} < t$. Therefore, what we really get in then end is
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}} \\ z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ z_{31} \in (\mathfrak{o} \cap \mathfrak{p}^{n_{31}-t}) / \mathfrak{p}^{n_{31}}}} \dot{\chi}_1\left(\frac{a}{b} z_{21} (\beta - \alpha) \right) \dot{\chi}_2 \left(\frac{b}{c} z_{32} (\gamma - \beta) \right)$$
Absorbing all units into the $z_{21}, z_{32}$ terms, we get $$vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}} \\ z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}}}} \dot{\chi}_1\left(\varpi^{-n_{21}} z_{21} \varpi^r \right) \dot{\chi}_2 \left(\varpi^{-n_{32}} z_{32} \varpi^{s} \right)$$ Making a change of variables, we get $$vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) \displaystyle\sum_{\substack{z_{21}' \in \mathfrak{p}^{-n_{21} + r} / \mathfrak{p}^{r} \\ z_{32}' \in \mathfrak{p}^{-n_{32} + s} / \mathfrak{p}^{s}}} \dot{\chi}_1\left(z_{21}' \right) \dot{\chi}_2 \left(z_{32}' \right) = $$ $$vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) \left( vol(\mathfrak{p}^r)^{-1} \displaystyle\int_{\mathfrak{p}^{-n_{21}+r} \cap \mathfrak{o}} \dot{\chi}_1(z_{21}') dz_{21}' \right) \left( vol(\mathfrak{p}^s)^{-1} \displaystyle\int_{\mathfrak{p}^{-n_{32}+s} \cap \mathfrak{o}} \dot{\chi}_2(z_{32}') dz_{32}' \right)$$ Therefore, since the integral of a nontrivial character over a group vanishes, we get
\begin{equation*} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \left\{ \begin{array}{rl} vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) q^{n_{21} + n_{32}} & \text{if } -n_{21} + r > 0 \ \mathrm{and} \ -n_{32} + s > 0 \ \\ 0 & \text{otherwise} \end{array} \right. \end{equation*}
which concludes Case 1).
Case 2): Suppose $r < t$. In this case, if one picks any random $z_{21}$ and $z_{32}$, one might not be able to find a $z_{31}$ that makes $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31}$ land in $\mathfrak{o}$. However, if one chooses $z_{21}, z_{32}$ such that $z_{21} z_{32} \in \mathfrak{p}^{t-r}$, then one can find a $z_{31}$ such that $\varpi^{-n_{31}+r} z_{21} z_{32} + \varpi^{-n_{31} + t} z_{31}$ will be in $\mathfrak{o}$. One can see that $z_{21} z_{32} \in \mathfrak{p}^{t-r}$ is the only additional condition that we need to add to the conditions in Case 1, so we get
$$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = $$ $$vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}}, z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ \mathrm{such \ that \ } z_{21} z_{32} \in \mathfrak{p}^{t-r}}} \dot{\chi}_1\left(\frac{a}{b} z_{21} (\beta - \alpha) \right) \dot{\chi}_2 \left(\frac{b}{c} z_{32} (\gamma - \beta) \right)$$
Absorbing all units into the $z_{21}, z_{32}$ terms, we get \begin{equation} vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) \displaystyle\sum_{\substack{z_{21} \in \mathfrak{o} / \mathfrak{p}^{n_{21}}, z_{32} \in \mathfrak{o} / \mathfrak{p}^{n_{32}} \\ \mathrm{such \ that \ } z_{21} z_{32} \in \mathfrak{p}^{t-r}}} \dot{\chi}_1\left(\varpi^{-n_{21}} z_{21} \varpi^r \right) \dot{\chi}_2 \left(\varpi^{-n_{32}} z_{32} \varpi^{s} \right) \ \ \label{xi} \end{equation} Since this sum is quite complicated, we compute this type of sum in full generality in a later section. We will instead denote the value of $\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1})$ by $\Xi(x)$ for the $x$ in this proposition, in the case that $r < t$. Moreover, since we will encounter the analogous type of sum in equation (\ref{xi}) for many other elements $x$, we will merely denote the value of $\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1})$ by $\Xi(x)$ for those $x$ as well. \qed
We note that in Case 1) above, if $n_{31} \leq t$, we get $vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) q^{n_{21} + n_{32}} = q^{n_{31} + n_{21} + n_{32}}$, which equals $q^{\ell(x)}$, as a simple calculation will show. This is how the terms of the form $q^{\ell(x)}$ appear in the character formula. If, on the other hand, $n_{31} > t$, we get $vol((\mathfrak{o} \cap \mathfrak{p}^{n_{31} - t}) / \mathfrak{p}^{n_{31}}) q^{n_{21} + n_{32}} = q^{t + n_{21} + n_{32}}$, which is what we have called $q^{\ell'(x)}$, as $t + n_{21} + n_{32}$ is a ``truncated length'' of $x$.
We now illustrate one more case in the $\mathcal{A}_{n_1}$ setting, which will show how the second type of Gauss sum in the character formula arises. Some of the notation in the next proposition will be explained in the proof.
\begin{proposition} Let $x = \matt{a}{0}{0}{0}{b}{0}{0}{0}{c}$. Suppose $n_{21} < 0, n_{31} < 0, n_{32} < 0$. Then
\begin{equation*} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \Gamma(x) \end{equation*} \end{proposition}
\proof We have a disjoint union $$H x H = \displaystyle\bigcup_{\substack{z_{21} \in \mathfrak{p} / \mathfrak{p}^{-n_{21}+1} \\ z_{31} \in \mathfrak{p} / \mathfrak{p}^{-n_{31}+1} \\ z_{32} \in \mathfrak{p} / \mathfrak{p}^{n_{32}+1}}} H x \matt{1}{0}{0}{z_{21}}{1}{0}{z_{31}}{z_{32}}{1}$$
Now, if $z = \matt{1}{0}{0}{z_{21}}{1}{0}{z_{31}}{z_{32}}{1}$, then $$xzgz^{-1}x^{-1} = \matt{\alpha}{0}{0}{\frac{b}{a} z_{21} (\alpha - \beta)}{\beta}{0}{\frac{c}{a} [ z_{32} z_{21} (\gamma - \beta) + z_{31} ( \alpha - \gamma) ]}{\frac{c}{b} z_{32} ( \beta - \gamma)}{\gamma}$$
\begin{equation} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \displaystyle\sum \dot{\chi}(\frac{c}{a} [ z_{32} z_{21} (\gamma - \beta) + z_{31} ( \alpha - \gamma) ] \ \ \label{gamma} \end{equation}
where the sum is over $z_{21} \in \mathfrak{p} / \mathfrak{p}^{-n_{21}+1}, z_{31} \in \mathfrak{p} / \mathfrak{p}^{-n_{31}+1}, z_{32} \in \mathfrak{p} / \mathfrak{p}^{n_{32}+1} \ \mathrm{such \ that} \ \frac{b}{a} z_{21} (\alpha - \beta) \in \mathfrak{p} \ \mathrm{and} \ \frac{c}{b} z_{32} ( \beta - \gamma) \in \mathfrak{p}$. Since this type of sum is quite complicated, we compute this type of sum in full generality in section \ref{morecalculations}. We will instead denote the value of $\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1})$ by $\Gamma(x)$ for the $x$ in this proposition. Moreover, since we will encounter the type of sum on the right hand side of the above equation (\ref{gamma}) for many other elements $x$, we will merely denote the value of $\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1})$ by $\Gamma(x)$ for those $x$ as well.
\subsection{The case of $x \in \mathcal{A}_{n_4}$} In this section we will consider the inner sums $\sigma(x)$ when $x \in \mathcal{A}_{n_4}$. This case is slightly different from that of $\mathcal{A}_{n_1}, \mathcal{A}_{n_2}$, and $\mathcal{A}_{n_3}$. In particular, the inequalities between $r,s,$ and $t$ that distinguished between Case 1)'s and Case 2)'s in the previous section are now shifted, as we shall show. This is why we need separate cases in the statement of the main theorem of the distribution character for $SL(3,F)$. We will show why the shifts occur in this section. The cases of $x \in \mathcal{A}_{n_5}$ and $x \in \mathcal{A}_{n_6}$ are similar to the case of $x \in \mathcal{A}_{n_4}$.
\begin{proposition} Let $x = \matt{a}{0}{0}{0}{0}{b}{0}{c}{0}$. Suppose $n_{31} < 0, n_{21} < 0, n_{23} < 0$. Then if $s \geq t -1$, we have $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) =$$
\begin{equation*} \left\{ \begin{array}{rl} vol((\mathfrak{p} \cap \mathfrak{p}^{-n_{21} - t + 1}) / \mathfrak{p}^{-n_{21}+1}) q^{-n_{31} - n_{23}-1} & \text{if } n_{23}+s+1 > 0 \ \mathrm{and} \ n_{31}+r > 0 \ \\ 0 & \text{otherwise} \end{array} \right. \end{equation*}
If $s < t-1$, then \begin{equation*} \displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \Xi(x) \end{equation*} \end{proposition}
\proof We have a disjoint union $$H x H = \displaystyle\bigcup_{\substack{z_{21} \in \mathfrak{p} / \mathfrak{p}^{-n_{21}+1} \\ z_{23} \in \mathfrak{p} / \mathfrak{p}^{-n_{23}} \\ z_{31} \in \mathfrak{p} / \mathfrak{p}^{-n_{31}+1}}} H x \matt{1}{0}{0}{z_{31}}{1}{0}{z_{21}}{z_{23}}{1}$$
Now, if $z = \matt{1}{0}{0}{z_{31}}{1}{0}{z_{21}}{z_{23}}{1}$, then $$xzgz^{-1}x^{-1} = \matt{\alpha}{0}{0}{\frac{b}{a}[z_{21}(\alpha - \gamma) + z_{31} z_{23} (\gamma - \beta)]}{\gamma}{\frac{b}{c} z_{23} (\beta - \gamma)}{\frac{c}{a} z_{31} (\alpha - \beta)}{0}{\beta}$$
We have to take into account the condition that $\frac{b}{a}[z_{21}(\alpha - \gamma) + z_{31} z_{23} (\gamma - \beta)] \in \mathfrak{p}$. We therefore wish to understand the condition $\varpi^{n_{21}+s} z_{31} z_{23} + \varpi^{n_{21} + t} z_{21} \in \mathfrak{p}$. We separate this into two cases, exactly in the way we did in Proposition \ref{firsttheorem}. Notice here that $z_{21} \in \mathfrak{p}, z_{23} \in \mathfrak{p}, z_{31} \in \mathfrak{p}$. Therefore, our first case is going to be
Case 1) Suppose $s \geq t -1$. Then an analogous computation as we have done before shows that $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) =$$
\begin{equation*} \left\{ \begin{array}{rl} vol((\mathfrak{p} \cap \mathfrak{p}^{-n_{21} - t + 1}) / \mathfrak{p}^{-n_{21}+1}) q^{-n_{31} - n_{23}-1} & \text{if } n_{23}+s+1 > 0 \ \mathrm{and} \ n_{31}+r > 0 \ \\ 0 & \text{otherwise} \end{array} \right. \end{equation*}
Case 2) Suppose $s < t - 1$. Then, using notation from the previous section, we have $$\displaystyle\sum_{y \in H \backslash H x H} \dot{\chi}(ygy^{-1}) = \Xi(x)$$ \qed
\section{Calculation of the Gauss sums}\label{morecalculations}
\subsection{The calculation of $\Gamma(x)$}\label{gammasections}
In this section we calculate the terms of the form $\Gamma(x)$ in a much more general context. In particular, we shall calculate $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j} } \chi(\varpi^{-a} xy + \varpi^{-b} z),$$ where $\chi$ is a character of $F$ that is zero on $F \setminus \mathfrak{o}$, and is $1$ on $\mathfrak{p}$, except when
i) $m + k < a, i < b$, and $n + \ell \leq a$
ii) $m + k < a, i < b$, and $j \leq b$
The reason that we may omit these cases is that the conditions $j \leq b$ and $n + \ell \leq a$ never occur in the $SL(3,F)$ calculations (as one can check, but we do not include these details here), so we can ignore them. We won't ignore every possible case where $j \leq b$ or $n + \ell \leq a$, since some of these cases are easy to write down. We will just ignore the above two special cases. We split up the calculation of the above sum into various cases.
\
Case 1) Suppose $m+k \geq a, i > b$. We then have that $\varpi^{-a} xy, \varpi^{-b} z \in \mathfrak{o}$. Therefore, $\chi(\varpi^{-a} xy + \varpi^{-b} z) = \chi(\varpi^{-a} xy) \chi( \varpi^{-b} z)$, so $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j} } \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy) \displaystyle\sum_{z \in \mathfrak{p}^i / \mathfrak{p}^j } \chi( \varpi^{-b} z)$$ After making a change of variables, we get $$\displaystyle\sum_{z \in \mathfrak{p}^i / \mathfrak{p}^j } \chi( \varpi^{-b} z) = \displaystyle\sum_{z' \in \mathfrak{p}^{i-b} / \mathfrak{p}^{j-b} } \chi( z') $$ $$=vol(\mathfrak{p}^{j-b})^{-1} \int_{\mathfrak{p}^{i-b} \cap \mathfrak{o}} \chi(z') dz' = vol(\mathfrak{p}^{j-b})^{-1} vol(\mathfrak{p}^{i-b}) = q^{j-i}$$ and so $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j} } \chi(\varpi^{-a} xy + \varpi^{-b} z) = q^{j-i} \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy)$$ We now consider two subcases : $m + k > a$ and $m + k = a$. Suppose that $m + k > a$. Then $\varpi^{-a} xy \in \mathfrak{p}$, and therefore $\chi(\varpi^{-a} xy) = 1 \ \forall x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}$ since $\chi$ is trivial on $\mathfrak{p}$. Therefore, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} 1 = vol(\mathfrak{p}^m / \mathfrak{p}^n) vol(\mathfrak{p}^k / \mathfrak{p}^{\ell}) = q^{n-m} q^{\ell-k}$$ Now suppose $m + k = a$. We argue by fixing values of $x$. Suppose $val(x) = m$. Then $$\displaystyle\sum_{y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}} \chi(\varpi^{-a} xy) = 0$$ since $\varpi^{-a} xy$ ranges over all elements of $\mathfrak{o} / \mathfrak{p}^{\ell-a+m}$. Therefore, the contributions to $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy)$$ from the elements $x$ that have valuation $m$ are all zeroes. Therefore, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^{m+1} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy)$$ But now in this sum, notice that since we have $x \in \mathfrak{p}^{m+1} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}$, we conclude that $\varpi^{-a} xy$ is always in $\mathfrak{p}$. Therefore, as before, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m+1} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^{m+1} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} 1 = q^{n - (m+1)} q^{\ell-k}$$
Case 2) Suppose $m+k \geq a, i = b$. This case is mostly analagous to Case 1), except we now have $$\displaystyle\sum_{z \in \mathfrak{p}^i / \mathfrak{p}^j } \chi( \varpi^{-b} z) = 0$$ since $i = b$, and therefore $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j} } \chi(\varpi^{-a} xy + \varpi^{-b} z) = 0$$
Case 3) Suppose $m+k \geq a, i < b$. We need to understand the condition $\varpi^{-a} xy + \varpi^{-b} z \in \mathfrak{o}$. We first assume that $j > b$. Thus, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^b / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z)$$ (note that $b < j$, so we can talk about $\mathfrak{p}^b / \mathfrak{p}^j$). But now note that since $z \in \mathfrak{p}^b$, we have that $\varpi^{-a} xy, \varpi^{-b} z \in \mathfrak{o}$. Therefore, $$ \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^b / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy) \displaystyle\sum_{ z \in \mathfrak{p}^b / \mathfrak{p}^j } \chi(\varpi^{-b} z) = 0$$ since the integral over a group of a nontrivial character vanishes.
If $j \leq b$, then in order for $\varpi^{-a} xy + \varpi^{-b} z$ to be in $\mathfrak{o}$, we require that $z = 0$, since $z$ is assumed to be in $\mathfrak{p}^i / \mathfrak{p}^j$. Therefore, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(\varpi^{-a} xy),$$ which may be rewritten as $$\displaystyle\sum_{\substack{x' \in \mathfrak{p}^{m-a} / \mathfrak{p}^{n-a} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(x'y)$$ after a change of variables. This type of sum will be handled in Case 4), which we now present.
Case 4) Suppose $m+k < a, i > b$. We consider two subcases. We therefore get $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}, z \in \mathfrak{p}^i / \mathfrak{p}^j \\ \mathrm{such \ that \ } xy \in \mathfrak{p}^a }} \chi(\varpi^{-a} xy + \varpi^{-b} z)$$ since if $xy \notin \mathfrak{p}^a$, then $\varpi^{-a} xy + \varpi^{-b} z \notin \mathfrak{o}$. Therefore, we get $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j} } \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}, z \in \mathfrak{p}^i / \mathfrak{p}^j \\ \mathrm{such \ that \ } xy \in \mathfrak{p}^a }} \chi(\varpi^{-a} xy + \varpi^{-b} z) =$$ $$ \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that \ } xy \in \mathfrak{p}^a}} \chi(\varpi^{-a} xy) \displaystyle\sum_{ z \in \mathfrak{p}^i / \mathfrak{p}^j } \chi(\varpi^{-b} z) = q^{j-i} \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that \ } xy \in \mathfrak{p}^a}} \chi(\varpi^{-a} xy)$$ We make a change of variables $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that \ } xy \in \mathfrak{p}^a}} \chi(\varpi^{-a} xy) = \displaystyle\sum_{\substack{x' \in \mathfrak{p}^{m-a} / \mathfrak{p}^{n-a}, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that \ } x'y \in \mathfrak{o}}} \chi(x'y)$$
Since $\chi$ vanishes outside $\mathfrak{o}$ $$\displaystyle\sum_{\substack{x' \in \mathfrak{p}^{m-a} / \mathfrak{p}^{n-a}, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that \ } x'y \in \mathfrak{o}}} \chi(x'y) = \displaystyle\sum_{\substack{x' \in \mathfrak{p}^{m-a} / \mathfrak{p}^{n-a} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(x'y)$$
After reindexing and relabeling, we are now interested in computing the following type of sum $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy)$$ We split this into two cases:
Case i) Suppose $n + \ell \geq -2$. Recall that $\chi$ vanishes outside $\mathfrak{o}$, so we are interested in when $xy \in \mathfrak{o}$. We will separate out the $x$ terms from the sum that have no chance of multiplying with an element of $y$ to land in $\mathfrak{o}$ unless $y$ is zero. We consider two subcases. Suppose $1 - \ell > m$. We write $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} - \mathfrak{p}^{1-\ell} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) + \displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy)$$ (note that we can write $x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} - \mathfrak{p}^{1-\ell} / \mathfrak{p}^n$ since we assumed $1 - \ell \geq m$). Let the first sum be denoted $A$ and the second sum be denoted $B$. For the first sum $A$, $xy$ can never be in $\mathfrak{o}$ unless $y = 0$. Therefore, we get $$A = \displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} - \mathfrak{p}^{1-\ell} / \mathfrak{p}^n \\ y = 0}} \chi(xy) = \displaystyle\sum_{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} - \mathfrak{p}^{1-\ell} / \mathfrak{p}^n} 1 = q^{n-m} - q^{n - (1 - \ell)}.$$ For $B$, we split up the sum as
$$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^{1 - \ell} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + 1} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) + \displaystyle\sum_{\substack{x \in \mathfrak{p}^{1 - \ell+1} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + 2} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy)$$ $$ + \displaystyle\sum_{\substack{x \in \mathfrak{p}^{1 - \ell+2} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + 3} / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) + ... + \displaystyle\sum_{\substack{x \in \mathfrak{p}^n / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy)$$
Note that if $val(x) = i$, we require $y \in \mathfrak{p}^{-i} / \mathfrak{p}^{\ell}$ in order to force $xy \in \mathfrak{o}$. Moreover, note that if $val(x) = i - \ell$, and $y$ ranges over $\mathfrak{p}^{\ell-i} / \mathfrak{p}^{\ell}$, then $xy$ ranges over $\mathfrak{o} / \mathfrak{p}^i$. Coupling this with the fact that the sum of a nontrivial character over a group vanishes, we may compute the above sums easily. We note that the evaluation of these sums can vary depending on $k, \ell, $ and $n$, as one can check.
We now assume $1 - \ell \leq m$. Then there is no analogous term $A$ as above that we need to evaluate, and so the evaluation of the sum $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy)$$ is analogous to the sum $B$ above.
Case ii) Assume $n + \ell < -2$. In this case, it's never possible that $xy \in \mathfrak{o}$ unless $x$ or $y$ is zero. Therefore, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) = \displaystyle\sum_{\substack{x = 0 \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) + \displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y = 0}} \chi(xy) - \chi(0*0)$$ We subtracted $\chi(0*0)$ since we have double counted the term $\chi(0*0)$ in the right hand side of the equality. Thus, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n} \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}}} \chi(xy) = \left(\displaystyle\sum_{ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}} 1 \right) + \left(\displaystyle\sum_{x \in \mathfrak{p}^{m} / \mathfrak{p}^{n}} 1 \right) - 1 = q^{n-m} + q^{\ell-k} - 1$$ This finishes the case $n + \ell < -2$.
Case 5) Suppose $m + k < a, i = b$. Then $$\displaystyle\sum_{ z \in \mathfrak{p}^i / \mathfrak{p}^j } \chi(\varpi^{-b} z) = 0$$ since $i = b$ (as in a previous case). Moreover, in order to get $\varpi^{-a} xy + \varpi^{-b} z$ to be in $\mathfrak{o}$ we need the negative valuation terms of $\varpi^{-a} xy$ to be zero, since $\varpi^{-b} z$ is always in $\mathfrak{o}$. Therefore, we are forced to take values of $x,y$ such that $\varpi^{-a} xy \in \mathfrak{o}$. Over these values of $x,y$, we get $\chi(\varpi^{-a} xy + \varpi^{-b} z) = \chi(\varpi^{-a} xy) \chi(\varpi^{-b} z)$. Therefore, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = 0$$
Case 6) Suppose $m+ k < a, i < b$. Recall from assumptions i) and ii) at the beginning of section \ref{gammasections}, we may assume that $n + \ell > a$ and $j > b$. Since $m + k < a, i < b$, we will have negative $\varpi$ powers in both $\varpi^{-a} xy$ and $\varpi^{-b} z$. Consider the negative valuation part of a term of the form $\varpi^{-a} xy + \varpi^{-b} z$. We need this negative valuation part to be zero. But once the negative valuation part of this is zero, we are free to let the rest of $x,y,z$ vary. Fix possible negative valuation parts, denoted $(xy)_{-}$ and $z_{-}$, of $\varpi^{-a} xy$ and $\varpi^{-b} z$, respectively. We compute the contribution to $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z)$$ of all terms $\chi(\varpi^{-a} xy + \varpi^{-b} z)$ such that $\varpi^{-a} xy$ has negative valuation $(xy)_{-}$ and $\varpi^{-b} z$ has negative valuation $(z)_{-}$, where $(xy)_{-} = -(z)_{-}$ (this last equality is forced upon us since otherwise $\chi$ will vanish). Since $(xy)_{-} = -(z)_{-}$, this contribution is $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}, z \in \mathfrak{p}^i / \mathfrak{p}^j \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^a, z \in \mathfrak{p}^b }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}, z \in \mathfrak{p}^i / \mathfrak{p}^j \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^a, z \in \mathfrak{p}^b }} \chi(\varpi^{-a} xy) \chi(\varpi^{-b} z) = 0.$$ This argument holds regardless of $(xy)_{-}$ and $(z)_{-}$. Thus, in the end, we are summing up a bunch of zeroes, so we finally get that $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ z \in \mathfrak{p}^i / \mathfrak{p}^j }} \chi(\varpi^{-a} xy + \varpi^{-b} z) = 0$$
\subsection{The calculation of $\Xi(x)$}
In this section we calculate the terms of the form $\Xi(x)$ in a much more general context. In particular, we shall calculate $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ We will calculate this sum in complete generality. We split up the calculation of the above sum into various cases.
Case 1): Suppose $n + \ell \leq c + 1$. Then it's never the case that $xy \in \mathfrak{p}^c$ unless at least one of $x,y$ are zero. Therefore, if $n + \ell \leq c + 1$, we get $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \ } \chi(y) + \displaystyle\sum_{y \in \mathfrak{p}^m / \mathfrak{p}^n \ } \chi(x) - 1$$ which is easily calculatable, each sum being $0$ or a power of $q$, depending on $k,\ell,m,n$. Note that we subtracted $\chi(0)\chi(0) = 1$ since this is $\chi(x) \chi(y)$ when $x = y = 0$ and we have double counted this term when we added the $x =0$ and $y = 0$ sums above.
Case 2): Now assume that $n + \ell > c+1$. Assume furthermore that $1 - \ell + c > m$. We will separate the sum into two parts. We write $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^m / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) + $$ $$\displaystyle\sum_{\substack{x \in ( \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n ), y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y).$$ (note that we can write $x \in ( \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n )$ since we assumed that $1 - \ell + c > m$). We first compute the second sum on the right hand side. In this sum, we will never have $xy \in \mathfrak{p}^c$ unless at least one of $x,y$ is zero. But $x$ can't be zero, since $x \in \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n $. Therefore, we must have $y = 0$. Therefore, the second sum is $$\displaystyle\sum_{\substack{x \in ( \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n ), y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{x \in ( \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n )} \chi(x) \chi(0) = $$ $$\displaystyle\sum_{x \in ( \mathfrak{p}^m / \mathfrak{p}^n)} \chi(x) - \displaystyle\sum_{x \in ( \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n) } \chi(x)$$ which is easy to calculate, each sum being $0$ or a power of $q$, depending on $m,n,\ell,c$.
So we now need to calculate the first sum $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ We separate this into cases:
Case A: $1 - \ell + c > 0$. We will calculate the sum by fixing the valuation of $x$. Namely, $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{\substack{x \in (\mathfrak{p}^{1 - \ell + c} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + c + 1} / \mathfrak{p}^n), y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) + $$ $$\displaystyle\sum_{\substack{x \in (\mathfrak{p}^{1 - \ell + c+1} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + c + 2} / \mathfrak{p}^n) \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) + $$ $$\displaystyle\sum_{\substack{x \in (\mathfrak{p}^{1 - \ell + c+2} / \mathfrak{p}^n - \mathfrak{p}^{1 - \ell + c + 3} / \mathfrak{p}^n) \\ y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) + ... + \displaystyle\sum_{\substack{x \in \mathfrak{p}^n / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$
Note that if $val(x) = 1 - \ell + c + i$, then in order for $xy$ to be in $\mathfrak{p}^c$, we must have $y \in \mathfrak{p}^{\ell-i-1}$. Coupling this with the fact that $\chi(x) = 1$ in every summation since we have assumed that $1 - \ell + c > 0$, we may compute the above sums easily. We note that the evaluation of these sums can vary depending on $c, k, \ell, $ and $n$, as one can check.
\
Case B: Suppose $1 - \ell + c \leq 0$ and $\ell > 0$. Suppose first that $n \leq 0$. Then $\chi(x) \chi(y) \neq 0$ iff $x = 0$ since $\chi$ is zero on elements of negative valuation. Therefore, if $n \leq 0$, $$\displaystyle\sum_{y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}} \chi(0) \chi(y) = \displaystyle\sum_{y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}} \chi(y)$$ which is easy to calculate. If $n > 0$, then again, since $\chi$ is zero on elements of negative valuation, and since $1 - \ell + c \leq 0$, so we get $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ In the case that $k \geq c$, $xy \in \mathfrak{p}^c$ always holds, so $$\displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{x \in \mathfrak{o} / \mathfrak{p}^n} \chi(x) \displaystyle\sum_{y \in \mathfrak{p}^k / \mathfrak{p}^{\ell}} \chi(y)$$ which is easy to calculate, depending on whether or not $n$ is zero.
Suppose that $k < c$. We break the sum $$\displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ into two sums :
$$\displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^c / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) + \displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in (\mathfrak{p}^k / \mathfrak{p}^{\ell} - \mathfrak{p}^c / \mathfrak{p}^{\ell}) \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ Note that we assumed that $1 - \ell + c \leq 0$, so we can talk about $\mathfrak{p}^c / \mathfrak{p}^{\ell}$. Moreover, since we assumed that $k < c$, we can talk about $y \in (\mathfrak{p}^k / \mathfrak{p}^{\ell} - \mathfrak{p}^c / \mathfrak{p}^{\ell})$. Call the first sum D and the second sum E. We first analyze E. Notice that since are subtracting all elements $y$ of valuation $\geq c$, if we take an element $x$ of valuation zero, $xy$ can never be in $\mathfrak{p}^c$. Therefore, $$E = \displaystyle\sum_{\substack{x \in \mathfrak{p} / \mathfrak{p}^n, y \in (\mathfrak{p}^k / \mathfrak{p}^{\ell} - \mathfrak{p}^c / \mathfrak{p}^{\ell}) \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ which equals $$\displaystyle\sum_{\substack{x \in \mathfrak{p} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) - \displaystyle\sum_{\substack{x \in \mathfrak{p} / \mathfrak{p}^n, y \in \mathfrak{p}^c / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y).$$ The first of these two sums can be handled by Case A, and the second sum equals $$\displaystyle\sum_{y \in \mathfrak{p}^c / \mathfrak{p}^{\ell}} \chi(y),$$ which is easy to calculate, and depends on $c$ and $\ell$.
We now handle the first sum D. It's always the case that $xy \in \mathfrak{p}^c$ in this sum, so we get $$\displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n, y \in \mathfrak{p}^c / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{\substack{x \in \mathfrak{o} / \mathfrak{p}^n \\ y \in \mathfrak{p}^c / \mathfrak{p}^{\ell}}} \chi(x) \chi(y) = \displaystyle\sum_{x \in \mathfrak{o} / \mathfrak{p}^n} \chi(x) \displaystyle\sum_{y \in \mathfrak{p}^c / \mathfrak{p}^{\ell}} \chi(y) = 0$$ since $$\displaystyle\sum_{x \in \mathfrak{o} / \mathfrak{p}^n} \chi(x) = 0$$ (recall that we are in the case that $n > 0$). Thus, $D = 0$.
Case C: Suppose $1 - \ell + c \leq 0$ and $\ell \leq 0$. Then since $\chi$ is zero on negative valuation terms, the $y$ terms don't contribute unless $y = 0$. Then $$\displaystyle\sum_{\substack{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n, y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y) = \displaystyle\sum_{x \in \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n} \chi(x) = \displaystyle\sum_{x \in \mathfrak{o} / \mathfrak{p}^n} \chi(x)$$ which is easy to calculate, depending on whether or not $n$ is zero.
\
Finally, the above analysis for Case 2) assumed that $1 - \ell + c > m$. The case that $1 - \ell + c \leq m$ is simpler and similar to the case of $1 - \ell + c \leq m$, as one can check. We note that there is no sum of the form $$\displaystyle\sum_{\substack{x \in ( \mathfrak{p}^m / \mathfrak{p}^n - \mathfrak{p}^{1-\ell+c} / \mathfrak{p}^n ), y \in \mathfrak{p}^k / \mathfrak{p}^{\ell} \\ \mathrm{such \ that} \ xy \in \mathfrak{p}^c}} \chi(x) \chi(y)$$ that we need to evaluate, in the case that $1 - \ell + c \leq m$.
\end{document} | arXiv |
\begin{document}
\title{Particle systems and kinetic equations modeling interacting agents in high dimension}
\renewcommand{\arabic{footnote}}{\fnsymbol{footnote}}
\footnotetext[2]{Faculty of Mathematics, Technical University of Munich, Boltzmannstrasse 3, D-85748 Garching, Germany} \footnotetext[3]{Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria}
\renewcommand{\arabic{footnote}}{\arabic{footnote}}
\begin{abstract} In this paper we explore how concepts of high-dimensional data compression via random projections onto lower-dimensional spaces can be applied for tractable simulation of certain dynamical systems modeling complex interactions. In such systems, one has to deal with a large number of agents (typically millions) in spaces of parameters describing each agent of high dimension (thousands or more). Even with today's powerful computers, numerical simulations of such systems are prohibitively expensive.
We propose an approach for the simulation of dynamical systems governed by functions of {\it adjacency matrices} in high dimension, by random projections via Johnson-Lindenstrauss embeddings, and recovery by compressed sensing techniques. We show how these concepts can be generalized to work for associated kinetic equations, by addressing the phenomenon of the delayed curse of dimension, known in information-based complexity for optimal numerical integration problems and measure quantization in high dimensions. \end{abstract}
\begin{keywords} Dimensionality reduction, dynamical systems, flocking and swarming, Johnson-Lindenstrauss embedding, compressed sensing, high-dimensional kinetic equations, delayed curse of dimension, optimal integration of measures in high dimension. \end{keywords}
\begin{AMS}
34C29,
35B35, 35Q91, 35Q94, 60B20,
65Y20.
\end{AMS}
\pagestyle{myheadings} \thispagestyle{plain} \markboth{M.~FORNASIER, J.~HA\v{S}KOVEC AND J.~VYB\'{I}RAL}{PARTICLE AND KINETIC MODELING IN HIGH DIMENSION}
\section{Introduction}\label{Sec:Introduction}
The dimensionality scale of problems arising in our modern information society has become very large and finding appropriate methods for dealing with them is one of the great challenges of today's numerical simulation. The most notable recent advances in data analysis are based on the observation that in many situations, even for very complex phenomena, the intrinsic dimensionality of the data is significantly lower than the ambient dimension. Remarkable progresses have been made in data compression, processing, and acquisition. We mention, for instance, the use of {\it diffusion maps} for data clouds and graphs in high dimension \cite{BN01,BN03,CLLMNWZ05a,CLLMNWZ05b,CL06,KMFK} in order to define low-dimensional local representations of data with small distance distortion, and meaningful automatic clustering properties. In this setting the embedding of data is performed by a {\it highly nonlinear} procedure, obtained by computing the eigenfunctions of suitable normalized diffusion kernels, measuring the probability of transition from one data point to another over the graph.
Quasi-isometrical {\it linear} embeddings of high-dimensional point clouds into low-dimen\-sional spaces of parameters are provided by the well-known Johnson-Lindenstrauss Lemma \cite{A,DG,JL}: any cloud of $\mathcal N$ points in $\mathbb R^d$ can be embedded by a random linear projection $M$ nearly isometrically into $\mathbb R^k$ with $k = \mathcal O(\varepsilon^{-2} \log(\mathcal N))$ (a precise statement will be given below). This embedding strategy is simpler than the use of diffusion maps, as it is linear, however it is ``blind'' to the specific geometry and local dimensionality of the data, as the embedding dimension $k$ depends exclusively on the number of points in the cloud. In many applications, this is sufficient, as the number of points $\mathcal N$ is supposed to be a power of the dimension $d$, and the embedding produces an effective reduction to $k = \mathcal O(\varepsilon^{-2} \log(\mathcal N)) = \mathcal O(\varepsilon^{-2} \log(d))$ dimensions. As clarified in \cite{BDDW,KW}, the Johnson-Lindenstrauss Lemma is also at the basis of the possibility of performing optimal compressed and nonadaptive acquisition of high-dimensional data. In {\it compressed sensing} \cite{carota06,do06-2} a vector $x \in \mathbb R^d$ is encoded in a vector $y\in\mathbb R^k$ by applying a random projection $M$, which is modeling a linear acquisition device with random sensors, i.e., $y = Mx$. From $y$ it is possible to decode $x$ approximately (see Theorem \ref{thmStrongRIP} below) by solving the convex optimization problem $$
x^{\#}=\arg \min_{M z =y} \left ( \| z\|_{\ell_1^d} := \sum_{i=1}^d |z_i| \right), $$ with the error distortion $$
\| x^{\#} -x\|_{\ell_1^d} \leq C \sigma_K(x)_{\ell_1^d}, $$
where $\sigma_K(x)_{\ell_1^d} = \inf_{z: \#{\rm supp\, }(z) \leq K} \|z -x\|_{\ell_1^d}$ and $K = \mathcal O(k/(\log(d/k)+1))$. We denote $\Sigma_K=\{z \in \mathbb R^d: \#{\rm supp\, }(z) \leq K\}$ the set of $K$-sparse vectors, i.e., the union of $K$-dimensional coordinate subspaces in $\mathbb R^d$. In particular, if $x \in \Sigma_K$, then $x^{\#} =x$. Hence, not only is $M$ a Johnson-Lindenstrauss embedding, quasi-isometrical on point clouds and $K$-dimensional coordinate subspaces, but also allows for the recovery of the most relevant components of high-dimensional vectors, from low-dimensional encoded information. A recent work \cite{BW09,W08} extends the quasi-isometrical properties of the Johnson-Lindenstrauss embedding from point clouds and $K$-dimensional coordinate subspaces to smooth compact Riemannian manifolds with bounded curvature. Inspired by this work, in \cite{IM11} the authors extend the principles of compressed sensing in terms of point recovery on smooth compact Riemannian manifolds.
Besides these relevant results in compressing and coding-decoding high-dimensional ``stationary'' data, dimensionality reduction of complex dynamical systems and high-dimensional partial differential equations is a subject of recent intensive research. Several tools have been employed, for instance, the use of diffusion maps for dynamical systems \cite{NLK06}, tensor product bases and sparse grids for the numerical solution of linear high-dimensional PDEs \cite{DSS,BG,GK,GO}, the reduced basis method for solving high-dimensional parametric PDEs \cite{BCDDPW,BMPPTxx,MPT02,RHP08,S08,VPRP03}. \\ In this paper we shall further explore the connection between data compression and {\it tractable} numerical simulation of dynamical systems. Eventually we address the solutions of associated high-dimensional kinetic equations. We are specially interested in dynamical systems of the type \begin{equation}\label{eq:dyn} \dot x_i(t)=f_i(\mathcal{D} x(t))+\sum_{j=1}^N f_{ij}(\mathcal{D} x(t))x_j(t), \end{equation} where we use the following notation: \begin{itemize} \item $N\in \mathbb{N}$ - number of agents, \item $x(t)=(x_1(t),\dots,x_N(t))\in\mathbb{R}^{d\times N}$, where $x_i:[0,T]\to \mathbb{R}^d$, $i=1,\dots,N$, \item $f_i:\mathbb{R}^{N\times N}\to \mathbb{R}^d,\quad i=1,\dots,N,$ \item $f_{ij}:\mathbb{R}^{N\times N}\to \mathbb{R}, \quad i,j=1,\dots,N$,
\item $\mathcal{D}:\mathbb{R}^{d\times N}\to \mathbb{R}^{N\times N}$, $\mathcal{D} x:=(\|x_i-x_j\|_{\ell_2^d})_{i,j=1}^N$ is the {\it adjacency matrix} of the point cloud $x$. \end{itemize} We shall assume that the governing functions $f_i$ and $f_{ij}$ are Lipschitz, but we shall specify the details later on. The system \eqref{eq:dyn} describes the dynamics of multiple complex agents $x(t)=(x_1(t),\dots,x_N(t))\in\mathbb{R}^{d\times N}$, interacting on the basis of their mutual ``social'' distance $\mathcal{D} x(t)$, and its general form includes several models for swarming and collective motion of animals and micro-organisms, aggregation of cells, etc. Several relevant effects can be included in the model by means of the functions $f_i$ and $f_{ij}$, in particular, fundamental binary mechanisms of {\it attraction, repulsion, aggregation, self-drive}, and {\it alignment} \cite{CCR,CFTV,CS1,CS2,DOrsogna,KS}. Moreover, possibly adding stochastic terms of random noise may also allow to consider {\it diffusion} effects \cite{BCC,CFTV}. However, these models and motion mechanisms are mostly derived borrowing a leaf from physics, by assuming the agents (animals, micro-organisms, cells etc.) as pointlike and exclusively determined by their spatial position and velocity in $\mathbb R^{d}$ for $d=3+3$. In case we wished to extend such models of social interaction to more ``sophisticated'' agents, described by many parameters ($d \gg 3 + 3$), the simulation may become computationally prohibitive. Our motivation for considering high-dimensional situations stems from the modern development of communication technology and Internet, for which we witness the development of larger and larger communities accessing information (interactive databases), services (financial market), social interactions (social networks) etc. For instance, we might be interested to simulate the behavior of certain subsets of the financial market where the agents are many investors, who are characterized by their portfolios of several hundreds of investments. The behavior of each individual investor depends on the dynamics of others according to a suitable social distance determined by similar investments. Being able to produce meaningful simulations and learning processes of such complex dynamics is an issue, which might be challenged by using suitable compression/dimensionality reduction techniques.\\ The idea we develop in this paper is to randomly project the system and its initial condition by Johnson-Lindenstrauss embeddings to a lower-dimensional space where an independent simulation can be performed with significantly reduced complexity. We shall show that the use of multiple projections and parallel computations allows for an approximate reconstruction of the high-dimensional dynamics, by means of compressed sensing techniques. After we explore the tractable simulation of the dynamical systems \eqref{eq:dyn} when the dimension $d$ of the parameter space is large, we also address the issue of whether we can perform tractable simulations when the number $N$ of agents is getting very large. Unlike the control of a finite number of agents, the numerical simulation of a rather large population of interacting agents ($N \gg 0$) can constitute a serious difficulty which stems from the accurate solution of a possibly very large system of ODEs. Borrowing the strategy from the kinetic theory of gases \cite{cip}, we want instead to consider a density distribution of agents, depending on their $d$-parameters, which interact with stochastic influence (corresponding to classical collisional rules in kinetic theory of gases) -- in this case the influence is ``smeared'' since two individuals may interact also when they are far apart in terms of their ``social distance'' $\mathcal{D} x$. Hence, instead of simulating the behavior of each individual agent, we shall describe the collective behavior encoded by a density distribution $\mu$, whose evolution is governed by one sole mesoscopic partial differential equation. We shall show that, under realistic assumptions on the concentration of the measure $\mu$ on sets of lower dimension, we can also acquire information on the properties of the high-dimensional measure solution $\mu$ of the corresponding kinetic equation, by considering random projections to lower dimension. Such approximation properties are determined by means of the combination of optimal numerical integration principles for the high-dimensional measure $\mu$ \cite{GL,Gruber} and the results previously achieved for particle dynamical systems.
\subsection{Fundamental assumptions}
We introduce the following notation for $\ell_p$-norms of vectors $v \in \mathbb R^d$, $$
\| v\|_{\ell_p^d} := \left( \sum_{i=1}^d |v_i|^p \right )^{1/p}\qquad \mbox{for } 1\leq p < \infty, $$ and $$
\| v\|_{\ell_\infty^d}:=\max_{i=1,\dots,d} |v_i|. $$ For matrices $x \in \mathbb R^{n \times m}$ we consider the mixed norm $$
\|x\|_{\ell_p^m(\ell_q^n)} := \| (\|x_i\|_{\ell_p^n})_{i=1}^m \|_{\ell_q^m}, $$ where $x_i \in \mathbb R^n$ is the $i^{th}$-column of the matrix $x$.
For the rest of the paper we impose three fundamental assumptions about Lipschitz and boundedness properties of $f_i$ and $f_{ij}$, \begin{align}
\label{eq:condf1} |f_i(a)-f_i(b)|&\le L\|a-b\|_{\ell_\infty^N(\ell_\infty^N)}, \quad i=1,\dots,N\\
\label{eq:condf2}\max_{i=1,\dots,N}\sum_{j=1}^N|f_{ij}(a)|&\le L',\\
\label{eq:condf3}\max_{i=1,\dots,N}\sum_{j=1}^N|f_{ij}(a)-f_{ij}(b)|&\le L''\|a-b\|_{\ell_\infty^N(\ell_\infty^N)}, \end{align} for every $a,b\in\mathbb{R}^{N\times N}$. Unfortunately, models of real-life phenomena would not always satisfy these conditions, for instance models of financial markets or socio-economic interactions can be expected to exhibit severely discontinuous behavior. However, these assumptions are reasonable in certain regimes and allow us to prove the concept we are going to convey in this paper, i.e., the possibility of simulating high-dimensional dynamics by multiple independent simulations in low dimension.
\subsection{Euler scheme, a classical result of stability and convergence, and its complexity} We shall consider the system of ordinary differential equations of the form \eqref{eq:dyn} with the initial condition \begin{equation}\label{eq:dyn2} x_i(0) = x_i^0 \,,\qquad i=1,\dots,N \,. \end{equation} The Euler method for this system is given by \eqref{eq:dyn2} and \begin{equation}\label{eq:eul1} x_i^{n+1}:=x_i^n+h\left[f_i(\mathcal{D} x^n)+\sum_{j=1}^N f_{ij}(\mathcal{D} x^n)x^n_j\right],\quad n=0,\dots,n_0-1. \end{equation} where $h>0$ is the time step and $n_0:=T/h$ is the number of iterations. We consider here the {\it explicit} Euler scheme exclusively for the sake of simplicity, for more sophisticated integration methods might be used.
The simulation of the dynamical system \eqref{eq:eul1} has a complexity which is at least the one of computing the {\it adjacency matrix} $\mathcal{D} \tilde x^n$ at each discrete time $t^n$, i.e., $\mathcal O( d \times N^2)$. The scope of the next sections is to show that, up to an $\varepsilon$-distortion, we can approximate the dynamics of \eqref{eq:dyn} by projecting the system into lower dimension and by executing in parallel computations with reduced complexity. Computation of the adjacency matrix in the new dimension requires only $\mathcal O( \varepsilon^{-2} \log (N)\times N^2)$ operations. Especially if the distortion parameter $\varepsilon>0$ is not too small and the number of agents is of a polynomial order in $d$, we reduce the complexity of computing the adjacency matrix to $\mathcal O(\log (d)\times N^2)$.
\section{Projecting the Euler method: dimensionality reduction of discrete dynamical systems}\label{Sec:Euler}
\subsection{Johnson-Lindenstrauss embedding}
We wish to project the dynamics of \eqref{eq:dyn} into a lower-dimensional space by employing a well-known result of Johnson and Lindenstrauss \cite{JL}, which we informally rephrase for our purposes as follows.
\begin{lemma}[Johnson and Lindenstrauss] \label{JLlem} Let $\mathcal P$ be an arbitrary set of $\mathcal N$ points in $\mathbb{R}^d$. Given a distortion parameter $\varepsilon>0$, there exists a constant $$k_0= \mathcal O(\varepsilon^{-2}\log (\mathcal N)),$$ such that for all integers $k\ge k_0$, there exists a $k\times d$ matrix $M$ for which \begin{equation}\label{eq:JLlem}
(1-\varepsilon)\|x-\tilde x\|_{\ell_2^d}^2\le \|Mx - M\tilde x\|_{\ell_2^k}^2\le (1+\varepsilon)\|x-\tilde x\|_{\ell_2^d}^2, \end{equation}
for all $x,\tilde x\in \mathcal P$. \end{lemma} It is easy to see that the condition \begin{equation}\label{eq:RIP1}
(1-\varepsilon)\|p\|_{\ell_2^d}^2\le \|M p\|_{\ell_2^k}^2\le (1+\varepsilon)\|p\|_{\ell_2^d}^2, \quad p \in \mathbb R^d, \end{equation} implies \begin{equation}\label{eq:RIP2}
(1-\varepsilon)\|p\|_{\ell_2^d} \le \|M p\|_{\ell_2^k} \le (1+\varepsilon)\|p\|_{\ell_2^d}, \quad p \in \mathbb R^d, \end{equation} for $0<\varepsilon<1$, which will be used in the following sections. On the other hand, \eqref{eq:RIP2} implies \eqref{eq:RIP1} with $3\varepsilon$ instead of $\varepsilon$.
Our aim is to apply this lemma to dynamical systems. As the mapping $M$ from Lemma \ref{JLlem} is linear and almost preserves distances between the points (up to the $\varepsilon>0$ distortion as described above), we restrict ourselves to dynamical systems which are quasi-linear or whose non-linearity depends only on the mutual distances of the points involved, as in \eqref{eq:dyn}.
Let us define the additional notation, which is going to be fixed throughout the paper:
\begin{itemize} \item $d\in \mathbb{N}$ - dimension (large), \item $\varepsilon>0$ - the distortion parameter from Lemma \ref{JLlem}, \item $k\in \mathbb{N}$ - new dimension (small), \item $M\in \mathbb{R}^{k\times d}$ - randomly generated matrix as described below. \end{itemize}
The only constructions of a matrix $M$ as in Lemma \ref{JLlem} known up to now are stochastic, i.e., the matrix is randomly generated and has the quasi-isometry property \eqref{eq:JLlem} with high probability. We refer the reader to \cite{DG} and \cite[Theorem 1.1]{A} for two typical versions of the Johnson-Lindenstrauss Lemma.
We briefly collect below some well-known instances of random matrices, which satisfy the statement of Lemma \ref{JLlem} with high probability: \begin{itemize} \item $k \times d$ matrices $M$ whose entries $m_{i,j}$ are independent realizations of Gaussian random variables $$ m_{i,j} \sim \mathcal N\left (0, \frac{1}{k} \right ); $$ \item $k \times d$ matrices $M$ whose entries are independent realizations of $\pm$ Bernoulli random variables $$ m_{i,j}:=\left \{ \begin{array}{ll} +\frac{1}{\sqrt k},& \text{ with probability } \frac{1}{2}\\ -\frac{1}{\sqrt k},& \text{ with probability } \frac{1}{2} \end{array} \right . $$
\end{itemize}
Several other random projections suitable for Johnson-Lindenstrauss embeddings can be constructed following Theorem \ref{KWthm} recalled below, and we refer the reader to \cite{KW} for more details.
\subsection{Uniform estimate for a general model}
If $M\in \mathbb{R}^{k\times d}$ is a matrix, we consider the projected Euler method in $\mathbb{R}^k$ associated to the high-dimensional system \eqref{eq:dyn2}-\eqref{eq:eul1}, namely \begin{align} \label{eq:eul2}y_i^0&:=Mx_i^0,\\ \label{eq:eul3}y_i^{n+1}&:=y_i^n+h\left[Mf_i(\mathcal{D}' y^n)+\sum_{j=1}^N f_{ij}(\mathcal{D}' y^n)y^n_j\right],\quad n=0,\dots,n_0-1. \end{align}
We denote here $\mathcal{D}':\mathbb{R}^{k\times N}\to \mathbb{R}^{N\times N}$, $\mathcal{D}' y:=(\|y_i-y_j\|_{\ell_2^k})_{i,j=1}^N$, the {\it adjacency matrix} of the agents $y=(y_1,\dots,y_N)$ in $\mathbb R^{k \times N}$. The first result of this paper reads as follows. \begin{theorem}\label{thm1} Let the sequences $$ \{x_i^n, i=1,\dots,N \ \text{and}\ n=0,\dots,n_0\} \quad \text{and}\quad \{y_i^n, i=1,\dots,N \ \text{and}\ n=0,\dots,n_0\} $$ be defined by \eqref{eq:dyn2}-\eqref{eq:eul1} and \eqref{eq:eul2}-\eqref{eq:eul3} with $f_i$ and $f_{ij}$ satisfying \eqref{eq:condf1}--\eqref{eq:condf3} and a matrix $M\in\mathbb{R}^{k\times d}$ with \begin{gather}
\label{eq:condM1}\left\|Mf_i(\mathcal{D}' y^n)-M f_i(\mathcal{D} x^n)\right\|_{\ell_2^k}\le (1+\varepsilon) \left\|f_i(\mathcal{D}' y^n)-f_i(\mathcal{D} x^n)\right\|_{\ell_2^d},\\
\label{eq:condM2}\|Mx_j^n\|_{\ell_2^k}\le (1+\varepsilon) \|x_j^n\|_{\ell_2^d},\\
\label{eq:condM3}(1-\varepsilon) \|x_i^n-x_j^n\|_{\ell_2^d}\le \|Mx_i^n-Mx_j^n\|_{\ell_2^k}\le (1+\varepsilon) \|x_i^n-x_j^n\|_{\ell_2^d} \end{gather} for all $i,j=1,\dots, N$ and all $n=0,\dots,n_0$. Moreover, let us assume that \begin{align}
\alpha&\ge \max_j\|x^n_j\|_{\ell_2^d}\quad \text{for all}\quad n=0,\dots,n_0,\quad j=1,\dots,N. \label{alpha1} \end{align} Let \begin{equation}\label{eq:defe}
e_i^n:=\|y_i^n-Mx_i^n\|_{\ell_2^k},\ i=1,\dots,N \ \text{and}\ n=0,\dots,n_0 \end{equation} and set $\mathcal{E}^n:=\max_ie_i^n$. Then \begin{equation}\label{eq:unifstab} {\mathcal E}^n\le \varepsilon hn B \exp(hnA), \end{equation} where $A:=L'+2(1+\varepsilon)(L+\alpha L'')$ and $B:=2\alpha(1+\varepsilon)(L+\alpha L'')$. \end{theorem}
We remark that conditions \eqref{eq:condM1}-\eqref{eq:condM3} are in fact satisfied as soon as $M$ is a suitable Johnson-Lindenstrauss embedding as in Lemma \ref{JLlem}, for the choice $\mathcal N = 2 N n_0$ and $k=\mathcal O(\varepsilon^{-2} \log(\mathcal N))$.
\begin{proof} Using \eqref{eq:defe} and \eqref{eq:dyn2}-\eqref{eq:eul1} and \eqref{eq:eul2}-\eqref{eq:eul3} combined with \eqref{eq:condM1} and \eqref{eq:condM2}, we obtain \begin{align*}
e_i^{n+1}&\le e_i^n+h\left \|Mf_i(\mathcal{D}' y^n)-M f_i(\mathcal{D} x^n)\right\|_{\ell_2^k}
+h\left\|\sum_{j=1}^N f_{ij}(\mathcal{D}' y^n)y_j^n-f_{ij}(\mathcal{D} x^n)Mx_j^n\right\|_{\ell_2^k}\\
&\le e_i^{n}+h(1+\varepsilon)\left\|f_i(\mathcal{D}' y^n)-f_i(\mathcal{D} x^n)\right\|_{\ell_2^d}\\
&\quad +h\sum_{j=1}^N \Bigl(\|f_{ij}(\mathcal{D}' y^n)y_j^n-f_{ij}(\mathcal{D}' y^n)Mx_j^n\|_{\ell_2^k}+\|f_{ij}(\mathcal{D}' y^n)Mx_j^n-f_{ij}(\mathcal{D} x^n)Mx_j^n\|_{\ell_2^k}\Bigr)\\
&\le e_i^{n}+h(1+\varepsilon)\left\|f_i(\mathcal{D}' y^n)-f_i(\mathcal{D} x^n)\right\|_{\ell_2^d}\\
&\quad +h\sum_{j=1}^N \Bigl(|f_{ij}(\mathcal{D}' y^n)|e_j^n+(1+\varepsilon)\|x_j^n\|_{\ell_2^d}\cdot|f_{ij}(\mathcal{D}' y^n)-f_{ij}(\mathcal{D} x^n)|\Bigr). \end{align*}
Taking the maximum on both sides, this becomes \begin{align*}
\mathcal{E}^{n+1}&\le \mathcal{E}^n +h(1+\varepsilon) \max_i \|f_i(\mathcal{D}' y^n)-f_i(\mathcal{D} x^n)\|_{\ell_2^d}\\
&\quad +h\mathcal{E}^n\max_i\sum_{j=1}^N |f_{ij}(\mathcal{D}' y^n)|+h(1+\varepsilon)\alpha \cdot \max_i \sum_{j=1}^N|f_{ij}(\mathcal{D}' y^n)-f_{ij}(\mathcal{D} x^n)|. \end{align*} We use \eqref{eq:condf1}--\eqref{eq:condf3} for $a=\mathcal{D}'y^n$ and $b=\mathcal{D} x^n$ to estimate all the terms on the right-hand side. This gives \begin{align*}
\mathcal{E}^{n+1}&\le \mathcal{E}^n +h(1+\varepsilon) L \|\mathcal{D}' y^n- \mathcal{D} x^n\|_{\ell_\infty^N(\ell_\infty^N)}+h\mathcal{E}^nL'+h(1+\varepsilon)\alpha L''\|\mathcal{D}' y^n- \mathcal{D} x^n\|_{\ell_\infty^N(\ell_\infty^N)}\\
&\le \mathcal{E}^n(1+hL')+h(1+\varepsilon)(L+\alpha L'')\left[\|\mathcal{D}' y^n-\mathcal{D}' Mx^n\|_{\ell_\infty^N(\ell_\infty^N)} + \|\mathcal{D}' M x^n-\mathcal{D} x^n\|_{\ell_\infty^N(\ell_\infty^N)}\right]\\ &\le \mathcal{E}^n(1+hL')+2h(1+\varepsilon)(L+\alpha L'')(\mathcal{E}^n+\alpha\varepsilon), \end{align*} where we used \eqref{eq:condM3} in the last line. This, together with ${\mathcal E}^0=0$, leads to $$ {\mathcal E}^n\le \varepsilon hn B \exp(hnA), $$ where $A:=L'+2(1+\varepsilon)(L+\alpha L'')$ and $B:=2\alpha(1+\varepsilon)(L+\alpha L'')$. \end{proof}
\subsection{Uniform estimate for the Cucker-Smale model}\label{Subs:UniformCS} As a relevant example, let us now show that Theorem \ref{thm1} can be applied to the well-known Cucker-Smale model, introduced and analyzed in \cite{CS1,CS2}, which is described by \begin{align} \dot x_i&=v_i \in \mathbb R^d, \label{CS1} \\
\dot v_i&=\frac{1}{N}\sum_{j=1}^N g(\|x_i-x_j\|_{\ell_2^d})(v_j-v_i), \quad i=1,\dots,N. \label{CS2} \end{align} The function $g:[0,\infty)\to \mathbb{R}$ is given by $g(s)=\frac{G}{(1+s^2)^{\beta}}$, for $\beta>0$, and bounded by $g(0)=G>0.$ This model describes the {\it emerging of consensus} in a group of interacting agents, trying to {\it align} (also in terms of abstract consensus) with their neighbors. One of the motivations of the model from Cucker and Smale was to describe the formation and evolution of languages \cite[Section 6]{CS2}, although, due to its simplicity, it has been eventually related mainly to the description of the {\it emergence of flocking} in groups of birds \cite{CS1}. In the latter case, in fact, spatial and velocity coordinates are sufficient to describe a pointlike agent ($d=3+3$), while for the evolution of languages, one would have to take into account a much broader dictionary of parameters, hence a higher dimension $d\gg 3+3$ of parameters, which is in fact the case of our interest in the present paper.
Let us show that the model is indeed of the type \eqref{eq:dyn}. We interprete the system as a group of $2N$ agents in $\mathbb{R}^d$, whose dynamics is given by the following equations \begin{align*} \dot x_i&=\sum_{j=1}^N f^{x}_{ij}v_j \in \mathbb R^d, \\ \dot v_i&=\sum_{j=1}^N f^{v}_{ij}(\mathcal{D} x)v_j, \quad i=1,\dots,N \end{align*}
with $f^{x}_{ij}:=\delta_{ij}$, $\displaystyle f^v_{ii}(\mathcal{D} x):=-\frac{1}{N}\sum_{k=1}^N g(\|x_{i}-x_{k}\|_{\ell_2^d})$, and $\displaystyle f^v_{ij}(\mathcal{D} x):=\frac{1}{N} g(\|x_{i}-x_{j}\|_{\ell_2^d})$, for $i\neq j$.
The condition \eqref{eq:condf1} is empty, \eqref{eq:condf2} reads $$
L'\ge\max(1,2G)\ge \max_i\left\{1,\frac{2}{N}\sum_{k=1}^Ng(\|x^n_{i}-x^n_{k}\|_{\ell_2^d})\right\}. $$ Finally, \begin{align*}
\max_i\frac{2}{N}\sum_{j=1}^N &\Bigl|g(\|x^n_{i}-x^n_{j}\|_{\ell_2^d})-g(\|y^n_{i}-y^n_{j}\|_{\ell_2^k})\Bigr|\\
&\le \max_i\frac{2\|g\|_{{\rm Lip}}}{N}\cdot \sum_{j=1}^N \Bigl| \|x^n_{i}-x^n_{j}\|_{\ell_2^d}-\|y^n_{i}-y^n_{j}\|_{\ell_2^k}\Bigr|\\
&\le 2\|g\|_{{\rm Lip}}\cdot \|\mathcal{D}' y^n-\mathcal{D} x^n\|_{\ell_\infty^N(\ell_\infty^N)} \end{align*}
shows that $L''\le 2\|g\|_{{\rm Lip}}.$ The boundedness of the trajectories in the phase-space of \eqref{CS1}-\eqref{CS2} at finite time has been proved, for instance, in \cite{HL}, see also \cite[Theorem 4.6]{CCR}. The boundedness at finite time is clearly sufficient to define the constant $\alpha$ appearing in Theorem \ref{thm1}, also because we are mainly interested in the dynamics for short time, due to the error propagation. Of course the constant $\alpha$ might grow with time, but, for instance, for the Cucker-Smale system it grows at most linearly in time \cite{CFTV}; as in the error estimate \eqref{eq:unifstab} we have an exponential function in time appearing, the possible linear growth can be considered a negligible issue; moreover, as our numerical experiments show, see Section \ref{Sec:Numerics}, the situation is much better in practice, and suitable scaling, as indicated below, allows us to assume in several circumstances that the constant $\alpha$ is uniformly bounded for all times. In fact, even when we were interested in longer time or even asymptotical behavior, especially when pattern formation is expected, then we would observe the following additional facts: In the Cucker-Smale model the center of mass and the mean velocity are invariants of the dynamics. Moreover the rate of communication between particles is given by $g(s)=\frac{G}{(1+s^2)^{\beta}}$. When $\beta \leq 1/2$ it is know (see \cite{CFTV}) that the dynamics will converge to a flocking configuration. In this case one can translate at the very beginning the center of mass and the mean velocity to $0$, and the system will keep bounded for all times. Hence in this case the constant $\alpha$ can also be considered uniform for all times (not only bounded at finite time).
\subsection{Least-squares estimate of the error for the Cucker-Smale model} The formula \eqref{eq:unifstab} provides the estimate of the maximum of the individual errors,
i.e., $\mathcal E^n:=\| (y^n_i - M x^n_i)_{i=1}^N \|_{\ell_\infty^N(\ell_2^k)}$. In this section we address the stronger $\ell_2^N(\ell_2^k)$-estimate for the error. For generic dynamical systems \eqref{eq:dyn} such estimate is not available in general, and one has to perform a case-by-case analysis. As a typical example of how to proceed, we restrict ourselves to the Cucker-Smale model, just recalled in the previous section. The forward Euler discretization of~\eqref{CS1}--\eqref{CS2} is given by \begin{align} \label{eq:l2:1} x^{n+1}_i&=x^n_i+h v^n_i,\\
\notag v^{n+1}_i&=v^n_i+\frac{h}{N}\sum_{j=1}^N g(\|x^n_i-x^n_j\|_{\ell_2^d})(v^n_j-v^n_i) \end{align} with initial data $x_i^0$ and $v_i^0$ given. Let $M$ be again a suitable random matrix in the sense of Lemma \ref{JLlem}. The Euler method of the projected system is given by the initial conditions $y^0_i=Mx^0_i$ and $w^0_i=Mv^0_i$ and the formulas \begin{align} \label{eq:l2:2}y^{n+1}_i&=y^n_i+hw^n_i,\\
\notag w^{n+1}_i&=w^n_i+\frac{h}{N}\sum_{j=1}^N g(\|y^n_i-y^n_j\|_{\ell_2^k})(w^n_j-w^n_i). \end{align}
We are interested in the estimates of the following quantities \begin{align}
\label{eq:l2:e1}e_{x,i}^n&:=\|y_i^n-Mx_i^n\|_{\ell_2^k},\quad \mathcal{E}_{x}^n:=\sqrt{\frac{1}{N}\sum_{i=1}^N (e_{x,i}^n)^2}=
\frac{\|(y_i^n-Mx_i^n)_{i=1}^N\|_{\ell_2^N(\ell_2^k)}}{\sqrt N},\\
\label{eq:l2:e2}e_{v,i}^n&:=\|w_i^n-Mv_i^n\|_{\ell_2^k},\quad \mathcal{E}_{v}^n:=\sqrt{\frac{1}{N}\sum_{i=1}^N (e_{v,i}^n)^2}=
\frac{\|(w_i^n-Mv_i^n)_{i=1}^N\|_{\ell_2^N(\ell_2^k)}}{\sqrt N}. \end{align}
\begin{theorem} Let the sequences $\{x_i^n\}, \{v_i^n\}, \{y_i^n\}$, $\{w_i^n\}$, $\{e^n_{x,i}\}$ and $\{e^n_{v,i}\}$, $i=1,\dots,N$ and $n=1,\dots,n_0$ be given by \eqref{eq:l2:1}, \eqref{eq:l2:2}, \eqref{eq:l2:e1} and \eqref{eq:l2:e2}, respectively. Let $\varepsilon>0$ and let us assume, that the matrix $M$ satisfies \begin{align*}
(1-\varepsilon)\|x_i^n-x_j^n\|_{\ell_2^d}\le \|Mx_i^n-Mx_j^n\|_{\ell_2^k} &\le (1+\varepsilon) \|x_i^n-x_j^n\|_{\ell_2^d}\quad\text{and}\\
(1-\varepsilon) \|v_i^n-v_j^n\|_{\ell_2^d}\le \|Mv_i^n-Mv_j^n\|_{\ell_2^k} &\le (1+\varepsilon) \|v_i^n-v_j^n\|_{\ell_2^d} \end{align*} for all $i,j=1,\dots,N$ and $n=0,\dots,n_0$.
Then the error quantities $\mathcal{E}_{x}^n$ and $\mathcal{E}_{y}^n$ introduced in \eqref{eq:l2:e1} and \eqref{eq:l2:e2} satisfy \begin{equation}\label{eq:leastsqstab} \sqrt{({\mathcal E_x^{n}})^2+({\mathcal E_v^{n}})^2}
\le \varepsilon(1+\varepsilon)hn\|g\|_{\rm Lip}VX\exp(hn\|{\mathcal A}\|), \end{equation}
where $V:=\max_{i,j,n}\|v^n_i-v^n_j\|_{\ell_2^d}$, $X:=\max_{i,j,n}\|x^n_i-x^n_j\|_{\ell_2^d}$ and $$
{\mathcal A}=\left(\begin{matrix}0&1\\2(1+\varepsilon)\|g\|_{\rm Lip}V&2G\end{matrix}\right). $$
\end{theorem}
\begin{proof} Using \eqref{eq:l2:1} and \eqref{eq:l2:2}, we obtain $$ e_{x,i}^{n+1}\le e_{x,i}^n+he_{v,i}^n\quad \text{and}\quad \mathcal{E}_x^{n+1}\le \mathcal{E}_x^n+h\mathcal{E}_v^n. $$
To bound the quantity $\mathcal{E}^{n}_v$ we have to work more. We add and subtract the term $g(\|y_i^n-y_j^n\|_{\ell_2^k})(Mv_j^n-Mv_i^n)$ and apply \eqref{eq:l2:1} and \eqref{eq:l2:2}. This leads to \begin{align}
\notag e_{v,i}^{n+1}&\le e_{v,i}^n+\frac{h}{N}\sum_{j=1}^N \Bigl(\|g(\|y_i^n-y_j^n\|_{\ell_2^k})(w_j^n-w_i^n)\pm g(\|y_i^n-y_j^n\|_{\ell_2^k})(Mv_j^n-Mv_i^n)\\
\notag &\qquad-g(\|x_i^n-x_j^n\|_{\ell_2^d})(Mv_j^n-Mv_i^n)\|_{\ell_2^k}\Bigr)\\
\label{eq:l2:3}&\le e_{v,i}^n+\frac{h}{N}\sum_{j=1}^N g(\|y_i^n-y_j^n\|_{\ell_2^k})(e_{v,j}^n+e_{v,i}^n)\\
\notag &\qquad+\frac{(1+\varepsilon)h\|g\|_{\rm Lip}}{N}\cdot\sum_{j=1}^N\|v_j^n-v_i^n\|_{\ell_2^d}\cdot\bigl | \|x_i^n-x_j^n\|_{\ell_2^d}-\|y_i^n-y_j^n\|_{\ell_2^k}\bigr|. \end{align} We estimate the first summand in \eqref{eq:l2:3} $$
\frac{h}{N}\sum_{j=1}^N g(\|y_i^n-y_j^n\|_{\ell_2^k})(e_{v,j}^n+e_{v,i}^n)\le \frac{hG}{N}\bigl[Ne_{v,i}^n+\sum_{j=1}^Ne_{v,j}^n\bigr] =hG e_{v,i}^n+\frac{hG}{N}\sum_{j=1}^Ne_{v,j}^n $$ and its $\ell_2$-norm with respect to $i$ by H\"older's inequality \begin{equation}\label{eq:l2:4} h\sqrt{N}G\mathcal{E}_v^n+\frac{hG}{N}\Biggl(\sum_{i=1}^N\biggl(\sum_{j=1}^Ne_{v,j}^n\biggr)^2\Biggr)^{1/2} \le 2h\sqrt{N}G\mathcal{E}_v^n. \end{equation}
To estimate the second summand in \eqref{eq:l2:3} we make use of \begin{align*}
\bigl| \|x_i^n&-x_j^n\|_{\ell_2^d}-\|y_i^n-y_j^n\|_{\ell_2^k}\bigr|\\
&\le \bigl| \|x_i^n-x_j^n\|_{\ell_2^d}-\|Mx_i^n-Mx_j^n\|_{\ell_2^k}\bigr| + \bigl |\|Mx_i^n-Mx_j^n\|_{\ell_2^k}-\|y_i^n-y_j^n\|_{\ell_2^k}\bigr|\\
&\le \varepsilon \|x_i^n-x_j^n\|_{\ell_2^d}+e_{x,i}^n+e_{x,j}^n. \end{align*} We arrive at \begin{align*}
&\frac{ (1+\varepsilon)h \|g\|_{\rm Lip}}{N}\sum_{j=1}^N \|v_j^n-v_i^n\|_{\ell_2^d}(\varepsilon \|x_i^n-x_j^n\|_{\ell_2^d}+e_{x,i}^n+e_{x,j}^n)\\
&\qquad\le \frac{(1+\varepsilon)h \|g\|_{\rm Lip}V}{N}
\biggl\{\varepsilon \sum_{j=1}^N\|x_i^n-x_j^n\|_{\ell_2^d} + N e_{x,i}^n + \sum_{j=1}^N e_{x,j}^n\biggr\}. \end{align*} The $\ell_2$-norm of this expression with respect to $i$ is bounded by \begin{align}
\notag &\frac{(1+\varepsilon)h \|g\|_{\rm Lip}V}{N}\left\{\varepsilon\Bigl(\sum_{i=1}^N \Bigl(\sum_{j=1}^N\|x_i^n-x_j^n\|_{\ell_2^d}\Bigr)^2\Bigr)^{1/2} +N\Bigl(\sum_{i=1}^N (e_{x,i}^n)^2\Bigr)^{1/2} + \sqrt{N}\sum_{j=1}^N e_{x,j}^n\right\}\\
\label{eq:l2:5}&\le (1+\varepsilon)h \|g\|_{\rm Lip}V\sqrt N(\varepsilon X+2\mathcal{E}_x^n). \end{align} Combining \eqref{eq:l2:3} with \eqref{eq:l2:4} and \eqref{eq:l2:5} leads to the recursive estimate \begin{align} \label{eq:finest}\mathcal{E}_x^{n+1}&\le \mathcal{E}_x^n+h\mathcal{E}_v^n,\\
\notag\mathcal{E}_v^{n+1}&\le \mathcal{E}_v^n+2hG\mathcal{E}_v^n+ h (1+\varepsilon) \|g\|_{\rm Lip}V\left\{\varepsilon X+2\mathcal{E}_x^n\right\}, \end{align} which we put into the matrix form \begin{equation}\label{eq:l2:6} \left(\begin{matrix}\mathcal{E}_x^{n+1}\\ \mathcal{E}_v^{n+1}\end{matrix}\right) \le{\mathcal A'}\left(\begin{matrix}\mathcal{E}_x^{n}\\\mathcal{E}_v^{n}\end{matrix}\right)
+\left(\begin{matrix}0\\(1+\varepsilon)\varepsilon h\|g\|_{\rm Lip}V X\end{matrix}\right), \end{equation} where ${\mathcal A'}$ is a $2\times 2$ matrix given by $$ {\mathcal A'}={\mathcal Id}+h{\mathcal A}:=\left(\begin{matrix}1&0\\0&1\end{matrix}\right)
+h\left(\begin{matrix}0&1\\2(1+\varepsilon)\|g\|_{\rm Lip}V&2G\end{matrix}\right). $$
Taking the norms on both sides of \eqref{eq:l2:6} leads to $$
\sqrt{({\mathcal E_x^{n+1}})^2+({\mathcal E_v^{n+1}})^2}\le (1+h\|{\mathcal A}\|)
\sqrt{({\mathcal E_x^{n}})^2+({\mathcal E_v^{n}})^2}+\varepsilon(1+\varepsilon)h\|g\|_{\rm Lip}VX, $$ which gives the least-squares error estimate \eqref{eq:leastsqstab}.
\end{proof}
\section{Dimensionality reduction for continuous dynamical systems}\label{Sec:Cont}
\subsection{Uniform estimates for continuous dynamical systems}
In this section we shall establish the analogue of the above results for the continuous time setting of dynamical systems of the type~\eqref{eq:dyn}, \( \label{Cont1}
\dot x_i &=& f_i(\mathcal{D} x) + \sum_{j=1}^N f_{ij}(\mathcal{D} x)x_j \,,\qquad i=1,\dots,N \,,\\
x_i(0) &=& x_i^0 \,,\qquad i=1,\dots,N \,. \label{Cont2} \) We adopt again the assumptions about Lipschitz continuity and boundedness of the right-hand side made in Section~\ref{Sec:Euler}, namely~\eqref{eq:condf1}, \eqref{eq:condf2}, and \eqref{eq:condf3}.
\begin{theorem}\label{Thm:Cont} Let $x(t)\in\mathbb{R}^{d\times N}$, $t\in[0,T]$, be the solution of the system~\eqref{Cont1}--\eqref{Cont2} with $f_i$'s and $f_{ij}$'s satisfying~\eqref{eq:condf1}--\eqref{eq:condf3}, such that \( \label{alpha}
\max_{t\in[0,T]} \max_{i,j} \|x_i(t) - x_j(t)\|_{\ell_2^d} \leq \alpha \,. \) Let us fix $k\in\mathbb{N}$, $k \leq d$, and a matrix $M\in\mathbb{R}^{k\times d}$ such that \( \label{Cont-JL}
(1-\varepsilon) \|x_i(t) - x_j(t)\|_{\ell_2^d} \leq \|Mx_i(t) - Mx_j(t)\|_{\ell_2^k} \leq (1+\varepsilon) \|x_i(t) - x_j(t)\|_{\ell_2^d} \,, \) for all $t\in[0,T]$ and $i$, $j=1,\dots,N$. Let $y(t)\in\mathbb{R}^{k\times N}$, $t\in[0,T]$ be the solution of the projected system \begin{eqnarray}
\dot y_i &=& Mf_i(\mathcal{D}' y) + \sum_{j=1}^N f_{ij}(\mathcal{D}' y) y_j \,,\qquad i=1,\dots, N\,, \nonumber\\
y_i(0) &=& Mx_i^0 \,,\qquad i=1,\dots, N\,, \label{lowdimsys} \end{eqnarray} such that for a suitable $\beta>0$, \( \label{beta}
\max_{t\in[0,T]} \Norm{y(t)}_{\ell_\infty^N(\ell_2^d)} \leq \beta \,. \)
Let us define the column-wise $\ell_2$-error $e_i(t) := \|y_i-Mx_i\|_{\ell_2^k}$ for $i=1,\dots,N$ and \[
\mathcal{E}(t) := \max_{i=1,\dots,N} e_i(t) = \Norm{y-Mx}_{\ell_\infty^N(\ell_2^k)} \,. \end{eqnarray*} Then we have the estimate \begin{equation} \label{eq:unifstab2}
\mathcal{E}(t) \leq \varepsilon\alpha t (L\Norm{M} + L''\beta)
\exp\left[(2L\Norm{M} + 2\beta L'' + L') t\right] \,. \end{equation} \end{theorem}
\begin{proof} Due to~\eqref{eq:condf1}--\eqref{eq:condf3}, we have for every $i=1,\dots, N$ the estimate \[
\tot{}{t} e_i &=& \frac{\langle y_i-Mx_i,\tot{}{t}(y_i-Mx_i)\rangle}{\|y_i-Mx_i\|_{\ell_2^k}}\le \left\|\tot{}{t}(y_i-Mx_i)\right\|_{\ell_2^k}\\
&\leq& \|M f_i(\mathcal{D}' y) - M f_i(\mathcal{D} x)\|_{\ell_2^k} + \sum_{j=1}^N \|f_{ij}(\mathcal{D}' y)y_j - f_{ij}(\mathcal{D} x)Mx_j\|_{\ell_2^k} \\
&\leq& L \Norm{M} \Norm{\mathcal{D}' y - \mathcal{D} x}_{\ell_\infty^N(\ell_\infty^N)}
+ \sum_{j=1}^N \left( \|f_{ij}(\mathcal{D} x)(Mx_j - y_j)\|_{\ell_2^k} + \|(f_{ij}(\mathcal{D} x)-f_{ij}(\mathcal{D}' y)) y_j\|_{\ell_2^k} \right) \\
&\leq& L \Norm{M} \Norm{\mathcal{D}' y - \mathcal{D} x}_{\ell_\infty^N(\ell_\infty^N)}
+ L' \Norm{Mx - y}_{\ell_\infty^N(\ell_2^k)} + L'' \Norm{\mathcal{D} x-\mathcal{D}' y}_{\ell_\infty^N(\ell_\infty^N)} \Norm{y}_{\ell_\infty^N(\ell_2^k)} \,. \end{eqnarray*} The term $\Norm{\mathcal{D}' y - \mathcal{D} x}_{\ell_\infty^N(\ell_\infty^N)} \leq \Norm{\mathcal{D}' y - \mathcal{D}' Mx}_{\ell_\infty^N(\ell_\infty^N)} + \Norm{\mathcal{D}' Mx - \mathcal{D} x}_{\ell_\infty^N(\ell_\infty^N)}$ is estimated by \[
\Norm{\mathcal{D}' y - \mathcal{D} Mx}_{\ell_\infty^N(\ell_\infty^N)} &=& \max_{i,j} \biggl| \|y_i-y_j\|_{\ell_2^k} - \|Mx_i - Mx_j\|_{\ell_2^k} \biggr| \\
&\leq& \max_{i,j} \|y_i-Mx_i\|_{\ell_2^k} + \|y_j - Mx_j\|_{\ell_2^k}
\leq 2\mathcal{E}(t) \,, \end{eqnarray*} and, using the assumption~\eqref{Cont-JL}, \[
\Norm{\mathcal{D}' Mx - \mathcal{D} x}_{\ell_\infty^N(\ell_\infty^N)} = \max_{i,j} \biggl| \|Mx_i-Mx_j\|_{\ell_2^k} - \|x_i - x_j\|_{\ell_2^d} \biggr|
\leq \varepsilon \max_{i,j} \|x_i - x_j\|_{\ell_2^k} = \varepsilon \Norm{\mathcal{D} x}_{\ell_\infty^N(\ell_\infty^N)} \,. \end{eqnarray*} Finally, by the a priori estimate~\eqref{alpha} for $\Norm{\mathcal{D} x}_{\ell_\infty^N(\ell_\infty^N)}$ and~\eqref{beta} for $\Norm{y}_{\ell_\infty^N(\ell_2^d)}$, we obtain \[
\tot{}{t} e_i &\leq& L \Norm{M} (2\mathcal{E}(t) + \varepsilon\alpha) + L' \mathcal{E}(t) + L'' \beta (2\mathcal{E}(t) + \varepsilon\alpha) \\
&=& (2L\Norm{M} + 2\beta L'' + L')\mathcal{E}(t) + \varepsilon\alpha(L\Norm{M} + L''\beta) \,. \end{eqnarray*}
Now, let us split the interval $[0,T)$ into a union of finite disjoint intervals $I_j=[t_{j-1},t_j)$, $j=1,\dots,K$ for a suitable $K\in\mathbb{N}$, such that $\mathcal{E}(t) = e_{i(j)}(t)$ for $t\in I_j$. Consequently, on every $I_j$ we have \[
\tot{}{t} \mathcal{E}(t) = \tot{}{t} e_{i(j)}(t) \leq (2L\Norm{M} + 2\beta L'' + L')\mathcal{E}(t) + \varepsilon\alpha(L\Norm{M} + L''\beta) \,, \end{eqnarray*} and the Gronwall lemma yields \[
\mathcal{E}(t) \leq \left[\varepsilon\alpha(L\Norm{M} + L''\beta)(t-t_{j-1}) +\mathcal{E}(t_{j-1})\right]
\exp\left((2L\Norm{M} + 2\beta L'' + L')(t-t_{j-1})\right) \end{eqnarray*} for $t\in [t_{j-1},t_j)$. A concatenation of these estimates over the intervals $I_j$ leads finally to the expected error estimate \[
\mathcal{E}(t) \leq \varepsilon\alpha t (L\Norm{M} + L''\beta)
\exp\left[(2L\Norm{M} + 2\beta L'' + L') t\right] \,. \end{eqnarray*} \end{proof}
\subsection{A continuous Johnson-Lindenstrauss Lemma} Let us now go through the assumptions we made in the formulation of Theorem~\ref{Thm:Cont} and discuss how they restrict the validity and applicability of the result. First of all, let us mention that~\eqref{alpha} and~\eqref{beta} can be easily proven to hold for locally Lipschitz right-hand sides $f_i$ and $f_{ij}$ on finite time intervals. Obviously, the critical point for the applicability of Theorem \ref{Thm:Cont} is the question how to find a matrix $M$ satisfying the condition~\eqref{Cont-JL}, i.e., being a quasi-isometry along the trajectory solution $x(t)$ for {\it every} $t\in[0,T]$. The answer is provided by the following generalization of the Johnson-Lindenstrauss Lemma (Lemma~\ref{JLlem}) for rectifiable $\mathcal C^1$-curves, by a suitable continuity argument. Let us stress that our approach resembles the ``sampling and $\epsilon$-net'' argument in \cite{BDDW,BW09,W08} for the extension of the quasi-isometry property of Johnson-Lindenstrauss embeddings to smooth Riemmanian manifolds. From this point of view the following result can be viewed as a specification of the work \cite{BW09,W08}. \\ We first prove an auxiliary technical result:
\begin{lemma}\label{lem:neigh} Let $0<\varepsilon<\varepsilon'<1$, $a\in\mathbb{R}^d$ and let $M:\mathbb{R}^d\to \mathbb{R}^k$ be a linear mapping such that $$
(1-\varepsilon)\|a\|_{\ell_2^d}\le \|M a\|_{\ell_2^k}\le
(1+\varepsilon)\|a\|_{\ell_2^d}. $$ Let $x\in\mathbb{R}^d$ satisfy \begin{equation}\label{eq:tau}
\|a-x\|\le \frac{(\varepsilon'-\varepsilon)\|a\|_{\ell_2^d}}{\|M\|+1+\varepsilon'}. \end{equation} Then \begin{equation}\label{eq:neigh}
(1-\varepsilon')\|x\|_{\ell_2^d}\le \|Mx \|_{\ell_2^k}\le
(1+\varepsilon')\|x\|_{\ell_2^d}. \end{equation} \end{lemma} \begin{proof} If $a=0$, the statement is trivial. If $a\not=0$, we denote the right-hand side of \eqref{eq:tau} by $\tau>0$ and estimate by the triangle inequality \begin{align*}
\frac{\|Mx\|_{\ell_2^k}}{\|x\|_{\ell_2^d}}&=\frac{\|M(x-a)+Ma \|_{\ell_2^k}}{\|x-a+a\|_{\ell_2^d}}\le
\frac{\|M\|\cdot\|x-a\|_{\ell_2^d}+(1+\varepsilon)\|a\|_{\ell_2^d}}{\|a\|_{\ell_2^d}-\|x-a\|_{\ell_2^d}}\\
&\le \frac{\|M\|\cdot\tau+(1+\varepsilon)\|a\|_{\ell_2^d}}{\|a\|_{\ell_2^d}-\tau}\le 1+\varepsilon' \,. \end{align*} A similar chain of inequalities holds for the estimate from below. \end{proof}
Now we are ready to establish a continuous version of Lemma~\ref{JLlem}.
\begin{theorem}\label{thmcontJL} Let $\varphi:[0,1]\to \mathbb{R}^d$ be a ${\mathcal C}^1$ curve. Let $0<\varepsilon<\varepsilon'<1$, $$
\gamma:=\max_{\xi\in[0,1]}\frac{\|\varphi'(\xi)\|_{\ell_2^d}}{\|\varphi(\xi)\|_{\ell_2^d}}<\infty \quad \text{and}\quad \mathcal N\ge (\sqrt d+2)\cdot \frac{\gamma}{\varepsilon'-\varepsilon}. $$
Let $k$ be such that a randomly chosen (and properly normalized) projector $M$ satisfies the statement of the Johnson-Lindenstrauss Lemma~\ref{JLlem} with $\varepsilon, d, k$ and $\mathcal N$ arbitrary points with high probability. Without loss of generality we assume that $\|M\|\le \sqrt{d/k}$ within the same probability (this is in fact the case, e.g., for the examples of Gaussian and Bernoulli random matrices reported in Section \ref{Sec:Euler}).
Then \begin{equation}\label{eq:JLphi2}
(1-\varepsilon')\|\varphi(t)\|_{\ell_2^d}\le \|M\varphi(t)\|_{\ell_2^k}\le
(1+\varepsilon')\|\varphi(t)\|_{\ell_2^d},\mbox{ for all } t\in[0,1] \end{equation} holds with the same probability. \end{theorem}
\begin{proof} Let $t_i=i/\mathcal N$, $i=0,\dots,\mathcal N$ and put $$
T_i:={\rm arg \ max}_{\xi\in [t_i,t_{i+1}]} \|\varphi'(\xi)\|_{\ell_2^d},\quad i=0,\dots,\mathcal N-1. $$
Let $M:\mathbb{R}^d\to \mathbb{R}^k$ be the randomly chosen and normalized projector (see Lemma \ref{JLlem}). Hence $\|M\|\le \sqrt{d/k}$ and \begin{equation}\label{eq:JL1}
(1-\varepsilon')\|\varphi(T_i)\|_{\ell_2^d}\le \|M(\varphi(T_i))\|_{\ell_2^k}\le
(1+\varepsilon')\|\varphi(T_i)\|_{\ell_2^d},\qquad i=1,\dots,\mathcal N \end{equation} with high probability. We show that \eqref{eq:JLphi2} holds with (at least) the same probability.
This follows easily from \eqref{eq:JL1} and the following estimate, which holds for every $t\in[t_i,t_{i+1}]$, \begin{align*}
\|\varphi(t)-\varphi(T_i)\|_{\ell_2^d}&\le \int_{t}^{T_i} \|\varphi'(s)\|_{\ell_2^d}ds\le \frac{\|\varphi'(T_i)\|_{\ell_2^d}}{\mathcal N}\le
\frac{\|\varphi'(T_i)\|_{\ell_2^d}(\varepsilon'-\varepsilon)}{\gamma(\sqrt d+2)}\\
&\le \frac{\|\varphi(T_i)\|_{\ell_2^d}(\varepsilon'-\varepsilon)}{\sqrt d+2}
\le \frac{\|\varphi(T_i)\|_{\ell_2^d}(\varepsilon'-\varepsilon)}{\|M\|+1+\varepsilon'}. \end{align*} The proof is then finished by a straightforward application of Lemma \ref{lem:neigh}. \end{proof}
\begin{remark} We show now that the condition $$
\gamma:=\max_{\xi\in[0,1]}\frac{\|\varphi'(\xi)\|_{\ell_2^d}}{\|\varphi(\xi)\|_{\ell_2^d}}<\infty$$ is necessary, hence it is a restriction to the type of curves one can quasi-isometrically project. Let $d\ge 3$. It is known that there is a continuous curve $\varphi:[0,1]\to [0,1]^{d-1}$, such that $\varphi([0,1])=[0,1]^{d-1}$, i.e., $\varphi$ goes \emph{onto} $[0,1]^{d-1}.$ The construction of such a \emph{space-filling} curve goes back to Peano and Hilbert. After a composition with suitable dilations and $d$-dimensional spherical coordinates we observe that there is also a \emph{surjective} continuous curve $\varphi:[0,1]\to {\mathbb S}^{d-1}$, where ${\mathbb S}^{d-1}$ denotes the $\ell_2^d$ unit sphere in $\mathbb{R}^d.$
As $M$ was supposed to be a projection, \eqref{eq:JLphi2} cannot hold for all $t$'s with $\varphi(t)\in {\rm ker}\ M\not = \emptyset.$ \end{remark}
Obviously, the key condition for applicability of Theorem \ref{thmcontJL} for finding a projection matrix $M$ satisfying~\eqref{Cont-JL} is that \( \label{Cond-contJL}
\sup_{t\in[0,T]} \max_{i,j} \frac{\|\dot x_i - \dot x_j\|_{\ell_2^d}}{\|x_i-x_j\|_{\ell_2^d}} \leq \gamma <\infty \,. \) This condition is, for instance, trivially satisfied when the right-hand sides $f_i$'s and $f_{ij}$'s have the following Lipschitz continuity: \[
\|f_i(\mathcal{D} x) - f_j(\mathcal{D} x)\|_{\ell_2^d} \leq L''' \|x_i-x_j\|_{\ell_2^d} \qquad\mbox{for all } i,j=1,\dots,N \,,\\
|f_{i,k}(\mathcal{D} x) - f_{j,k}(\mathcal{D} x)| \leq L'''' \|x_i-x_j\|_{\ell_2^d} \qquad\mbox{for all } i,j,k=1,\dots,N. \end{eqnarray*} We will show in the examples below how condition~\eqref{Cond-contJL} is verified in cases of dynamical systems modeling standard social mechanisms of {\it attraction, repulsion, aggregation} and {\it alignment}.
\subsection{Applicability to relevant examples of dynamical systems describing social dynamics}
In this section we show the applicability of our dimensionality reduction theory to well-known dynamical systems driven by ``social forces'' of {\it alignment, attraction, repulsion, aggregation, and self-drive}. Although these models were proposed as descriptions of {\it group motion in physical space}, the fundamental social effects can be considered as building blocks in the more abstract context of many-parameter social dynamics. It has been shown \cite{CFTV,DOrsogna} that these models are able to produce meaningful {\it patterns}, for instance {\it mills} in two spatial dimensions (see Figure~\ref{fig:mills}), reproducing the behavior of certain biological species. \begin{figure}
\caption{Mills in nature and in models}
\label{fig:mills}
\end{figure} However, we should expect that in higher dimension the possible patterns produced by the combination of fundamental effects can be much more complex.
\subsubsection{The Cucker-Smale system (alignment effect)}\label{sec:cs} As shown in Section~\ref{Sec:Euler}, the Cucker and Smale flocking model~\eqref{CS1}--\eqref{CS2} is of the type~\eqref{eq:dyn}, satisfies the Lipschitz continuity assumptions~\eqref{eq:condf1}--\eqref{eq:condf3}, and it is bounded at finite time, as already discussed in Section \ref{Subs:UniformCS}. Therefore, to meet all the assumptions of Theorem~\ref{Thm:Cont}, we only need to check that it also satisfies the condition~\eqref{Cond-contJL}. However, for this we need to consider a slightly different framework than in Section~\ref{Subs:UniformCS}; instead of considering the $2N$ $d$-dimensional variables ($N$ position variables and $N$ velocity variables), we need to arrange the model as $N$ variables in $\mathbb{R}^{2d}$, each variable consisting of the position part (first $d$ entries) and of the velocity part (the other $d$ entries). We have then \[
\|\dot x_i-\dot x_j\|_{\ell_2^d} + \|\dot v_i -\dot v_j\|_{\ell_2^d} &\leq& \|v_i-v_j\|_{\ell_2^d}
+ \frac1N \sum_{k=1}^N \bigl |g(\|x_i-x_k\|_{\ell_2^d}) - g(\|x_j-x_k\|_{\ell_2^d})\bigr | \|v_k\|_{\ell_2^d} \\
&\leq& \|v_i-v_j\|_{\ell_2^d} + \frac{\Norm{g}_{Lip}}{N} \sum_{k=1}^N \bigl| \|x_i-x_k\|_{\ell_2^d} - \|x_j-x_k\|_{\ell_2^d}\bigr| \|v_k\|_{\ell_2^d} \\
&\leq& \|v_i-v_j\|_{\ell_2^d} + \frac{\Norm{g}_{Lip}}{N} \left( \sum_{k=1}^N \|v_k\|_{\ell_2^d}\right) \|x_i-x_j\|_{\ell_2^d} \\
&\leq& \|v_i-v_j\|_{\ell_2^d} + c \|x_i-x_j\|_{\ell_2^d} \,, \end{eqnarray*}
for a suitable constant $c$ depending on the initial data. We used here the a-priori boundedness of the term $\frac{1}{N} \left( \sum_{k=1}^N \|v_k\|_{\ell_2^d}\right)$, see~\cite{CS2} or~\cite{TH} for details. Consequently, we can satisfy~\eqref{Cond-contJL} with $\gamma=\max(1,c)$.
\subsubsection{Second order dynamic model with self-propulsion and pairwise interactions (self-drive, attraction, and repulsion effects)}\label{sec:dors} Another practically relevant model which fits into the class given by~\eqref{eq:dyn} is a second order dynamic model with self-propulsion and pairwise interactions,~\cite{LRC00,DOrsogna}: \(
\dot x_i &=& v_i \,, \label{DOrsogna1}\\
\dot v_i &=& (a-b\|v_i\|_{\ell_2^d}^2)v_i - \frac{1}{N} \sum_{j\neq i} \nabla_{x_i} U(\|x_i-x_j\|_{\ell_2^d}) \,,\qquad i=1,\dots,N,
\label{DOrsogna2} \) where $a$ and $b$ are positive constants and $U:[0,\infty)\to\mathbb{R}$ is a smooth potential. We denote $u(s) = U'(s)/s$ and assume that $u$ is a bounded, Lipschitz continuous function. We again arrange the model as a system of $N$ variables in $\mathbb{R}^{2d}$, each variable consisting of the position part (first $d$ entries) and of the velocity part (the other $d$ entries). Consequently, the model can be put into a form compliant with~\eqref{eq:dyn} as follows: \[
\dot x_i &=& \sum_{j=1}^N f^{xv}_{ij} v_j \,,\\
\dot v_i &=& \sum_{j=1}^N f^{vv}_{ij}(\mathcal{D} v) v_j + \sum_{j=1}^N f^{vx}_{ij}(\mathcal{D} x) x_j\,,\\ \end{eqnarray*} with $f^{xv}_{ij} = \delta_{ij}$,
$f^{vx}_{ii}(\mathcal{D} x) = - \frac{1}{N} \sum_{j\neq i} u(\|x_i-x_j\|_{\ell_2^d})$ and
$f^{vx}_{ij}(\mathcal{D} x) = \frac{1}{N} u(\|x_i-x_j\|_{\ell_2^d})$ for $i\neq j$. Moreover, we may set $f^{vv}_{ij}(\mathcal{D} v) = \delta_{ij} (a-b\|v_i\|_{\ell_2^d}^2)$ by introducing an auxiliary, noninfluential constant zero particle $(x_0,v_0)=(0,0)$ with null dynamics, i.e., $f_0^{*\star}=0$ and $f_{0j}^{*\star}=0$, where $*,\star \in \{x,v\}$. Then,~\eqref{eq:condf1} is void, while~\eqref{eq:condf2} is satisfied by \begin{align*}
\max_i \sum_j (|f^{xv}_{ij}(\mathcal{D} x,\mathcal{D} v)|&+
|f^{vx}_{ij}(\mathcal{D} x,\mathcal{D} v)|+|f^{vv}_{ij}(\mathcal{D} x,\mathcal{D} v)|)\\
& \leq 1 + a + b\max_i \|v_i\|_{\ell_2^d}^2 + 2 \Norm{u}_{L_\infty} \leq L' \,, \end{align*}
since the theory provides an apriori bound on $\beta_v:=\sup_{t\in[0,T]} \max_i \|v_i\|_{\ell_2^d}$, see~\cite{DOrsogna}. Condition~\eqref{eq:condf3} for $f^{xv}_{ij}$ is void, while for $f^{vv}_{ij}$ it is satisfied by \begin{align*}
\max_i \sum_j \left| f_{ij}^{vv}(\mathcal{D} v) - f_{ij}^{vv}(\mathcal{D} w) \right|
&\leq b\max_i \left| \|v_i\|_{\ell_2^d}^2 - \|w_i\|_{\ell_2^d}^2 \right| \\
&\leq b \max_i\left(\|v_i\|_{\ell_2^d}+\|w_i\|_{\ell_2^d}\right) \|v_i-w_i\|_{\ell_2^d}\\
&\leq L''\Norm{\mathcal{D} v - \mathcal{D} w}_{\ell_\infty^N(\ell_\infty^N)} \,, \end{align*} where we again use the apriori boundedness of $\beta_v$. For $f^{vx}_{ij}$ is~\eqref{eq:condf3} satisfied by \[
\max_i \sum_j \left| f_{ij}^{vx}(\mathcal{D} x) - f_{ij}^{vx}(\mathcal{D} y) \right|
&\leq& \max_i \frac{2}{N} \sum_{j\neq i} \left| u(\|x_i-x_j\|_{\ell_2^d}) - u(\|y_i-y_j\|_{\ell_2^d})\right| \\
&\leq& \max_i \frac{2}{N} \Norm{u}_{{\rm Lip}} \sum_{j\neq i} \left| \|x_i-x_j\|_{\ell_2^d} - \|y_i-y_j\|_{\ell_2^d} \right| \\
&\leq& 2 \Norm{u}_{{\rm Lip}} \Norm{\mathcal{D} x - \mathcal{D} y}_{\ell_\infty^N(\ell_\infty^N)}.
\end{eqnarray*}
Finally, it can be easily checked that condition~\eqref{Cond-contJL} is satisfied by \[
\|\dot x_i - \dot x_j\|_{\ell_2^d} + \|\dot v_i - \dot v_j\|_{\ell_2^d} \leq
(1+a+3b\beta_v^2) \|v_i-v_j\|_{\ell_2^d} + \left( \Norm{u}_{L_\infty} + 2 \beta_x \Norm{u}_{{\rm Lip}} \right) \|x_i-x_j\|_{\ell_2^d} \,, \end{eqnarray*}
where $\beta_x:=\sup_{t\in[0,T]} \max_i \|x_i\|_{\ell_2^d}$. We notice that also this model is bounded at finite time as shown in \cite[Theorem 3.10 (formula (22))]{CCR}, and therefore for any fixed horizon time $T$, there is a constant $\alpha=\alpha(T)>0$ such that \eqref{alpha1} and \eqref{alpha} hold. In the paper \cite{DOrsogna} it is shown that this model tends to produce patterns of different quality, in particular mills, double mills, and translating crystalline flocks (see also Figure \ref{fig:mills}). These patterns were further studied in~\cite{cop}. Starting from the Liouville equation for the many-body problem the authors derive the corresponding kinetic and macroscopic hydrodynamic equations. The kinetic theory approach leads to the identification of macroscopic structures otherwise not recognized as solutions of the hydrodynamic equations, such as double mills of two superimposed flows. The authors found conditions allowing for the existence of such solutions and compared them to the case of single mills. In~\cite{combc} the authors utilize the methods of classical statistical mechanics to connect the individual-based models of the type~\eqref{DOrsogna1}--\eqref{DOrsogna2} to their continuum formulations and determine criteria for the validity of the latter. They show that H-stability of the interaction potential plays a fundamental role in determining both the validity of the continuum approximation and the nature of the aggregation state transitions. They perform a linear stability analysis of the continuum model and compare the results to the simulations of the individual-based one.
Without entering into further details, let us stress that mills and double mills are uniformly bounded in time (and stable). Hence in these cases, we can assume that actually the constant $\alpha$ is again bounded for all times. Moreover, when the dynamics converges to a translating crystalline flocks, we may reason in a similar way as done for the Cucker-Smale model (although in this case the pattern in unstable).
\subsection{Recovery of the dynamics in high dimension from multiple simulations in low dimension}
The main message of Theorem \ref{Thm:Cont} is that, under suitable assumptions on the governing functions $f_i, f_{ij}$, the trajectory of the solution $y(t)$ of the {\it projected} dynamical system \eqref{lowdimsys} is at an $\varepsilon$ error from the trajectory of the {\it projection} of the solution $x(t)$ of the dynamical system \eqref{Cont1}-\eqref{Cont2}, i.e., \begin{equation}\label{approxk}
y_i (t) \approx M x_i(t) \mbox { or, more precisely, } \| Mx_i(t) - y_i(t) \|_{\ell_2^k} \leq C(t) \varepsilon, \quad t \in [0,T]. \end{equation}
We wonder whether this approximation property can allow us to ``learn'' properties of the original trajectory $x(t)$ in high dimension.
\subsubsection{Optimal information recovery of high-dimensional trajectory from low-dimensional projections}\label{CSsec}
In this section we would like to address the following two fundamental questions: \begin{itemize} \item[(i)] Can we quantify the best possible information of the high-dimensional trajectory one can recover from one or more projections in lower dimension? \item[(ii)] Is there any practical method which performs an optimal recovery? \end{itemize} The first question was implicitly addressed already in the 70's by Kashin and later by Garnaev and Gluskin \cite{ka77,gagl84}, as one can put in relationship the optimal recovery from linear measurements with Gelfand width of $\ell_p$-balls, see for instance \cite{codade09}. It was only with the development of the theory of {\it compressed sensing} \cite{carota06,do06-2} that an answer to the second question was provided, showing that $\ell_1$-minimization actually performs an {\it optimal recovery} of vectors in high dimension from random linear projections to low dimension. We address the reader to \cite[Section 3.9]{FR} for further details. In the following we concisely recall the theory of compressed sensing and we apply it to estimate the optimal information error in recovering the trajectories in high dimension from lower dimensional simulations.
Again a central role here is played by (random) matrices with the so-called {\it Restricted Isometry Property} RIP, cf. \cite{ca08}.
\begin{definition}[Restricted Isometry Property] A $k\times d$ matrix $M$ is said to have the Restricted Isometry Property of order $K\leq d$ and level $\delta\in (0,1)$ if $$
(1-\delta)\|x\|^2_{\ell_2^d} \leq \|M x\|^2_{\ell_2^k} \leq (1+\delta) \|x\|^2_{\ell_2^d} $$ for all $K$-sparse $x \in \Sigma_K=\{z \in \mathbb R^d: \#{\rm supp\, }(z) \leq K\}$. \end{definition}
Both the typical matrices used in Johnson-Lindenstrauss embeddings (cf. Lemma \ref{JLlem}) and matrices with RIP used in compressed sensing are usually generated at random. It was observed by \cite{BDDW} and \cite{KW}, that there is an intimate connection between these two notions. A simple reformulation of the arguments of \cite{BDDW} yields the following.
\begin{theorem}[Baraniuk, Davenport, DeVore, and Wakin]\label{BDDWthm} Let $M$ be a $k\times d$ matrix drawn at random which satisfies $$
(1-\delta/2)\|x\|^2_{\ell_2^d} \leq \|M x\|^2_{\ell_2^k} \leq (1+\delta/2) \|x\|^2_{\ell_2^d},\quad x\in {\mathcal P} $$ for every set ${\mathcal P}\subset \mathbb{R}^d$ with $\#{\mathcal P}\le \bigl(\frac{12ed}{\delta K}\bigr)^K$ with probability $0<\nu<1$. Then $M$ satisfies the Restricted Isometry Property of order $K$ and level $\delta/3$ with probability at least equal to $\nu$. \end{theorem}
Combined with several rather elementary constructions of Johnson-Lindenstrauss embedding matrices available in literature, cf. \cite{A} and \cite{DG}, this result provides a simple construction of RIP matrices. The converse direction, namely the way from RIP matrices to matrices suitable for Johnson-Lindenstrauss embedding was discovered only recently in \cite{KW}.
\begin{theorem}[Krahmer and Ward]\label{KWthm}
Fix $\eta > 0$ and $\varepsilon> 0$, and consider a finite set $\mathcal P\subset \mathbb R^d$ of cardinality $|\mathcal P| = \mathcal N$. Set $K \geq 40 \log \frac{4 \mathcal N}{\eta}$, and suppose that the $k \times d$ matrix $\tilde M$ satisfies the Restricted Isometry Property of order $K$ and level $\delta \leq \varepsilon/4$. Let $\xi \in \mathbb R^d$ be a Rademacher sequence, i.e., uniformly distributed on $\{-1, 1\}^d$ . Then with probability exceeding $1 -\eta$, \begin{equation*}
(1-\varepsilon)\|x\|_{\ell_2^d}^2\le \|M x\|_{\ell_2^k}^2\le (1+\varepsilon)\|x\|_{\ell_2^d}^2. \end{equation*} uniformly for all $x \in \mathcal P$, where $M:= \tilde M \operatorname{diag}(\xi)$, where $\operatorname{diag}(\xi)$ is a $d\times d$ diagonal matrix with $\xi$ on the diagonal. \end{theorem}
We refer to \cite{ra10} for additional details.
\begin{remark} Notice that $M$ as constructed in Theorem \ref{KWthm} is both a Johnson-Lindenstrauss embedding and a matrix with RIP, because \begin{align*}
(1-\delta)\|x\|_{\ell_2^d}^2 &= (1-\delta)\|\operatorname{diag}(\xi) x\|_{\ell_2^d}^2 \leq \|\underbrace{\tilde M \operatorname{diag}(\xi)}_{:=M} x\|_{\ell_2^k}^2 \\
&\leq (1+\delta) \|\operatorname{diag}(\xi) x\|_{\ell_2^d}^2 = (1+\delta) \| x\|_{\ell_2^d}^2. \end{align*} The matrices considered in Section \ref{Sec:Euler} satisfy with high probability the RIP with $$ K = \mathcal O \left ( \frac{k}{1+ \log( d/k)} \right). $$
\end{remark}
Equipped with the notion of RIP matrices we may state the main result of the theory of compressed sensing, as appearing in \cite{fo09}, which we shall use for the recovery of the dynamical system in $\mathbb{R}^d.$ \begin{theorem}\label{thmStrongRIP} Assume that the matrix $M \in \mathbb R^{k \times d}$ has the RIP of order $2 K$ and level $$ \delta_{2 K} < \frac{2}{3+\sqrt{7/4}} \approx 0.4627. $$
Then the following holds for all $x \in \mathbb R^d$. Let the low-dimensional approximation $y = M x + \eta$ be given with $\|\eta\|_{\ell_2^k}\leq C \varepsilon$. Let $x^\#$ be the solution of \begin{equation}\label{P1:eta}
\min_{z \in \mathbb R^d} \|z\|_{\ell_1^d} \quad \mbox{ subject to } \|M z - y\|_{\ell_2^k} \leq \|\eta\|_{\ell_2^k}. \end{equation} Then \[
\|x - x^\#\|_{\ell_2^d} \leq C_1 \varepsilon + C_2 \frac{\sigma_K(x)_{\ell_1^d}}{\sqrt{K}} \end{eqnarray*}
for some constants $C_1,C_2 > 0$ that depend only on $\delta_{2K}$, and $\sigma_K(x)_{\ell_1^d} = \inf_{z: \#{\rm supp\, }(z) \leq K} \|z -x\|_{\ell_1^d}$ is the best-$K$-term approximation error in $\ell_1^d$. \end{theorem} \\ This result says that provided the stability relationship \eqref{approxk}, we can approximate the individual trajectories $x_i(t)$, for each $t \in [0,T]$ fixed, by a vector $x^\#_i(t)$ solution of an optimization problem of the type \eqref{P1:eta}, and the accuracy of the approximation depends on the best-$K$-term approximation error $\sigma_K(x_i(t))_{\ell_1^d}$. Actually, the results in \cite{carota06,do06-2} in connection with \cite{codade09,ka77,gagl84}, state also that this is asymptotically the best one can hope for.
One possibility to improve the recovery error is to increase the dimension $k$ (leading to a smaller distortion parameter $\varepsilon>0$ in the Johnson-Lindenstrauss embedding). But we would like to explore another possibility, namely projecting and simulating {\it in parallel and independently} the dynamical system $L$-times in the lower dimension $k$ \begin{equation}\label{eq:paral1} \dot y^\ell_i = M^\ell f_i(\mathcal{D}' y^\ell) + \sum_{j=1}^N f_{ij}(\mathcal{D}' y^\ell) y_j^\ell\,,\qquad y_i^\ell(0) = M^\ell x_i^0 \,,\quad \ell=1,\dots, L. \end{equation} Let us give a brief overview of the corresponding error estimates. The number of points needed in each of the cases is ${\mathcal N}\approx N\times n_0$, where $N$ is the number of agents and $n_0=T/h$ is the number of iterations. \begin{itemize} \item We perform 1 projection and simulation in $\mathbb{R}^k$: Then $\varepsilon={\mathcal O}\Bigl(\sqrt{\frac{\log {\mathcal N}}{k}}\Bigr)$, $K=\mathcal O \left ( \frac{k}{1+ \log( d/k)} \right)$ and an application of Theorem \ref{thmStrongRIP} leads to \begin{equation}\label{eq:paral2}
\|x_i(t) - x_i^\#(t)\|_{\ell_2^d} \le C'(t)\left( \sqrt{\frac{\log {\mathcal N}}{k}} + \frac{\sigma_K(x_i(t))_{\ell_1^d}}{\sqrt{K}}\right). \end{equation} Here, $C'(t)$ combines both the constants from Theorem \ref{thmStrongRIP} and the time-dependent $C(t)$ from \eqref{approxk}. So, to reach the precision of order $C'(t)\epsilon>0$, we have to choose $k\in\mathbb{N}$ large enough, such that $\sqrt{\frac{\log {\mathcal N}}{k}}\le \epsilon$ and $\frac{\sigma_K(x_i(t))_{\ell_1^d}}{\sqrt{K}}\le \epsilon$. We then need $k\times N^2$ operations to evaluate the adjacency matrix.
\item We perform 1 projection and simulation in $\mathbb{R}^{L\times k}$: Then $\varepsilon'={\mathcal O}\Bigl(\sqrt{\frac{\log {\mathcal N}}{Lk}}\Bigr)$ and $K'=\mathcal O \left ( \frac{Lk}{1+ \log(d/Lk)} \right)$ and an application of Theorem \ref{thmStrongRIP} leads to \begin{equation}\label{eq:paral4}
\|x_i(t) - x_i^\#(t)\|_{\ell_2^d} \le C'(t)\left( \sqrt{\frac{\log {\mathcal N}}{Lk}} + \frac{\sigma_{K'}(x_i(t))_{\ell_1^d}}{\sqrt{K'}}\right). \end{equation}
The given precision of order $C'(t)\epsilon>0$, may be then reached by choosing $k,L\in\mathbb{N}$ large enough, such that $\sqrt{\frac{\log {\mathcal N}}{Lk}}\le \epsilon$ and $\frac{\sigma_{K'}(x_i(t))_{\ell_1^d}}{\sqrt{K'}}\le \epsilon$. We then need $Lk\times N^2$ operations to evaluate the adjacency matrix.
\item We perform $L$ independent and parallel projections and simulations in $\mathbb{R}^k$: Then we assemble the following system corresponding to \eqref{eq:paral1} $$ {\mathcal M}x=\left ( \begin{array}{l} M^1 \\hbox{\rlap{I}\kern.16em\rlap{I}M}^2\\\dots\\\dots \\ M^{L} \end{array} \right ) x_i = \left ( \begin{array}{l} y^1_i \\y^2_i\\\dots\\\dots \\y^{L}_i \end{array} \right ) - \left ( \begin{array}{l} \eta^1_i \\\eta^2_i\\\dots\\\dots \\\eta^{L}_i \end{array} \right ), $$ where for all $\ell=1,\dots, L$ the matrices $M^\ell \in \mathbb R^{k \times d}$ are (let us say) random matrices with each entry generated independently with respect to the properly normalized Gaussian distribution as described in Section \ref{Sec:Euler}. Then ${\mathcal M}/\sqrt{L}$ is a $Lk\times d$ matrix with Restricted Isometry Property of order $K'=\mathcal O \left ( \frac{Lk}{1+ \log(d/Lk)} \right)$ and level $\delta < 0.4627$. The initial distortion of each of the projections is still $\varepsilon={\mathcal O}\Bigl(\sqrt{\frac{\log {\mathcal N}}{k}}\Bigr)$. Therefore, by applying Theorem \ref{thmStrongRIP}, we can compute $x_i^\#(t)$ such that \begin{equation}\label{eq:paral3}
\|x_i(t) - x^\#_i(t)\|_{\ell_2^d} \le C'(t)\left( \sqrt{\frac{\log {\mathcal N}}{k}} + \frac{\sigma_{K'}(x_i(t))_{\ell_1^d}}{\sqrt{K'}}\right). \end{equation} Notice that the computation of $x_i^\#(t)$ can also be performed in parallel, see, e.g., \cite{fo07}. The larger is the number $L$ of projections we perform, the larger is $K'$ and the smaller is the second summand in \eqref{eq:paral3}; actually $\sigma_{K'}(x_i(t))_{\ell_1^d}$ vanishes for $K'\geq d$. Unfortunately, the parallelization can not help to reduce the initial distortion $\varepsilon>0$. To reach again the precision of order $C'(t)\epsilon>0$, we have to choose $k\in\mathbb{N}$ large enough, such that $\sqrt{\frac{\log {\mathcal N}}{k}}\le \epsilon$. Then we chose $L\ge 1$ large enough such that $\frac{\sigma_{K'}(x_i(t))_{\ell_1^d}}{\sqrt{K'}}\le \epsilon$. We again need $k\times N^2$ operations to evaluate the adjacency matrix. \end{itemize}
In all three cases, we obtain the estimate \begin{equation}\label{eq:paral5}
\|x_i(t) - x_i^\#(t)\|_{\ell_2^d} \le C'(t)\left( \varepsilon + \frac{\sigma_{K}(x_i(t))_{\ell_1^d}}{\sqrt{K}}\right), \end{equation} where the corresponding values of $\varepsilon>0$ and $K$ together with the number of operations needed to evaluate the adjacency matrix may be found in the following table.
\vskip.5cm \begin{center}
\begin{tabular}{|c|c|c|c|} \hline &$\varepsilon$&$K$& number of operations\\ \hline \vphantom{$\biggl($}1 projection into $\mathbb{R}^k$ & ${\mathcal O}\Bigl(\sqrt{\frac{\log {\mathcal N}}{k}}\Bigr)$ & $\mathcal O \left ( \frac{k}{1+ \log( d/k)} \right)$ & $k\times N^2$\\ \hline \vphantom{$\biggl($}1 projection into $\mathbb{R}^{L\times k}$ & ${\mathcal O}\Bigl(\sqrt{\frac{\log {\mathcal N}}{Lk}}\Bigr)$ & $\mathcal O \left ( \frac{Lk}{1+ \log(d/Lk)} \right)$& $Lk\times N^2$\\ \hline \vphantom{$\biggl($}$L$ projections into $\mathbb{R}^k$ & ${\mathcal O}\Bigl(\sqrt{\frac{\log {\mathcal N}}{k}}\Bigr)$ & $\mathcal O \left ( \frac{Lk}{1+ \log(d/Lk)} \right)$ & $k\times N^2$\\ \hline \end{tabular} \end{center}
\subsubsection{Optimal recovery of trajectories on smooth manifolds}
In recent papers \cite{BW09,W08,IM11}, the concepts of compressed sensing and optimal recovery were extended to vectors on smooth manifolds. These methods could become very useful in our context if (for any reason) we would have an apriori knowledge that the trajectories $x_i(t)$ keep staying on or near such a smooth manifold. Actually this is the case, for instance in molecular dynamics, where simulations, e.g. in the form of the coordinates of the atoms in a molecule as a function of time, lie on or near an intrinsically-low-dimensional set in the high-dimensional state space of the molecule, and geometric properties of such sets provide important information about the dynamics, or about how to build low-dimensional representations of such dynamics \cite{RZMC,ZRMC}. In this case, by using appropriate recovery methods as described in \cite{IM11}, we could recover high-dimensional vectors from very low dimensional projections with much higher accuracy. However, this issue will be addressed in a following paper.
\subsection{Numerical experiments}\label{Sec:Numerics} In this section we illustrate the practical use and performances of our projection method for the Cucker-Smale system~\eqref{CS1}--\eqref{CS2}.
\subsubsection{Pattern formation detection in high dimension from lower dimensional projections} As already mentioned, this system models the emergence of consensus in a group of interacting agents, trying to align with their neighbors.
The qualitative behavior of its solutions is formulated by this well known result~\cite{CS1, CS2, TH}:
\begin{theorem} Let $(x_i(t), v_i(t))$ be the solutions of~\eqref{CS1}--\eqref{CS2}. Let us define the fluctuation of positions around the center of mass $x_c(t) = \frac{1}{N} \sum_{i=1}^N x_i(t)$, and, resp., the fluctuation of the rate of change around its average $v_c(t) = \frac{1}{N} \sum_{i=1}^N v_i(t)$ as \[
\Lambda(t) = \frac1N \sum_{i=1}^N \|x_i(t) - x_c(t)\|_{\ell_2^d}^2 \,,\qquad
\Gamma(t) = \frac1N \sum_{i=1}^N \|v_i(t) - v_c(t)\|_{\ell_2^d}^2 \,. \end{eqnarray*} Then if either $\beta\leq 1/2$ or the initial fluctuations $\Lambda(0)$ and $\Gamma(0)$ are small enough (see~\cite{CS1} for details), then $\Gamma(t)\to 0$ as $t\to\infty$. \end{theorem}
The phenomenon of $\Gamma(t)$ tending to zero as $t\to\infty$ is called \emph{flocking} or {\it emergence of consensus}. If $\beta > 1/2$ and the initial fluctuations are not small, it is not known whether a given initial configuration will actually lead to flocking or not, and the only way to find out the possible formation of {\it consensus patterns} is to perform numerical simulations. However, these can be especially costly if the number of agents $N$ and the dimension $d$ are large; the algorithmic complexity of the calculation is $\mathcal{O}(d\times N^2)$. Therefore, a significant reduction of the dimension $d$, which can be achieved by our projection method, would lead to a corresponding reduction of the computational cost.
\begin{figure}
\caption{Numerical results for $\beta=1.5$: First row shows the evolution of $\Gamma(t)$ of the system projected to dimension $k=100$ (left) and $k=10$ (right) in the twenty realizations, compared to the original system (bold dashed line). Second row shows the initial values $\Gamma(t=0)$ and final values $\Gamma(t=30)$ in all the performed simulations.}
\label{Numfig1}
\end{figure}
\begin{figure}
\caption{Numerical results for $\beta=1.62$: First row shows the evolution of $\Gamma(t)$ of the system projected to dimension $k=100$ (left) and $k=25$ (right) in the twenty realizations, compared to the original system (bold dashed line). Second row shows the initial values $\Gamma(t=0)$ and final values $\Gamma(t=30)$ in all the performed simulations.}
\label{Numfig2}
\end{figure}
\begin{figure}
\caption{Numerical results for $\beta=1.7$: First row shows the evolution of $\Gamma(t)$ of the system projected to dimension $k=100$ (left) and $k=10$ (right) in the twenty realizations, compared to the original system (bold dashed line). Second row shows the initial values $\Gamma(t=0)$ and final values $\Gamma(t=30)$ in all the performed simulations.}
\label{Numfig3}
\end{figure}
We illustrate this fact by a numerical experiment, where we choose $N=1000$ and $d=200$, i.e., every agent $i$ is determined by a $200$-dimensional vector $x_i$ of its state and a $200$-dimensional vector $v_i$ giving the rate of change of its state. The initial datum $(x^0, v^0)$ is generated randomly, every component of $x^0$ being drawn independently from the uniform distribution on $[0,1]$ and every component of $v^0$ being drawn independently from the uniform distribution on $[-1,1]$. We choose $\beta = 1.5$, $1.62$ and $1.7$, and with every of these values we perform the following set of simulations: \begin{enumerate} \item Simulation of the original system in $200$ dimensions. \item Simulations in lower dimensions $k$: the initial condition $(x^0, v^0)$ is projected into the $k$-dimensional space by a random Johnson-Lindenstrauss projection matrix $M$ with Gaussian entries. The dimension $k$ takes the values $150$, $100$, $50$, $25$, $10$, $5$, and $2$. For every $k$, we perform the simulation twenty times, each time with a new random projection matrix $M$. \end{enumerate} All the simulations were implemented in MATLAB, using $1500$ steps of the forward Euler method with time step size $0.02$. The paths of $\Gamma(t)$ from the twenty experiments with $k=100$ and $k=25$ or $k=10$ are shown in the first rows of Figs.~\ref{Numfig1},~\ref{Numfig2} and, resp.,~\ref{Numfig3} for $\beta = 1.5$, $1.62$ and, resp., $1.7$.
The information we are actually interested in is whether flocking takes place, in other words, whether the fluctuations of velocities $\Gamma(t)$ tend to zero. Typically, after an initial phase, the graph of $\Gamma(t)$ gives a clear indication either about exponentially fast convergence to zero (due to rounding errors, ``zero'' actually means values of the order $~10^{-30}$ in the simulations) or about convergence to a positive value. However, in certain cases the decay may be very slow and a very long simulation of the system would be needed to see if the limiting value is actually zero or not. Therefore, we propose the following heuristic rules to decide about flocking from numerical simulations: \begin{itemize} \item If the value of $\Gamma$ at the final time $t=30$ is smaller than $10^{-10}$, we conclude that flocking took place. \item If the value of $\Gamma(30)$ is larger than $10^{-3}$, we conclude that flocking did not take place. \item Otherwise, we do not make any conclusion. \end{itemize} In the second rows of Figs.~\ref{Numfig1},~\ref{Numfig2} and~\ref{Numfig3} we present the initial and final values of $\Gamma$ of the twenty simulations for all the dimensions $k$, together with the original dimension $d=200$. In accordance with the above rules, flocking takes place if the final value of $\Gamma$ lies below the lower dashed line, does not take place if it lies above the upper dashed line, otherwise the situation is not conclusive. The results are summarized in Table~\ref{Numtable1}.
\begin{table}[htb] \footnotesize \begin{tabular}{ccccc} $\beta = 1.5$ & & $\beta = 1.62$ & & $\beta = 1.7$ \\
\begin{tabular}[h]{c|ccc} dim & pos & neg & n/a \\ \hline 200 & 1 & 0 & 0 \\ 150 & 20 & 0 & 0 \\ 100 & 20 & 0 & 0 \\ 50 & 20 & 0 & 0 \\ 25 & 20 & 0 & 0 \\ 10 & 14 & 0 & 6 \\ 5 & 4 & 4 & 12 \\ 2 & 3 & 8 & 9 \end{tabular} & &
\begin{tabular}[h]{c|ccc} dim & pos & neg & n/a \\ \hline 200 & 1 & 0 & 0 \\ 150 & 20 & 0 & 0 \\ 100 & 20 & 0 & 0 \\ 50 & 13 & 0 & 7 \\ 25 & 1 & 1 & 18 \\ 10 & 0 & 18 & 2 \\ 5 & 0 & 19 & 1 \\ 2 & 0 & 18 & 2 \end{tabular} & &
\begin{tabular}[h]{c|ccc} dim & pos & neg & n/a \\ \hline 200 & 0 & 1 & 0 \\ 150 & 0 & 20 & 0 \\ 100 & 0 & 20 & 0 \\ 50 & 0 & 20 & 0 \\ 25 & 0 & 20 & 0 \\ 10 & 0 & 20 & 0 \\ 5 & 0 & 20 & 0 \\ 2 & 0 & 20 & 0 \end{tabular} \end{tabular} \caption{Statistics of the flocking behaviors of the systems in the original dimension $d=200$ and in the projected dimensions. With $\beta=1.5$ and $\beta=1.62$, the original system ($d=200$) exhibited flocking behavior. With $\beta=1.5$, even after random projections into $25$ dimensions, the system exhibited flocking in all $20$ repetitions of the experiment, and still in $14$ cases in dimension $10$. With $\beta=1.62$, the deterioration of the flocking behavior with decreasing dimension was much faster, and already in dimension $25$ the situation was not conclusive. This is related to the fact that the value $\beta=1.62$ was chosen to intentionally bring the system close to the borderline between flocking and non-flocking. Finally, with $\beta=1.7$, the original system did not flock, and, remarkably, all the projected systems (even to two dimensions) exhibit the same behavior. } \label{Numtable1} \end{table}
Experience gained with a large amount of numerical experiments shows the following interesting fact: The flocking behavior of the Cucker-Smale system is very stable with respect to the Johnson-Lindenstrauss projections. Usually, the projected systems show the same flocking behavior as the original one, even if the dimension is reduced dramatically, for instance from $d=200$ to $k=10$ (see Figs~\ref{Numfig1} and~\ref{Numfig3}). This stability can be roughly explained as follows: Since the flocking behavior depends mainly on the initial values of $\Gamma$ and $\Lambda$, which are statistical properties of the random distributions used for the generation of initial data, and since $N$ is sufficiently large, the concentration of measure phenomenon takes place. Its effect is that the initial values of the fluctuations of the projected data are very close to the original ones, and thus the flocking behavior is (typically) the same. There is only a narrow interval of values of $\beta$ (in our case this interval is located around the value $\beta= 1.62$), which is a borderline region between flocking and non-flocking, and the projections to lower dimensions spoil the flocking behavior, see Fig~\ref{Numfig2}. Let us note that in our simulations we were only able to detect cases when flocking took place in the original system, but did not take place in some of the projected ones. Interestingly, we never observed the inverse situation, a fact which we are not able to explain satisfactorily. In fact, one can make other interesting observations, deserving further investigation. For instance, Figs.~\ref{Numfig1} and~\ref{Numfig2} show that if the original system exhibits flocking, then the curves of $\Gamma(t)$ of the projected systems tend to lie above the curve of $\Gamma(t)$ of the original one. The situation is reversed if the original system does not flock, see Fig.~\ref{Numfig3}.
From a practical point of view, we can make the following conclusion: To obtain an indication about the flocking behavior of a highly dimensional Cucker-Smale system, it is typically satisfactory to perform a limited number of simulations of the system projected into a much lower dimension, and evaluate the statistics of their flocking behavior. If the result is the same for the majority of simulations, one can conclude that the original system very likely has the same flocking behavior as well.
\begin{figure}
\caption{Numerical results showing the time evolution of the relative error of projection (left panel) and relative error of recovery via $\ell_1$-minimization (right panel) of the $v$-variables.
}
\label{Numfig4}
\end{figure} \subsubsection{Numerical validation of the high and low dimensional approximation properties} Finally, we show how the relative error of projection and recovery evolves in time. We consider an initial datum $(x^0,v^0)\in\mathbb{R}^{N\times d}\times\mathbb{R}^{N\times d}$ for the Cucker-Smale system with $N=d=200$ and randomly generated entries from the normal distribution. The parameter $\beta=0.4$, therefore, the system will exhibit flocking. First we project the system into $k=20,40,60,100,140,180$ dimensions and calculate the relative error of the projection of the $v$-variables, given by \[
\left(\frac{\sum_{i=1}^N \|Mv_i-v_j\|_{\ell_2^k}^2}{\sum_{i=1}^N \|Mv_i\|_{\ell_2^k}^2} \right)^{1/2} \,. \end{eqnarray*} We observe that the maximal relative error (for $k=20$) is around $14\%$, which we consider as a very good result. Moreover, in all $9$ cases, the error first increases, but after $t\simeq 22$ it starts decreasing, which is a consequence of the flocking behavior and concentration of measure, see the graphics in Figure~\ref{Numfig4} on the left. This clearly shows that the worst-case estimate of Theorem~\ref{thm1} with exponential growth in time is overly pessimistic.
In our second experiment, we take a randomly generated initial condition with $N=d=200$ and $80\%$ of the entries set to zero. Then, we take $L$ projections of the system into $20$ dimensions, with $L=1,2,3,5,7,9$, and reconstruct the $v$-trajectories using $\ell_1$-minimization, as described in Section~\ref{CSsec}. In the graphics in Figure~\ref{Numfig4} on the right, we plot the relative errors, given by \[
\left(\frac{\sum_{i=1}^N \|\tilde v_i-v_i\|_{\ell_2^k}^2}{\sum_{i=1}^N \|v_i\|_{\ell_2^k}^2} \right)^{1/2} \,, \end{eqnarray*} where $\tilde v_i$ are the recovered trajectories. Again, we observe that the errors grow much slower than exponentially, and after $t\simeq 15$ they even tend to stay constant or slightly decrease.
\section{Mean-field limit and kinetic equations in high dimension} In the previous sections we were concerned with tractable simulation of the dynamical systems of the type~\eqref{eq:dyn} when the dimension $d$ of the parameter space is large. Another source of possible intractability in numerical simulations appears in the situation where the number of agents $N$ is very large. In general, large $N$ imposes even a much more severe limitation than large $d$, since the computational complexity of~\eqref{eq:dyn} is $\mathcal{O}(d\times N^2)$. Therefore, in the next sections we consider the so-called {\it mean-field limit} of~\eqref{eq:dyn} as $N\to\infty$, where the evolution of the system is described by time-dependent probability measures $\mu(t)$ on $\mathbb R^d$, representing the density distribution of agents, and satisfying mesoscopic partial differential equations of the type \eqref{genkin}. This strategy originated from the kinetic theory of gases, see \cite{cip} for classical references. We show how our projection method can be applied for dimensionality reduction of the corresponding kinetic equations and explain how the probability measures can be approximated by atomic measures. Using the concepts of {\it delayed curse of dimension} and {\it measure quantization} known from optimal integration problems in high dimension, we show that under the assumption that the measure concentrates along low-dimensional subspaces (and more generally along low-dimensional sets or manifolds), it can be approximated by atomic measures with sub-exponential (with respect to $d$) number of atoms. Through such approximation, we shall show that we can approximate suitable random averages of the solution of the original partial differential equation in high dimension by tractable simulations of corresponding solutions of lower-dimensional kinetic equations.
Another interesting approach to the problem of efficient numerical simulation of large group dynamics is the so-called ``equation-free'' approach, see e.g.~\cite{mnllk}. Here, convenient coarse-grained variables that account for rapidly developing correlations during initial transients are chosen, in order to perform efficient computations of coarse-grained steady states and their bifurcation analysis. The big advantage of the equation-free approach is that the coarse-grained dynamics can be explored without the assumption of the continuum limit equation as we consider here. The premise of the method is that coarse-grained governing equations conceptually exist, but are not explicitly available in closed form. The main idea is that short bursts of appropriately initialized microscopic (fine-scale) simulations and the projection of the results onto coarse-grained variables result in time-steppers (mappings) for those variables (which is effectively the same as the discretization of the unavailable equations). One then processes the results of the short simulations to estimate various coarse-grained quantities (such as time derivatives, action of Jacobians, residuals) to perform relevant coarse-grained level numerical computations, as if those quantities were obtained from coarse-grained governing equations.
\subsection{Formal derivation of mean-field equations} In this section we briefly explain how the mean-field limit description corresponding to~\eqref{eq:dyn} can be derived. This is given, under suitable assumptions on the family of the governing functions $\mathcal F_N =\{f_i, f_{ij}: i,j =1, \dots N\}$, by the general formula \begin{equation}\label{genkin} \frac{\partial \mu}{\partial t} + \nabla \cdot ( \mathcal H_{\mathcal F}[\mu] \mu) =0, \end{equation} where $\mathcal H_{\mathcal F}[\mu]$ is a field in $\mathbb{R}^d$, determined by the sequence $\mathcal F=(\mathcal F_N)_{N \in \mathbb N}$.
In order to provide an explicit example, we show how to formally derive the mean field limit of systems of the type \begin{eqnarray}
\dot x_i &=& v_i \,, \label{eq:dyn4}\\
\dot v_i &=& \sum_{j=1}^N f^{vv}_{ij}(\mathcal{D} x, \mathcal{D} v) v_j + \sum_{j=1}^N f^{vx}_{ij}(\mathcal{D} x) x_j\,,\label{eq:dyn5} \end{eqnarray} with \[
f^{vx}_{ij}(\mathcal{D} x) &=& - \frac{\delta_{ij}} {N}\sum_{k\neq i} u(\|x_i-x_k\|_{\ell_2^d})
+ \frac{1- \delta_{ij}} {N} u(\|x_i-x_j\|_{\ell_2^d}) \,,\\
f^{vv}_{ij}(\mathcal{D} x,\mathcal{D} v) &=& \delta_{ij} \left ( h(\|v_i\|_{\ell_2^d}^2)
- \frac{1} {N} \sum_{k=1}^N g(\|x_i-x_k\|_{\ell_2^d}) \right)
+ \frac{1- \delta_{ij}} {N} g(\|x_i-x_j\|_{\ell_2^d}) \,. \end{eqnarray*} Note that for suitable choices of the functions $h,g,u$ this formalism includes both the Cucker-Smale model~\eqref{CS1}--\eqref{CS2} and the self-propulsion and pairwise interaction model~\eqref{DOrsogna1}--\eqref{DOrsogna2}. We define the empirical measure associated to the solutions $x_i(t)$, $v_i(t)$ of \eqref{eq:dyn4}--\eqref{eq:dyn5} as $$
\mu^N(t):=\mu^N(t,x,v)= \frac{1}{N} \sum_{i=1}^N \delta_{x_i(t)}(x) \delta_{v_i(t)}(v) \,. $$ Taking a smooth, compactly supported test function $\xi\in C^\infty_0(\mathbb{R}^{2d})$ and using \eqref{eq:dyn4}--\eqref{eq:dyn5}, one easily obtains by a standard formal calculation (see~\cite{CFTV}) \begin{eqnarray}
\frac{d}{dt} \langle \mu^N(t) , \xi \rangle
&=& \frac{d}{dt} \left (\frac{1}{N} \sum_{i=1}^N \xi(x_i(t),v_i(t)) \right) \label{BBGKY}\\
&=& \int_{\mathbb{R}^{2d}} \nabla_x\xi(x,v)\cdot v \,\mathrm{d} \mu^N(t,x,v)
+ \int_{\mathbb{R}^{2d}} \nabla_v\xi(x,v)\cdot \mathcal{H}[\mu^N(t)](x,v) \,\mathrm{d} \mu^N(t,x,v) \nonumber \,, \end{eqnarray} with \[
\mathcal{H}[\mu](x,v) = h(\|v\|_{\ell_2^d}) v
+ \int_{\mathbb{R}^{2d}} g(\|x-y\|_{\ell_2^d}) (w-v) \,\mathrm{d}\mu(y,w)
+ \int_{\mathbb{R}^{2d}} u(\|x-y\|_{\ell_2^d}) (y-x) \,\mathrm{d}\mu(y,w) \,. \end{eqnarray*} We now assume weak convergence of a subsequence of $(\mu^N(t))_{N \in \mathbb N}$ to a time-dependent measure $\mu(t)=\mu(t,x,v)$ and boundedness of its first order moment, which indeed can be established rigorously for the Cucker-Smale and the self-propulsion and pairwise interaction systems (see~\cite{TH},~\cite{DOrsogna}). Then, passing to the limit $N\to\infty$ in~\eqref{BBGKY}, one obtains in the strong formulation that $\mu$ is governed by $$
\frac{\partial \mu}{\partial t}(t,x,v) + v \cdot \nabla_x \mu(t,x,v)
+ \nabla_v \cdot \left ( \mathcal H[\mu(t)](x,v) \mu(t,x,v) \right ) = 0 \,, $$ which is an instance of the general prototype \eqref{genkin}.
Using the same formal arguments as described above, one can easily derive mean field limit equations corresponding to~\eqref{eq:dyn} with different choices of the family $\mathcal F$.
\subsection{Monge-Kantorovich-Rubinstein distance and stability} In several relevant cases, and specifically for the Cucker-Smale and the self-propulsion and pairwise interaction systems \cite{CCR}, solutions of equations of the type \eqref{genkin} are stable with respect to suitable distances. We consider the space $\mathcal P_1(\mathbb R^d)$, consisting of all probability measures on $\mathbb R^d$ with finite first moment. In $\mathcal P_1(\mathbb R^d)$ and for solutions of \eqref{genkin}, a natural metric to work with is the so-called {\it Monge-Kantorovich-Rubinstein distance} (also called {\it Wasserstein distance}) ~\cite{Villani}, \begin{equation}\label{wasserstein}
W_1(\mu,\nu) := \sup \{ \left | \langle \mu - \nu, \xi \rangle \right |= \left| \int_{\mathbb R^d} \xi(x) d(\mu-\nu)(x) \right |, \xi \in {\rm Lip}(\mathbb R^d), {\rm Lip}(\xi) \leq 1 \}. \end{equation} We further denote $\mathcal P_c(\mathbb R^d)$ the space of compactly supported probability measures on $\mathbb R^d$. In particular, throughout the rest of this paper, we will assume that for any compactly supported measure valued weak solutions $\mu(t),\nu(t) \in C([0,T],\mathcal P_c(\mathbb R^d))$ of \eqref{genkin} we have the following stability inequality \begin{equation}\label{stability} W_1(\mu(t), \nu(t)) \leq C(t) W_1(\mu(0), \nu(0)), \quad t \in [0,T], \end{equation} where $C(t)$ is a positive increasing function of $t$ with $C(0)>0$, independent of the dimension $d$. We address the interested reader to \cite[Section 4]{CCR} for a sample of general conditions on the vector field $\mathcal H[\mathcal F](\mu)$ which guarantee stability \eqref{stability} for solutions of equations \eqref{genkin}.
\subsection{Dimensionality reduction of kinetic equations} Provided a high-dimensional measure valued solution to the equation \begin{equation}\label{genkin2} \frac{\partial \mu}{\partial t} + \nabla \cdot ( \mathcal H_{\mathcal F}[\mu] \mu) =0, \quad \mu(0)= \mu_0 \in \mathcal P_c(\mathbb R^d) \,, \end{equation} we will study the question whether its solution can be approximated by suitable projections in lower dimension.
Given a probability measure $\mu \in \mathcal P_1(\mathbb R^d)$, its projection into $\mathbb{R}^k$ by means of a matrix $M:\mathbb R^d \to \mathbb R^k$ is given by the {\it push-forward} measure $\mu_M := M \# \mu$, \begin{equation}\label{pf} \langle \mu_M, \varphi \rangle := \langle \mu, \varphi(M \cdot) \rangle \quad \mbox{for all } \varphi \in {\rm Lip}(\mathbb R^{k}). \end{equation} Let us mention two explicit and relevant examples: \begin{itemize} \item If $\mu^N= \frac{1}{N}\sum_{i=1}^N \delta_{x_i}$ is an atomic measure, we have $\langle \mu_M^N, \varphi \rangle = \langle \mu^N, \varphi(M \cdot) \rangle = \frac{1}{N}\sum_{i=1}^N \varphi(M x_i)$. Therefore, \begin{equation}\label{projatom} \mu_M^N = \frac{1}{N}\sum_{i=1}^N \delta_{M x_i}\,. \end{equation} \item If $\mu$ is absolutely continuous with respect to the Lebesgue measure, i.e., it is a function in $L^1(\mathbb R^d)$, the calculation requires a bit more effort: Let us consider $M^\dagger$ the pseudo-inverse matrix of $M$. Recall that $M^\dagger = M^* (M M^*)^{-1}$ is a right inverse of $M$, and $M^\dagger M$ is the orthogonal projection onto the range of $M^*$. Moreover, $x= M^\dagger M x + \xi_x$, where $\xi_x \in \ker M$ for all $x \in \mathbb R^d$. According to these observations, we write \begin{eqnarray*} \int_{\mathbb R^d} \varphi (M x) \mu (x)dx &=& \int_{\mathbb R^d} \varphi (M x) \mu (M^\dagger M x + \xi_x)dx\\ &=& \int_{{\rm ran} M^* \oplus \ker M} \varphi (M x) \mu (M^\dagger M x + \xi_x)dx\\ &=& \int_{{\rm ran} M^* } \int_{\ker M} \varphi (M v) \mu (M^\dagger M v + v^\perp)dv^\perp dv \end{eqnarray*}
Note now that $M_{| {\rm ran} M^*} : {\rm ran} M^* \to {\rm ran} M \eqsim \mathbb R^k$ is an isomorphism, hence $y = M v$ implies the change of variables $dv= \det(M_{| {\rm ran} M^*})^{-1} d y = \det(M M^*)^{-1/2} d y$. Consequently, we have \begin{eqnarray*} \int_{\mathbb R^d} \varphi (M x) \mu (x)dx &=& \int_{\mathbb R^d} \varphi (M x) \mu (M^\dagger M x + \xi_x)dx\\ &=& \int_{{\rm ran} M^* } \int_{\ker M} \varphi (M v) \mu (M^\dagger M v + v^\perp)dv^\perp dv\\ &=& \int_{\mathbb R^k} \left (\frac{1}{\det(M M^*)^{1/2}} \int_{\ker M} \mu (M^\dagger y + v^\perp)dv^\perp \right) \varphi (y) dy \,, \end{eqnarray*} and $$ \mu_M(y) = \frac{1}{\det(M M^*)^{1/2}} \int_{\ker M} \mu (M^\dagger y + v^\perp)dv^\perp. $$ \end{itemize} According to the notion of push-forward, we can consider the measure valued function $\nu \in C([0,T],\mathcal P_c (\mathbb R^k))$, solution of the equation \begin{equation}\label{genkin3} \frac{\partial \nu}{\partial t} + \nabla \cdot ( \mathcal H_{\mathcal F_M}[\nu] \nu) =0, \quad \nu(0)= (\mu_0)_M \in \mathcal P_c(\mathbb R^k), \end{equation} where $(\mu_0)_M = M \# \mu_0$ and $\mathcal F_M= (\{M f_i, f_{ij}, i,j=1,\dots, N\})_{N \in \mathbb N}$. As for the dynamical system \eqref{lowdimsys}, also equation \eqref{genkin3} is fully defined on the lower-dimensional space $\mathbb R^k$ and depends on the original high-dimensional problem exclusively by means of the initial condition.
The natural question at this point is whether the solution $\nu$ of \eqref{genkin3} provides information about the solution $\mu$ of \eqref{genkin2}. In particular, similarly to the result of Theorem~\ref{Thm:Cont}, we will examine whether the approximation $$ \nu(t) \approx \mu_M(t), \quad t \in [0,T], $$ in Monge-Kantorovich-Rubinstein distance is preserved in finite time. We depict the expected result by the following diagram: \begin{equation*} \begin{matrix} &\mu(0) & \stackrel{t}{\longrightarrow} & \mu(t) & \cr &\downarrow M & &\downarrow M& \cr & \nu(0)=(\mu_0)_M&\stackrel{t}{\longrightarrow} &\nu(t) \approx \mu_M(t)&. \end{matrix} \end{equation*} This question will be addressed by approximation of the problem by atomic measures and by an application of Theorem~\ref{Thm:Cont} for the corresponding dynamical system, as concisely described by \begin{equation*} \begin{matrix} &\mu & \stackrel{W_1(\mu,\,\mu^N) \lesssim \varepsilon}{\longrightarrow} & \mu^N & \cr & \downarrow M & & \downarrow M& \cr & \nu \approx \mu_M &\stackrel{W_1(\nu,\,\nu^N) \lesssim \varepsilon}{\longrightarrow} & \nu^N \approx \mu^N_M & \end{matrix} \end{equation*} Let us now recall the framework and general assumptions for this analysis to be performed. We assume again that for all $N\in \mathbb N$ the family $\mathcal F_N =\{f_i, f_{ij}: i,j =1, \dots N\}$ is composed of functions satisfying \eqref{eq:condf1}-\eqref{eq:condf3}. Moreover, we assume that associated to $\mathcal F = (\mathcal F_N)_{N \in \mathbb N}$ and to \begin{equation}\label{eq:dyn6} \dot x_i(t)=f_i(\mathcal{D} x(t))+\sum_{j=1}^N f_{ij}(\mathcal{D} x(t))x_j(t), \end{equation} we can define a mean-field equation \begin{equation}
\frac{\partial \mu}{\partial t} + \nabla \cdot ( \mathcal H[\mathcal F](\mu) \mu) =0, \quad \mu(0)=\mu_0 \in \mathcal P_c(\mathbb R^d), \end{equation} such that for any compactly supported measure valued weak solutions $\mu(t),\nu(t) \in C([0,T],\mathcal P_c(\mathbb R^d))$ of \eqref{genkin} we have the following stability \begin{equation}\label{stability2} W_1(\mu(t), \nu(t)) \leq C(t) W_1(\mu(0), \nu(0)), \quad t \in [0,T], \end{equation} where $C(t)$ is a positive increasing function of $t$, independent of the dimension $d$. We further require that corresponding assumptions, including stability, hold for the projected system \eqref{eq:eul3} and kinetic equation \eqref{genkin3}. Then we have the following approximation result:
\begin{theorem}\label{Thm:Kin} Let us assume that $\mu_0 \in \mathcal P_c(\mathbb R^d)$ and there exist points $\{x_1^0, \dots, x_N^0\} \subset \mathbb R^d$, for which the atomic measure $\mu_0^N= \frac{1}{N} \sum_{i=1}^N \delta_{x_i^0}$ approximates $\mu_0$ up to $\varepsilon>0$ in Monge-Kantorovich-Rubinstein distance, in the following sense \begin{equation}\label{epskin} W_1(\mu_0, \mu_0^N) \leq \varepsilon,\quad N=\mathcal N^{\overline k(\varepsilon)}\mbox{ for } \overline k(\varepsilon) \leq d \mbox{ and } \overline k(\varepsilon) \to d \mbox{ for } \varepsilon \to 0. \end{equation} Requirement \eqref{epskin} is in fact called the \emph{delayed curse of dimension} as explained below in detail in Section \ref{delay}. Depending on $\varepsilon>0$ we fix also $$ k = k(\varepsilon) =\mathcal O(\varepsilon^{-2} \log(N)) = \mathcal O(\varepsilon^{-2} \log(\mathcal N) \overline k(\varepsilon)). $$ Moreover, let $M:\mathbb R^d \to \mathbb R^k$ be a linear mapping which is a \emph{continuous Johnson-Lindenstrauss embedding} as in \eqref{Cont-JL} for continuous in time trajectories $x_i(t)$ of \eqref{eq:dyn6} with initial datum $x_i(0)=x_i^0$. Let $\nu \in C([0,T], \mathcal P_c(\mathbb R^k))$ be the weak solution of \begin{eqnarray}\label{genkin4} &&\frac{\partial \nu}{\partial t} + \nabla \cdot ( \mathcal H[\mathcal F_M](\nu) \nu) =0, \\ && \nu(0)= (\mu_0)_M \in \mathcal P_c(\mathbb R^k),\label{genkin5} \end{eqnarray} where $(\mu_0)_M = M \# \mu_0$. Then \begin{equation}\label{eq:approxkin}
W_1(\mu_M(t),\nu(t))\leq \mathcal C(t) \|M\| \varepsilon, \quad t \in [0,T], \end{equation} where $\mathcal C(t)$ is an increasing function of $t$, with $\mathcal C(0) >0$, which is at most polynomially growing with the dimension $d$. \end{theorem} \begin{proof} Let us define $\nu^N(t)$ the solution to equation \eqref{genkin4} with initial datum $\nu^N(0)=(\mu_0^N)_M$, or, equivalently, thanks to \eqref{projatom} $$ \nu^N(t) = \frac{1}{N} \sum_{i=1}^n \delta_{y_i(t)}, $$ where $y_i(t)$ is the solution of \begin{eqnarray*}
\dot y_i &=& f_i(\mathcal{D}' y) + \sum_{j=1}^N f_{ij}(\mathcal{D}' y) y_j \,,\qquad i=1,\dots, N\,, \nonumber\\
y_i(0) &=& Mx_i^0 \,,\qquad i=1,\dots, N\,. \end{eqnarray*} We estimate $$ W_1(\mu_M(t), \nu(t)) \leq W_1(\mu_M(t),(\mu^N(t))_M) + W_1((\mu^N(t))_M, \nu^N(t))+ W_1(\nu^N(t), \nu(t)). $$ By using the definition of push-forward \eqref{pf} and \eqref{epskin}, the first term can be estimated by \begin{eqnarray*} W_1(\mu_M(t),(\mu^N(t))_M) &=& \sup\{\langle \mu_M(t) - (\mu^N(t))_M,\varphi \rangle: {\rm Lip}(\varphi) \leq 1\} \\ &=& \sup\{\langle \mu(t) - \mu^N(t),\varphi(M \cdot) \rangle: {\rm Lip}(\varphi) \leq 1\}\\
&\leq& \|M\| W_1(\mu(t), \mu^N(t)) \leq \|M\| C(t) \varepsilon. \end{eqnarray*} We estimate now the second term \begin{eqnarray*} W_1((\mu^N(t))_M, \nu^N(t)) &=& \sup\{\langle (\mu^N(t))_M- \nu^N(t),\varphi \rangle: {\rm Lip}(\varphi) \leq 1\} \\ &=& \sup\{\frac{1}{N} \sum_{i=1}^N (\varphi(M x_i(t)) - \varphi(y_i(t))): {\rm Lip}(\varphi) \leq 1\}\\
&\leq& \frac{1}{N} \sum_{i=1}^N \| Mx_i(t) - y_i(t)\|_{\ell_2^k}. \end{eqnarray*} We recall the uniform approximation of Theorem~\ref{Thm:Cont}, $$
\| Mx_i(t) - y_i(t)\|_{\ell_2^k} \leq D(t) \varepsilon\,,\qquad i=1,\dots, N, $$ where $D(t)$ is the time-dependent function on the right-hand-side of \eqref{eq:unifstab2}. Hence $$ W_1(\mu_M(t),(\mu^N(t))_M) \leq D(t) \varepsilon. $$ We address now the upper estimate of the third term, by the assumed stability of the lower dimensional equation \eqref{genkin3} \begin{eqnarray*} W_1(\nu^N(t), \nu(t)) &\leq& C(t) W_1(\nu^N(0), \nu(0)) \\ &= & C(t) W_1((\mu^N_0)_M, (\mu_0)_M) \\
&\leq& C(t) \|M\| W(\mu^N_0,\mu_0) \leq C(t) \|M\| \varepsilon. \end{eqnarray*}
We can fix $\mathcal C(t) = 2 C(t) \|M\| + D(t)$, and, as observed in Theorem \ref{thmcontJL}, we can assume without loss of generality that $\|M\| \leq \sqrt{\frac{d}{k}}$. Hence, $\mathcal C(t)$ depends at most polynomially with respect to the dimension $d$. \end{proof}
\subsection{Approximation of probability measures by atomic measures and optimal integration}\label{sec:appr}
In view of the fundamental requirement \eqref{epskin} in Theorem \ref{Thm:Kin}, given $\mu_0 \in \mathcal P_c(\mathbb R^d)$, we are interested to establish an upper bound to the best possible approximation in Monge-Kantorovich-Rubinstein distance by means of atomic measures $\mu^N_0 = \frac{1}{N}\sum_{i=0}^{N-1} \delta_{x_i^0}$ with $N$ atoms, i.e., \begin{eqnarray} \mathcal E_N(\mu_0)&:=&\inf_{\mu^N_0= \frac{1}{N}\sum_{i=0}^{N-1} \delta_{x_i^0}} W_1(\mu_0,\mu^N_0) \label{optint} \\
&=& \inf_{\{x_0^0,\dots, x_{N-1}^0\} \subset \mathbb R^d} \sup \big \{ | \int_{\mathbb R^d} \xi(x) d\mu_0(x) - \frac{1}{N} \sum_{i=0}^{N-1} \xi(x_i^0) |: \xi \in {\rm Lip}(\mathbb R^d), {\rm Lip}(\xi) \leq 1 \big \} \nonumber. \end{eqnarray} In fact, once we identify the optimal points $\{x_0^0,\dots, x_{N-1}^0\}$, we can use them as initial conditions $x_i(0)=x_i^0$ for the dynamical system \eqref{eq:dyn6}, and by using the stability relationship \eqref{stability}, we obtain \begin{equation}\label{stability_red} W_1(\mu(t), \mu^N(t)) \leq C(T) W_1(\mu_0, \mu^N_0), \quad t \in [0,T] \,, \end{equation} where $\mu^N(t) = \frac{1}{N} \sum_{i=0}^{N-1} \delta_{x_i(t)}$, meaning that the solution of the partial differential equation \eqref{genkin} keeps optimally close to the particle solution of \eqref{eq:dyn6} also for successive time $t>0$. Note that estimating \eqref{optint} as a function of $N$ is in fact a very classical problem in numerical analysis well-known as {\it optimal integration} with its high-dimensional behaviour being a relevant subject of the field of {\it Information Based Complexity} \cite{NW,TWW}.
The numerical integration of Lipschitz functions with respect to the Lebesgue measure and the study of its high-dimensional behaviour goes back to Bakhvalov \cite{Bakh}, but much more is known nowadays. We refer to \cite{GL} and \cite{Gruber} for the state of the art of quantization of probability distributions.
The scope of this section is to recall some facets of these estimates and to reformulate them in terms of $W_1$ and ${\mathcal E}_N$. We emphasize that here and in what follows, we consider generic compactly supported probability measures $\mu$, not necessarily absolutely continuous with respect to the Lebesgue measure. We start first by assuming $d=1$, i.e., we work with a univariate measure $\mu \in \mathcal P_c(\mathbb R)$ with support ${\rm supp\, } \mu \subset [a, b]$ and $\sigma := b - a >0$. We define the points $x_0,\dots,x_{N-1}$ as the {\it quantiles} of the probability measure $\mu$, i.e., $x_0:=a$ and \begin{equation}\label{quantiles} \frac{i}{N} = \int_{-\infty}^{x_i} d\mu(x), \quad i=1,\dots, N-1. \end{equation} This is notationally complemented by putting $x_N:=b$. Note that by definition $\int_{x_{i}}^{x_{i+1}} d\mu(x) =\frac{1}{N}, i=0,\dots, N-1$, and we have \begin{eqnarray}
\notag\left|\int_{\mathbb R} \xi(x) d\mu(x) - \frac{1}{N} \sum_{i=0}^{N-1} \xi(x_i) \right | &=& \left | \sum_{i=0}^{N-1} \int_{x_i}^{x_{i+1}} (\xi(x)- \xi(x_i)) d\mu(x) \right |\\
\label{quantiles2}&\leq& \sum_{i=0}^{N-1} \int_{x_i}^{x_{i+1}}\left |\xi(x)- \xi(x_i)\right | d\mu(x) \\ \notag&\leq& \frac{{\rm Lip}(\xi)}{N} \sum_{i=0}^{N-1} (x_{i+1}-x_i) = \frac{{\sigma} \rm Lip(\xi)}{N}. \end{eqnarray} Hence it is immediate to see that $$ \mathcal E_N(\mu)=\inf_{\mu^N= \frac{1}{N}\sum_{i=0}^{N-1} \delta_{x_i^0}} W_1(\mu,\mu^N) \leq \frac{{\sigma}}{N}. $$ We would like to extend this estimate to higher dimension $d >1$. However, for multivariate measures $\mu$ there is no such an easy upper bound, see \cite{GL} and \cite{Gruber} for very general statements, and for the sake of simplicity we restrict here the class of measures $\mu$ to certain special cases. As a typical situation, we address tensor product measures and sums of tensor products. \begin{lemma}\label{lem:W:tensors} Let $\mu^1,\dots,\mu^d\in \mathcal P_1(\mathbb{R})$ with $W_1(\mu^j,\mu^{j,N_j})\le \varepsilon_j,$ $j=1,\dots, d$ for some $N_1,\dots,N_d\in\mathbb{N}$, $\varepsilon_1,\dots,\varepsilon_d >0$ and $\mu^{j,N_j}:=\frac{1}{N_j}\sum_{i=0}^{N_j-1}\delta_{x^j_{i}}$. Let $N=\prod_{i=1}^d N_i$. Then \begin{equation*} W_1(\mu^1\otimes\dots\otimes \mu^d,\mu^{N})\le \sum_{j=1}^d \varepsilon_j, \end{equation*} where \begin{equation*} \mu^N:=\frac{1}{N}\sum_{x\in X}\delta_x\quad\text{and}\quad X:=\prod_{j=1}^d\{x^j_0,\dots,x^j_{N_j-1}\}. \end{equation*} \end{lemma} \begin{proof} The proof is based on a simple argument using a telescopic sum. For $j=1,\dots,d+1$ we put \begin{align*} V_j&:=\frac{1}{\prod_{i=j}^d N_{i}}\sum_{i_{j}=0}^{N_{j}-1}\dots\sum_{i_d=0}^{N_d-1}\int_{\mathbb{R}^{j-1}} \xi(x_1,\dots,x_{j-1},x^{j}_{i_{j}},\dots,x^d_{j_d}) d\mu^1(x_1)\dots d\mu^{j-1}(x_{j-1}). \end{align*} Of course, if $j=1$, then the integration over $\mathbb{R}^{j-1}$ is missing and if $j=d+1$ then the summation becomes empty. Now $$ \int_{\mathbb{R}^d}\xi(x) d\mu(x)-\frac{1}{\prod_{i=1}^d N_i}\sum_{i_1=0}^{N_1-1}\dots\sum_{i_d=0}^{N_d-1}\xi(x^{1}_{i_{1}},\dots,x^d_{i_d}) =\sum_{j=1}^{d}(V_{j+1}-V_j) $$
together with the estimate $|V_{j+1}-V_j|\le \varepsilon_j$ finishes the proof. \end{proof}
Lemma \ref{lem:W:tensors} says, roughly speaking, that the tensor products of sampling points of univariate measures are good sampling points for the tensor product of the univariate measures. Next lemma deals with sums of measures.
\begin{lemma}\label{lem:W:sums} Let $\mu_1,\dots,\mu_L\in \mathcal{P}_1(\mathbb{R}^d)$ with $W_1(\mu_l,\mu_l^{N})\le \varepsilon_l$, $l=1,\dots, L$ for some $N\in\mathbb{N}$, $\varepsilon_1,\dots,\varepsilon_L >0$ and $\mu_l^{N}:=\frac{1}{N}\sum_{i=0}^{N-1}\delta_{x_{l,i}}$. Then \begin{equation*} W_1\Bigl(\frac{\mu_1+\dots+ \mu_L}{L},\mu^{LN}\Bigr)\le \frac{1}{L}\sum_{l=1}^L \varepsilon_l, \end{equation*} where \begin{equation*} \mu^{LN}:=\frac{1}{LN}\sum_{x\in X}\delta_x=\frac{1}{L}\sum_{l=1}^L \mu_l^{N}\quad\text{and}\quad X:=\bigcup_{l=1}^L\{x_{l,0},\dots,x_{l,N-1}\}. \end{equation*} \end{lemma} \begin{proof} We use the homogeneity of the Monge-Kantorovich-Rubinstein distance $W_1(a\mu,a\nu)=aW_1(\mu,\nu)$ for $\mu,\nu\in\mathcal P_1(\mathbb{R}^d)$ and $a\ge 0$ combined with its subadditivity $W_1(\mu_1+\mu_2,\nu_1+\nu_2)\le W_1(\mu_1,\nu_1)+W_1(\mu_2,\nu_2)$ for $\mu_1,\mu_2,\nu_1,\nu_2\in\mathcal P_1(\mathbb{R}^d)$. We obtain \begin{equation*} W_1\Bigl(\frac{\mu_1+\dots+ \mu_L}{L},\frac{\mu_1^{N}+\dots+\mu_L^{N}}{L}\Bigr)\le \frac{1}{L}\sum_{l=1}^L W_1(\mu_l,\mu_l^{N})\le \frac{1}{L}\sum_{l=1}^L \varepsilon_l. \end{equation*} \end{proof}
Next corollary follows directly from Lemma \ref{lem:W:tensors} and Lemma \ref{lem:W:sums}. \begin{corollary}\label{cor1} (i) Let $\mu^1,\dots,\mu^d\in \mathcal P_1(\mathbb{R})$ and $N_1,\dots,N_d\in\mathbb{N}$.
Then \begin{equation*} \mathcal E_{N}(\mu^1\otimes\dots\otimes\mu^d)\le \sum_{j=1}^d \mathcal E_{N_j}(\mu^j),\quad \text{where} \quad N:=N_1\cdots N_d. \end{equation*} (ii) Let $\mu_1,\dots,\mu_L\in \mathcal{P}_1(\mathbb{R}^d)$ and $N\in\mathbb{N}$. Then \begin{equation*} {\mathcal E}_{LN}\Bigl(\frac{\mu_1+\dots+ \mu_L}{L}\Bigr)\le \frac{1}{L}\sum_{l=1}^L {\mathcal E}_N(\mu_l). \end{equation*} \end{corollary}
\subsection{Delayed curse of dimension}\label{delay}
Although Lemma \ref{lem:W:tensors}, Lemma \ref{lem:W:sums} and Corollary \ref{cor1} give some estimates of the Monge-Kantorovich-Rubinstein distance between general and atomic measures, the number of atoms needed may still be too large to allow the assumption \eqref{epskin} in Theorem \ref{Thm:Kin} to be fulfilled. Let us for example consider the case, where $\mu^1=\dots=\mu^d$ in Lemma \ref{lem:W:tensors} and $\varepsilon_1=\dots=\varepsilon_d=:\varepsilon.$ Then, of course, $N_1=\dots=N_d=:{\mathcal N}$ and we observe, that the construction given in Lemma \ref{lem:W:tensors} gives an atomic measure, which approximates $\mu$ up to the error $d\varepsilon$ using ${\mathcal N}^d$ atoms, hence with an exponential dependence on the dimension $d$. This effect is another instance of the well-known phenomenon of the {\it curse of dimension}.
However, in many real-life high-dimensional applications the objects of study (in our case the measure $\mu\in{\mathcal P}_c(\mathbb{R}^d)$) concentrate along low-dimensional subspaces (or, more general, along low-dimensional manifolds) \cite{BN01,BN03,CLLMNWZ05a,CLLMNWZ05b,CL06}. The number of atoms necessary to approximate these measures behaves in a much better way, allowing the application of \eqref{epskin} and Theorem \ref{Thm:Kin}. To clarify this effect, let us consider $\mu=\mu^1\otimes\dots\otimes \mu^d$ with ${\rm supp\, }\mu^j\subset [a_j,b_j]$ and define $\sigma_j=b_j-a_j$. Let us assume, that $\sigma_1\ge \sigma_2\ge \dots\ge \sigma_d > 0$ is a rapidly decreasing sequence. Furthermore, let $\varepsilon>0$. Then we define $\overline k:=\overline k(\varepsilon)$ to be the smallest natural number, such that $$ \sum_{k=\overline k(\varepsilon)+1}^{d}\sigma_k\le \varepsilon/2 $$ and put $N_k=1$ for $k\in\{\overline k(\varepsilon)+1,\dots,d\}$. The numbers $N_1=\dots=N_{\overline k(\varepsilon)}={\mathcal N}$ are chosen large enough so that $$ \frac{1}{\mathcal N}\sum_{k=1}^{\overline k(\varepsilon)} {\sigma_k}\le \varepsilon/2. $$ Then Lemma \ref{lem:W:tensors} together with \eqref{quantiles} state that there is an atomic measure $\mu^N$ with $N={\mathcal N}^{\overline k(\varepsilon)}$ atoms, such that \begin{equation}\label{delayed} W_1(\mu,\mu^N)\le \sum_{k=1}^d \frac{\sigma_k}{N_k}\le \varepsilon/2+\varepsilon/2. \end{equation} Hence, at the cost of assuming that the tensor product measure $\mu$ is concentrated along a $\overline k(\varepsilon)$-dimensional coordinate subspace, we can always approximate the measure $\mu$ with accuracy $\varepsilon$ by using an atomic measure supported on points whose number depends exponentially on $\overline k=\overline k(\varepsilon) \ll d$. However, if we liked to have $\varepsilon \to 0$, then $\overline k(\varepsilon) \to d$ and again we are falling under the curse of dimension. This delayed kicking in of the need of a large number of points for obtaining high accuracy in the approximation \eqref{delayed} is in fact the so-called {\it delayed curse of dimension}, expressed by assumption \eqref{epskin}, a concept introduced first by Curbera in \cite{C00}, in the context of optimal integration with respect to Gaussian measures in high dimension.
Let us only remark, that the discussion above may be easily extended (with help of Lemma \ref{lem:W:sums}) to sums of tensor product measures. In that case we obtain as atoms the so-called \emph{sparse grids}, cf. \cite{BG}. Using suitable change of variables, one could also consider measures concentrated around (smooth) low-dimensional manifolds, but this goes beyond the scope of this work, see \cite{GL} for a broader discussion.
\subsubsection*{Acknowledgments} We acknowledge the financial support provided by the START award ``Sparse Approximation and Optimization in High Dimensions'' no. FWF~Y~432-N15 of the Fonds zur F\"orderung der wissenschaftlichen Forschung (Austrian Science Foundation). We would also like to thank Erich Novak for a discussion about optimal integration and for pointing us to some of the references given in Section \ref{sec:appr}.
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\end{document} | arXiv |
Analysis of associations between emotions and activities of drug users and their addiction recovery tendencies from social media posts using structural equation modeling
Deeptanshu Jha1 &
Rahul Singh ORCID: orcid.org/0000-0003-3900-42981
Addiction to drugs and alcohol constitutes one of the significant factors underlying the decline in life expectancy in the US. Several context-specific reasons influence drug use and recovery. In particular emotional distress, physical pain, relationships, and self-development efforts are known to be some of the factors associated with addiction recovery. Unfortunately, many of these factors are not directly observable and quantifying, and assessing their impact can be difficult. Based on social media posts of users engaged in substance use and recovery on the forum Reddit, we employed two psycholinguistic tools, Linguistic Inquiry and Word Count and Empath and activities of substance users on various Reddit sub-forums to analyze behavior underlining addiction recovery and relapse. We then employed a statistical analysis technique called structural equation modeling to assess the effects of these latent factors on recovery and relapse.
We found that both emotional distress and physical pain significantly influence addiction recovery behavior. Self-development activities and social relationships of the substance users were also found to enable recovery. Furthermore, within the context of self-development activities, those that were related to influencing the mental and physical well-being of substance users were found to be positively associated with addiction recovery. We also determined that lack of social activities and physical exercise can enable a relapse. Moreover, geography, especially life in rural areas, appears to have a greater correlation with addiction relapse.
The paper describes how observable variables can be extracted from social media and then be used to model important latent constructs that impact addiction recovery and relapse. We also report factors that impact self-induced addiction recovery and relapse. To the best of our knowledge, this paper represents the first use of structural equation modeling of social media data with the goal of analyzing factors influencing addiction recovery.
Substance use constitutes a major contemporary health epidemic. There were 70,237 substance use overdose deaths in 2017, which was a 9.6% increase from 2016 [1]. In the US, abuse of alcohol and other illicit drugs is estimated to lead to a monetary impact of over $740 billion annually because of increased expenses related to loss of work productivity, health care, and crime [2]. Substance use can also increase the risk for liver [3], or lung diseases [4], and especially infectious diseases such as Hepatitis B, or C, and HIV/AIDS [5].
Drug addiction was usually considered a moral or character flaw. This view has undergone a significant change and addiction is now considered a chronic illness characterized by health deterioration, poor social functioning, and loss of control over substance use [6]. Substance use has also been established to change the brain function and makes a user crave drugs. The substance use journey typically begins with experimentation and because of the perceived positive effects, a person gets addicted. After an individual decides to break the addiction cycle, they typically experience physical and emotional withdrawals that are manifested through sadness, restlessness, anxiety, nausea, vomiting, sweating, and cramping. Depending on factors such as the substances used as well as the amount and duration of use, such symptoms typically last for 3–5 days and can be managed by medications, vitamins, and exercise [2]. The notion of "recovery" is polysemous in that it may be considered as an ongoing process or as a granular event [7]. Regardless, recovery is a long-term process requiring continuous effort and diligence [2]. Substance withdrawal management regimes that can lead to recovery from addiction involve managing both physical and emotional symptoms experienced by individuals as they give up drugs. To manage these symptoms, individuals are typically recommended to focus on self-development [8, 9] with the help of their families, and friends [2]. Many individuals however, relapse into drug use because they fail to follow substance use disorder treatment regimens [10].
Though managing emotional and physical symptoms during drug withdrawals is manifestly important, these constructs are multifarious, latent (i.e. not directly observable), and difficult or impossible to directly measure. In this paper, we have proposed the use of structural equation modeling (SEM)—a multivariate latent variable modeling technique to estimate critical latent constructs (italicized hereafter) such as emotional distress, physical pain, self-development, and relationships by analyzing social media activities of substance users. Social media has generated recent interest as a novel source of information in drug abuse epidemiology [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. Being semi-anonymous, social media consists of unfiltered and self-reported conversations and activities of an individual. Of the different social media platforms, we used drug use and recovery data available on Reddit. This social media forum is the fifth most visited website in the USA and has over 330 million active users [26]. Reddit is a community-based social media forum where the communities (called subreddits) are created based on common interest. Members of the subreddit can post, vote, and comment in the subreddit. Each subreddit has moderators who ensure that the content posted by the members of the subreddit are topically focused. At the time of writing, there are more than 138,000 subreddits on Reddit [26], with a number of subreddits focusing on recreational drug use (RDU) and drug addiction recovery (DAR).
Problem formulation and overview of proposed approach
Our aim was to determine the effect of emotional distress, physical pain, self-development efforts, relationships (of the drug user), and geographic disparities on drug addiction recovery and relapse, using SEM as a rigorous modeling methodology. Solving this problem required addressing the following sub-problems: first, we needed to identify and determine the instances of emotional distress, physical pain, self-development efforts, relationships, and geographic disparities in the social media posts and activity of the drug users. Then, we had to come up with a model to infer the relationships between the unobserved constructs (emotional distress, physical pain, self-development efforts, and relationships) and the observable construct drug addiction recovery (determined by observing if a user posted in a drug addiction recovery forum). Our approach consisted of the following steps: (1) we used two psychometrically validated dictionaries, namely, Linguistic Inquiry and Word Count (LIWC) and Empath, to identify instances of emotional distress, physical pain, relationships, self-development efforts, and geographic disparities present in the posts of the drug user. (2) We also utilized the forum activity of the users on Reddit to identify the instances of self-development efforts and relationships. (3) We applied SEM to identify and quantify the relationship between emotional distress, physical pain, self-development, relationships, and geographic disparities on one hand and drug addiction recovery and relapse on the other.
A number of recent works have utilized data from social media in conjunction with methods from machine learning and natural language processing to study and understand patterns associated with a diverse set of health-related issues, such as influenza [27], mental health [28], and suicidal ideation [29]. In terms of studying substance abuse, early works focused on manual identification of themes and tonality of the drug use posts on social media [12, 13]. The image-based social media platform Instagram was analyzed to conduct content analysis for codeine misuse in [14]. Studies have also investigated the use of social media for examining geographic differences in opioid-related discussions [15] and identified topics related to substance delivery methods, drug types, and other factors associated with recreational drug use [16]. In [17] transductive classification was applied to identify opioid addicts on Twitter. Other works have identified opioid use related tweets [18] and studied information sharing amongst drug users on Reddit [19]. Drug addiction recovery has been the focus of far fewer works. Among the latter, in our previous work Eshleman et al. [20], random forests were used with subreddit activity as features to identify users open to addiction recovery interventions in a predictive setting. The Gini impurity criterion, which measures how often a random element from a set would be labeled incorrectly if labeled according to the distribution of labels in the set, was used to rank the different subreddits on the basis of their importance. This analysis found correlations amongst subreddit categories, such as, mental health, spirituality, and relationships with addiction recovery behavior. The SEM model in the current work was developed using two latent variables—"relationships" and "mental and physical well-being", both of which were directly inspired by findings reported in [20]. In particular, we used user activity in the following subreddits: "relationships", "relationship_advice", "parenting", and "childfree" to reflect the latent variable "relationships". Similarly, we used subreddits, such as, "meditation", "yoga", "gainit", "bodyweightfitness", and "running" to estimate the latent variable "mental and physical well-being". In other works, MacLean et al. [21], used a trans-theoretical model of behavior change to predict the stages of addiction recovery and relapse. Lu et al. [22], used the cox regression model to identify transitions to addiction recovery subreddits. Chancellor et al. [23], studied recovery-related posts on Reddit to identify clinically unverified treatments for drug withdrawal popular amongst drug users on Reddit. Rubya et al. [24]., investigated how users in online recovery communities enact anonymity Finally, Tamersoy et al. [25], studied Reddit forums to characterize smoking and drinking abstinence and were able to predict long-term and short-term abstinence.
The current work addresses two outstanding issues in this problem domain at the state-of-the-art: first, drug addiction recovery and relapse involves (latent) variables that cannot be directly measured and have to be inferred from observable variables. Second, the addiction and recovery processes involve complex interplay of relationships between the observed and latent variables, which needs to be characterized. Current methods in the area involve variables that have to be explicitly measured and consequently are incapable of addressing these two issues. We demonstrate how SEM can be a powerful framework to test, evaluate, and characterize multivariate causal relationships in addiction recovery and relapse where both observable and latent factors are involved.
The withdrawal management model obtained using LIWC variables
In Table 1 and Fig. 1 we present the correlations between the LIWC indicators in the withdrawal management model. From this data we observe that the majority of the LIWC variables are positively correlated with each other. We also observe some correlations that are not so obvious. For example, we see that the second (0.78) and third highest (0.72) correlations were for the categories "swear" and "sexual", and "anger" and "sexual". As displayed in Table 2, the high correlation was due to common expletives in these categories. We also see that the LIWC category "health" had high correlation values with categories, such as "negative emotion" (0.39), "sad" (0.25), and "anxiety" (0.28). This indicates that users in our dataset usually talked about health (physical symptoms) in the context of negative emotions- as may be expected for users experiencing withdrawals.
Table 1 Correlation matrix of the LIWC variables present in the withdrawal management model
Table 2 LIWC variables in our model for users who display or do not display addiction recovery tendencies
Correlation diagram of the LIWC variables present in the withdrawal management model (see also Table 1). Positive correlations are color-coded in blue and negative correlations in red. The size of each square represents the magnitude of the correlations. As this visualization indicates, every variable-pair in the model was positively correlated. The two highest correlations values were observed for the variable pairs "anger" and "swear" followed by "anger" and "sexual"
In Table 2 we compare the values of the indicators for "emotional distress", and "physical pain" between the users who posted or did not post in DAR subreddits. The corresponding table for Empath variables is presented in Additional file 1: Table S1. We used LIWC to determine the value of each indicator for the posts of drug users in our dataset. Then the distributions of the values of indicators for the set of users who posted in a DAR subreddit was compared with the set of users who did not post in a DAR subreddit with the null hypothesis being that there was no difference between the distributions. The Mann–Whitney U-test [30], a non- parametric test, was used to compare the distributions and we observe statistically significant differences between the two set of users for each observable variable.
The values of the indicators of the latent variable "emotional distress" were found to be higher for the users who displayed addiction recovery behavior. Posts corresponding to addiction recovery behavior typically consisted of higher values for the LIWC categories: "feel" (20%, p < 0.005), "anger" (22.2%, p < 0.005), "authentic" (9%, p < 0.005), "sexual" (13.3%, p < 0.005), "negative emotion" (20%, p < 0.005), "sad" (25%, p < 0.005), "affect" (7.5%, p < 0.005), "anxiety" (26.0%, p < 0.005), and "swear" (16.6%, p < 0.005) as compared to the other LIWC categories used by us (Table 2).
Similarly, the values for the indicators of the latent variable "physical pain" were higher for the users who displayed addiction recovery behavior. Accordingly, our data shows that drug users complained about their health and physical discomforts during the withdrawal phase. Correspondingly, these posts were found to have higher values for the relevant LIWC categories: "body" (5.4%, p < 0.005), "health" (30.3%, p < 0.005), "biology" (13.3%, p < 0.005), and "death" (28.5%, p < 0.005) (Table 2).
Path analysis
Figure 2 displays the final LIWC withdrawal management model with factor loadings (the value for correlations are not displayed in the figure to maintain clarity). In Fig. 2, the effect of the variables "emotional distress" and "physical pain" on drug addiction recovery behavior is studied. We estimated the latent variable "emotional distress" with nine LIWC categories: "negative emotion", "sad", "anger", "anxiety", "feel", "affect", "swear", "sexual", and "authentic". The latent variable "physical pain" was estimated using four indicators "biology", "death", "health", and "body". All of the paths in the model were found to be statistically significant. Both "emotional distress" and "physical pain" were found to influence addiction recovery behavior. However, "emotional distress" was found to be more evident in withdrawal as compared to "physical pain"; all of the indicator variables for "emotional pain" were found to have a strong effect on withdrawal, with the LIWC categories "anger" and "swear" being the two most significant indicators.
The LIWC withdrawal management model. Ellipses indicate latent variables, rectangles represent observed variables, straight line with one arrowhead represents a direct effect, and a curved line represents covariance. As indicated by this model emotional and physical pain positively affects the recovery propensity of a drug user. However, for the LIWC indicators emotional factors were found to be more important than physical factors
RMSEA, SRMR, CFI, and TLI were used to assess the model fit. The results based on the hypothesized model indicated a decent fit with RMSEA = 0.08, TLI = 0.90, CFI = 0.95, and SRMR = 0.07. The relatively higher value observed for the RMSEA was due to the covariance between the LIWC categories. These covariances increased the number of paths that had to be estimated in the model, reduced the degrees of freedom of the model, and led to relatively higher RMSEA values. The values for the TLI, CFI, and SRMR indices all indicate high-quality model fit. Table 3 summarizes the results of the final SEM model.
Table 3 Latent variable structure, direct effects, and covariances the final LIWC withdrawal management SEM model
The withdrawal management model obtained using Empath variables
In Table 4 and Fig. 3 we present the correlations between the Empath indicators for the withdrawal management model. Similar to the LIWC variables, all of the Empath variables in the model were also found to be positively correlated with each other with the categories "pain" and "shame" (0.89) followed by "suffering" and "hate" (0.71) having the highest correlation values. The Empath category "suffering" was also found to be correlated with "medical_emergency" (0.22), "weakness" (0.25), "health" (0.34), and "pain" (0.69) indicating that users in the withdrawal phase discussed physical symptoms in the context of distress. In Additional file 1: Table S1 we compare the values of the Empath based indicators for "emotional distress", and "physical pain" between the users who post and do not post in DAR subreddits.
Table 4 Correlation matrix of the Empath variables present in the withdrawal management model
Correlation diagram of the Empath variables present in the withdrawal management model (see also Table 4). Positive correlations are color-coded in blue and negative correlations in red. The size of each square represents the magnitude of the correlations. As this visualization indicates, every variable-pair in the model is positively correlated. The two highest correlation values were observed for the variable-pairs "pain" and "shame" followed by "suffering" and "hate"
Figure 4 displays the Empath indicator-based withdrawal management model with factor loadings (the value for correlations are not displayed in the figure to maintain clarity). In this figure, the effect of "emotional distress" and "physical pain" on drug addiction recovery behavior is studied. We estimated the latent variable "emotional distress" with four Empath categories: "negative_emotion", "hate", "shame", and "suffering" The latent variable "physical pain" was estimated using four indicators ""pain", "medical_emergency", "weakness", and "health". All of the paths in the model were found to be statistically significant. As was the case for the model built using LIWC indicators, both "emotional distress" and "physical pain" were found to influence addiction recovery behavior. All of the indicators for "emotional distress" had a strong positive effect, with "shame" and "suffering" being the most contributory. Similarly, all of the indicators for the "physical pain" had a strong positive effect, with "pain" having the highest effect. As opposed to the LIWC model, however, "physical pain" was found to be more evident in withdrawal as compared to "emotional distress". The model quality was determined using RMSEA, SRMR, CFI, and TLI. The hypothesized model indicated a good fit with RMSEA = 0.07, TLI = 0.96, CFI = 0.98, and SRMR = 0.03. Similar to the LIWC model, the relatively higher value observed for the RMSEA was due to the covariance between the Empath categories. The values for the TLI, CFI, and SRMR indices all indicate high-quality model fit. Table 5 summarizes this SEM model.
The Empath indicators-based withdrawal management model. Ellipses indicate latent variables, rectangles represent observed variables, straight line with one arrowhead represents a direct effect, and a curved line represents covariance. As indicated by this model, emotional and physical pain were found to positively influence the propensity of a drug user to recover. Unlike the model built using LIWC indicators, for the Empath indicators physical factors were found to be more important than emotional factors in recovery
Table 5 Latent variable structure, direct effects, and covariances of the Empath withdrawal management SEM model
The recovery efforts model obtained using subreddit activities
Analysis of subreddit activities
In Fig. 5 and Additional file 1: Table S4 we present the correlations between the forum activity used in the SEM model for recovery efforts. From the figure and table, we observed that unlike the LIWC variables the correlation values between the forum activity displayed across different subreddits was low. The highest correlation was between the forums "careerguidance" and "resumes" (0.3), followed by "entrepreneur" and "careerguidance" (0.2).
Correlation diagram of the subreddit activities variables present in the recovery model (see also Additional file 1: Table S4). Positive correlations are color-coded in blue and negative correlations in red. The size of each square represents the magnitude of the correlations. As this visualization indicates, every variable-pair in the model is positively correlated. The two highest correlation values were observed for the variable-pairs "career-guidance" and "resume" followed by "career-guidance" and "Entrepreneur"
The comparison of the forum activity for the users who posted and did not post in a DAR subreddit was conducted in a manner similar to that described in the withdrawal management model (Table 6). The values of the subreddit activities corresponding to the latent variable "mental and physical well-being" were higher for users who displayed addiction recovery behavior. Some of these subreddits were: "fitness" (66.6%, p < 0.005), "meditation" (85.7%, p < 0.005), "yoga" (85.7%, p < 0.005), "gainit" (66.6%, p < 0.005), "bodyweightfitness" (100%, p < 0.005), and "running" (75.8%, p < 0.005) (Table 6). Similarly, the values for the subreddit activities corresponding to the latent variable "career" were higher for users who displayed addiction recovery behavior. Some of these subreddits were: "jobs" (96.2%, p < 0.005), "entrepreneur" (66.6%, p < 0.005), "careerguidance" (66.6%, p < 0.005), and "resumes" (66.6%, p < 0.005). Finally, the values of the subreddit activities corresponding to the latent variable "relationships" were also found to be higher for users who displayed addiction recovery behavior. Examples of subreddits for which enhanced activity was observed included: "relationships" (66.6%, p < 0.005), "relationship_advice" (50%, p < 0.005), "parenting" (50%, p < 0.005), and "childfree" (66.6%, p < 0.005) (Table 6).
Table 6 Comparison of normalized values for different variables in our model using subreddit activities for people who show and do not show addiction recovery behavior
Figure 6 shows the subreddit activity-based recovery model with factor loadings (the value for correlations are not displayed in the figure to maintain clarity). In it, the effect of "mental and physical well-being", "career" and "relationships" on drug addiction recovery behavior is studied. We estimated the latent variable "mental and physical well-being" with six indicators: "fitness", "meditation", "yoga", "gainit", "bodyweightfitness", and "running". The latent variable "career" was estimated using four indicators "jobs", "entrepreneur", "careerguidance", "resumes". Finally, the latent variable "relationships" was estimated using the following four indicators: "relationship_advice", "relationships", "parenting", and "childfree". The effect of "mental and physical well-being" and "relationships" on addiction recovery behavior was found to be statistically significant and positive, whereas, the effect of "career" on addiction recovery behavior was negative and statistically insignificant. All of the indicator variables for "mental and physical well-being" had a strong positive effect, with "fitness" and "bodyweightfitness" being the most contributory. Similarly, the indicator variables for "relationships" also had a strong positive effect on "relationships" (except "childfree" which was statistically insignificant). "relationship_advice" had highest effect on "relationships" followed by the subreddit "relationships". Between "relationships", and "mental and physical well-being", "relationships" was found to be more important for addiction recovery behavior. The fit indices for the final model indicated a good fit with the fit indices being: RMSEA = 0.02, TLI = 0.90, CFI = 0.92, and SRMR = 0.02. Table 7 summarizes the SEM model.
The SEM model for addiction recovery using subreddit activities. Mental and physical well-being (MPWB) and relationships were found to positively influence addiction recovery behavior. Career/job prospects negatively effects recovery behavior, however, its effect was statistically insignificant
Table 7 Latent variable factor structure, direct effects, and covariances the final subreddit activity based recovery SEM model
Relapse modeling using LIWC
In Table 8 and Fig. 7 we present the correlations observed between the LIWC indicators in the relapse model. All of the LIWC variables were found to be positively correlated with each other with the highest correlation observed for the categories "you΄" and "female΄" (0.76) followed by "you΄" and "male΄" (0.72). In Additional file 1: Table S3 we compare the values of the LIWC based indicators for "anti-social", "motion΄" (lack of physical activity), and "religion΄" (lack of religious) between the users who relapse and who do not relapse.
Table 8 Correlation matrix of the LIWC variables present in the LIWC relapse model
Correlation diagram of the LIWC variables present in the LIWC relapse model (see also Table 8). Positive correlations are color-coded in blue and negative correlations in red. The size of each square represents the magnitude of the correlations. As this visualization indicates, every variable-pair in the model is positively correlated. The two highest correlation values were observed for the variable-pairs "you΄" and "female΄" followed by "you΄" and "male΄"
Figure 8 shows the final LIWC based relapse model with factor loadings (the value for correlations are not displayed in the figure to maintain clarity). In this figure, the effect of "anti-social", "motion΄" (lack of physical activity), and "religion΄" (lack of religious) on relapse behavior is studied. We estimated the latent variable "anti-social" using the negation of the following six LIWC categories: "friend", "we", "shehe", "you", "male", "female". The effect of "anti-social" and the negation variables "motion΄", and "religion΄" were found to increase relapse behavior and were statistically significant. The effect of the negation variable "tone΄" (lack of positive emotion) on recovery was negative and statistically insignificant. All of the indicator variables for "anti-social" had a strong positive effect, with "you΄" and "male΄" being the most contributory. "Anti-social" was found to have the highest effect on the relapse behavior. The fit indices for the final model indicated a good fit with the fit indices being: RMSEA = 0.07, TLI = 0.96, CFI = 0.98, and SRMR = 0.03. Table 9 summarizes the model.
Final model of factors for the LIWC relapse model. "Anti-Social", "religion΄", and "motion΄" were found to positively influence relapse behavior. Tone΄ negatively affected relapse behavior, however, its effect was statistically insignificant
Table 9 Latent variable structure, direct effects, and covariances of the LIWC-based SEM model for relapse
Relapse modeling using Empath
In Additional file 1: Table S2 we compare the values of the Empath based indicators for the negation variables "positive emotion΄" (lack of positive emotion), "career΄" (lack of career interests), and "urban΄" (lack of urban facilities) between the users who relapse and who do not. In Table 10 and Fig. 9 we present the correlations between the Empath indicators present in the relapse model. Similar to the LIWC variables, all of the Empath variables in the model were found to be positively correlated with each other with the categories "joy΄" and "zest΄" (0.95) followed by "white_collar_job΄" and "blue_collar_job΄" (0.69) having the highest correlation values.
Table 10 Correlation matrix of the Empath variables present in the Empath relapse model
Correlation diagram of the Empath variables present in the Empath relapse model (see also Table 10). Positive correlations are color-coded in blue and negative correlations in red. The size of each square represents the magnitude of the correlations. As this visualization indicates, every variable-pair in the model was positively correlated. The two highest correlation values were observed for the variable-pairs "joy΄" and "zest΄" followed by "white_collar_job΄" and "blue_collar_job΄"
Figure 10 displays the Empath indicator-based relapse model with factor loadings (the value for correlations are not displayed in the figure to maintain clarity). In this figure, the effect of "positive emotion΄", "career΄" and "urban΄" on relapse behavior is shown. We estimated the latent variable "positive emotion΄" with the negation of the following Empath indicators: "joy", "zest", "cheerfulness", and "positive emotion". The latent variable "career" was estimated using the negation of three Empath indicators: "blue_collar_job", "white_collar_job", and "office". All of the path models were found to be statistically significant. The effect of "positive emotion΄", "career΄", and "urban΄" were found to be lead to relapse and were statistically significant. The indicator variables for "positive emotion΄" were found to have a strong effect, with "joy΄΄" and "zest΄΄" being the most contributory. Similarly, all of the indicators for "career΄" also had a strong effect, with "white_collar_job΄" and "office΄" being the most contributory. The fit indices indicated a good fit for this model: RMSEA = 0.04, TLI = 0.98, CFI = 0.99, and SRMR = 0.07. This model is summarized in Table 11.
The Empath indicator-based relapse model. Ellipses indicate latent variables, rectangles represent observed variables, straight line with one arrowhead represents a direct effect, and a curved line represents covariance. As indicated by this model, "positive emotion΄", "career΄", and "urban΄" were found to positively influence the relapse behavior of a drug user
Table 11 Latent variable structure, direct effects, and covariances the Empath relapse SEM model
The role of emotional distress and physical pain in withdrawal management
We observed that both emotional distress and physical pain played a significant role for redditors who display addiction recovery and relapse related behavior. To understand the reason behind this observation we further investigated the posts from individuals discussing their withdrawals from drugs. We observed that users typically experienced both physical pain and emotional distress during withdrawal. Also, we often observed users to have employed chemical treatments such as methadone and suboxone, alternative therapies such as kratom, xanax, and loperamide, as well as other supplements known to suppress physical symptoms of withdrawal. Interventions for assuaging emotional distress were found by us to be less prevalent. In Table 12 we present example posts describing some of the measures taken by individuals to suppress physical pain and discomfort. Interestingly, many users who had successfully managed their withdrawal process and were well into recovery, were observed by us to display a sense of loss after giving up their drug of choice. Paraphrased examples of posts describing such behavior are shown in Table 13.
Table 12 Paraphrased posts discussing different therapies utilized by the drug users to suppress physical discomforts during withdrawals
Table 13 Example paraphrased posts displaying drug craving and emotional distress for drug users in addiction recovery
Mental and physical well-being
Both mental and physical well-being were found to have a positive effect of addiction recover behavior. Physical activities are known to increase the production dopamine, noradrenaline, and serotonin and can act as mechanisms for a natural high [31,32,33,34,35,36,37,38,39]. Many initiatives such as "lace- 'em-up" have demonstrated the importance of physical activity for recovering addicts [40]. Our work confirms that similar conclusions can be drawn by analyzing social media data. In Table 14 we display paraphrased excerpts from posts demonstrating the positive effects of mental and physical activities on addiction recovery behavior.
Table 14 Example paraphrased posts displaying participation of users in differen mental and physical well being activities while inaddiction recovery
We found that "relationships" had a positive effect on addiction recovery. Unsurprisingly, friends and family play an important role in the addiction recovery efforts of an individual. There are many reasons that underlie this finding. First, the stigma associated with drug use causes an individual to feel shame and fear discrimination. Consequently, they don't feel safe to discuss their issues with co-workers, or strangers. It has been shown that addicts and recovering addicts feel comfortable in sharing their addictions and recovery journey with friends and family [41]. Research has also highlighted the willingness and positive outcomes of users undergoing addiction recovery efforts with the help and support drug-free friends, family members, and significant others [42]. Our analysis of social media data led to similar conclusions. In Table 15 we share excerpts from posts depicting the different ways friends and family affect the addiction recovery behavior.
Table 15 Example paraphrased posts displaying the role of family and friends in addiction recovery
We observed a negative, albeit statistically insignificant, effect of career/job opportunities on addiction recovery behavior. As noted in the "Research design and methods" section, the addiction literature is ambiguous on the effect of profession on addiction recovery. To highlight this point, we present example posts showing both the negative and positive aspects of profession on addiction recovery in Table 16.
Table 16 Example paraphrased posts discussing positive and negative impacts of focusing of career during addiction recovery
Supporting addiction recovery and personalized addiction recovery care
Personalized addiction recovery treatments have been found to be essential for successful abstinence [43, 44]. Our results identifying the impact of family and friends, self-development efforts, emotional distress and physical pain on addiction recovery can be utilized to provide direction for a person's recovery. For example, an individual in the initial stages of abstinence may be asked to focus on mental and physical well-being, and at least for some time stay away from high pressure situations (new jobs or returning to a previous stressful job). Their family and friends could also be made aware about their role in an individual's recovery and how they provide a safe non-judgmental space for the afflicted individual. Additionally, efforts could be made to manage emotional pains and cravings during and after the withdrawal period.
In this paper, we have described a framework that uses SEM to analyze and quantify latent constructs using SEM for modelling addiction recovery behavior using data from social media. The paper presents different SEM models to quantify the relationship between a number of observable and latent variables and their link to substance addiction.
To the best of our knowledge, this is the first study to utilize social media data and SEM to measure the latent constructs associated with substance abuse and recovery. Our results underscore the value of information present on social media platforms like Reddit to the study of substance misuse and design of interventions.
Data source and participants
We used a set of 117 recreational drug use (RDU) subreddits, and 29 drug addiction recovery (DAR) subreddits reported in our prior works to identify users discussing drug use and recovery on Reddit [20, 45]. In [20] we had utilized the word2vec algorithm [46] to create a term embedding space. In this space related terms were grouped using an iterative set expansion technique to construct drug-use and addiction-recovery lexicons. These lexicons were subsequently employed to characterize the different subreddits following which bi-clustering was used to cluster the different RDU and DAR subreddits. These bi-clusters were further manually curated to arrive at two RDU, and DAR subreddits sets. For this paper, we further identified 170,097 unique users discussing their drug use and recovery from these two RDU and DAR subreddit sets. For each of these users we retrieved their 1000 most recent posts (the specific number of retrieved posts was platform imposed) using the praw api [47]. Finally, we filtered out those users who had less than five nonempty posts in the RDU and DAR subreddit sets. As a consequence of this filtering, we ended up with a set of 7025 users consisting of 2679 users who posted in both RDU and DAR subreddit, and 4346 users who posted only in an RDU subreddit. In Table 17 we present example posts in different RDU and DAR subreddits.
Table 17 Example Reddit posts from the recreation and addiction recovery forums
Overview of modeling and analysis
In Fig. 11 we display the key steps of our analysis process. We used LIWC or Empath to analyze the posts of the users in our dataset to extract language features, such as, negative emotions, anxiety, and pain, associated with recovery/relapse behavior of drug users. We next hypothesized certain unobserved (latent) variables for the observed features as well as the relationship between observed and latent variables. The model and its goodness of fit was iteratively analyzed and refined using SEM to obtain the final path diagram displaying the interrelationships between latent and observed variables and recovery/relapse behavior. In the following, we describe each of the modeling steps.
Overview of modeling and analysis process: the posts of every user were analyzed using the LIWC and Empath dictionaries to generate the matrix \({M}_{n x m+1}\), where \(n\) represented the number of users and \(m\) the number of LIWC or Empath categories of interest. Each cell \({M}_{ij}\) in the matrix represents LIWC and Empath generated values of category j for the ith user. The column labeled 'RECOVERY' contained a binary flag representing if the user posted in a recovery forum. The data in \(M\) was subsequently analyzed using SEM
Linguistic feature specification using LIWC and Empath
LIWC [48] and Empath [49] are text analysis tools developed to measure psychological, cognitive, emotional, and behavioral components in a given text sample using human-validated dictionaries. Given a piece of text, these dictionaries can be utilized to make complex determinations, such as, calculating the percentage of terms related to sadness, religion, finance, negative emotions, or physical activity. In particular, LIWC outputs the percentage of total words that belong to 90 unique categories defined therein. Empath operates similarly and uses over 200 categories. Empath can also be used to create new categories by defining appropriate seed terms. Our research used the existing categories of Empath.
Basic concepts and definitions of structural equation modeling
In this section we describe the essential terms and concepts used in SEM. SEM is also referred to as the analysis of co-variance structure as model fitting is accomplished by utilizing the observed co-variances of the variables. For a detailed explanation of SEM, the reader is referred to [50]. SEM models are represented as a graphical representation of variable relationships and are called path diagrams. In SEM terminology observed variables (manifest variables) are those variables that are present in the dataset and can be measured. These variables are represented as rectangles in a path diagram. By contrast latent variables are not directly observable. Latent variables can be interpreted as the causes of manifest variables and are represented as ovals in the path diagram. In these diagrams, putative relationships between two variables are represented as directed edges (paths) weighted by path coefficients that are analogous to regression coefficients. Latent variables or error terms that co-vary are joined by curved arrows in the path diagram. SEM designates two other sets of variables: exogenous variables are determined to be outside of the model and have no paths pointing to them while endogenous variables are determined by the system of equations and have at least one path pointing to them. Both exogenous and endogenous variables can be observable or latent. Finally, for a specific model, its degrees of freedom (d), denotes the number of model parameters that are allowed to vary. Specifically, d is the difference between the number of possible parameters that can be estimated and number of actual parameters estimated. The number of possible parameters is quadratic in p -the number of observed variables while the number of estimated variables consists of all the paths (direct effects, correlations, error terms) being estimated in the model. A model is considered to be under-identified, just-identified, or over-identified if d < 0, d = 0, and d > 0 respectively. To estimate and evaluate the relationships in the model correctly we need to have d > 0.
It is important to clarify the relationship between SEM and another popular graph-based probabilistic reasoning framework, called Bayesian Networks (BN). We begin by noting that SEM does not denote a single technique; it refers to a family of related procedures. This family can be broadly characterized in terms of taking three inputs and generating three outputs [51]. The inputs being: (1) one or more qualitative causal hypotheses, (2) a set of questions about causal relations among variables of interest, and (3) a model instance. The outputs of SEM are: (1) estimates of model parameters for hypothesized effects, (2) a set of logical implications of the model that can be tested in the data, and (3) a measure of how well the testable implications of the model are supported by the data. The point of SEM is to test a theory by specifying a model that represents predictions of the aforementioned theory from among plausible constructs measured with appropriate observed variables. BN represent dependencies among sets of random variables as (causal) graphs which are traversed to update conditional probabilities of events. The ideas underlying BN have been extended to the broader problem of causal inference under a framework called the structural causal model (SCM), which is subsumed under the umbrella of SEM [52]. In our problem context, a direct application of BN entails limitations. In particular, BN cannot differentiate between causal and non-causal relationships without intervention from a domain expert [53]. Furthermore, it is non-trivial to employ BN while differentiating between latent and observed variables—a core requirement in our research. Finally, the output of BN is known not to be well suited for theoretical explanations [54].
The process of structural equation modeling
SEM is an iterative process and involves the following steps: (1) Model specification: At this step a researcher hypothesizes the latent variables, the observed variables, and the relationships between them. (2) Estimation: The proposed model structure is estimated by using covariance analysis to solve a system of equations representing the interrelationships in the system. (3) Evaluation of model fit: The model fit can be evaluated using a variety of measures, such as, the comparative fit index (CFI), the Tucker Lewis index (TLI), root mean square error of approximation (RMSEA), and standardized root mean square residual (SRMR). (4) Model re-specification: If the initial fit is not deemed to be adequate, the model is modified and the above steps iterated.
SEM estimation
In the estimation step the difference between the sample covariance (\(C\)) and the model-predicted covariance (\(\tilde{C} \left( \theta \right)\)) is minimized. The underlying idea is that the covariance matrix of the observed variables is a function of a set of parameters. If the parameters are correctly estimated (i.e. the model is correct) then the population covariance matrix will be exactly reproduced as shown in Eq. (1), where \(\theta\) denotes the vector of model parameters.
$$C = \tilde{C} \left( \theta \right)$$
The standard form of the structural equation relating the endogenous and exogenous variable is:
$${\varvec{y}} = {\varvec{By}} + \user2{ \Gamma x} + \user2{ \zeta }$$
In Eq. (2), \(\user2{y }\left( {n \times 1} \right)\) denotes the n dependent or endogenous variables, \(\user2{x }\left( {m \times 1} \right)\) denotes the m exogenous variables, and \({\varvec{\zeta}} \left( {n \times 1} \right)\) denotes the specification errors. The matrix B \(\left( {n \times n} \right)\) denotes the coefficients of the regression of y variables on other y variables with zeros on the diagonal which implies a variable cannot cause itself. The matrix \({\varvec{\varGamma}}\) \(\left( {n \times m} \right)\) denotes the coefficients of regression of the endogenous variables on the exogenous variables. A maximum likelihood function is used to fit the structural model equations by minimizing the fitting function (FML) shown in Eq. (3):
$$F_{ML} = \log \left| {{\varvec{C}}\left( {\varvec{\theta}} \right)} \right| + tr\left( {{\varvec{S}}{\mathbf{C}}^{ - 1} \left( {\varvec{\theta}} \right)} \right) - \log \left| {\varvec{S}} \right| - \left( {m + n} \right)$$
In Eq. (3), S is the sample covariance matrix, |.| denotes the determinant, and tr (.) denotes the trace of a matrix. Additionally, in SEM, it is assumed that \({\varvec{C}}\left( {\varvec{\theta}} \right)\), and S are positive-definite which means they are non-singular.
Employing SEM for social media data modeling: an operational explanation
In this section, we explain the progression of our analysis-process from Reddit posts to a final SEM model. As the specific context, we describe the withdrawal management modeling process using LIWC indicators. To generate this model, we had used 209,804 posts from 7025 drug users. The withdrawal management model involved nine LIWC categories: "negative emotion", "sad", "anger", "anxiety", "feel", "affect", "swear", "sexual", and "authentic" which were postulated to capture the emotive underpinnings of a post. Similarly, the four LIWC categories: "biology", "death", "health", and "body" were postulated to describe physical discomfort. In Table 2 we present example posts and the terms identified by LIWC for the aforementioned categories. We also present post-specific LIWC category values in the table. Also, Additional file 2: Table S1 contains the LIWC category values for a sample set of 1000 users engaged in substance use. Finally, the (binary) variable "recovery" was the outcome variable of the model; it was set to 1 if an individual posted in a DAR subreddit else it was set to 0. As explained in Fig. 11, the posts of these users were analyzed using LIWC to generate the matrix \(M_{7025 x 14}\).
In SEM, variables that can be measured constitute the observable variables. In our context (Fig. 2) this role was fulfilled by the thirteen LIWC categories listed above (these variables are represented as rectangles in the path diagram shown in Fig. 2). Our hypothesis was that the latent variables (represented as ovals in Fig. 2): "emotional distress" could be measured using the LIWC categories: "negative emotion", "sad", "anger", "anxiety", "feel", "affect", "swear", "sexual", and "authentic", while the latent variable "physical pain" could be measured via the LIWC categories: "biology", "death", "health", and "body". Finally, we hypothesized that these two latent variables had a direct effect on the recovery behavior as reflected by the Reddit posts of drug users. We measured the recovery behavior (observed variable) by using a binary variable "recovery" which was set to 1 if a user was found to have posted in drug addiction recovery forum. Alternatively, this variable was set to 0. The reader may also note that "emotional distress", and "physical pain" were the only endogenous variables in the model; the rest of the variables being exogenous.
Next, in the SEM estimation step the difference between the population covariance (\(C\)), i.e., the covariance observed in LIWC variables and the "recovery" variable for the population of 7025 drug users and the hypothesized-model-predicted covariance (\(\tilde{C} \left( \theta \right)\)) was minimized. For our dataset, the standard form of the structural equation (Eq. (2)) relating the endogenous and exogenous variable took the following form:
$${\varvec{y}}_{{14\user2{ x }1}} = {\varvec{B}}_{{14\user2{ x }14}} \user2{ y}_{{14\user2{ x }1}} + \user2{ \Gamma }_{{14\user2{ x }2}} \user2{ x}_{{2\user2{ x }1}} + \user2{ \zeta }_{{14\user2{ x }1}}$$
In Eq. (4), \(\user2{y }\left( {14 \times 1} \right)\) denotes the 14 exogenous variables (13—LIWC categories and 1—recovery variable), \(\user2{x }\left( {2 \times 1} \right)\) denotes the 2 endogenous variables ("emotional distress" and "physical pain"), and \({\varvec{\zeta}} \left( {14 \times 1} \right)\) denotes the specification errors. The matrix B \(\left( {14 \times 14} \right)\) denotes the effect of the exogenous variables on other exogenous variables while the matrix \({\varvec{\varGamma}}\) \(\left( {14 \times 2} \right)\) denotes the coefficients of regression of the LIWC variables on the endogenous variables. The maximum likelihood function explained in Eq. (3) is used to fit the structural model equations by minimizing the fitting function (FML) and obtain the model shown graphically in Fig. 2.
Model evaluation
In SEM, the model fit is evaluated by examining difference between the sample covariance (\(C\)) and the covariance (\(\tilde{C} \left( \theta \right)\)) computed using the model. The goal is to minimize the difference between \(C\) and \(\tilde{C} \left( \theta \right)\). The simplest fitting function for SEM models is the Chi-square fit \(\chi^{2}\) = \(\left( {N - 1} \right)F_{ML}\). However, this function is affected by sample size; large sample sizes may increase the χ2 value even if the difference between \(C\) and \(\tilde{C} \left( \theta \right)\) is small and small sample sizes may lead to Type II errors [50]. The \(\chi^{2}\) function however, is used as part of other fitting functions. Typically, these fitting functions are of three types: relative goodness-of-fit functions, parsimony functions, and functions that determine absolute (standalone) fit.
Examples of relative goodness-of-fit functions include the CFI (Eq. 5) and TLI (Eq. 6) measures. These measures compare the proposed model against a baseline model where all variables are allowed to have a variance, but none are allowed to co-vary. For both CFI and TLI, goodness of fit values above 0.90 denote high-quality agreement [55].
$$CFI = 1 - \frac{{\max \left[ {\chi_{I}^{2} - d_{I} , 0} \right]}}{{\max \left[ {\chi_{I}^{2} - df_{I} ,\chi_{B}^{2} - d_{B} ,0} \right]}}$$
$$TLI = \frac{{{\raise0.7ex\hbox{${\chi_{B}^{2} }$} \!\mathord{\left/ {\vphantom {{\chi_{B}^{2} } {d_{B} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${d_{B} }$}} - {\raise0.7ex\hbox{${\chi_{I}^{2} }$} \!\mathord{\left/ {\vphantom {{\chi_{I}^{2} } {d_{I} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${d_{I} }$}}}}{{{\raise0.7ex\hbox{${\chi_{B}^{2} }$} \!\mathord{\left/ {\vphantom {{\chi_{B}^{2} } {d_{B} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${d_{B} }$}} - 1}}$$
In Eqs. (5) and (6), the baseline model is indicated by the subscript B while the subscript I denotes the proposed model. The degree of freedom is denoted by d.
The RMSEA [see Eq. (7)] constitutes an example of a parsimony-based fitting measure. The RMSEA takes into the account the complexity of the model by penalizing models with lower degrees of freedom since such models lead to higher values of RMSEA. RMSEA values less than 0.01, 0.05, and 0.08 are respectively considered to indicate excellent, good, or mediocre fit [55].
$$RMSEA = \sqrt {\frac{{\chi_{I}^{2} - d_{I} }}{{\left( {d_{I} } \right)\left( {n - 1} \right)}}}$$
In the above equation, n denotes the sample size.
Finally, SRMR [see Eq. (8)] is an example of an absolute fit index. SRMR is the average of standardized residuals between the observed and the model computed covariance matrices. An advantage of using SRMR over CFI, TLI, and RMSEA is that it is independent of the sample size.
$$SRMR = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{p} \mathop \sum \nolimits_{j = 1}^{i} \left[ {{\raise0.7ex\hbox{${{\text{C}}_{ij} - \tilde{C} \left( \theta \right)_{ij} }$} \!\mathord{\left/ {\vphantom {{{\text{C}}_{ij} - \tilde{C} \left( \theta \right)_{ij} } {{\text{C}}_{ii} {\text{C}}_{jj} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${{\text{C}}_{ii} {\text{C}}_{jj} }$}}} \right]^{2} }}{{{\raise0.7ex\hbox{${p\left( {p + 1} \right)}$} \!\mathord{\left/ {\vphantom {{p\left( {p + 1} \right)} 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}}$$
In the above equation \({\text{C}}_{ii}\) and \({\text{C}}_{jj}\) are the observed standard deviations and p is the number of observed variables. Usually, SRMR values of less than 0.08 are considered to denote models of adequate quality [55].
Modeling withdrawal management and recovery
Withdrawal from drug addiction is accompanied by physical discomforts and negative emotions. Sedatives, opioids, and alcohol are known to cause intense physical discomforts during withdrawals, while withdrawal from substances such as marijuana, and stimulants cause emotional negativity [56]. Physical symptoms during the process of withdrawal include a variety of symptoms such as muscle aches, runny nose, dilated pupils, piloerection, insomnia, sweating, yawning, shivering, pain, cramps, weight loss, toothache, colds, and sometimes even mortality [57,58,59]. Emotional distress and negativity during withdrawal is characterized by aggression, anxiety, and loss of temper [60,61,62]. The medical approach to manage withdrawal symptoms typically involves gradually tapering doses of drug agonists to diminish the bodily discomforts and prevent a relapse. However, there are no clear methods to measure, and compare the intensity of either emotional distress or physical pain during withdrawal. In the following we describe the development of SEM models to determine the effect and importance of "emotional distress", and "physical pain" in withdrawal management using linguistic features determined using both LIWC and Empath.
Determining observed variables using LIWC
We used nine LIWC categories: "negative emotion", "sad", "anger", "anxiety", "feel", "affect", "swear", "sexual", and "authentic" to measure the latent variable "emotional distress". Examples of terms in each of the categories are presented in Table 18. The categories "negative emotion", "sad", "anger", and "anxiety" consisted of terms that had a negative connotation or valance and reflected negative thoughts. The category "feel" consisted of terms related to bodily sensations, while the category "affect" consisted of terms having both a negative and a positive connotation. We included the LIWC category "swear" as one of the indicators for "emotional distress" because we noticed that it was common for drug users to employ expletives to express their physical and emotional anguish. We also included the LIWC category "sexual" as one of our indicators for "emotional distress" because of analogous reasons. "Authentic" was a summary variable and was calculated as a single value for a given text input. The algorithm in LIWC for determining the authenticity of a text was developed based on the studies on deceptive and truthful communications [48, 63]; it determines the openness, honesty, and disclosure of a given body of text. Consequently, there are no example terms for "authentic" in Table 18. To reflect the latent variable "physical pain", we used the following four LIWC categories: "biology", "death", "health", and "body". Example terms in each of these categories are presented in Table 18. The category "biology" contained terms related to human biology and biological activities. Terms representing death were present in the category "death" (bury, coffin, kill). The category "health" consisted of a number of terms related to medicine and health of an individual. The category "body" consisted of terms related to body parts and bodily functions. Additional file 2: Table S1 contains the LIWC category values for a sample set of users engaged in substance use. Finally, the (binary) variable "recovery" was the outcome variable of the model; it was set to 1 if an individual posted in a DAR subreddit else it was set to 0.
Table 18 Example terms present in different LIWC and Empath categories
Determining observed variables using Empath
We used four Empath categories: "negative_emotion", "hate", "shame", and "suffering" to measure the latent variable "emotional distress". Examples of terms in each of the categories are presented in Table 18. The categories "negative_emotion", "hate", "shame", and "suffering" all consisted of terms that had a negative undertone and reflected negative feelings. To reflect the latent variable "physical pain", we used the following four Empath categories: "pain", "medical_emergency", "health", and "weakness" (see Table 18 for examples). The category "pain" contained terms related to physical discomfort. Terms representing a medical emergency were present in the category "medical_emergency". The category "health" consisted of a number of terms related to the health of an individual and the category "weakness" consisted of terms related to lack of strength of an individual. Again, the (binary) variable "recovery" was the outcome variable of the model; it was set to 1 if an individual posted in a DAR subreddit else it was set to 0.
The SEM model for withdrawal management
The SEM modeling was conducted using the lavaan package [64]. Here, we estimate the effect of "emotional distress" and "physical pain" on drug addiction recovery behavior using LIWC and Empath. As mentioned before, drug addiction recovery behavior was measured using an observed variable ("recovery"). The reader may note that the LIWC model was based on our estimation of the latent variable "emotional distress", using nine indicators [(1)"negative emotion", (2) "authentic", (3) "sad", (4) "affect", (5) "anger", (6) "anxiety", (7) "sexual", (8) "feel", and (9) "swear"]. Similarly, the latent variable and "physical pain", was estimated with four indicators [(1) "health", (2) "biology", (3) "death", and (4) "body"]. The Empath model estimated the latent variable "emotional distress", using four indicators [(1)"negative_emotion", (2) "hate", (3) "suffering", and (4) "shame"]. Similarly, the latent variable "physical pain", was estimated with four indicators [(1) "health", (2) "weakness", (3) "pain", and (4) "medical_emergency"]. It may be noted that the LIWC and Empath categories were not exclusive in that terms could simultaneously belong to different categories. We also observed that terms of certain categories frequently co-occurred. For example, in posts describing effects of withdrawal, expression of negative emotions or terms describing sadness would usually co-occur with terms associated with health. Consequently, such variables were allowed to co-vary in our models. The specific models obtained using the LIWC and Empath variables are described in the "Results" section.
The SEM model for recovery
Self-development efforts and relationships have been found to be indispensable for drug addiction recovery [65]. Family support, especially for adolescents in long term residential programs has been proven to be necessary for successful recovery from addiction [66]. Studies have also showed that having a strong social and family resource improves the chances of addiction recovery [67,68,69,70].
Self-development efforts encompassing activities that lead to mental and physical well-being, such as regular exercise, meditation, and yoga have been observed to help heal the body and mind [71, 72]. Such activities have also been shown to address psychological and physiological needs of a recovering addict by reducing negative feelings and preventing weight gain following abstinence. Additionally, regular exercise is known to alleviate physical and mental stress. It is also known to positively alter the brain chemistry as it releases endorphins and creates a natural high, similar to ones released when an individual uses drugs. Studies have shown that addition of exercise as a lifestyle change leads to abstinence or reduction in drug use [31,32,33,34]. Mediation and yoga has also been proved to help individuals in their withdrawals and addiction by acting a calming effect during their period of struggles [35,36,37]. Professional activities constitute another aspect of self-development. However, the literature on the importance of jobs, and career on addiction recovery is ambiguous: some sources suggest that a stable job helps provide the recovering addicts with income and health benefits, improved mental health, and a purpose in their life. For example, Flynn et al. [72], found job/career to be one of the fundamental personal motivations for a recovering addict to stay sober. The importance of vocational rehabilitation and job search as one of the services in the social model of recovery has also been noted [73]. Other works have found that employed individuals undergoing recovery are more engaged in recovery activities and are more likely to abstain from substance use [74,75,76,77]. However, studies also have found that returning to old jobs, or stress experienced at work can lead to drug use and relapse [76]. Amongst these, Buczkowski et al., identified smoking environment at work as one of the triggers for relapse of smoking [77]. The stress associated with changing jobs has been cited to lead to substance use relapse [78,79,80,81,82]. Furthermore, the social stigma associated with drug addiction has been found to play a major role in the unwillingness of working individuals to opt for recovery interventions [83]. Finally, since employers are prejudiced against recovering addicts applying for jobs, such situations can also lead to a relapse or unwillingness to come out as an addict [83].
Because of the aforementioned reasons self-development efforts and relationships play a pivotal role in withdrawal management and drug addiction recovery. We therefore construct a SEM model to determine the effect and importance of the latent variables "mental and physical well-being", "career", and "relationships" in drug addiction recovery. To estimate these latent variables, we utilized forum activity of the drug users in multiple subreddits related to self-development efforts and relationships. We used the number of times an individual posted in the following eight subreddits: "fitness", "meditation", "yoga", "gainit", "bodyweightfitness", and "running" to estimate the latent variable "mental and physical well-being". Similarly, we used the posts in the subreddits: "jobs", "entrepreneur", "careerguidance", and "resumes" to estimate latent variable "career". As indicator variable for "relationships" we used the posts in the four subreddits: "relationship_advice", "relationships", "parenting", and "childfree". Finally, our outcome variable for the model was "recovery". The SEM model captures the effect of these variables on addiction recovery.
Modeling addiction relapse
As described above, the variables "emotional distress", "physical pain", "relationships", and "self-development" were found to play a critical role in addiction recovery. In addition to these factors, religion and geographic disparities were also found by us to influence the process of recovery. These results are supported by previous work in the field of relapse where it was found that recovering individuals display higher levels of religious faith [84,85,86,87]. Similarly, researchers have observed that addicts living in a rural setting have a higher chance for relapse as compared to their urban counterparts [88,89,90,91] because of limited access to relapse prevention facilities and preventive medications. In the following, we describe models that study the effect of the aforementioned latent variables along with demographic setting for drug users who undergo relapse. We defined relapse as the event of an individual posting in an RDU subreddit after posting in a DAR subreddit. Individuals who never posted in an RDU subreddit after posting in a DAR subreddit were defined to be in (continued) recovery. Based on these definitions 2363 individuals in our dataset were found to have relapsed, while 1355 users displayed continued recovery. To study users who relapsed while minimizing the impact of stray postings, we investigated only those users who had at least five posts in succession in a DAR subreddit before they were defined to have relapsed. Similarly, to study users who displayed signs of continued recovery we investigated only who had at least five posts in DAR subreddits before they stopped posting. As a consequence of this filtering, we ended up with a total of 174 users of whom 108 were identified to have relapsed while 66 users were identified to have continued their recovery journey till our observations concluded. Also, to extract relapse specific information, we scaled the values for LIWC and Empath categories by dividing them by the number of days between the post under investigation and the day when the user was defined to have relapsed.
Determining observed variables using LIWC for modeling relapse
While modeling users who relapsed we observed a limitation of using psycholinguistic dictionaries such as LIWC and Empath. Anti-social behavior, lack of religious expression, physical exercise, and positive emotion increases the chances of a relapse. However, using these dictionaries we could only obtain a value for the presence of such categories, i.e., the absence of such psycholinguistic information was not represented via any appropriate categories. To overcome this weakness and to build a model for relapse using LIWC, we generated values for such (absent, in LIWC or Empath) variables by subtracting the numeric weight of the corresponding LIWC/Empath categories from 1. For example, if a post had a value of 0.2 for the category "friends", we calculated the value of "friends΄" (i.e. the negation of the category "friends") to be 0.8 (hereafter, such variables are referred to as negated variables and denoted by a prime). We used negation of the following six LIWC categories "friend", "we", "shehe", "you", "male", and "female" to represent and study the latent variable "anti-social". To model lack of physical exercise and religious expression we used the negation of LIWC categories "motion" and "religion". The (binary) variable "relapse" was the outcome variable in our model; it was set to 1 if an individual relapsed else it was set to 0.
We used Empath to model the relapse behavior as a consequence of lack of positive emotion, career interests, and urban facilities. Similar to obtaining the values of LIWC categories for modeling relapse, we used negation of the following four Empath categories "joy", "zest", "cheerfulness", and "positive emotion" to study the latent variable "positive emotion΄" (lack of positive emotion). To model "career΄" (and lack of career development), we used the negation of the following three Empath categories: "blue_collar_job", "white_collar_job", and "office". Finally, to model "urban΄" (i.e., the lack of an urban setting and facilities) we used the negation of LIWC category "urban". The (binary) variable "relapse" was the outcome variable in our model; it was set to 1 if an individual relapsed else it was set to 0.
The SEM model for relapse of addiction
In this model we estimated the effect of factors including the social and physical activities of a drug user, their positive or negative emotions, recourse to religion, career-related activities, and location (urban or rural) on relapse by employing linguistic characteristics determined using LIWC and Empath. The relapse behavior was itself measured using the observed variable "relapse". The latent variable "anti-social" was estimated using six negated LIWC categories ("friend΄", "we΄", "shehe΄", "you΄", "male΄", and "female΄") and two observed negated variables "motion΄" and "religion΄". The Empath model estimated the latent negation variable "positive emotion΄" using four negated categories ("joy΄", "zest΄", "cheerfulness΄", and "positive emotion΄"). Similarly, the latent negated variable "career΄" was estimated using three negated categories ("blue_collar_job΄", "white_collar_job΄", and "office΄"). Finally, the variable "urban΄" corresponding to the location of the user was an observed variable in the model. The models obtained using the LIWC and Empath variables are described in the "Results" section.
Any investigation of the type reported by us must take cognizance of user privacy concerns. In our case, the data used in this paper was publicly available (via Reddit) and the authors did not have personal interaction with any of the users. Even though this data is publicly available, to ensure user privacy, we anonymized the data and all examples presented in the paper were paraphrased.
The primary data used in this paper is publicly available from Reddit. We are unable to act as a secondary source of this data because of the rules and regulations as enforced by Reddit. According to these rules, Reddit account holders own their content and data. Reddit grants the developers a non-exclusive, non-transferable, and revocable license which does not allow for further data sharing and sub-licensing. The LIWC category values for a sample set of substance users is presented in Additional file 2: Table S1.
SEM:
RDU:
Recreational drug use
Drug addiction recovery
LIWC:
Linguistic Inquiry and Word Count
BN:
SCM:
Structural causal model
CFI:
Comparative fit index
TLI:
Tucker Lewis index
RMSEA:
Root mean square error of approximation
SRMR:
Standardized root mean square residual
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The authors would like to thank the reviewers for their comments and suggestions.
This article has been published as part of BMC Bioinformatics Volume 21 Supplement 18, 2020: Proceedings from the 8th Workshop on Computational Advances in Molecular Epidemiology (CAME 2019). The full contents of the supplement are available online at https://bmcbioinformatics.biomedcentral.com/articles/supplements/volume-21-supplement-18.
This research and its publication was funded in part by the National Institutes for Health through the Grant 1R25MD011714, the National Science Foundation through grant IIS-1817239, and a seed grant from the Center for Computing in Life Sciences at San Francisco State University. The funding bodies had no role in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript.
Department of Computer Science, San Francisco State University, 1600 Holloway Ave., San Francisco, CA, 94132, USA
Deeptanshu Jha & Rahul Singh
Deeptanshu Jha
RS formulated the problem and provided technical guidance and mentoring. DJ was responsible for data collection, modeling, and coding. Model analysis was conducted by RS and DJ. The paper was written by RS and DJ. All authors read and approved the final manuscript.
Correspondence to Rahul Singh.
The authors declare that there are no competing interests.
Additional file 1
. Supplementary Tables and Analysis.
. LIWC category values for a sample set of 1000 users.
Jha, D., Singh, R. Analysis of associations between emotions and activities of drug users and their addiction recovery tendencies from social media posts using structural equation modeling. BMC Bioinformatics 21, 554 (2020). https://doi.org/10.1186/s12859-020-03893-9
Personalized interventions
Substance misuse disorder | CommonCrawl |
\begin{document}
\title{Quantum contextuality for rational vectors}
\author{Ad\'an Cabello}
\email{[email protected]}
\affiliation{Departamento de F\'{\i}sica Aplicada II, Universidad de
Sevilla, E-41012 Sevilla, Spain}
\author{Jan-{\AA}ke Larsson}
\email{[email protected]}
\affiliation{Institutionen f\"or Systemteknik, Link\"opings
Universitet, SE-581 83 Link\"oping, Sweden}
\date{\today}
\begin{abstract} The Kochen-Specker theorem states that noncontextual hidden variable models are inconsistent with the quantum predictions for every yes-no question on a qutrit, corresponding to every projector in three dimensions. It has been suggested [D. A. Meyer, Phys. Rev. Lett. \textbf{83}, 3751 (1999)] that the inconsistency would disappear when restricting to projectors on unit vectors with rational components; that noncontextual hidden variables could reproduce the quantum predictions for rational vectors. Here we show that a qutrit state with rational components violates an inequality valid for noncontextual hidden-variable models [A. A. Klyachko {\em et al.}, Phys. Rev. Lett. \textbf{101}, 020403 (2008)] using rational projectors. This shows that the inconsistency remains even when using only rational vectors. \end{abstract}
\pacs{03.65.Ta,
03.65.Ud}
\maketitle
The Kochen-Specker theorem from 1967 \cite{KS67} states that the quantum predictions from a three-dimensional quantum system (a qutrit) are inconsistent with noncontextual hidden variables. The proof uses 117 directions in three dimensions, arranged in a pattern such that they cannot be colored in a particular manner, see \cite{KS67} for details. Later proofs use less directions, but one common feature (in the three-dimensional versions) is that the set of unit vectors includes irrational components. It was noted in \cite{Meyer99} that the Kochen-Specker proof needs these irrational vectors to be completed. Indeed, when using only the rational subset of vectors, the set is colorable in the manner required by quantum mechanics. It was also suggested \cite{Meyer99} that, for this reason, the inconsistency between quantum mechanics and noncontextual hidden variables disappears, and that quantum mechanics can be imitated by noncontextual hidden variable models restricted to rational vectors.
It has been recently shown \cite{KCBS08} that the following inequality is a necessary and sufficient condition for qutrit noncontextual hidden variables, for measurements $A_i$ with possible outcomes $-1$ and $+1$, such that $A_i$ and $A_{i+1}$ (modulo 5) are compatible: \begin{equation} \sum_{i=0}^4 \langle A_i A_{i+1} \rangle \ge -3.
\label{KCBS} \end{equation} Using the rational qutrit state \begin{equation}
\langle\psi| = \left(\frac{354}{527},\frac{357}{527},-\frac{158}{527}\right), \end{equation} and the observables \begin{equation}
A_i = 2 |v_i\rangle \langle v_i| - \openone, \end{equation} associated to the rational vectors \begin{subequations} \label{rationalvectors} \begin{align}
\langle v_0| = & \left(1,0,0\right),\\
\langle v_1| = & \left(0,1,0\right),\\
\langle v_2| = & \left(\frac{48}{73},0,-\frac{55}{73}\right),\\
\langle v_3| = & \left(\frac{1925}{3277},\frac{2052}{3277},\frac{1680}{3277}\right),\\
\langle v_4| = & \left(0,\frac{140}{221},-\frac{171}{221}\right), \end{align} \end{subequations} we obtain a value of $-3.941$ for the left-hand side of \eqref{KCBS}, which deviates very little from the maximum violation at $-3.944$. Thus, even when using only rational vectors, the inconsistency is not nullified. The violation shows that the (physical content of) the Kochen-Specker theorem remains, namely, that the quantum-mechanical predictions cannot be reproduced by noncontextual hidden variables.
\end{document} | arXiv |
\begin{document}
\title{Structure of cell decompositions in Extremal Szemer\'edi-Trotter examples} \author{Nets Hawk Katz\thanks{California Institute of Technology, Pasadena CA, supported by a Simons Investigatorship},\and Olivine Silier\thanks{University of California Berkeley}}
\maketitle \begin{abstract} The symmetric case of the Szemer\'edi-Trotter theorem says that any configuration of $N$ lines and $N$ points in the plane has at most $O(N^{4/3})$ incidences. We describe a recipe involving just $O(N^{1/3})$ parameters which sometimes (that is, for some choices of the parameters)
produces a configuration of N point and N lines. (Otherwise, we say the recipe fails.) We show that any near-extremal example for Szemer\'edi Trotter is densely related to a successful instance of the recipe. We obtain this result by getting structural information on cell decompositions for extremal Szemer\'edi-Trotter examples. We obtain analogous results for unit circles. \end{abstract}
\section{Introduction}\label{introSection}
If $l$ is a line and $p$ a point in the real plane ${\mathbb R}^2$, we say that $(l,p)$ is an incidence if $p \in l$. The most fundamental result in the theory of incidences between points and lines in the plane is the Szemer\'edi-Trotter theorem \cite{ST} which bounds their number:
\begin{thm}[Szemer\'edi-Trotter 1983] Let $\mathcal{L}$ be a set of $n$ lines in the plane and $\mathcal{P}$ be a set of $m$ points. Then if $I(\mathcal{L},\mathcal{P})$ is the set of incidences between lines of $\mathcal{L}$ and points of $\mathcal{P}$, we have the bound
$$|I(\mathcal{L},\mathcal{P})| \lesssim n^{{2 \over 3}} m^{{2 \over 3}} + n + m.$$ \end{thm}
One thing that is remarkable about the Szemer\'edi-Trotter theorem is that as far as the exponents are concerned, it is sharp. A number of examples are known, but we are far from classifying all possible examples. To do so remains one of the central open problems in incidence geometry of the plane.\cite{G} We restrict ourselves to the symmetric case where $n=m$, although the question is interesting whenever $\sqrt{n} < m < n^2$.
{\bf Inverse Szemer\'edi Trotter problem} Let $\mathcal{L}$ be a set of $n$ lines and $\mathcal{P}$ be a set of $n$ points with $$I(\mathcal{L}, \mathcal{P}) \geq n^{{4 \over 3}-} .$$ What can be said about the structure of $\mathcal{L}$ and $\mathcal{P}$?
A related question which motivates this study is the unit distance problem.
{\bf Unit distance problem} Let $\mathcal{P}$ be a set of $n$ points in the plane. Let
$U(\mathcal{P})$ be the set of pairs of points in $\mathcal{P}$ which are at Euclidean distance 1. What upper bound can one put on $|U(\mathcal{P})|$?
The conjectured bound in the unit distance problem is $n^{1+}$ but the best known bound is $n^{{4 \over 3}}$. This is not a coincidence [forgive the pun]. Unit distances are incidences between the points of $\mathcal{P}$ and the unit circles centered at those points. Now unit circles are not lines, but they do share some properties in common. Each unit circle is defined by two parameters and while it is not the case that unit circles intersect in at most one point, they do intersect in at most two. Essentially every technique which has been used in a proof of the Szemer\'edi-Trotter theorem can be applied in the case of unit distances and this is the source of the $n^{{4 \over 3}}$ bound.
A connection between the unit distance problem and the inverse Szemer\'edi-Trotter problem is that if one had an inverse theorem for unit distances at the exponent ${4 \over 3}$, one could gain a small improvement in the exponent by showing that the inverse cases don't exist. This is, in fact, a big part of our motivation which is why we don't mind restricting to the symmetric case in Szemer\'edi-Trotter .
To illustrate the source of the difficulty in obtaining an inverse Szemer\'edi-Trotter theorem, we describe a simpler, related problem in which the inverse theorem is fairly straightforward to obtain. We note that the Szemer\'edi-Trotter theorem uses a great deal more about the structure of the plane than the fact that two lines intersect at a simple point. If we had restricted ourselves to using only that fact, we would have obtained this weaker result.
\begin{thm} [Cauchy-Schwarz] Let $\mathcal{L}$ be a set of $n$ lines in the plane and $\mathcal{P}$ be a set of $m$ points. Then if $I(\mathcal{L},\mathcal{P})$ is the set of incidences between lines of $\mathcal{L}$ and points of $\mathcal{P}$, we have the bound
$$|I(\mathcal{L},\mathcal{P})| \lesssim n^{{1 \over 2}} m + n .$$ \end{thm}
The inverse Cauchy-Schwarz problem is to describe all sets of $n$ lines and $n$ points with $n^{{3 \over 2}}$ incidences. ``It's easy," the reader should exclaim, ``there are none by Szemer\'edi-Trotter." But we will suspend disbelief and nevertheless try to describe them despite their nonexistence. What follows is a sketch.
In a configuration of $n$ points and $n$ lines with $n^{{3 \over 2}-}$ incidences, the typical point is incident to $n^{{1 \over 2}-}$ lines and the typical line is incident to $n^{{1 \over 2}-}$ points. We pick an initial point $p_1$. Let $B(p_1)$, the ``bush" of $p_1$ be the set of points incident to one of the lines incident to $p_1$. We should have
$$|B(p_1)| \gtrsim n^{1-}.$$ Already, a substantial proportion of the point set $\mathcal{P}$ belongs to $B(p_1)$ and lies on the lines going through the point $p_1$. We can go one step further and do this twice. We can choose points $p_1$ and $p_2$ so that
$$|B(p_1) \cap B(p_2)| \gtrsim n^{1-}.$$ In other words, a substantial proportion of the point set consists of points lying on a line incident to $p_1$ and a line incident to $p_2$. After a projective transformation sending $p_1$ and $p_2$
to points at infinity, we get that a substantial portion of the point set lies on a product set $A \times B$ with each of $|A|$ and $|B|$ of size $n^{{1 \over 2}-}$. This is not yet an inverse theorem, but it is what we refer to as a \textbf{proto-inverse theorem}. Recall that an inverse theorem gives a complete characterization of the solution set to the inverse problem. A proto-inverse theorem on the other hand gives a looser characterization which must include all solutions to the inverse theorem but may also include non-examples. We have parametrized (a substantial portion of) the point set and whereas {\it a priori} we needed $O(n)$ parameters to describe the point set, we now need just $O(n^{{1 \over 2}})$ parameters. This is an important step which has hitherto not been available in the case of Szemer\'edi-Trotter.
To go from the proto-inverse theorem for Cauchy Schwarz to an actual inverse theorem we consider the case of $n^{1-}$ lines having at least $n^{{1 \over 2}-}$ incidences each with a product set $A \times B$ with each of $A$ and $B$ having size at most $n^{{1 \over 2}}$. By rescaling, we can have $0,1 \in A$ with $n^{1-}$ lines having an incidence with each of $\{0\} \times B$ and $\{1\} \times B$. Thus the lines are identified with pairs of points to which they are incident $(0,b_1), (1,b_2)$. If the same line is incident to a point of $\{a\} \times B$, we get that $a b_2 + (1-a) b_1 \in B$. Thus for a typical $a \in A$, the quotient ${1-a \over a}$ has $n^{{3 \over 2}-}$ representations as a member of ${B-B \over B-B}$. This is true for at least $n^{{1 \over 2}-}$ choices of $a$. For subsets of the reals, this phenomenon is ruled out by the sum-product theorem. In other settings, (finite fields, the $\delta$-discretized setting) things are bit more delicate because the Szemer\'edi-Trotter theorem isn't true. Inverse Cauchy-Schwarz, although it didn't go by that name, played an important role in the development of sum-product theory in those settings. (See \cite{KT} and \cite{BKT}.)
In this paper, we obtain the first, to our knowledge, proto-inverse theorem for Szemer\'edi-Trotter and for the unit distance problem at the exponent ${4 \over 3}$.
\begin{thm} There is collection $A$ of $N^{{1 \over 3}}$ parameters and maps $\mathcal{L}$ and $\mathcal{P}$ so that for some values of the parameters $A$, $(\mathcal{L}(A),\mathcal{P}(A))$ is a configuration of at least $N^{1-}$ and at most $N$ lines and points. If $(\mathcal{L},\mathcal{P})$ is an extremal configuration of between $N^{1-}$ and $N$ lines and points for Szemer\'edi-Trotter then so is $(\mathcal{L} \cap \mathcal{L}(A), \mathcal{P} \cap \mathcal{P}(A))$. The analogous result is true for unit circles. \end{thm}
We obtain the parametrization in the theorem (which appears in Theorems \ref{protoinverseST} and \ref{protoinverseSTcircles} in the body of the paper) by a deep study of the cell decompositions which prove the Szemer\'edi-Trotter theorem. There is a rather strong analogy to the proto-inverse theorem for Cauchy-Schwarz mentioned above. In the Cauchy-Schwarz setting most of the points lie on a pair of bushes, or after a projective transformation, a product set. In the Szemer\'edi-Trotter setting, it is the cell decomposition which is given by two bushes. We give this as Theorem \ref{doublebushmixing} and the analogous result for unit circles as Theorem \ref{doublebushmixingcircle}. The main idea is that we combine the ideas of cell decomposition by choosing random lines and of the crossing number inequality. By counting the crossings inside cells we are able to organize extremal examples using heuristics suggested by random selections.
\section{Extremal examples and Cell decompositions}\label{cellSection}
We shall be concerned in this paper with ``extremal examples" for the Szemer\'edi-Trotter theorem. Our examples will be (almost) symmetric consisting of approximately $N$ lines and approximately $N$ points. We will use the notation that the inequality $$A \lesssim B,$$ between two non-negative quantities $A$ and $B$ will mean that there is a constant $C$ independent of $N$ so that $$A \leq CB.$$
We would like to allow ourselves losses of small powers of $N$. We choose at the beginning of the paper a small exponent $\epsilon_0$. Implicitly, at each line of the paper, there will be a different exponent $\epsilon$ depending on the line of the paper, with each $\epsilon$ having the property $\epsilon \lesssim \epsilon_0$. We will abbreviate $A \lesssim N^{O(\epsilon_0)}$ by $A \lesssim N^{+}$ or $A \lesssim N^{0+}$ . Similarly we introduce $A \gtrsim N^{-}$ to mean $AN^{O(\epsilon_0)} \gtrsim 1 $ and $A \sim N^{\pm}$ to mean $$N^{-} \lesssim A \lesssim N^+.$$ We let exponents add in the natural way.
Our definition of an extremal example for the Szemer\'edi Trotter theorem will allow $N^+$ errors. This will be slightly unusual for the study of point-line incidences in the plane. The reason is that the Szemer\'edi Trotter theorem is totally sharp in the $\lesssim$ sense. For that reason, the main tools used in studying the problem have been honed to be sharp in the $\lesssim$ sense.
However, there are two reasons we will allow $N^+$ errors. The first is that what we're really after are inverse theorems and these will be stronger and more useful if they apply to examples that fail to be sharp by $N^+$.
The second reason is that we will be studying the properties of probabilistically constructed cell decompositions. While these have been refined to the $\lesssim$ level, the probabilistic construction for doing that is a bit more sophisticated, and we will be taking advantage of the ease of use of the simpler one.
\begin{defn}[Extremal configuration] \label{extreme} We say that with ${\cal L}$ a collection of at most $N$ and at least $N^{1-}$ lines in the plane and ${\cal P}$ a collection of at most $N$ and at least $N^{1-}$ points in the plane and with ${\cal I} ({\cal L},{\cal P})$ denoting the set of incidences (that is pairs $(L,P)$ of a line from ${\cal L}$ and a point from ${\cal P}$ with the point on the line), we say that $({\cal L},{\cal P})$ is an {\bf extremal configuration} if
$$|{\cal I} ({\cal L},{\cal P})| \gtrsim N^{{4 \over 3}-}.$$ \end{defn}
Sometimes, we wish to restrict our attention to only a large subset of the incidences of an extremal configuration. We introduce the notion of an extremal partial configuration.
\begin{defn}[Extremal partial configuration] \label{partial} We say that with ${\cal L}$ a collection of at most $N$ and at least $N^{1-}$ lines in the plane and ${\cal P}$ a collection of at most $N$ and at least $N^{1-}$ points in the plane and with ${\cal I} ({\cal L},{\cal P})$ denoting the set of incidences (that is pairs $(L,P)$ of a line from ${\cal L}$ and a point from ${\cal P}$ with the point on the line) and with $${\cal J}({\cal L},{\cal P}) \subset {\cal I}({\cal L},{\cal P})$$ we say that $({\cal L},{\cal P}, {\cal J}(\mathcal{L},\mathcal{P}))$ is an {\bf extremal partial configuration} if
$$|{\cal J} ({\cal L},{\cal P})| \gtrsim N^{{4 \over 3}-}.$$ \end{defn}
Any pair $({\cal L},{\cal P})$ wth $\mathcal{L}$ consisting of lines in the plane and with $N^{1-} \lesssim |\mathcal{L}| \leq N$
and with $\mathcal{P}$ consisting of points in the plane with $N^{1-} \lesssim |\mathcal{P}| \leq N$ will be called a {\bf configuration}.
\begin{defn}[Cell decomposition, line weighted] \label{celll} Given a configuration $({\cal L},{\cal P})$, we say that a partition of ${\cal P}$ into $r^2$ disjoint subsets (called cells) $C_1,\dots , C_{r^2}$ is a \textbf{line weighted cell decomposition} if no line in $L \in {\cal L}$ is incident to points in $\gtrsim r$ cells and no cell has $\gtrsim {N^{1+} \over r}$ lines of ${\cal L}$ incident to any of its points. A decomposition having all these properties except the bound on the number of cells a line can be incident to points in will be called a \textbf{provisionally line weighted cell decomposition}. \end{defn}
\begin{defn}[Cell decomposition, point weighted] \label{cellp} Given a configuration $({\cal L},{\cal P})$, we say that a partition of ${\cal P}$ into $r^2$ disjoint subsets (called cells) $C_1,\dots , C_{r^2}$ is a \textbf{point weighted cell decomposition} if no line in $L \in {\cal L}$ is incident to points in $\gtrsim r$ cells and no cell contains $\gtrsim {N^{1+} \over r^2}$ points of ${\cal P}$. \end{defn}
Next we show that any extremal configuration with a line weighted cell decomposition into approximately $N^{{2/3}}$ parts can be refined into an extremal configuration with a point weighted cell decomposition using the same partition. This is true because cells with too many points do not produce enough incidences per point because of the bound on the number of lines and the Szemer\'edi Trotter theorem. We simply remove the points of those cells.
\begin{thm} \label{refine1} Let $({\cal L},{\cal P})$ be an extremal configuration with a line weighted cell decomposition $C_1,\dots C_{r^2}$ with $N^{{1 \over 3}-} \lesssim r \lesssim N^{{1 \over 3}}$. Then there is a subset ${\cal P}^{\prime}$
of ${\cal P}$ with $|{\cal P}^{\prime}| \gtrsim N^{1-}$ and $({\cal L},{\cal P}^{\prime})$ an extremal configuration so that the nonempty elements of the list $C_1 \cap {\cal P}^{\prime}, \dots, C_{r^2} \cap {\cal P}^{\prime}$ is a point weighted cell decomposition. \end{thm}
\begin{proof} For the remainder of this proof, we fix the value of $\epsilon$ which corresponds to the current line of the paper. We have $|I(\mathcal{L},\mathcal{P})| \gtrsim N^{{4 \over 3}-\epsilon}$, we have
$N^{1-\epsilon} \lesssim |\mathcal{P}|,|\mathcal{L}| \leq N$ and we have $N^{{1 \over 3} - \epsilon} \lesssim r \lesssim N^{ {1 \over 3}}$. We divide the cells into two classes
${\cal C}_{big}$ and ${\cal C}_{notsobig}$ where $C_j$ is placed into ${\cal C}_{big}$ if $|C_j|> N^{ {1 \over 3} + 10\epsilon}$ and into ${\cal C}_{notsobig}$ otherwise. It suffices to take $$\mathcal{P}_{\epsilon}^{\prime} = \bigcup_{C \in {\cal C}_{notsobig}} C,$$ and show that
$$| \mathcal{P}_{\epsilon}^{\prime} | \gtrsim N^{1-20\epsilon},$$ and
$$|I(\mathcal{L},\mathcal{P}_{\epsilon}^{\prime} )| \gtrsim N^{ {4 \over 3} - \epsilon}.$$ [This is because at the end of the proof, we can reset the value of $\epsilon$ to $20\epsilon$.] We calculate
$$|I(\mathcal{L},\mathcal{P})|=\sum_{C \in {\cal C}_{big}}| I(\mathcal{L},C) |+ \sum_{C \in {\cal C}_{notsobig}}| I(\mathcal{L},C)|.$$ To bound the first term, we apply the Szemer\'edi-Trotter theorem to each big cell using the fact that there are at most $N^{{2 \over 3}+\epsilon}$ lines going through each cell obtaining
$$\sum_{C \in {\cal C}_{big}} |I(\mathcal{L},C)| \lesssim \sum_{C \in {\cal C}_{big}} N^{{4 \over 9} + {2 \over 3}\epsilon}
|C|^{{2 \over 3}}$$
$$ \lesssim \sum_{C \in {\cal C}_{big}} N^{{1 \over 3} - {8 \over 3} \epsilon} |C| \lesssim N^{{4 \over 3 }- {8 \over 3}\epsilon}.$$
Here the penultimate inequality uses that each $|C|$ is at least $N^{{1 \over 3} + 10 \epsilon}$ and the last inequality uses that $|\mathcal{P}| \lesssim N$.
Now we know that $$N^{{4 \over 3} -\epsilon} \lesssim I(\mathcal{L},\mathcal{P}_{\epsilon}^{\prime} ) ,$$
and we need only show that this implies a good lower bound on $|\mathcal{P}_{\epsilon}^{\prime} |$. But this follows immediately from the Szemer\'edi-Trotter theorem and the extremality of the example.
\end{proof}
Next, we will show that for any extremal configuration together with a point weighted cell decomposition with $N^{{1 \over 3}-} \lesssim r \lesssim N^{{1 \over 3}}$ there is a refinement of the set of lines preserving extremality so that each line is incident to points in $\gtrsim N^{{1 \over 3}-}$ cells. This is a direct application of the Cauchy Schwarz inequality. The lines we remove don't account for many incidences.
\begin{thm} \label{refine2} Let $(\mathcal{L},\mathcal{P})$ be an extremal configuration. Let $C_1, \dots , C_{r^2}$ be a point weighted cell decomposition with $r \sim N^{{1 \over 3} \pm}$. Then there is a refinement $\mathcal{L}^{\prime} \subset \mathcal{L}$ so that
$|I(\mathcal{L}^{\prime},\mathcal{P})| \gtrsim N^{{4 \over 3}-}$ and every $L \in \mathcal{L}^{\prime}$ is incident to points in $\gtrsim N^{{1 \over 3}-}$ cells. \end{thm}
\begin{proof} For the remainder of the proof, we fix the value of $\epsilon$ corresponding to this line, with $I(\mathcal{L},\mathcal{P}) \gtrsim N^{{4 \over 3}-\epsilon}$.
Consider $\mathcal{L}_{\epsilon}$, the set of lines intersecting fewer than $r^{1-20\epsilon}$ cells. It
suffices to show that $|I(\mathcal{L}_{\epsilon},\mathcal{P})|$ is considerably smaller than $N^{{4 \over 3}-\epsilon}$. For each line $L$, let $C_L$ denote the set of cells in which $L$ is incident to a point. For $L$ a line and $P$ a point, we let $I_{L,P}$ be the indicator function of incidence, namely $I_{L,P}=1$ if $P$ is incident to $L$ and $0$ otherwise.
We calculate
\begin{flalign*}
|I(\mathcal{L}_{\epsilon},\mathcal{P})| &=\sum_{L \in \mathcal{L}_{\epsilon}} \sum_{C \in C_L} \sum_{P \in C} I_{LP} &\\
&\lesssim N^{{1 \over 2}} r^{{1-20 \epsilon \over 2}} (
\sum_{L \in \mathcal{L}_{\epsilon}} \sum_{C \in C_L}
(\sum_{P \in C} I_{LP})^2 )^{{1 \over 2}} &\\
&\lesssim N^{{1 \over 2}} r^{{1-20 \epsilon \over 2}} (
I(\mathcal{L}_{\epsilon},\mathcal{P}) + \sum_C \sum_{P_1 \in C} \sum_{P_2 \in C , P_2 \neq P_1} \sum_{L}
I_{LP_1} I_{LP_2} )^{{1 \over 2}} &\\
&\lesssim N^{{1 \over 2}} r^{{1-20 \epsilon \over 2}} (N^{{4 \over 3}+2 \epsilon})^{{1 \over 2}} &\\
&\lesssim N^{{4 \over 3} -{4 \epsilon \over 3}+O(\epsilon^2)} \end{flalign*} \end{proof}
The goal of the next theorem is to say that for an extremal example having a point weighted cell decomposition with $r$ just on the low side of $N^{{1 \over 3}}$, most of the incidences come from cells having around ${N \over r^2}$ points, and around $ {N \over r}$ lines making just a few incidences with these points, but at least two. As a corollary, we will obtain a kind of inverse theorem for the lines having a few incidences with the points of a cell that will prove useful later. (The idea is that for any set of points, the number of lines intersecting two of them is controlled by the square of the number of points.)
\begin{thm} \label{refine3}
Let $(\mathcal{L},\mathcal{P})$ be an extremal configuration. Specifically let $|I(\mathcal{L},\mathcal{P})|=N^{{4 \over 3}-\epsilon}$ with $\epsilon$ fixed.
Let $C_1,\dots , C_{r^2}$ be a point-weighted cell decomposition for $(\mathcal{L},\mathcal{P})$
with $N^{{1 \over 3} - 5\epsilon} \leq r \leq { |I(\mathcal{L},\mathcal{P})| \over 100 |\mathcal{L}|}$. Then there is a set of incidences $J(\mathcal{L},\mathcal{P}) \subset I(\mathcal{L},\mathcal{P})$ so that $|J(\mathcal{L},\mathcal{P})| \gtrsim N^{{4 \over 3}-\epsilon}$, but for every line $L$ and cell $C$ for which there is $P \in C$ with $(L,P) \in J(\mathcal{L},\mathcal{P})$, we have that
$$2 \leq | I( \{L\},C)| \lesssim N^+.$$ \end{thm}
\begin{proof} The way this proof will work is that we will remove from $I(\mathcal{L},\mathcal{P})$ all incidences that would violate the conditions for $J(\mathcal{L},\mathcal{P})$ and observe that we have removed less than half of the set $I(\mathcal{L},\mathcal{P})$.
First, for any point P for which there are more than $N^{{1 \over 3} +10 \epsilon}$ lines incident, we can remove all these incidences and applying Szemer\'edi-Trotter, we see that since there are $\lesssim N^{1-10\epsilon}$ many such points, we have removed $\lesssim N^{{4 \over 3} - {20 \epsilon \over 3} + }$ incidences.
For any line $L$ and cell $C$ for which there is a unique point $P$ with $(L,P)$ an incidence, we remove these incidences and we have removed at most $r |\mathcal{L}|$ incidences since each line has incidences with at most $r$ cells.
Finally for the incidences which remain, for any cell $C$, all lines are incident to at least two points of the cell
(which define the line) so there are at most $|C|^2$ such lines. We can remove all incidences from cells that do not contribute at least $\gtrsim N^{-} |C|^2 $ incidences. We note that especially rich lines cannot contribute most of the incidences. The number of lines passing through $k$ of the points is at most
${|C|^2 \over k^3}$ contributing at most ${C^2 \over k^2}$ incidences (this follows from Szemer\'edi--Trotter), so we remove these for $k$ which are not $\lesssim N^+$.
\end{proof}
\begin{cor} \label{structuredcells} Let $(\mathcal{L},\mathcal{P})$ be an extremal configuration. Let $C_1,\dots , C_{r^2}$ be a point-weighted cell decomposition for $(\mathcal{L},\mathcal{P})$
with $r \geq N^{{1 \over 3}-}$ but $r \leq { |I(\mathcal{L},\mathcal{P})| \over 100 |\mathcal{L}|}$. Then there is a set ${\cal C}$ of $\gtrsim r^{2-}$ cells so that for each $C \in {\cal C}$, there is a set of lines $\mathcal{L}_C$ with
$$|\mathcal{L}_C| \gtrsim |C|^{2-},$$ and with each $L \in \mathcal{L}_C$ incident to at least $2$ but $\lesssim N^+$ points in $C$. Each set $\mathcal{L}_C$ has density $\gtrsim N^-$ in the set of lines intersecting two points in $C$.
\end{cor}
\section{The Probabilistic method and cell decompositions} \label{probSection}
There are several different methods known for producing cell decompositions for proving the Szemer\'edi-Trotter theorem. The most modern is polynomial partitioning. In that method, the boundaries of cells are given by the zero set of a polynomial. If a cell decomposition is thus obtained, it is naturally point-weighted. This is called the cellular case. The alternative is that the points all lie in the zero set of a fairly low-degree polynomial. This is called the structured case. In this case we obtain the Szemer\'edi-Trotter theorem by bounding the intersection of a curve of bounded degree and a line by the curve's degree. However, in the structured case we don't have a cell decomposition of the set of points. The method of polynomial partition was first introduced by Larry Guth and the first author in resolving the Erd\"os distinct distances problem in the plane. \cite{GK}
An older approach is to define a cell decomposition by randomly selecting lines from the configuration's line set. That approach most naturally produces a cell decomposition that is line-weighted. This seems to have been first developed in the seminal paper of Clarkson {\it et. al.} \cite{CEGSW} as a simplification and improvement of a deterministic construction found in the original paper of Szemer\'edi and Trotter. \cite{ST}. This always produces a cell decomposition, at least if we started with a configuration containing no overly rich lines..
Our present aim is to use cell decompositions in order to learn about the properties of extremal configurations. For this purpose, the deterministic nature of polynomial partitioning is unhelpful. The cells are chosen by the Borsuk-Ulam theorem in a somewhat mysterious way. Not many choices are available. We will work with the probabilistic method where almost every selection of lines yields an acceptable cell decomposition. The fact that an extremal configuration behaves much
the same for each selection of lines seems to yield a lot of information on extremal configurations. Our original plan for how to carry out our arguments used this observation heavily. But for technical reasons, it turns out to be beneficial to use a different classic approach to the Szemer\'edi-Trotter theorem, the crossing number inequality. Principally, we will use this inequality to say that the lines in a typical point weighted cell of an extremal configuration behave in the way that you would expect from a random selection of lines.
We largely follow Terry Tao's blogpost \cite{T} on probabilistic constructions of cell decompositions. Given $\mathcal{L}$, a collection of $N$ lines, we choose a random subset $\mathcal{L}_r$ with each line of $\mathcal{L}$ chosen independently with probability ${r \over N}$.
We use one probabilistic calculation repeatedly:
\begin{lem} \label{consecutivelines} Let ${\cal S} \subset \mathcal{L}$ be any subset of ${C N \log N \over r}$ lines with $C$ a sufficiently large constant. The probability that none of the lines in ${\cal S}$ is selected is bounded by ${1 \over N^{100}}$. \end{lem}
\begin{proof} By independence of the individual lines, the probability that no line in ${\cal S}$ is selected is exactly $$(1-{r \over N})^{{C N \log N \over r}}.$$ \end{proof}
We would like to choose a random selection $\mathcal{L}_r$ containing at least ${r \over 2}$ lines and no more than $2r$ lines. By Chernoff bounds, the probability that this fails is exponentially small in $N$. Moreover, we'll make a list of $O(N^4)$ events controlled by Lemma \ref{consecutivelines} that we would like not to occur. We obtain a set of lines which satisfies our requirements with probability at least $1-O(N^{-96})$.
This might be a good moment to review how Chernoff bounds work so that when we later need to use them a bit more seriously, we'll be better prepared.
\begin{lem} \label{Chernoff} Let $X_1, \dots , X_M$ be independent Bernoulli variables equal to $1$ with probability $p$ and zero otherwise. Let $pM> M^{\delta}$ for some $\delta>0$ then $P$, the probabilitiy that $X$ is larger than $10 pM$, where $$X=X_1+X_2+ \dots X_M,$$ is bounded by $$ e^{-8pM}$$ \end{lem}
\begin{proof} Observe that since the $X_j$'s are independent and identically distributed, we have $$E(e^X)=E(e^{X_1})^M=(1+p(e-1))^M \sim e^{p(e-1)M}.$$ But $P e^{10 pM} \leq E(e^X) \sim e^{(e-1) p M}.$ Hence $$P \lesssim e^{-8pM}.$$ \end{proof}
For each line $l \in \mathcal{L}$, we establish an ordering on $\mathcal{L} \backslash \{l\}$ based on the position of their intersection with $l$. (It might be at infinity.) The ordering is ill-defined when multiple lines are concurrent at a point of $l$, but we order concurrent lines arbitrarily. For each choice of $l^{\prime} \in \mathcal{L}$, with $l^{\prime} \neq l$ we would like to exclude the event that none of the ${C N \log N \over r}$ consecutive lines following $l^{\prime}$ in the order induced by $l$ are selected for $\mathcal{L}_r$. This is a list of $O(N^2)$ events governed by Lemma \ref{consecutivelines}.
One then chooses a direction different from that of all lines in $\mathcal{L}$ which we will refer to as vertical. At each point $p$ of intersection of two of the lines of $\mathcal{L}$, the vertical line at $p$ induces an order on the lines of $\mathcal{L}$. We would like to exclude the case that none of the first ${C N \log N \over r}$ lines above $p$ are chosen for $\mathcal{L}_r$ and none of the first ${C N \log N \over r}$ lines below $p$ are chosen for $\mathcal{L}_r$. This is a list of $O(N^2)$ events governed by Lemma \ref{consecutivelines}.
For future reference, we would also like to rule out one other kind of event. Given two points $p_1$ and $ p_2$ each of which are at the intersection of two lines of $\mathcal{L}$ for which at least ${C N \log N \over r}$ lines of $\mathcal{L}$ intersect the open line segment between $p_1$ and $p_2$, at least one of those lines will be selected for $\mathcal{L}_r$. This is a list of $O(N^4)$ events governed by Lemma \ref{consecutivelines}.
With probability $1-O(N^{-96})$, a random selection of lines satisfies our specifications. Let $\mathcal{L}_r$ be such a selection.
\begin{defn} \label{funneldecomp} A \textbf{funnel decomposition} is obtained from a cell decomposition by breaking each cell into trapezoids by taking a vertical line segment from each vertex of the cell until the point where it intersects some edge. (These trapezoids are called funnels in \cite{CEGSW} where this construction was invented.) \end{defn}
\begin{lem} \label{probconstruction} With $r \sim N^{{1 \over 3} \pm}$, the funnel decomposition produces a provisionally line-weighted cell decomposition with probability at least $1-O(N^{-96})$. \end{lem}
\begin{proof}
We let $\mathcal{L}_r$ be a selection obeying the above specifications. We start with the cells given by the configuration $\mathcal{L}_r$ and break it into a funnel decomposition. Because none of the adverse events occur, at most ${C N \log N \over r}$ lines enter any given funnel from each of its 4 sides.
\end{proof}
For our purposes, both the above construction and the slightly refined one of Matou\v{s}ek \cite{M} seem unsatisfactory because of the introduction of edges which are not on lines of $\mathcal{L}$. We would like to be able to recognize the lines entering a cell as lines that intersect a fixed selected line of intersection $L$ between consecutive points of intersection with other selected lines. For this reason, we find cell decompositions that come from just selecting $r$ random lines and making no further adjustments most natural. This is problematic because such a decomposition is no longer line weighted due to cells with too many edges. We will deal with this by bounding the number of points that can be contained in such cells and refining the point set $\mathcal{P}$ to only include points in cells with $\lesssim N^+$ edges.
The main ingredient in our bound will be the theorem of Clarkson {\it et. al.} \cite{CEGSW} on the complexity of cell decompositions given by general families of lines.
\begin{thm} \label{cellcomplexity} Let $\mathcal{L}$ be a set of $r$ lines. It divides projective space into $O(r^2)$ cells. Let ${\cal C}$ be any subcollection of $m$ of these cells. Then the total number of edges of cells in ${\cal C}$ is $O(r^{{2 \over 3}} m^{{2 \over 3}} + r)$. \end{thm}
\begin{cor} \label{numberofbigcells} Let $\mathcal{L}_r$ be any set of $r$ lines and let ${\cal C}$ be the set of cells which they define having $>s$ and $\leq 2s$ edges. Then if $s \leq r^{{1 \over 2}}$, we have
$$|{\cal C}| \lesssim {r^2 \over s^3},$$ and if $s \geq r^{{1 \over 2}}$
$$|{\cal C}| \lesssim {r \over s}.$$ \end{cor}
Having now obtained a bound on the number of cells with a certain number of edges, we now control the number of rich points in such a cell. Let $(\mathcal{L},\mathcal{P})$ be an extremal configuration in which each point $p$ of $\mathcal{P}$ is at least $N^{{1 \over 3}-}$ rich.
\begin{lem} \label{pointsinacell} Let $K$ be a cell coming from an acceptable selection of lines $\mathcal{L}_{N^{{1 \over 3}}}$. Suppose $K$ has $s$ sides. Then $K$ contains at most $sN^{{1 \over 3}+}$ points of $\mathcal{P}$. \end{lem}
\begin{proof} Following the construction in \cite{M}, for each cell we choose a vertex and divide the cell into triangles by adding edges between the chosen vertex and all non-adjacent cell vertices. Then $C$ is divided into $s-2$ triangles $T$. Each triangle has at most ${3 N^{{2 \over 3}} \log N \over 2}$ lines entering it. Suppose there are $P$ points of $\mathcal{P}$ in the triangle $T$. Then there are at least $P N^{{1 \over 3}-}$ incidences in $T$. The Szemer\'edi Trotter theorem guarantees that $P \leq N^{{1 \over 3}+}$. Thus the total number of points in $K$ is at most $(s-2) N^{{1 \over 3}+}$ points of $\mathcal{P}$ which was to be shown. \end{proof}
To extract structure from our configurations we will chose to throw out undesirable points and lines keeping only those that enjoy the desirable properties.
\begin{defn}
Given an extremal partial configuration $(\mathcal{L}, \mathcal{P}, \cal{J(\mathcal{L}, \mathcal{P})})$ we say $(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})$ is a \textbf{refinement} if $|\mathcal{L}| \sim |\mathcal{L}^{\prime}|$ and $|\mathcal{P}| \sim |\mathcal{P}|^{\prime}$ and $|\cal{J(\mathcal{L},\mathcal{P})}| \sim |\cal{J} (\mathcal{L}^{\prime},\mathcal{P}^{\prime})|$. \end{defn}
Finally, we combine Corollary \ref{numberofbigcells} with Lemma \ref{pointsinacell} to bound the number of points of $\mathcal{P}$ contained in cells with between $s$ and $2s$ sides. If $s \leq r^{{1 \over 2}}$ with $r=N^{{1 \over 3}}$ , we obtain the bound ${r^2 \over s^3} sN^{{1 \over 3}+} \leq {N^{1+ } \over s^2}$. If $s \geq r^{{1 \over 2}}$, we obtain the bound ${r \over s} sN^{{1 \over 3}+} \leq N^{{2 \over 3}+}$. As long as $s$ is much bigger than $N^+$, we do not capture a significant number of points. We conclude the following theorem.
\begin{thm} \label{nicerefinement} Let $(\mathcal{L},\mathcal{P})$ be an extremal configuration with each point of $\mathcal{P}$ being at least $N^{{1 \over 3}-}$ rich. Then for each acceptable random selection $\mathcal{L}_{N^{{1 \over 3}}}$ there is a refinement $\mathcal{P}^{\prime} \subset
\mathcal{P}$ with $|\mathcal{P}^{\prime}| \geq {1 \over 2} |\mathcal{P}|$ so that no point of $\mathcal{P}^{\prime}$ is contained in a cell with more than $N^{+}$ sides, and in light of Lemma \ref{pointsinacell}, each such cell has at most $N^{{1 \over 3}+}$ points of $\mathcal{P}^{\prime}$ so that we have obtained a point weighted decomposition for the extremal configuration $(\mathcal{L},\mathcal{P}^{\prime})$ \end{thm}
The main power of Theorem \ref{nicerefinement} for us will be that we can use it to deduce strong properties of extremal examples without reference to any cell decomposition.
Next we're going to use Corollary \ref{structuredcells} to get structuring result for extremal configurations where many lines have points bounding intervals where approximately $N^{{2 \over 3}}$ lines intersect. Furthermore large subsets of these groups of about $N^{{2 \over 3}}$ lines are structured: they intersect two of a set of not much more than $N^{{1 \over 3}}$ points. A key ingredient in proving this will be the standard crossing number inequality which we state here.
\begin{lem} \label{crossingnumber} \cite{S} Let $G(V,E)$ be a planar graph with $v$ vertices and $e$ edges. Suppose that $e \geq 10v$. Then the number of crossings between edges is $\gtrsim {e^3 \over v^2}$. \end{lem}
We will write down a corollary describing the usual way that we use this. Any cell where a large number of lines are incident to at least two points must have a lot of crossings.
\begin{cor} \label{crossingcells} Let $\mathcal{P}_c$ be a collection of $N^{{1 \over 3}+ \delta_1}$ points in a convex region $R$ of the plane. Let $\mathcal{L}_c$ be a collection of $N^{{2 \over 3} - \delta_2}$ lines each of which is incident to at least $M+1$ points of $\mathcal{P}_c$. Then $\gtrsim M^3 N^{{4 \over 3} -\delta_1 - 3 \delta_2}$ pairs of lines from $\mathcal{L}_c$ cross in the region $R$. \end{cor}
\begin{proof} Define a graph $G$ with $\mathcal{P}_c$ as the point set and for each line, pick $M$ consecutive pairs of points that the line is incident to to be edges. Because the set $R$ is convex, the edges lie in $R$. Now apply Lemma \ref{crossingnumber}. \end{proof}
We make a definition of a structured set of lines.
\begin{defn} We say that a set $\mathcal{L}_1$ of at least $N^{{2 \over 3}-}$ lines is \textbf{structured} if there is a set $\mathcal{P}_1$ of at most $N^{{1 \over 3}+}$ points so that each line of $\mathcal{L}_1$ is incident to at least two points of $\mathcal{P}_1$. We call this set of points \textbf{structuring}. \end{defn}
\begin{figure}
\caption{Six structuring points structure the set of black lines. A generic pair of structuring points must define a line in the structured set but not all the pairs must.}
\label{fig:structured_lines}
\end{figure}
Note that since $\lesssim N^{\frac{2}{3}+}$ lines go through at least two among $\lesssim N^{\frac{1}{3}+}$ points, the structuring points essentially define the structured lines. Now we're ready to state our structuring theorem.
\begin{thm} \label{verynicespacing} Let $(\mathcal{L},\mathcal{P}, \cal{J})$ be an extremal partial configuration. Then there is a refinement $(\mathcal{L}^{\prime},\mathcal{P},\cal{J}')$ and so that for each line $l \in \mathcal{L}^{\prime}$ there are points $p_1, \dots , p_M$ of $\mathcal{P}$ with $(l,p_j) \in \cal{J}'$ and the $p_j$'s in order of their position on $l$ and with $M \gtrsim N^{{1 \over 3}-}$ so that for each consecutive pair of points $p_j,p_{j+1}$, there is a structured set of lines $\mathcal{L}_j$ so that each $l^{\prime}$ in $\mathcal{L}_j$ intersects $l$ in the open interval bounded by the points $p_j$ and $p_{j+1}$. We say the lines in $\mathcal{L}^{\prime}$ \textbf{organize} $\mathcal{P}$. \end{thm}
\begin{figure}
\caption{An organizing line $l \in \mathcal{L}'$ is shown with two intervals bounded respectively by points $p_1,p_2$ and $p_2, p_3$. Each of these intervals is crossed by a structured set of (gray) lines. The two gray points on $l$ are elements of $\mathcal{P}$ which are not included in our refined construction as the incidence between $l$ and the gray points are not included in $\cal{J}'$.}
\label{fig:organizing_line}
\end{figure}
\begin{proof} Starting with the extremal partial configuration $(\mathcal{L},\mathcal{P},\cal{J})$ we remove all points with fewer than $N^{{1 \over 3}-}$ incidences in $\cal{J}$ obtaining a refinement $\mathcal{P}^{\prime} \subset \mathcal{P}$ so that $(\mathcal{L},\mathcal{P}^{\prime}, \cal{J}^{\prime})$ is still an extremal partial configuration (with $\cal{J}^{\prime}$ the intersection of $\cal{J}$ with the Cartesian product of $\mathcal{L}$ and $\mathcal{P}^{\prime}$) but satisfies the hypotheses of Theorem \ref{nicerefinement}. We apply Theorem \ref{nicerefinement} to obtain a further refinement $\mathcal{P}^{\prime \prime} \subset \mathcal{P}^{\prime}$ so that we have a point weighted cell decomposition ${\cal C} = \{ C_1, \dots, C_{r^2} \}$ for the extremal configuration $(\mathcal{L}, \mathcal{P}^{\prime \prime})$ with $r \gtrsim N^{{1 \over 3}-}$ so that $(\mathcal{L},\mathcal{P}^{\prime \prime})$
together with ${\cal C}$ satisfy the hypotheses of Corollary \ref{structuredcells}. From Corollary \ref{structuredcells} we obtain a refinement ${\cal C}^{\prime}$ of ${\cal C}$ with $|{\cal C}^{\prime}| \gtrsim N^{{2 \over 3}-}$ and with a subset $\mathcal{L}_C$ of the lines of $\mathcal{L}$ going through any cell $C$ of ${\cal C}^{\prime}$ being a structured set. Hence, any subset $\mathcal{L}_1 \subset \mathcal{L}_C$ with
$|\mathcal{L}_1| \gtrsim N^{{2 \over 3}-}$ is also a structured set.
We now examine a fixed cell $C \in {\cal C}^{\prime}$, the set of points $\mathcal{P}_C$ consisting of points of $\mathcal{P}^{\prime \prime}$ which lie inside the cell $C$ and the structured set $\mathcal{L}_C$ from the previous paragraph. We define a graph $G(V,E)$ whose vertex set consists of the points of $\mathcal{P}_C$ and whose edges are the pairs of consecutive points of $\mathcal{P}_C$ which are incident to any line of $\mathcal{L}_C$. Because $\mathcal{L}_C$ is a structured set (structured by the point set $\mathcal{P}_C$ because of the application of Corollary \ref{structuredcells}), there is at least one edge of $G$ for each line of $\mathcal{L}_C$. Thus,
$v=|V| \lesssim N^{{1 \over 3}+}$ while $e=|E| \gtrsim N^{{2 \over 3}-}$. We conclude from Lemma \ref{crossingnumber} that the number of crossings for the graph is $\gtrsim {e^3 \over v^2} \gtrsim N^{{4 \over 3}-}$. But the crossings of the graph are nothing other than intersections between lines of $\mathcal{L}_C$ which occur inside the cell $C$. For each line $l$ in $\mathcal{L}_C$ which intersects at least $N^{{2 \over 3}-}$ lines of $\mathcal{L}_C$, we associate the interval $I$ which is the intersection of the line with the cell. This interval contains at least one (in fact, two) points of $\mathcal{P}^{\prime \prime}$ and intersects a structured subset $\mathcal{L}_I$ from $\mathcal{L}_C$. We will count pairs $(l,I)$. Because the lines of $\mathcal{L}_C$ all must intersect the cell $C$ which has fewer than $N^{+}$ edges, it must be that $|\mathcal{L}_C| \lesssim N^{{2 \over 3}+}$ and therefore each cell $C$ of ${\cal C}^{\prime}$ generates at least $N^{{2 \over 3}-}$ pairs $(l,I)$. Summing over all the cells of ${\cal C}^{\prime}$, we obtain at least $N^{{4 \over 3}-}$ many such pairs. For each line $l$, the different intervals $I$ for which $(l,I)$ are such pairs are disjoint. Since each interval is crossed by a structured set of lines, there are at most $N^{{1 \over 3}+}$ intervals $I$ for each line $l$. Thus there must be at least $N^{1-}$ lines $l$ with at least $N^{{1 \over 3}-}$ pairs $(l,I)$. We will call this set of lines $\mathcal{L}^{\prime}$.
For each line $l \in \mathcal{L}^{\prime}$, we order the intervals $I$ for which $(l,I)$ is a pair as $I_1,I_2, \dots ,I_M$. For each $j =2k-1$ odd, we pick a point $p$ from $\mathcal{P}$ from the at least two which lie in the interval and call it $p_k$. For each $j=2k$ odd, we pick the structured set $\mathcal{L}_k$ which intersects $I_j$.
\end{proof}
We'd now like to take advantage of our result. Theorem \ref{verynicespacing} is a method of associating to each extremal configuration a refinement which is rather nicely parametrized. We will do this by applying point-line duality to the result of Theorem \ref{verynicespacing}. The result of the theorem gives us many lines $l$ which are incident to particular sets of points $p_1, \dots ,p_M$ with $M \gtrsim N^{{1 \over 3}-}$ so that we have $\gtrsim N^{{2 \over 3}-}$ lines intersecting $l$ between adjacent points which are structured. We use point--line duality, applying Theorem \ref{verynicespacing} to lines and points instead of points and lines.
\begin{thm} \label{dualverynicespacing} Let $(\mathcal{L},\mathcal{P}, \cal{J})$ be an extremal partial configuration. Then there is a refinement $(\mathcal{L},\mathcal{P}^{\prime}, \cal{J}')$ so that for each point $p \in \mathcal{P}^{\prime}$ there are lines $l_1, \dots , l_M$ of $\mathcal{L}$ which are incident to $p$ in $\cal{J}'$ in order of their direction and with $M \gtrsim N^{{1 \over 3}-}$ so that each sector bounded by consecutive pairs of lines $l_j,l_{j+1}$ contains a structured set of $\gtrsim N^{\frac{2}{3}-}$ points $\mathcal{P}_j$. We say the points in $\mathcal{P}^{\prime}$ \textbf{organize} $\mathcal{L}$. \end{thm}
\begin{figure}
\caption{An organizing point $p \in \mathcal{P}'$ is shown with two sectors $s_1$ and $s_2$ bounded respectively by lines $l_1,l_2$ and $l_2, l_3$. Sectors $s_1$ and $s_2$ each contain a structuring set of (gray) lines. The two dotted black lines through $p$ are elements of $\mathcal{L}$ which are not included in our refined construction as the incidence between $p$ and the dotted lines are not included in $\cal{J}'$.}
\label{fig:organizing_point}
\end{figure}
For any point $p \in \mathcal{P}^{\prime}$ with lines $l_1, \dots l_M$ incident to it, there are $\gtrsim N^{{2 \over 3}-}$ points in each of the $\gtrsim N^{{1 \over 3}-}$ sectors for a total of $N^{1-}$ points. We take this set of points as a refinement $\mathcal{P}^{\prime}$ of our original set of points $\mathcal{P}$. What is particularly pleasant about this structure is that each of the $\gtrsim N^{{2 \over 3}-}$ points of $\mathcal{P}^{\prime}$ between two adjacent lines $l_j$ and $l_{j+1}$ lie on at least two of the $\gtrsim N^{{1 \over 3}-}$ structuring lines.
Structuring lines seem very odd precisely because all of the points on them lie in a particular sector between an $l_j$ and $l_{j+1}$. But this is not as odd as it seems. We see from the proof of Theorem \ref{verynicespacing}, that the structuring lines for $p$ are dual to the points of cells through which the line dual to $p$ passes. Every point lies in a cell, and every cell has $N^{2/3 \pm}$ lines going through it so by duality the set of $N^{1/3}$ structuring lines define the $N^{2/3 \pm}$ points in a sector.
We're going to show that for any choice of $p \in \mathcal{P}^{\prime}$ a typical line $l$ will have incidences in most of the sectors between consecutive lines $l_j$ and $l_{j+1}$. To do this we first need to introduce a refinement of the configuration endowed with a cell decomposition whose boundary lines include the bush through $p$.
\begin{thm}[bush construction] \label{bushconstruction} For any extremal configuration $(\mathcal{L}, \mathcal{P})$ there exists a subset $\mathcal{P}^{\prime}$ of $\gtrsim N^{1-}$ points in $\mathcal{P}$ which are organizing with $\sim N^{{1 \over 3} \pm}$ sectors and a refined configuration $(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})$ such that the $\gtrsim N^{1-}$ lines in $\mathcal{L}^{\prime}$ organize $\mathcal{P}^{\prime}$. Also for any $p\in \mathcal{P}^{\prime}$ the refinement $(\mathcal{L}^{\prime}, \mathcal{P}_p)$ where $\mathcal{P}_p$ are the points in $\mathcal{P}$ organized by $p$, has a refinement $(\mathcal{L}^{\prime}, \mathcal{P}_p^{\prime})$ which is an extremal configuration with a point-weighted cell decomposition where each cell is contained in a sector. Moreover, any line $l \in \mathcal{L}^{\prime}$ which crosses exactly $N^{{2 \over 3}+\alpha}$ lines of $\mathcal{L}^{\prime}$ within the sector $s$ with $\alpha>k\epsilon$ for $k$ sufficiently large will not enter more than $N^{\alpha+}$ cells in $s$.
\end{thm}
\begin{proof}
\textit{Refinement properties:} First we keep only points from $\mathcal{P}$ which are $\sim N^{{1 \over 3} \pm}$ rich. This does not significantly affect the number of incidences. We then apply Theorem \ref{dualverynicespacing} and obtain the refinement $(\mathcal{L}, \mathcal{P}^{\prime})$ of organizing points. Then we apply Theorem \ref{verynicespacing} to $(\mathcal{L}, \mathcal{P}^{\prime})$ obtaining the refinement $(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})$ where the lines in $\mathcal{L}^{\prime}$ organize $\mathcal{P}^{\prime}$. Now we have shown the first claim of the theorem.
\textit{Cell decomposition:} Let $p \in \mathcal{P}^{\prime}$ and $\mathcal{P}_p$ be the set of points in $\mathcal{P}$ organized by $p$. Note there are $\gtrsim N^{1-}$ organized points each $\gtrsim N^{{1 \over 3}-}$ rich so $(\mathcal{L}^{\prime}, \mathcal{P}_p)$ is an extremal configuration which we work in for this paragraph. We label the bush of $M \sim N^{{1\over 3}\pm}$ lines intersecting $p$ as $l_1,\dots, l_M$. Now we pick random $l^{\prime}_1,\dots , l^{\prime}_K$ with $K \sim N^{{1 \over 3}}$ from $\mathcal{L}$. Our cell decomposition will be made from the lines $l_1,\dots,l_M$ together with the lines $l^{\prime}_1, \dots, l^{\prime}_K$. So each cell is contained in a simple sector.
Now, the randomly selected lines $l^{\prime}_1, \dots , l^{\prime}_K$ can be chosen to be in the very likely case where they separate the points on each structuring line of a sector into mostly distinct cells.
For each structuring line $l_s$, there are points $p_1, \dots ,p_L$ from $\mathcal{P}_p$ with $L \gtrsim N^{{1 \over 3}-}$ so that between each consecutive pair $p_k,p_{k+1}$ there are $\gtrsim N^{{2 \over 3}-}$ lines of $\mathcal{L}$ which cross the line $l_s$. Applying Lemma \ref{consecutivelines}, we get a bound of $N^{-100}$ on the probability that any $N^{{2 \over 3}+}$ consecutive lines of $\mathcal{L}^{\prime}$ in the order they intersect $l_s$ don't include one of the $l^{\prime}$'s. We can select the $l^{\prime}$'s so that none of these events happen. Thus each cell contains at most $N^{+}$ points per structuring line. Since there are $\lesssim N^{{1 \over 3}+}$ structuring lines in each sector, our cell decomposition is point weighted.
To get the claim about lines $l$ with $N^{{2 \over 3}+\alpha}$ crossings simply apply Lemma \ref{Chernoff} to control the probability that more than 10 times the expected number of lines crossing $l$ are selected as random lines.
\end{proof}
When trying to show that a typical line in a configuration $(\mathcal{L},\mathcal{P})$ will have incidences in most of the sectors of an organizing point, the enemy case is lines which take too many points in a given sector. A typical line has $N^{2/3}$ crossings between each pair of points. So if we show that lines with too many crossings in a sector do not contribute significantly to the total number of incidences in that sector, then most incidences come from lines taking $\lesssim N^{+}$ points in that sector and we win.
\begin{thm} \label{sectorincidences} There exist $\gtrsim N^{1-}$ organizing points $p$ in $(\mathcal{L}, \mathcal{P})$ with $\sim N^{{1 \over 3}\pm}$ sectors and some integer $k$ such that for every sector $\gtrsim N^{1-}$ of its incidences come from lines taking fewer than $N^{k \epsilon}$ points in that sector. \end{thm}
\begin{proof} We choose a point $p$ to be the center of our bush and use the bush construction $(\mathcal{L}', \mathcal{P}_p')$ from Theorem \ref{bushconstruction}. For a large enough constant $k$, we toss out sectors that have $\gtrsim N^{{5 \over 3}+k_1\epsilon}$ line line crossings. There are at most $N^{{1 \over 3}-k_1\epsilon}$ such sectors because $\gtrsim N^{2-}$ line-line crossings total. So we still kept $\sim N^{{1 \over 3}\pm}$ sectors which each have $\sim N^{{5 \over 3} \pm}$ line-line crossings and each sector contributes $\gtrsim N^{1-}$ incidences. So this refinement still yields a configuration $(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})$. Similarly, our cell decomposition from Theorem \ref{bushconstruction} has $\lesssim N^{{2 \over 3}}$ cells, so we can remove any sectors with more than $N^{{1 \over 3} + k_1 \epsilon}$ cells.
By Theorem \ref{refine3} we may chose a subset $J(\mathcal{L}^{\prime}, \mathcal{P}^{\prime}) \subset I(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})$ such that $|J(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})| \gtrsim N^{{4 \over 3}-}$ and every line has $\lesssim N^{+}$ incidences from $J(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})$ per cell.
\begin{defn}[Fast lines] \label{fastlines} We say that a line is $\alpha-$fast for a sector $s$ if it has $\sim N^{{2 \over 3}+\alpha\pm}$ crossings with lines in $\mathcal{L}$. \end{defn}
This is the enemy case. Our goal is to show these do not contribute significantly to the number of incidences in the sector. Note that since a fast line crosses $N^{{2 \over 3}+\alpha \pm}$ lines of $\mathcal{L}$ in $s$, it must cross at least $N^{\alpha-}$ of the randomly selected lines $l^{\prime}$ and therefore must enter at least $N^{\alpha-}$ cells. Similarly by Theorem \ref{bushconstruction} each $\alpha$ fast line enters no more than $N^{\alpha+}$ cells in $s$.
\begin{defn}[Slow lines] \label{slowlines} We say that a line is slow for a sector $s$ if it has $\lesssim N^{{2 \over 3}+k_2\epsilon\pm}$ crossings with lines in $\mathcal{L}$. \end{defn}
We want to show slow lines contribute $\gtrsim N^{1-k_2 \epsilon}$ incidences. Assume to the contrary for some $\alpha$ the $\alpha-$fast lines contribute $\gtrsim N^{1-{\alpha \over 10^{10}}}$ incidences. Note that there are $N^{{5 \over 3}}$ line-line crossings in $s$ so there are $\lesssim N^{1-\alpha}$ $\alpha-$ fast lines which each have $\sim N^{{2 \over 3}+\alpha \pm}$ crossings. If not, the sum of $\alpha-$fast contributions is $N \sum_{\alpha=k_2}^{1/(3\epsilon)} N^{-\alpha \epsilon/10^{10}} \sim N^{1-k_2\epsilon/10^{10}}$ so by choosing constant $k_2$ large enough, the $\alpha-$fast lines contribute a vanishingly small fraction of the incidences and we also conclude that slow lines contribute most incidences.
If the $\alpha$ fast lines contribute $\gtrsim N^{1-{\alpha \over 10^{10}}}$ incidences in $s$, then there must be at $\gtrsim N^{{1 \over 3} - {\alpha \over 10^{10}}}$ cells in which $\alpha$ lines contribute $N^{{2 \over 3} - {\alpha \over 10^{10}}}$ incidences each. Since each structuring line $l_s$ in the sector $s$ passes through $N^{{1 \over 3}-}$ of the $N^{{1 \over 3}+}$ of the cells in $s$, it must be that there is some structuring line $l_s$ in the sector so that the $\alpha$ fast lines contribute $N^{{2 \over 3} - {\alpha \over 10^{10}}}$ incidences in each of $N^{{1 \over 3} - {\alpha \over 10^{10}}-}$ cells through which $l_s$ passes. The cells through which $l_s$ passes are ordered according to their intersections with $l_s$ and each $\alpha-$fast line passes through a set of $N^{\alpha \pm}$ of these. Since there are $\lesssim N^{{1-\alpha}+}$ $\alpha$-fast lines, we can find an interval of length $N^{\alpha+}$ of the cells that intersect $l_s$ with only $N^{{2 \over 3}+}$ $\alpha$-fast lines (belonging to a set we'll call $\mathcal{L}_{fast}$) intersecting them with the property that there are $N^{\alpha ( 1- 10^{-10} ) -}$ cells where the lines of $\mathcal{L}_{fast}$ make $N^{{2 \over 3}- \alpha (10^{-10}) -}$ incidences.
We need a quick lemma affirming the union of an interval of cells is in fact a convex set.
\begin{lem} \label{consecutivecells} Let $l_s$ be a structuring line and $C_1,\dots, C_k$ be
consecutive polygonal cells through which the line $l_s$ passes. Then $$R=\bigcup_{t=1}^k C_t,$$ is a convex region. \end{lem}
\begin{proof} The cells in the sector $s$ are bounded by the two lines that bound the sector and the $l^{\prime}$'s which we selected randomly. The cells $C_1,\dots , C_k$ are divided by consecutive $l^{\prime}$'s intersecting $l_s$, namely $l_{j_1}^{\prime}, \dots, l_{j_{k-1}}^{\prime}$. Simply removing $l_{j_1}^{\prime}, \dots, l_{j_{k-1}}^{\prime}$ from our list of bounding lines, $R$ is the cell defined by the remaining lines which contains each $C_j$. Therefore it is convex. \end{proof}
We let $R$ be the union of the $N^{\alpha+}$ consecutive cells above. There are $N^{{2 \over 3}+}$ many $\alpha$-fast lines going through $R$, each making $N^{\alpha ( 1- 2(10^{-10}) ) -}$ incidences with the $N^{{1 \over 3}+\alpha+}$ many points in $R$. We apply Corollary \ref{crossingcells} to conclude that there must be $N^{{4 \over 3} + \alpha (1- 6(10^{-10})) -}$ crossings among $\alpha$-fast lines in $R$, but this is a contradiction because there are at most $N^{{4 \over 3}+}$ many such crossings.
\end{proof}
It is worth emphasizing that Theorem \ref{sectorincidences} says that most (up to $\epsilon$ loss in the exponent) lines will take $O(1)$ points from each cell they crosses. In other words, points on a line are evenly spaced among cells. This is the lynchpin fact in our main result (see Section \ref{protoinverseST}).
\begin{thm} \label{doublebushmixing} Let $(\mathcal{L},\mathcal{P})$ be an extremal configuration. Then there are two organizing points $p_1$ and $p_2$, and two bushes $l_{1,1},\dots, l_{M_1,1}$ incident to $p_1$ and $l_{1,2}, \dots, l_{M_2,2}$ incident to $p_2$ with $M_1,M_2 \gtrsim N^{{1 \over 3}-}$ and a refinement $\mathcal{P}^{\prime} \subset \mathcal{P}$ so that the two bushes break $\mathcal{P}^{\prime}$ into $M_1 M_2$ cells which are point weighted and so that each sector $s$ of the bush at $p_1$ having at least $N^{{2 \over 3}-}$ points of $\mathcal{P}^{\prime}$ has at least $N^{1-}$ lines of $\mathcal{L}$ incident to at least one point of the sector. \end{thm}
\begin{proof} Apply Theorem \ref{verynicespacing}, we find a point $p_1$ with bush $l_{1,1},\dots, l_{M,1}$ and structuring lines holding in total $N^{1-}$ points. We call this set of points $\mathcal{P}_1$. Then $(\mathcal{L}, \mathcal{P}_1)$ is an extremal configuration. We apply Theorem \ref{sectorincidences} to find a refinement of the set of incidences
$J(\mathcal{L},\mathcal{P}_1)$ with $|J(\mathcal{L},\mathcal{P}_1)| \gtrsim N^{{4 \over 3}-}$ so that each line of $\mathcal{L}$ takes only $N^{+}$ incidences of $J(\mathcal{L},\mathcal{P}_1)$ in each sector of the bush at $p_1$.
We restrict to those points in $\mathcal{P}_1$ which are at least $N^{{1 \over 3}-}$ rich in incidences of $J(\mathcal{L},\mathcal{P}_1)$. We refer to that set as $\mathcal{P}_2$. The set $(\mathcal{L},\mathcal{P}_2)$. is an extremal configuration. We refine the set of lines to $\mathcal{L}_1$ which take $N^+$ incidences in $N^{{1 \over 3}-}$ sectors of the bush at $p_1$. We let $\mathcal{P}_3$ be the set of points of $\mathcal{P}_2$ that are $N^{{1 \over 3}-}$ rich with respect to lines of $\mathcal{L}_1$ incident to only $N^+$ other points in the same sector. The pair $(\mathcal{L}_1,\mathcal{P}_3)$ is an extremal configuration. Pick an organizing point $p_2$ with bush $l_{1,2} \dots l_{M_2,2}$ and structuring lines for each sector of the bush from $\mathcal{L}_1$. From the structuring lines keep only the points which occur in groups of at most $N^{+}$ in the first bush. Call this set of points $\mathcal{P}^{\prime}$. Each structuring line has at most $N^{+}$ points in each sector of the first bush. But since there are only at most $N^{{1 \over 3}+}$ structuring lines in each sector of the second bush, the cell decomposition given by the two bushes is point weighted.
Since each sector of the first bush has $N^{1-}$ incidences coming from lines making at most $N^{+}$ incidences in the sector, there must be $N^{1-}$ lines incident to points of the sector.
\end{proof}
\section{A proto inverse Szemer\'edi Trotter theorem} \label{protoiSTSection}
A proto inverse Szemer\'edi Trotter theorem will be a recipe for constructing a configuration of points and lines which may not terminate or may not yield an extremal configuration but so that a large portion of every extremal example can be obtained using this recipe.
When considering such a recipe, an important piece of information is how many parameters one needs to specify to obtain an instantiation of the recipe.
There is a trivial recipe taking $O(N)$ parameters. Namely use $4N$ parameters to
completely specify a set of $N$ lines, $\mathcal{L}$ and a set of $N$ points $\mathcal{P}$. Examine the set of incidences between these lines and points $I(\mathcal{L},\mathcal{P})$. If it happens to be that $|I(\mathcal{L},\mathcal{P})| \gtrsim N^{{4 \over 3}-}$, then we have constructed an extremal configuration and in fact, every extremal configuration can be constructed in this way. This recipe and its proto inverse Szemer\'edi Trotter theorem amount to really just the definition of extremal configuration, and nothing has been gained.
Now, however, we describe a recipe using just $O(N^{{1 \over 3}})$ parameters. Our recipe will be based on a cell decomposition consisting of a grid of axis parallel rectangles. $a_1 < a_2 < \dots < a_{N^{{1 \over 3}}}$ will be real numbers representing the $x$ coordinates of the grid. $b_1 < b_2 < \dots < b_{N^{{1 \over 3}}}$ will be the $y$ coordinates of the grid. The final ingredients will be a set of lines $l_{s,1} \dots l_{s,N^{{1 \over 3}}}$ which will serve as the structuring lines for the strip of cells between $x$ coordinates $a_1$ and $a_2$.
We declare the recipe to have failed if the lines $l_s$ do not have at least $N^{{2 \over 3}-}$ crossings with $x$-coordinate between $a_1$ and $a_2$. Otherwise, we declare the recipe to have failed unless there are at least $N^{{1 \over 3}-}$ values of $j$ so that at least $N^{{ 1 \over 3}-}$ and no more than $N^{{1 \over 3}+}$ of the crossings with $x$-coordinates between $a_1$ and $a_2$ have $y$ coordinates between $b_j$ and $b_{j+1}$. Otherwise, we define these crossings to be the points of the cell $[a_1,a_2] \times [b_j,b_{j+1}]$ For each cell, we find all lines which are incident to two points. We declare this set of lines to be $\mathcal{L}$. We say that the recipe has failed unless there are at least $N^{{2 \over 3}-}$ choices of $(j,k)$ so that $N^{{2 \over 3} \pm}$ lines of $\mathcal{L}$ cross the cell $[a_j,a_{j+1}] \times [b_k, b_{k+1}]$. Otherwise, we say that the recipe has failed unless for at least $N^{{2 \over 3}-}$ of these choices $(j,k)$ the lines going through the $(j,k)$th cell are structured. If they are structured, we refer to the structuring points as the points of the $(j,k)$th cell. And combining all of these structuring points, we get the set $\mathcal{P}$ and we declare that the construction has succeeded. In this case, $(\mathcal{L},\mathcal{P})$ is an extremal configuration. We denote the output of this recipe as a function of its inputs as $(\mathcal{L}(a,b,l_s), \mathcal{P}(a,b,l_s))$.
\begin{thm} \label{protoinverseST} Let $(\mathcal{L},\mathcal{P})$ be an extremal configuration. Then there is a choice of the $O(N^{{1 \over 3}})$ parameters $(a,b,l_s)$ and a projective transformation $P$ so that $(P(\mathcal{L}) \cap \mathcal{L}(a,b,l_s),P(\mathcal{P}) \cap \mathcal{P}(a,b,l_s))$ is an extremal configuration (of $N^{1-}$ lines $N^{1-}$ points and $N^{{4 \over 3}-}$ incidences.) \end{thm}
\begin{proof} This is essentially a consequence of Theorem \ref{doublebushmixing} and its proof. We will choose $P$ to be a projective transformation sending the points $p_1$ and $p_2$ to the points at infinity corresponding to the $x$ direction and $y$ direction. We find a sector through $p_1$ through which $N^{1-}$ lines make $N^{1-}$ incidences with the structured points and making at least two incidences per cell. (Possibly by removing all but $N^{-}$ density of elements of each bush.) Then we choose $a,b$ so that they give all the cells of the double bush construction with $a_1,a_2$ corresponding to the sector chosen above and we let $l_s$ be the structuring lines for the sector. \end{proof}
\begin{thm} \label{dualprotoinverseST} Let $(\mathcal{L},\mathcal{P})$ be an extremal configuration. Then there are $N^{1-}$ organizing lines such that any pair of them will organize a subconfiguration $(\mathcal{L}',\mathcal{P})$ such that a strip of lines is the set of size $N^{2/3 \pm}$ in $\mathcal{L}'$ that intersects an organizing line at a given interval between adjacent points on the organizing line. Then cells are intersections of strips. Furthermore each of the strips are structured by $N^{1/3 \pm}$ structuring points. \end{thm}
\begin{proof} This is the dual case of Theorem \ref{doublebushmixing} \end{proof}
\begin{thm}[Mixing] \label{mixing} Let $(\mathcal{L},\mathcal{P})$ be an extremal configuration with the cell decomposition given by Theorem \ref{protoinverseST}. Then $\gtrsim N^{{4 \over 3}-}$ pairs of cells share $\gtrsim N^{{1 \over 3}-}$ lines which take at least two incidences of $J'$ in each of the two cells. \end{thm}
\begin{figure}
\caption{Two organizing lines $l_1,l_2$ go through several cells including cells $\mathcal{C}_{1},\mathcal{C}_{2}$ respectively. These cells are shown to share many lines which each take two points in $\mathcal{C}_{1}$ and in $\mathcal{C}_{2}$.}
\label{fig:mixing}
\end{figure}
\begin{proof}
Assume we have the cell decomposition from Theorem \ref{protoinverseST}. An application of Theorem \ref{dualprotoinverseST} gives us that $\gtrsim N^{2-}$ pairs of lines are organizing.
Consider a pair of organizing lines $l_1, l_2$. We take a refinement of the line set such that each line has at least two incidences in a cell that $l_1$ has incidences in and at least two incidences in a cell that $l_2$ has incidences in. Note this must yield an extremal refinement for $\gtrsim N^{2-}$ pairs of organizing lines $l_1,l_2$ because each of the $\gtrsim N^{1-}$ organizing lines in $\mathcal{L}'$ has at least 2 incidences in $\gtrsim N^{{1 \over 3}-}$ cells which each have $\gtrsim N^{{2 \over 3}-}$ lines going through it which each have at least two incidences in that cell. So each organizing line contributes $\gtrsim N^{1-}$ lines (out of a total of $< N$ lines) so $\gtrsim N^{1-}$ other organizing lines must share $\gtrsim N^{1-}$ regular lines that take at least two points in one of each of their cells.
From our initial application of Theorem \ref{dualprotoinverseST} we know that $\gtrsim N^{{2 \over 3}-}$ pairs of intervals between adjacent points on $l_1$ and adjacent points on $l_2$ share $\gtrsim N^{{1 \over 3}-}$ lines. (Note this is still true after our refinement because we kept $\gtrsim N^{1-}$ lines). Furthermore lines take $\lesssim N^{+}$ incidences in a single cell so $\gtrsim N^{{2 \over 3}-}$ pairs of cells that $l_1$ and $l_2$ take incidences in share $\gtrsim N^{{1 \over 3}-}$ lines out of $\lesssim N^{{2 \over 3}}$ total pairs of cells.
Since this result holds for any generic pair of $\gtrsim N^{2-}$ pairs of organizing lines, we conclude that any generic pair of cells must share $\gtrsim N^{{1 \over 3}-}$ lines. \end{proof}
So we conclude that our two bush cell decomposition can be chosen to be constructed using two sets of $\sim N^{{1 \over 3} \pm}$ parallel lines (projective transformation) which form a grid. $\gtrsim N^{{2 \over 3}-}$ of the rectangles in the grid are cells which contain $\sim N^{{1 \over 3}\pm}$ points and $\sim N^{{2 \over 3}\pm}$ lines which take about 2 points in the cell. Finally we have the mixing property that generic pairs of cells share $\sim N^{{1 \over 3} \pm}$ lines.
\begin{figure}\label{fig:two_bush}
\end{figure}
\section{Cell decompositions for extremal configurations in the unit distances problem}
We adapt the argument from Sections \ref{cellSection} to \ref{protoiSTSection} to the incidence problem between points and unit circles centered at the points, which is equivalent to the unit distance problem.
Section \ref{cellSection} only contains counting arguments that rely on the property that two lines intersect in at most one point. Since two unit circles intersect in at most two points, the counting arguments all still hold in the unit circle case. That said, to tackle some complications due to circle cells lacking convexity, we will slightly modify the chosen number of cells and points per cell. Furthermore the first few probability results in Section \ref{probSection} also rely on counting arguments so also hold for unit circles.
\subsection{Cell Decomposition Preliminaries}
As before, the unit circle cell decompositions that currently exist in the literature \cite{CEGSW} do not satisfy our purpose because they add additional boundary components to the cells which are not part of our original set of circles. A main ingredient in our proof will be the theorem of Clarkson et. al. \cite{CEGSW} on the complexity of cell decompositions given by general families of unit circles.
\begin{thm} \label{cellcomplexitycircle} Let $\mathcal{C}$ be a set of $r$ unit circles. It divides $\mathbb{R}^2$ into $O(r^2)$ connected components called \textbf{cells}. Let ${\cal S}$ be any subcollection of $m$ of these cells. Then the total number of edges of cells in ${\cal S}$ is $O(r^{{2 \over 3}} m^{{2 \over 3}} \beta(r) + r)$ where $\beta(r)$ is the inverse of the Ackermann function. \end{thm}
Note the inverse Ackermann function grows notoriously slowly (much slower than $\log(n)$). \cite{A}
\begin{cor} \label{numberofbigcellscircle} Let $\mathcal{L}_r$ be any set of $r$ unit circles and let ${\cal C}$ be the set of cells which they define having $>s$ and $\leq 2s$ edges. Then if $s \leq r^{{1 \over 2}}/\beta(r)^{3/2}$, we have
$$|{\cal C}| \lesssim \beta(r)^3{r^2 \over s^3},$$ and if $s \geq r^{{1 \over 2}}/\beta(r)^{3/2}$
$$|{\cal C}| \lesssim {r \over s}.$$ \end{cor}
To extract structure from our configurations we will chose to throw out undesirable points and lines keeping only those that enjoy the desirable properties.
\begin{defn}
Given an extremal partial unit circle configuration $(\mathcal{L}, \mathcal{P}, \cal{J(\mathcal{L}, \mathcal{P})})$ we say $(\mathcal{L}^{\prime}, \mathcal{P}^{\prime}, \cal{J^{\prime}(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})} )$ is a \textbf{refinement} if $|\mathcal{L}| \sim |\mathcal{L}^{\prime}|$ and $|\mathcal{P}| \sim |\mathcal{P}|^{\prime}$ and $|\cal{J(\mathcal{L},\mathcal{P})}| \sim |\cal{J}^{\prime}(\mathcal{L}^{\prime},\mathcal{P}^{\prime})|$. \end{defn}
We also exclude certain very low probability events. For each circle $c \in \mathcal{L}$ we establish an ordering on $\mathcal{L}_c \subset \mathcal{L}$, the set circles that intersect $c$ in two points, based on the position of their intersection with $c$. We do this by choosing a reference point on $c$ and a direction (clockwise or counter clockwise) and then ordering the circles of $\mathcal{L}_c$ by the order of their first point of intersection with $c$ starting at the chosen reference point and going in the chosen direction. The ordering is ill-defined when multiple circles are concurrent at a point of $c$, but we order concurrent circles arbitrarily. For each choice of $c^{\prime} \in \mathcal{L}$, with $c^{\prime} \neq c$ we consider the order induced by $c$ with reference point equal to a point of intersection of $c$ and $c'$. We exclude the event that none of the ${C N \log N \over r}$ consecutive circles following $c^{\prime}$ in this order are selected for $\mathcal{L}_r$. This is a list of $O(N^2)$ events governed by Lemma \ref{consecutivelines}.
At each point $p$ of intersection of two circles of $\mathcal{L}$, the vertical line going through $p$ induces an order on the circle of $\mathcal{L}$ which is just the order of the first points of intersection between circles and the vertical line (either in ascending or descending order). We would like to exclude the case that none of the first ${C N \log N \over r}$ circles above $p$ are chosen for $\mathcal{L}_r$ and none of the first ${C N \log N \over r}$ circles below $p$ are chosen for $\mathcal{L}_r$. This is a list of $O(N^2)$ events. So by Lemma \ref{consecutivelines} a uniform random selection of circles yields a cell decomposition that excludes the above undesirable events with probability $1-O(N^{-98})$. We call a selection of unit circles which excludes the above events {\bf acceptable}.
\begin{lem} \label{pointsinacellcircle} Let $K$ be a cell coming from an acceptable selection of unit circles $\mathcal{L}_{N^{{1 \over 3}-\alpha}}$ where $\alpha>0$ is very small. (But $\alpha$ large compared to the current valueof $\epsilon$.) Suppose $K$ has $s$ sides. Then $K$ contains at most $sN^{{1 \over 3}+2 \alpha +}$ many $N^{{1 \over 3}-}$-rich points of $\mathcal{P}$. \end{lem}
\begin{proof} Following the construction in \cite{CEGSW}, at every intersection of pairs of circles in $\mathcal{L}_{N^{{1 \over 3}-\alpha}}$ draw the vertical segment going up until it hits the next circle from $\mathcal{L}_{N^{{1 \over 3}-\alpha}}$ and the vertical segment going down until it hits the next circle from $\mathcal{L}_{N^{{1 \over 3}-\alpha}}$. The connected components bounded by circles from $\mathcal{L}_{N^{{1 \over 3}-\alpha}}$ and vertical segments are called \textbf{funnels}. Each funnel is bounded by at most two circle arcs (top and bottom) and two vertical segments (left and right). Since we chose an acceptable set of unit circles $\mathcal{L}_{N^{{1 \over 3}-\alpha}}$ each boundary element has at most ${\frac{N}{N^{{1 \over 3}-\alpha}} \log(N)}=N^{{2 \over 3}+\alpha} \log(N)$ circles entering it. Suppose there are $P$ points of $\mathcal{P}$ in the funnel $F$. Then there are at least $P N^{{1 \over 3}-}$ incidences in $F$. The Szemer\'edi Trotter theorem guarantees that $P \leq N^{{1 \over 3}+2 \alpha +}$.
If $K$ has $s$ sides, then it has at most $s$ funnels so the total number of points in $K$ is at most $s N^{{1 \over 3}+2 \alpha +}$ points of $\mathcal{P}$ which was to be shown. \end{proof}
Finally, we combine Corollary \ref{numberofbigcellscircle} with Lemma \ref{pointsinacellcircle} to bound the number of rich points of $\mathcal{P}$ contained in cells with between $s$ and $2s$ sides. If $s \leq r^{{1 \over 2}}/ \beta(r)^{{3 \over 2}}$ with $r \leq N^{{1 \over 3}-\alpha}$, we obtain the bound $\beta(r)^3{r^2 \over s^3} sN^{{1 \over 3}+2\alpha+} \leq {N^{1+ } \over s^2}$. If $s \geq r^{{1 \over 2}}/\beta(r)^{{3 \over 2}}$, we obtain the bound ${r \over s} sN^{{1 \over 3}+2\alpha+} \leq N^{{2 \over 3}+\alpha+}$. As long as $s$ is much bigger than $N^+$, we do not capture a significant number of rich points. We conclude the following theorem.
\begin{thm} \label{nicerefinementcircle} Let $(\mathcal{L},\mathcal{P})$ be an extremal configuration with each point of $\mathcal{P}$ being at least $N^{{1 \over 3}-}$ rich. Then for each acceptable random selection $\mathcal{L}_{N^{{1 \over 3}-\alpha}}$, where $\alpha > 0$ is very small, there is a refinement $\mathcal{P}^{\prime} \subset
\mathcal{P}$ with $|\mathcal{P}^{\prime}| \geq {1 \over 2} |\mathcal{P}|$ so that no point of $\mathcal{P}^{\prime}$ is contained in a cell with more than $N^{+}$ sides, and in light of Lemma \ref{pointsinacellcircle}, each such cell has at most $N^{{1 \over 3}+2\alpha+}$ points of $\mathcal{P}^{\prime}$ so that we have obtained a point weighted decomposition for the extremal configuration $(\mathcal{L},\mathcal{P}^{\prime})$ \end{thm}
\begin{thm} \label{boundedsidescircle}
Let $(\mathcal{L},\mathcal{P})$ be an extremal unit circle configuration. Specifically let $|I(\mathcal{L},\mathcal{P})|=N^{{4 \over 3}-\epsilon}$ with $\epsilon$ fixed.
Let $C_1,\dots , C_{r^2}$ be a point-weighted cell decomposition for $(\mathcal{L},\mathcal{P})$
with $N^{{1 \over 3} - 5\epsilon} \leq r \leq N^{{1 \over 3} - 4\epsilon}$. Then there is a set of incidences $J(\mathcal{L},\mathcal{P}) \subset I(\mathcal{L},\mathcal{P})$ so that $|J(\mathcal{L},\mathcal{P})| \gtrsim N^{{4 \over 3}-\epsilon}$, but for every circle $L$ and cell $C$ for which there is $P \in C$ with $(L,P) \in J(\mathcal{L},\mathcal{P})$, we have that
$$2 \leq | I( \{L\},C)| \lesssim N^+$$ and there exists some other point $P'$ in $C$ such that $P$ and $P'$ are adjacent on $L$ and the circle arc $(P,P')$ is entirely contained in $C$. \end{thm}
\begin{proof} The way this proof will work is that we will remove from $I(\mathcal{L},\mathcal{P})$ all incidences that would violate the conditions for $J(\mathcal{L},\mathcal{P})$ and observe that we have removed less than half of the set $I(\mathcal{L},\mathcal{P})$.
For any circle $L$ and cell $C$ for which $L$ has less than $N^{2\epsilon}$ incidences, we remove these incidences and we have removed at most $r |\mathcal{L}| N^{2\epsilon} \lesssim N^{{4 \over 3}-2\epsilon}$ incidences since each circle has incidences with at most $r$ cells. By Theorem \ref{nicerefinementcircle} each cell has at most $N^{+} \sim N^{\epsilon}$ sides. Furthermore, every circle intersects a cell side in at most two points. Thus for every circle $L$ and cell $C$, $L$ has at most $N^{\epsilon}$ disjoint connected circle arcs contained in $C$. By the previous refinement, if $L$ has an incidence with a point in $C$ in $J(\mathcal{L}, \mathcal{P})$ then $L$ has at least $N^{2\epsilon}$ incidences in $C$ so by pigeonhole principle, at least two incidences with points $P$ and $P'$ must occur on the same connected circle arc in $C$.
For the incidences which remain, for any cell $C$, all circles are incident to at least two points of the cell
(which define the line) so there are at most $|C|^2$ such circles. We can remove all incidences from cells that do not contribute at least $\gtrsim N^{-} |C|^2 $ incidences. We note that especially rich circles cannot contribute most of the incidences. The number of circles passing through $k$ of the points is at most
${|C|^2 \over k^3}$ contributing at most ${|C|^2 \over k^2}$ incidences in $C$ (this follows from Szemer\'edi--Trotter). By Theorem \ref{nicerefinementcircle} we have $|C| \lesssim N^{{1 \over 3}+2(4\epsilon)+}$ since $\alpha < 4\epsilon$ so the total number of $k$ rich incidences is at most $r^2 {|C|^2 \over k^2} \lesssim (N^{{1 \over 3} -4\epsilon +{1 \over 3}+2(4\epsilon)+}/k)^2=N^{{4 \over 3}+8 \epsilon+}/k^2$. So we may remove incidences for $k \lesssim N^{5\epsilon}$. Thus $N^{2\epsilon} \lesssim | I( \{L\},C)| \lesssim N^{5\epsilon}$.
\end{proof}
We obtain the following corollary.
\begin{cor} \label{structuredcellscircle} Let $(\mathcal{L},\mathcal{P})$ be an extremal unit circle configuration. Let $C_1,\dots , C_{r^2}$ be a point-weighted cell decomposition for $(\mathcal{L},\mathcal{P})$ with $r \sim N^{{1 \over 3}-}$. Then there is a set ${\cal C}$ of $\gtrsim r^{2-}$ cells so that for each $C \in {\cal C}$, there is a set of circles $\mathcal{L}_C$ with
$$|\mathcal{L}_C| \gtrsim |C|^{2-},$$ and with each $L \in \mathcal{L}_C$ incident to at least $2$ but $\lesssim N^+$ points in $C$ such that for at least two of these points adjacent on $L$, the circle arc between them is entirely contained in $C$. Each set $\mathcal{L}_C$ has density $\gtrsim N^-$ in the set of lines intersecting two points in $C$.
\end{cor}
\subsection{Crossing Numbers and Structuring Circles}
As before, a key ingredient in proving this will be the crossing number inequality which we state here. This is nearly the same result as the standard crossing number inequality from Lemma \ref{crossingnumber} with the key difference that edges are unit circle arcs.
\begin{defn}
A \textit{crossing} between unit circle arcs is a point in their intersection. \end{defn}
Note that a pair of circle arcs can have zero, one, or two crossings.
\begin{lem}\cite{S} \label{crossingnumbercircle} Let $\mathcal{P}$ be a set of $n$ points in the plane and $\mathcal{L}$ be a set of unit circles. Let $G$ be the multigraph where $\mathcal{P}$ is the vertex set and for every circle $c$ in $\mathcal{L}$ and pair of adjacent points $(p,p')$ on $c$ we add an edge between $p$ and $p'$. Let $e$ be the number of edges in $G$. Then the number of crossings between edges is $\gtrsim {e^3 \over n^2}$. \end{lem}
\begin{wrapfigure}[20]{r}{0.4\textwidth}
\centering
\includegraphics[width=0.4\textwidth]{structured_circles.png}
\caption{Five structuring points structure the set of black unit circles. A generic pair of structuring points must define a circle in the structured set but not all the pairs must.}
\label{fig:structured_circles} \end{wrapfigure}
We make a definition of a structured set of circles.
\begin{defn} We say that a set $\mathcal{L}_1$ of at least $N^{{2 \over 3}-}$ unit circles is \textbf{structured} if there is a set $\mathcal{P}_1$ of at most $N^{{1 \over 3}+}$ points so that each circle of $\mathcal{L}_1$ is incident to at least two points of $\mathcal{P}_1$. We call this set of points \textbf{structuring}. \end{defn}
Note that since $\lesssim N^{\frac{2}{3}+}$ circles go through at least two among $\lesssim N^{\frac{1}{3}+}$ points, the structuring points essentially define the structured circles. Now we're ready to state our structuring theorem.
\begin{thm} \label{verynicespacingcircle} Let $(\mathcal{L},\mathcal{P}, \cal{J})$ be an extremal partial unit circle configuration. Then there exists a point weighted cell decomposition with $\gtrsim N^{{2 \over 3}-}$ cells and a refinement of the configuration $(\mathcal{L}^{\prime},\mathcal{P},\cal{J}')$ so that for each circle $c \in \mathcal{L}^{\prime}$ there are points $p_0, \dots , p_M$ of $\mathcal{P}$ with $(c,p_j) \in \cal{J}'$ and the $p_j$'s in order of their position on $c$ and with $M \gtrsim N^{{1 \over 3}-}$ so that for each consecutive pair of points $p_{2k},p_{2k+1}$, the circle arc bounded by $p_{2k},p_{2k+1}$ is entirely contained in a cell, and there is a structured set of circles $\mathcal{L}_k$ so that each $c^{\prime}$ in $\mathcal{L}_k$ intersects $c$ in the open circle arc bounded by the points $p_{2k}$ and $p_{2k+1}$ exactly once. We say the circles in $\mathcal{L}^{\prime}$ \textbf{organize} $\mathcal{P}$. \end{thm} \begin{figure}
\caption{An organizing circle $c$ has two circle arcs bounded by $p_0,p_1$ and $p_2,p_3$ respectively, where both arcs are crossed by a structured set of circles (dotted light gray with crossing arcs in solid black). The dual formulation is shown: the centers of the structured circles form a set of structured points (black) contained in the interior of the sectors defined below (hashed gray region bounded by dotted black circles which intersect at the organizing point $p$: center of $c$).}
\label{fig:organizing_circle}
\end{figure}
Note that we restrict our attention to crossings where the circle arcs share only one point in common instead of two.
\begin{defn}
We say two circle arcs share a \textbf{simple crossing} if they have exactly one point in their intersection. We say two circle arcs share a \textbf{double crossing} if they have exactly two points in their intersection. Similarly, two circles have a simple crossing if they each contain an arc bounded by two adjacent points such that these arcs share a simple crossing. Likewise for double crossing. \end{defn}
\begin{proof} First consider an extremal partial unit circle configuration $(\mathcal{L},\mathcal{P}, \cal{J})$. We consider the point weighted cell decomposition $C_1, \dots , C_{N^{{2 \over 3}-}}$ guaranteed by Corollary \ref{structuredcellscircle}. We obtain the refined unit circle incidence multigraph $G$ where the vertex set is $\mathcal{P}$ and for every pair of vertices, we add an edge if the corresponding points are adjacent on a unit circle, and the unit circle arc that they bound has length $\lesssim N^{-\epsilon}$ and is entirely contained inside a cell and such that each cell $\lesssim N^{{2 \over 3}+}$ edges. By Corollary \ref{structuredcellscircle} we know each cell has $\gtrsim N^{{2 \over 3}-}$ edges of $G$ and $\lesssim N^{{1 \over 3}+}$ points.
We know from Lemma \ref{crossingnumbercircle} that there are $\gtrsim N^{{4 \over 3}-}$ crossings per cell. Now we must show that $\gtrsim N^{{4 \over 3}-}$ of them are $\textit{simple}$ crossings.
\begin{lem}\label{simple_crossings}
Every cell in a cell decomposition from Corollary \ref{structuredcellscircle} has $\gtrsim N^{{4 \over 3}-}$ simple crossings.
\end{lem}
\begin{proof}
Since there are $\sim N^{{2 \over 3} \pm}$ segments per cell then $\gtrsim N^{{2 \over 3}-}$ of them must have $N^{{2 \over 3}-}$ crossings. Assume one of these edges does not contribute $\gtrsim N^{{2 \over 3}-}$ simple crossings. Then it must have $\gtrsim N^{{2 \over 3}-}$ double crossings. Recall all edges are circle arcs of length less than $N^{-\epsilon}$ so double crossings correspond to circles being tangent with small error. Quantitatively, given a fixed edge that has double crossings with two other edges, their centers are at distance $\lesssim N^{-\epsilon}$ (small angle approximation).
Then also by small angle approximation, the length of the the circle arc spanning the two intersections between circles whose centers are at distance $\lesssim N^{-\epsilon}$ is $\sim 1$ i.e. much larger than $N^{-\epsilon}$. Thus no edge can span both the crossings of a pair of circles with close centers. So if two edges form double crossings with a third edge, then the first two must form a simple crossing. Furthermore this simple crossing must be contained in the same cell since all edges are entirely contained in a cell.
So if one of these crossing rich edges $e$ in a cell does not contribute $\gtrsim N^{{2 \over 3}-}$ simple crossings, then every pair of edges that form double crosses with $e$ must simple cross each other. But $e$ is crossing rich so it must have $\gtrsim N^{{2 \over 3}-}$ double crossings which implies $\gtrsim N^{{4 \over 3}-}$ simple crossings in the cell. Thus every cell has $\gtrsim N^{{4 \over 3}-}$ simple crossings.
\end{proof}
Each cell has $\lesssim N^{{1 \over 3}}$ points so $\lesssim N^{{2 \over 3}}$ circle arcs between pairs of points so dividing the number of simple crossings we obtained from the above Lemma by the number of circle arcs we get that there are $\gtrsim N^{{2 \over 3}-}$ circle arcs with $\gtrsim N^{{2 \over 3}-}$ simple crossings. We call these sector arcs because their $\gtrsim N^{{2 \over 3}-}$ simple crossings come from a set of $\gtrsim N^{{2 \over 3}-}$ structured circles which are structured by the set of points in the cell.
There are $\gtrsim N^{{2 \over 3}-}$ cells so $\gtrsim N^{{4 \over 3}-}$ sector arcs total. There are $\lesssim N$ circles in the configuration each of which (under our refinements) has no more than one circle arc per cell so has $\lesssim N^{{1 \over 3}}$ sector arcs. Thus $\gtrsim N^{1-}$ circles must have $\gtrsim N^{{1 \over 3}-}$ sector arcs. \end{proof}
We'd now like to take advantage of our result. Theorem \ref{verynicespacingcircle} is a method of associating to each extremal unit circle configuration a refinement which is rather nicely parametrized. We will do this by applying point-circle duality to the result of Theorem \ref{verynicespacingcircle}. The result of the theorem gives us many circles $c$ which are incident to particular sets of points $p_1, \dots ,p_M$ with $M \gtrsim N^{{1 \over 3}-}$ so that we have $\gtrsim N^{{2 \over 3}-}$ circles intersecting $c$ between adjacent points which are structured. We use point--circle duality which maps a point to the unit circle centered at it and vise-versa, applying Theorem \ref{verynicespacingcircle} to circles and points instead of points and circles.
\begin{wrapfigure}[20]{r}{0.5\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{circle_sectors.png}
\caption{$p_b$ is in the sector $s_i$ and $p_b'$ is not. Note that $c_b$ crosses the circle arc of $c_a$ between $p_1,p_2$ exactly once whereas $c_b'$ crosses it twice.}
\label{fig:circle_sectors} \end{wrapfigure}
\begin{defn}
Consider a point $p$ with a bush $c_1, \dots, c_M$ of circles going through it ordered according to the direction of their tangent at $p$. Let $d_1, \dots, d_M$ be the interiors of the circles $c_i$. This bush of circles defines $M$ \textbf{sectors} where we define the $i^{th}$ sector $s_i = (d_i \cap d_{i+1}^c) \cup (d_i^c \cap d_{i+1})$ where $d^c$ is the complement of the disk $d$. \end{defn}
We analyze what it means for a point to be in the interior of a circle in terms of circle crossings.
\begin{lem}\label{circleduality}
A point $p$ is in the sector $s_i$ of an organizing circle $c$ if and only if the unit circle centered at $p$ crosses the arc of $c$ between $p_i$ and $p_{i+1}$ exactly once. \end{lem}
\begin{proof}
Given two circles $c_a$ and $c_b$ and points $p_1$ and $p_2$ on $c_a$, the property that $c_b$ crosses the circle arc of $c_a$ contained between $p_1$ and $p_2$ exactly once is equivalent to saying that exactly one of $p_1$ or $p_2$ is in the interior of $c_b$. Taking the dual which exchanges unit circles with their center points and vise-versa, this statement is equivalent to saying that $p_b$, the center of $c_b$, is in exactly one of the interiors of $c_1$ or $c_2$, the circles centered at $p_1$ and $p_2$ respectively. So the statement is dual to saying that the point $p_b$ (dual to $c_b$) is in the sector $s_1$ centered at the point $p_a$.
Taking $c_a$ to be the organizing circle from the lemma statement and $c_b$ to be the circle centered at $p$ from the lemma statement and $p_a$, $p_b$ to be $p_i$ and $p_{i+1}$ respectively we obtain the statement. \end{proof}
This motivates the our need to distinguish between \textit{simple crossings} and \textit{double crossings}. We obtain the dual of Theorem \ref{verynicespacingcircle}:
\begin{thm} \label{dualverynicespacingcircle} Let $(\mathcal{L},\mathcal{P}, \cal{J})$ be an extremal partial unit circle configuration. Then there is a refinement $(\mathcal{L},\mathcal{P}^{\prime}, \cal{J}')$ so that for each point $p \in \mathcal{P}^{\prime}$ there are circles $c_0, \dots ,c_M$ of $\mathcal{L}$ which are incident to $p$ in $\cal{J}'$ in order of their direction and with $M \gtrsim N^{{1 \over 3}-}$ so that each sector $s_j$ bounded by consecutive pairs of circles $c_{2j},c_{2j+1}$ contains a structured set of $\gtrsim N^{\frac{2}{3}-}$ points $\mathcal{P}_j$. We say the points in $\mathcal{P}^{\prime}$ \textbf{organize} $\mathcal{L}$. \end{thm}
\begin{proof}
We apply Lemma \ref{circleduality} to Theorem \ref{verynicespacingcircle}.
\end{proof}
For any point $p \in \mathcal{P}^{\prime}$ with circles $c_1, \dots c_M$ incident to it, there are $\gtrsim N^{{2 \over 3}-}$ points in each of the $\gtrsim N^{{1 \over 3}-}$ sectors for a total of $N^{1-}$ points. We take this set of points as a refinement $\mathcal{P}^{\prime}$ of our original set of points $\mathcal{P}$. What is particularly pleasant about this structure is that each of the $\gtrsim N^{{2 \over 3}-}$ points of $\mathcal{P}^{\prime}$ in a sector $s_j$ lie on at least two of the $\gtrsim N^{{1 \over 3}-}$ structuring circles.
Structuring circles seem very odd precisely because all of the points on them lie in a particular sector $s_j$. But this is not as odd as it seems. We see from the proof of Theorem \ref{verynicespacingcircle}, that the structuring circles for $p$ are dual to the points in cells through which the circle dual to $p$ passes. Every point lies in a cell, and every cell has $N^{{2 \over 3} \pm}$ circles going through it so by duality the set of $N^{{1 \over 3}}$ structuring circles define the $N^{{2 \over 3} \pm}$ points in a sector.
We're going to show that for any choice of organizing point $p \in \mathcal{P}^{\prime}$ a typical circle $c$ will have incidences in most sectors $s_j$. To do this we first need to introduce a refinement of the configuration endowed with a cell decomposition whose boundary circles include the bush through $p$.
\begin{thm}[bush construction] \label{bushconstructioncircle} For any extremal unit circle configuration $(\mathcal{L}, \mathcal{P})$ there exists a subset $\mathcal{P}^{\prime}$ of $\gtrsim N^{1-}$ points in $\mathcal{P}$ which are organizing with $\sim N^{{1 \over 3} \pm}$ sectors and a refined configuration $(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})$ such that the $\gtrsim N^{1-}$ circles in $\mathcal{L}^{\prime}$ organize $\mathcal{P}^{\prime}$. Also for any $p\in \mathcal{P}^{\prime}$ the refinement $(\mathcal{L}^{\prime}, \mathcal{P}_p)$ where $\mathcal{P}_p$ are the points in $\mathcal{P}$ organized by $p$, has a refinement $(\mathcal{L}^{\prime}, \mathcal{P}_p^{\prime})$ which is an extremal configuration with a point-weighted cell decomposition where each cell is contained in a sector. Moreover, any circle $c \in \mathcal{L}^{\prime}$ which has exactly $N^{{2 \over 3}+\alpha}$ simple crossings with circles of $\mathcal{L}^{\prime}$ within the sector $s$ with $\alpha>k\epsilon$ for $k$ sufficiently large will not enter more than $N^{\alpha+}$ cells in $s$.
\end{thm}
Note as before we split our circles into circle arcs between adjacent points so we say two circles have a simple crossing if they each have a circle arc between a pair of adjacent points such that the intersection of these circle arcs is exactly one point.
\begin{proof}
\textit{Refinement properties:} We apply Theorem \ref{dualverynicespacingcircle} and obtain the refinement $(\mathcal{L}, \mathcal{P}^{\prime})$ of organizing points. Then we apply Theorem \ref{verynicespacingcircle} to $(\mathcal{L}, \mathcal{P}^{\prime})$ obtaining the refinement $(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})$ where the circles in $\mathcal{L}^{\prime}$ organize $\mathcal{P}^{\prime}$. Now we have shown the first claim of the theorem.
\textit{Cell decomposition:} Let $p \in \mathcal{P}^{\prime}$ and $\mathcal{P}_p$ be the set of points in $\mathcal{P}$ organized by $p$. Note there are $\gtrsim N^{1-}$ organized points each $\gtrsim N^{{1 \over 3}-}$ rich so $(\mathcal{L}^{\prime}, \mathcal{P}_p)$ is an extremal configuration which we work in for this paragraph. We label the bush of $M \sim N^{{1\over 3}\pm}$ circles intersecting $p$ as $c_0,\dots, c_M$. Now we pick random $c^{\prime}_1,\dots , c^{\prime}_K$ with $K \sim N^{{1 \over 3}}$ from $\mathcal{L}^{\prime}$. Our ``cell decomposition" will be made from the circles $c_0,\dots,c_M$ together with the circles $c^{\prime}_0, \dots, c^{\prime}_K$. But we will not think of this collection of circles as giving a cell decomposition in the conventional sense. Recall that the set of points $E_c$ in ${\bf R}^2$ which are the centers of circles that simple cross $c_0$ are double-covered by the sectors $s_i = (d_i \cap d_{i+1}^c) \cup (d_i^c \cap d_{i+1})$ defined by the adjacent circles $c_i$ and $c_{i+1}$. For each sector $s_i$ we let the circles $c^{\prime}_1, \dots,c^{\prime}_K$ subdivide $s_i$ into cells. We have now obtained a collection of cells which double cover $E$. This will essentially be all right for us. We will sometimes over count incidences, but we will at most double count them.
The randomly selected circles $c^{\prime}_1, \dots , c^{\prime}_K$ can be chosen to be in the very likely case where they separate the points on the structuring circles of each sector into (essentially) distinct cells. For each structuring circle $c_s$, there are points $p_1, \dots ,p_L$ from $\mathcal{P}$ with $L \gtrsim N^{{1 \over 3}-}$ so that between each consecutive pair $p_k,p_{k+1}$ there are $\gtrsim N^{{2 \over 3}-}$ circles of $\mathcal{L}$ which cross the circle $c_s$. Applying Lemma \ref{consecutivelines}, we get a bound of $N^{-100}$ on the probability that any $N^{{2 \over 3}+}$ consecutive circles of $\mathcal{L}$ in the order they intersect $c_s$ don't include one of the $c^{\prime}$'s. We can select the $c^{\prime}$'s so that none of these events happen. Thus each cell contains at most $N^{+}$ points per structuring circle. Since there are $\lesssim N^{{1 \over 3}+}$ structuring circles in each sector, our cell decomposition is point weighted.
We would like to obtain one further property of our double-covering cell decomposition, so we refine it a bit further. The circles $c_1^{\prime} ,\dots, c_K^{\prime}$ have a total of at most $2K^2 \sim N^{{2 \over 3}}$ crossings. We want to restrict to sectors in which there are no more than $N^{{1 \over 3}+}$ crossings. This restrictions requires us to remove a small proportion of the sectors. Now in each sector $s_j$ that we have retained we will remove some circles from the list $c_1^{\prime}, \dots, c_K^{\prime}$. [These are removed only for consideration in the sector $s_j$] We choose the circles to remove so that each remaining circle $c_j^{\prime}$ has only $N^{+}$ crossings with other $c^{\prime}$'s in the sector $s_j$. The effect of the removals is that we have joined together a number of cells in the sector $s_j$. This ruins the point weightedness of the cells. We remove the cells having more than $N^{{1 \over 3}+}$ points, but we have not removed as many as $N^{{2 \over 3}-}$ points because on each structuring circle of $s_j$, a removed point was adjacent to a removed circle $c_k^{\prime}$. Thus we retain $N^{{1 \over 3}-}$ points per structuring circle. Now we have a new ``cell decomposition" which covers $N^{1-}$ points at most twice and which is subordinate to the sectors, so that in each retained sector $s_j$, none of the retained dividing circles $c_k^{\prime}$ intersect more than $N^{+}$ other dividing circles so each bounds at most $N^+$ cells in that sector.
To get the claim about circles $c$ with $N^{{2 \over 3}+\alpha}$ simple crossings apply Lemma \ref{Chernoff} to control the probability that more than 10 times the expected number of circles that share a simple crossing with $c$ are selected as random circles.
\end{proof}
When trying to show that a typical circle in a configuration $(\mathcal{L},\mathcal{P})$ will have incidences in most of the sectors of an organizing point, the enemy case is circles which take too many points in a given sector. A typical circle has $N^{2/3}$ simple crossings between each pair of points. So if we show that circles with too many simple crossings in a sector do not contribute significantly to the total number of incidences in that sector, then most incidences come from circles taking $\lesssim N^{+}$ points in that sector and we win.
\begin{thm} \label{sectorincidencescircle} There exist $\gtrsim N^{1-}$ organizing points $p$ in $(\mathcal{L}, \mathcal{P})$ with $\sim N^{{1 \over 3}\pm}$ sectors and some integer $k$ such that for every sector $\gtrsim N^{1-}$ of its incidences come from circles taking fewer than $N^{k \epsilon}$ points in that sector. \end{thm}
\begin{proof} We use the bush construction $(\mathcal{L}', \mathcal{P}_p')$ from Theorem \ref{bushconstructioncircle} and chose a point $p$ to be the center of our bush. For a large enough constant $k$, we toss out sectors that have $\gtrsim N^{{5 \over 3}+k_1\epsilon}$ simple circle arc crossings. There are at most $N^{{1 \over 3}-k_1\epsilon}$ such sectors because $\gtrsim N^{2-}$ simple circle arc crossings total by Lemma \ref{simple_crossings} (because there are $\gtrsim N^{{2 \over 3}-}$ cells with $\gtrsim N^{{4 \over 3}-}$ simple crossings each). So we still kept $\sim N^{{1 \over 3}\pm}$ sectors which each have $\sim N^{{5 \over 3} \pm}$ simple circle arc crossings and each sector contributes $\gtrsim N^{1-}$ incidences. So this refinement still yields a configuration $(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})$. Similarly, our cell decomposition from Theorem \ref{bushconstructioncircle} has $\lesssim N^{{2 \over 3}}$ cells, so we can remove any sectors with more than $N^{{1 \over 3} + k_1 \epsilon}$ cells.
By Corolarry \ref{structuredcellscircle} we may chose a subset $J(\mathcal{L}^{\prime}, \mathcal{P}^{\prime}) \subset I(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})$ such that $|J(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})| \gtrsim N^{{4 \over 3}-}$ and every circle has $\lesssim N^{+}$ incidences from $J(\mathcal{L}^{\prime}, \mathcal{P}^{\prime})$ per cell.
\begin{defn}[Fast lines] \label{fastlinescircle} We say that a circle is $\alpha-$fast for a sector $s$ if it has $\sim N^{{2 \over 3}+\alpha\pm}$ simple crossings with circles in $\mathcal{L}$. \end{defn}
This is the enemy case. Our goal is to show these do not contribute significantly to the number of incidences in the sector. Note that since a fast circle crosses $N^{{2 \over 3}+\alpha \pm}$ circles of $\mathcal{L}$ in $s$, it must cross at least $N^{\alpha-}$ of the randomly selected circles $l^{\prime}$ and therefore must enter at least $N^{\alpha-}$ cells. Similarly by Theorem \ref{bushconstructioncircle} each $\alpha$ fast line enters no more than $N^{\alpha+}$ cells in $s$. So $\alpha-$fast circles go through $\sim N^{\alpha \pm}$ cells in this sector.
\begin{defn}[Slow lines] \label{slowlinescircle} We say that a circle is slow for a sector $s$ if it has $\lesssim N^{{2 \over 3}+k_2\epsilon\pm}$ simple crossings with circles in $\mathcal{L}$. \end{defn}
We want to show slow circles contribute $\gtrsim N^{1-k_2 \epsilon}$ incidences. Assume to the contrary for some $\alpha$ the $\alpha-$fast circles contribute $\gtrsim N^{1-{\alpha \over 10^{10}}}$ incidences. Note that there are $N^{{5 \over 3}}$ simple circle arc crossings in $s$ so there are $\lesssim N^{1-\alpha}$ $\alpha-$ fast circles which each have $\sim N^{{2 \over 3}+\alpha \pm}$ crossings. If not, the sum of $\alpha-$fast contributions is $N \sum_{\alpha=k_2}^{1/(3\epsilon)} N^{-\alpha \epsilon/10^{10}} \sim N^{1-k_2\epsilon/10^{10}}$ so by choosing constant $k_2$ large enough, the $\alpha-$fast circles contribute a vanishingly small fraction of the incidences and we also conclude that slow circles contribute most incidences.
If the $\alpha-$fast circles contribute $\gtrsim N^{1-{\alpha \over 10^{10}}}$ incidences in $s$, then there must be $\gtrsim N^{{1 \over 3} - {\alpha \over 10^{10}}}$ cells in which $\alpha-$fast circles contribute $N^{{2 \over 3} - {\alpha \over 10^{10}}}$ incidences each. We can pick a structuring line of the sector $s$ and use it to ``double"-order the retained dividing circles $c_k^{\prime}$ which were retained for the sector $s$. This is an ordered list in which each $c_k^{\prime}$ appears at most twice. Within the sector $s$, this ordering does not change much depending on which structuring line we pick since dividing lines intersect at most $O(N^+)$ other dividing lines in $s$. Thus if an $\alpha$- fast line $c$ intersects a particular $c_k^{\prime}$ any $c_l^{\prime}$ must be in one of $4$ intervals of length $N^{\alpha+}$ in the ordering for $s$. (It's 4 intervals because each dividing line appears twice in the ordering and because there are two disconnected peieces of the sector.) Each interval is associated to the cells which its $c_k^{\prime}$ bound. In this way, we can find an interval of length $N^{\alpha+}$ associated to $N^{{2 \over 3}+}$ $\alpha$-fast lines.
We let $R$ be the union of the $N^{\alpha+}$ consecutive cells above. There are $N^{{2 \over 3}+}$ many $\alpha$-fast circles going through $R$, each making $N^{\alpha ( 1- 2(10^{-10}) ) -}$ incidences with the $N^{{1 \over 3}+\alpha+}$ many points in $R$. We apply Szemer\'edi--Trotter (for unit circles) and obtain a contradiction.
\end{proof}
It is worth emphasizing that Theorem \ref{sectorincidencescircle} says that most (up to $\epsilon$ loss in the exponent) circles will take $O(1)$ points from each cell they crosses. In other words, points on a circle are evenly spaced among cells. This is the linchpin fact in our main result (see Section \ref{protoinverseSTcircle}).
\begin{thm} \label{doublebushmixingcircle} Let $(\mathcal{L},\mathcal{P})$ be an extremal unit circle configuration. Then there are $\gtrsim N^{2-}$ pairs of organizing points $p_1$ and $p_2$, with bushes $c_{1,1},\dots, c_{M_1,1}$ incident to $p_1$ and $c_{1,2}, \dots, c_{M_2,2}$ incident to $p_2$ with $M_1,M_2 \sim N^{{1 \over 3}\pm}$ and a refinement $\mathcal{P}^{\prime} \subset \mathcal{P}$ so that the two bushes break $\mathcal{P}^{\prime}$ into $M_1 M_2$ cells which are point weighted and so that each sector $s$ of the bush at $p_1$ having at least $N^{{2 \over 3}-}$ points of $\mathcal{P}^{\prime}$ has at least $N^{1-}$ circles of $\mathcal{L}$ incident to at least two points in $s$. \end{thm}
\begin{proof} Apply Theorem \ref{verynicespacingcircle}, we find a point $p_1$ with bush $c_{1,1},\dots, c_{M,1}$ and structuring circles holding in total $N^{1-}$ points. We call this set of points $\mathcal{P}_1$. Then $(\mathcal{L}, \mathcal{P}_1)$ is an extremal configuration. We apply Theorem \ref{sectorincidencescircle} to find a refinement of the set of incidences
$J(\mathcal{L},\mathcal{P}_1)$ with $|J(\mathcal{L},\mathcal{P}_1)| \gtrsim N^{{4 \over 3}-}$ so that each circle of $\mathcal{L}$ takes only $N^{+}$ incidences of $J(\mathcal{L},\mathcal{P}_1)$ in each sector of the bush at $p_1$.
We restrict to those points in $\mathcal{P}_1$ which are at least $N^{{1 \over 3}-}$ rich in incidences of $J(\mathcal{L},\mathcal{P}_1)$. We refer to that set as $\mathcal{P}_2$. The set $(\mathcal{L},\mathcal{P}_2)$. is an extremal configuration. We refine the set of circles to $\mathcal{L}_1$ which take $N^+$ incidences in $N^{{1 \over 3}-}$ sectors of the bush at $p_1$. We let $\mathcal{P}_3$ be the set of points of $\mathcal{P}_2$ that are $N^{{1 \over 3}-}$ rich with respect to circles of $\mathcal{L}_1$ incident to only $N^+$ other points in the same sector. The pair $(\mathcal{L}_1,\mathcal{P}_3)$ is an extremal configuration. Pick an organizing point $p_2$ with bush $c_{1,2} \dots c_{M_2,2}$ and structuring circles for each sector of the bush from $\mathcal{L}_1$. From the structuring circles keep only the points which occur in groups of at most $N^{+}$ in the first bush. Call this set of points $\mathcal{P}^{\prime}$. Each structuring circle has at most $N^{+}$ points in each sector of the first bush. But since there are only at most $N^{{1 \over 3}+}$ structuring circles in each sector of the second bush, the cell decomposition given by the two bushes is point weighted.
Since each sector of the first bush has $N^{1-}$ incidences coming from circles making at most $N^{+}$ incidences in the sector, there must be $N^{1-}$ circles incident to points of the sector.
\end{proof}
\subsection{A proto inverse theorem for unit distances} \label{protoinverseSTcircle}
Here we just state the analog for unit circles of Theorem \ref{protoinverseST}
\begin{thm} \label{protoinverseSTcircles} Let $(\mathcal{L},\mathcal{P})$ be an extremal configuration. Then there is a choice of the $O(N^{{1 \over 3}})$ parameters $(a,b,l_s)$ so that $(\mathcal{L} \cap \mathcal{L}(a,b,l_s),\mathcal{P} \cap \mathcal{P}(a,b,l_s))$ is an extremal configuration (of $N^{1-}$ circles $N^{1-}$ points and $N^{{4 \over 3}-}$ incidences.) \end{thm}
The proof is just the same as the proof of Theorem \ref{protoinverseST}. We let $a$ and $b$ encode the location of two bushes for the two bush cell decomposition and let $l_s$ encode the sturcturing lines for one of the sectors of the first bush.
\end{document} | arXiv |
Nanoscale Research Letters
December 2017 , 12:13 | Cite as
Enhancement of Electrochemical Performance by the Oxygen Vacancies in Hematite as Anode Material for Lithium-Ion Batteries
Peiyuan Zeng
Yueying Zhao
Yingwu Lin
Xiaoxiao Wang
Jianwen Li
Wanwan Wang
Zhen Fang
Nano Express
The application of hematite in lithium-ion batteries (LIBs) has been severely limited because of its poor cycling stability and rate performance. To solve this problem, hematite nanoparticles with oxygen vacancies have been rationally designed by a facile sol–gel method and a sequential carbon-thermic reduction process. Thanks to the existence of oxygen vacancies, the electrochemical performance of the as-obtained hematite nanoparticles is greatly enhancing. When used as the anode material in LIBs, it can deliver a reversible capacity of 1252 mAh g−1 at 2 C after 400 cycles. Meanwhile, the as-obtained hematite nanoparticles also exhibit excellent rate performance as compared to its counterparts. This method not only provides a new approach for the development of hematite with enhanced electrochemical performance but also sheds new light on the synthesis of other kinds of metal oxides with oxygen vacancies.
Hematite Oxygen vacancies Calcination Lithium-ion batteries
The online version of this article (doi: 10.1186/s11671-016-1783-0) contains supplementary material, which is available to authorized users.
Because of its high theoretical capacity, natural abundance, and environmental friendliness, hematite (α-Fe2O3) has been regarded as a promising anode material for lithium-ion batteries (LIBs) [1, 2, 3, 4]. However, the practical application of hematite is greatly limited because of its low conductivity, large volume variation, and easy aggregation during the discharge/charge process [5, 6, 7, 8, 9]. To overcome these drawbacks, two main methods are employed. The first method concerns on the synthesis of nano-sized iron oxides with different structures, which will shorten the transportation distances of electron and Li+. The second method focuses on elevating the conductivity of hematite, which is mainly realized by forming the composite between hematite and materials with high electronic conductivity [10, 11, 12, 13, 14]. Despite these progresses, a simpler method for the preparation of hematite with enhanced electrochemical performance is still needed when considering its practical uses.
The introduction of oxygen vacancies into metal oxides has been proved to be an effective method to modulate the intrinsic electrochemical properties of the metal oxides [15, 16]. The existence of oxygen vacancies could effectively change the electronic structure of the metal oxides, reduce the energy requirement for electron or ion diffusion, and lower the resistance, which could be beneficial to improve the electrochemical performances of the metal oxides [17]. What is more, previous reports also clearly indicate that the existence of oxygen vacancies could facilitate the phase transition and reduce the stress during Li+ insertion/depletion, which will be helpful to improve the rate performance as well as the cycling stabilities of the electrode materials. Oxygen vacancies could also provide more physical space for Li+ storage thus improving the specific capacity of the materials [18, 19]. For this reason, a large number of efforts have been devoted to the synthesis of electrode material with oxygen vacancies, all of which have shown enhanced electrochemical performance when used in LIBs. For example, Li3VO4−δ was synthesized by annealing Li3VO4 powders in vacuum, and the introduction of oxygen vacancies lead to the enhanced initial coulombic efficiency, reversible capacity, and cycling stability [20]. The as-synthesized Li3VO4−δ delivers a reversible capacity of 247 mAh g−1 after 400 cycles at 500 mA g−1, which is much higher than the corresponding value of pristine Li3VO4 (64 mAh g−1). MoO3−x nanosheets were synthesized by oxidizing Mo powers in the atmosphere containing H2O2 and absolute ethanol. The as-prepared materials exhibit fascinating reversible capacity and long-term cycling stability (179.3 mAh g−1 at 1 A g−1) when used as anode materials for sodium ion batteries [18]. Anatase TiO2−δ–carbon nanotubes (CNTs) composites were prepared by a two-step CVD method. The CNT grown on TiO2 leads to the formation of oxygen vacancies under the reducing atmosphere, which greatly enhanced the electrochemical performance, especially the rate performance. The half cells cycled at 30 C can still deliver a capacity of more than 40 mAh g−1 [21]. V2O5 nanosheets with oxygen vacancies were also prepared by a hydrothermal reaction [22]. The as-prepared H-V2O5 electrode exhibits excellent cycling stability and improved rate capability, which could be mainly attributed to the introduction of oxygen vacancies. Tong and his co-workers proposed a facile method to generate oxygen vacancies into the materials by slight nitridation in NH3 atmosphere [23, 24]. Using this method, hematite and titanium dioxide with oxygen vacancies had been successfully synthesized and delivered enhanced cyclability and rate performance. Additionally, TiO2 heterostructured nanosheet was synthesized by hydrogenation process. This kind of heterostructured nanosheet delivered a fascinating electrochemical performance. When it was used as anode material in full battery, the full battery could achieve high energy and power density [25].
Thus, it is reasonable to believe that the electrochemical performance of hematite in LIBs could be effectively enhanced by the introduction of oxygen vacancies. However, few report concerning on the effect of oxygen vacancies in hematite has been published in the field of LIBs to date. Meanwhile, the reported method for the preparation of oxygen defect α-Fe2O3 are usually based on the thermal decomposition of FeOOH in the inert gas or in vacuum, which usually needs tedious procedure and complex equipment. In this work, we present a facile method for the synthesis of α-Fe2O3 with oxygen vacancies via a two-step process incorporating a sol–gel synthesis of the precursor and thermal annealing of the precursor in air. In this synthetic route, the precursor was synthesized by a sol–gel method and then calcined in air to yield α-Fe2O3 nanoparticles with oxygen vacancies. The partial reduction of Fe(III) during the carbon-thermic process leads to the formation of oxygen vacancies in the final product, which has also been reported for the synthesis of titanium dioxide with oxygen vacancies [21, 26]. Compared with the previous reports, the preparation of α-Fe2O3 nanoparticles with oxygen vacancies is more simple, which will lower the cost during the production process. What is more, this method can be easily scaled up by simply increasing the initial amount of the starting material. These two fascinating characteristics make this method suitable for the large-scale application in the future. Thanks to the oxygen vacancies, the electrochemical performance of α-Fe2O3 is greatly promoted. Remarkably, the as-prepared Fe2O3−δ still maintained a reversible capacity of 1252 mAh g−1 at 2 C after 400 cycles. Meanwhile, the as-obtained Fe2O3−δ also exhibit excellent rate performance. Even being cycled at 40 C, the as-prepared electrode material can still deliver a discharge capacity of 188 mAh g−1, which is much higher than the corresponding value than the reported α-Fe2O3 electrode material. This synthetic method not only provides a new method for the enhancement of hematite-based electrode materials but also sheds a new light for the preparation of metal oxides with oxygen vacancies.
Synthesis of Fe2O3−δ Nanoparticles
In a typical procedure, 2 mmol FeCl3·6H2O and 4 mmol urea were dissolved in 46 mL distilled water with continuous stirring. Then, 4 mL acrylic acid was added into the as-formed yellow solution. In the next step, the mixed solution was transferred into a 70-mL Teflon-lined stainless steel autoclave and maintained at 140 °C for 12 h. After cooling down to room temperature, the gel-like product was collected by centrifugation, washed with distilled water and absolute alcohol several times and then dried in an oven at 80 °C overnight. To obtain hematite with oxygen vacancies, the as-formed precursor was calcined at 350 °C for 1.5 h in air with a heating rate of 2 °C min−1.
Sample Characterizations
The morphology and structure of the sample was investigated by transmission electron microscopy (TEM, Hitachi HT 7700) and high-resolution TEM (HRTEM, JEOL-2010). X-ray diffraction patterns were obtained using a Bruker D8 Advance with Cu-Kα radiation. X-ray photoelectron spectra (XPS) of the samples were recorded on an ESCALAB 250. The thermogravimetric analysis (TGA) was carried out on SDT 2960 with a heating rate of 10 °C min−1 from 20 to 600 °C. The BET surface area was determined on an ASAP 2460 sorption apparatus. All the as-prepared samples were degassed at 150 °C for 10 h prior to nitrogen adsorption measurements. Electron paramagnetic resonance (EPR) tests were carried out on a Bruker A300 spectrometer (X-band, frequency 9.43 GHz) equipped with Bruker ER4141VTM liquid nitrogen system. The microwave power was 0.595 mW and modulation amplitude 3.0 G. The samples were measured at 90 K with center field 3500 G and sweep width 5000 G.
Electrochemical Measurements
The electrochemical measurements were performed on coin-type cells (CR2032). The electrode was prepared using active material, acetylene black (Super P), and polyvinylidene fluoride (PVDF) in a weight ratio of 6:2:2. The electrolyte was a solution of 1 M LiPF6 in a mixture of EC:DEC (1:1 by volume). The cells were assembled in in an argon-filled glovebox (Mikrouna, Super (1220/750/900)) with both moisture and oxygen concentrations below 0.1 ppm. The galvanostatic discharge/charge characteristics were tested in the potential window of 0.01 to 3.0 V using a Neware battery tester. Cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS) were tested on a CHI660E electrochemical workstation.
The hydrothermal process at 140 °C for 12 h will lead to the formation of the gel-like precursor, which will be used as the starting material for the preparation of hematite with oxygen vacancies. The corresponding XRD pattern of the precursor (Additional file 1: Figure S1a) clearly indicates that the precursor is mainly composed of FeOOH (JCPDS No. 29–0713), which is obtained by the hydrolysis of Fe3+ in the solution. The TEM observation of the precursor (Additional file 1: Figure S1b) further confirms that the as-formed nanoparticles are wrapped in a gel-like matrix. To get further insight into the composition of the precursor, FT-IR was employed and the corresponding result is shown in Additional file 1: Figure S2. The corresponding result clearly indicates the formation of polyacrylic acid (PAA). The absorption bands centering at 1634 and 984 cm−1 could be attributed to the C=C and =CH2, respectively. And the absorption band at 1705 cm−1 can be assigned to the C=O double bond vibration of COOH groups [27, 28]. And the formation of PAA could be proved according to the disappearance of the absorption bands centering at 1634 and 984 cm−1, which has been widely reported in the previous reports [29]. Thus, the as-obtained precursor could be regarded as a nanocomposite forming by dispersing the FeOOH nanoparticles in the matrix of PAA.
For the formation of hematite with oxygen vacancies, an in situ carbon-thermic process was employed. This process can be divided into two steps: (i) the carbonization process of PAA; (ii) the transformation of FeOOH to hematite and thermal reduction of hematite with the as-formed carbon in the first step. To understand good control of the in situ carbon-thermic process, thermogravimetric analysis was employed as a guide here (Fig. 1). The total weight loss during the heating process is about 76%, indicating the high content of organic compounds in the precursor. The first stage of weight loss below 150 °C is about 12%, which can be ascribed to the evaporation of water molecules in the gel-like precursor. A major weight loss can be observed in the temperature range between 200 and 400 °C, which can be ascribed to the carbonization of PAA, the decomposition of FeOOH, the partial reduction of the as-formed hematite, and combustion of the as-formed carbon, respectively. Less than 1% weight loss can be observed as the temperature is higher than 450 °C, indicating the burnout of carbon. The DTA analysis curve has two exothermic peaks locating at 273 and 350 °C. The first exothermic peak can be ascribed to the carbonization of the organic component, while the second exothermic peak may correspond to the formation of hematite and carbon-thermic reduction of hematite. During the heating progress, the corundum crucible was filled with CO2, which would provide a hypoxic environment and lead to the formation of α-Fe2O3 with oxygen vacancies.
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TG–DTA analysis curves of the as-prepared precursor
According to the above analysis, the carbon-thermic reduction process during the calcination process may also lead to the formation of impurities such as Fe3O4 or carbon in the final product. To exclude the existence of Fe3O4 or carbon, both XRD and Raman spectra were employed. Figure 2a is the XRD pattern of the as-prepared sample, on which all the diffraction peaks can be indexed to be the rhombohedral-phased α-Fe2O3 (JCPDS No. 33–0664). No other diffraction peaks belonging to C or Fe3O4 is detected, indicating high purity of the sample. To further exclude the existence of C or Fe3O4, the Raman spectrum was employed and the result is shown in Fig. 2b. The peaks locating at 227, 293, 408, 496, 608, 658, and 1315 cm−1 are the typical peaks for α-Fe2O3, indicating the formation of rhombohedral-phased α-Fe2O3. The peaks centering at 227 and 496 cm−1 correspond to the A1g modes of α-Fe2O3, while the peaks centered at 293, 408, and 608 cm−1 can be assigned to the Eg modes of α-Fe2O3 [30]. The peak locating at 658 cm−1 could be attributed to the disorder effects or the presence of hematite nanocrystals [31, 32]. The broad peak centered at 1315 cm−1 can be assigned to the two-magnon scattering which results from the interaction between two magnons [33]. No peaks can be found at 1350 and 1580 cm−1, which indicates the absence of carbon in the sample [34]. Meanwhile, the typical Raman peaks for Fe3O4 are also not detected, indicating that the as-obtained sample is α-Fe2O3.
a XRD pattern of the obtained sample. b Raman spectra of Fe2O3−δ
The overall XPS spectrum of the as-prepared sample is shown as Fig. 3a, which clearly reveals the presence of oxygen vacancies in the as-obtained product. As it is shown in Fig. 3b, the peak centering at 710.8 and 724.3 eV are the characteristic peaks of Fe3+ in hematite [35, 36, 37, 38]. The existence of oxygen vacancies in the as-prepared sample can be proved according to the XPS spectrum of O1s (Fig. 3c). The peak locating at 529.4 eV could be ascribed to the lattice oxygen of hematite, while the peak centering at 532.1 eV is associated with the oxygen vacancies in hematite [39, 40, 41, 42]. To further confirm the existence of oxygen vacancies in the as-prepared materials, EPR spectra of Fe2O3−δ was employed (Fig. 3d). For comparison purpose, the EPR spectrum of the commercial α-Fe2O3 is also recorded. As it is shown in Fig. 3d, the commercial α-Fe2O3 shows EPR signals at g = 2.0 and g = 4.3, which could be attributed to Fe3+ ions coupled by exchange interactions and Fe3+ ions in rhombic and axial symmetry sites, respectively [43, 44]. Because Fe2+ ions are not directly involved in the EPR absorption, it only shows a single broad resonance line centered at about g = 3.6. This phenomenon can be attributed to the interactions between Fe2+ ions and Fe+3 ions, which will influence the lines shape as a result. Based on the above characterizations, the existence of oxygen vacancies can be clearly proved. Thus, the chemical formula of the as-obtained product could be described as α-Fe2O3−δ [43, 44].
a The wide-survey, b Fe 2p, and c O 1s XPS spectra of Fe2O3−δ. d EPR spectra of Fe2O3−δ collected at 90 K, where H is the magnetic field
According to the TEM observation (Fig. 4a), the as-prepared sample is composed of a large number of nanoparticles, with diameters ranging from ~5 to 20 nm. Meanwhile, the as-prepared α-Fe2O3−δ sample is mesoporous according to the TEM observation. The typical lattice distance is determined to be 0.27 nm for the as-prepared sample, which corresponds to the \( \left(10\overline{1}4\right) \) lattice plane (Fig. 4b). This result further confirms that the as-prepared sample is rhombohedral phased α-Fe2O3. The porous nature of the as-prepared α-Fe2O3−δ sample is further proved by the nitrogen adsorption–desorption experiment. The BET surface area of the as-prepared α-Fe2O3−δ sample is determined to be 54.58 m2 g−1. And the BJH pore size distribution centers at about 6 nm, corresponding to the interspaces between these nanoparticles (Additional file 1: Figure S3).
a TEM image of the as-prepared Fe2O3−δ and b the corresponding HRTEM image
The electrochemical behavior of the as-prepared α-Fe2O3−δ was firstly studied by the CV measurements at a scanning rate of 0.1 mV s−1, with the potential window from 0.01 to 3.0 V (Fig. 5a). In the first cycle, two cathodic peaks at 1.6 and 0.72 V can be observed, which corresponds to the insertion of Li+ into α-Fe2O3−δ and the reduction of Fe2O3−δ into metallic Fe. In the anodic progress, a broaden peak (between 1.6 and 1.9 V) and an ambiguous peak (at 2.3 V) can be observed, corresponding to the electrochemical oxidation reaction of metallic Fe to Fe2+ and Fe3+, respectively [34]. In the second and third cycles, the cathodic peak shifts from 0.72 to 0.79 V, and the intensity greatly decreases, which may result from the decomposition of the electrolyte and formation of the solid electrolyte interphase (SEI) layer in the first cycle [45, 46, 47, 48, 49]. The CV plots overlap in the following cycles, indicating the excellent reversibility of the materials. The electrochemical reaction of this process can be expressed as follows:
a CV curves of the as-prepared Fe2O3−δ. b The initial three galvanostatic charge/discharge profiles of Fe2O3−δ. c Cycling performance and coulombic efficiencies of Fe2O3−δ and commercial Fe2O3 at 2 C. d Typical charge/discharge curves of Fe2O3−δ sample during long-term cycles at 2 C. e Rate performance of Fe2O3−δ at various current densities. f Charge/discharge curves of Fe2O3−δ sample at different current densities
$$ {\mathrm{Fe}}_2{\mathrm{O}}_{3-\updelta} + x\ {\mathrm{Li}}^{+} + x\ {\mathrm{e}}^{-}\to {\mathrm{Li}}_x{\mathrm{Fe}}_2{\mathrm{O}}_{3-\updelta} $$
$$ {\mathrm{Li}}_x{\mathrm{Fe}}_2{\mathrm{O}}_{3-\updelta} + \left(2-x\right)\ {\mathrm{Li}}^{+} + \left(2-x\right)\ {\mathrm{e}}^{-}\to\ {\mathrm{Li}}_2{\mathrm{Fe}}_2{\mathrm{O}}_{3-\updelta} $$
$$ {\mathrm{Li}}_2{\mathrm{Fe}}_2{\mathrm{O}}_{3-\updelta} + \left(4-2\updelta \right)\ {\mathrm{Li}}^{+} + \left(4-2\updelta \right)\ {\mathrm{e}}^{-}\to\ 2{\mathrm{Fe}}^0 + \left(3-\updelta \right)\ {\mathrm{Li}}_2\mathrm{O} $$
Figure 5b is the initial three discharge/charge curves of α-Fe2O3−δ at a current density of 2 C. The initial discharge/charge capacities for the as-prepared α-Fe2O3−δ are 1863/1296 mAh g−1, respectively. According to previous reports, oxygen vacancies are easily re-oxidized over time and the high conductivity also gradually diminishes [16]. Nevertheless, the as-prepared Fe2O3−δ sample in this work exhibits excellent cycling stability during the charge/discharge process (Fig. 5c). On the contrary, commercial Fe2O3 delivers a low initial coulombic efficiency and poor cyclability. Only about 250 mAh g−1 discharge capacity could be maintained after 20 cycles under identical condition. In initial several cycles, the electrodes of Fe2O3−δ show a slight decrease in capacity, which can be ascribed to the slow formation rate of complete SEI layer at high current density. Typical charge/discharge curves of the Fe2O3−δ sample during long-term cycles is shown in Fig. 5d. Only slight capacity decay could be found in the whole test process. And after 400 cycles at 2 C, the discharge capacity is about 1252 mAh g−1, which is higher than the theoretical value of hematite (1007 mAh g−1). The excessive capacity can be explained from several aspects. On one hand, the materials were obtained by calcination, which will lead to the formation of lattice defects in the typical nanostructure. These lattice defects will provide more active sites for Li+ insertion/extraction, which could improve the specific capacity of the materials [50]. On the other hand, the decomposition and reformation of the SEI layer will also lead to the increase in capacity [51], but the central aspect is that the introduction of oxygen vacancies in the materials, which will provide more physical space for Li+ storage, changes the intrinsic property of the sample and leads to the higher specific capacity than theoretical value.
The as-prepared α-Fe2O3−δ also exhibits fascinating rate performance during the charge/discharge cycles when the current density increased from 0.5 to 40 C in a stepwise manner and then returned to 0.5 C (Fig. 5e). The average reversible capacities of α-Fe2O3−δ were 1549, 1389, 1258, 995, 848, and 556 mAh g−1 at the discharge rate of 0.5, 1, 2, 5, 10, and 20 C, respectively. It is worth noting that the as-obtained α-Fe2O3−δ can still deliver a reversible capacity of 198 mAh g−1 at a high current density of 40 C. As the current density increased, the discharge/charge plat becomes ambiguous, indicating the redox reaction mainly occurred on the surface of the electrode materials other than the inside of the material (Fig. 5f). An average discharge capacity as high as 1590 mAh g−1 can maintain when the current rate returned to 0.5 C. This result clearly demonstrates that the as-obtained α-Fe2O3−δ is a good candidate for the potential application as high-rate anode materials for LIBs.
To further understand the discharge/charge storage mechanism of the as-prepared materials, CV measurements of α-Fe2O3−δ cells after 50 cycles were carried out at different scan rates, and the corresponding is shown in Fig. 6a. As the scan rates increase, the cathodic and anodic peaks shift to lower and higher potentials with increasing peak currents. The migratory peaks indicate the kinetics of Li+ insertion/extraction at the electrode–electrolyte interfaces. However, the increasing peak currents are not proportional to the square root of the scan rate, which indicates that the discharge/charge progresses are composed of non-Faradaic and Faradaic behavior [52, 53, 54]. And the relationship between peak current (i) and scan rate (v) can be expressed as follows:
a CV curves at different scan rates after 50 cycles. b Log (i) versus log (v) plots at different redox states of the as-prepared Fe2O3−δ
$$ i = a{v}^b $$
$$ \log (i) = \mathrm{blog}(v) + \log (a), $$
where i is the peak current, v is the scan rate, and a and b are the adjustable parameters. The type of discharge/charge progresses can be determined by the value of b. When b = 1, the progresses mainly rely on pseudo-capacitive control, and when b = 0.5, the progresses are dependent on ionic diffusion. The linear relationship between log (i) and log (v) is shown in Fig. 6b. The b values (the slopes of the four lines) of the four redox peaks are 0.97, 0.86, 0.99, and 0.77, which means the electrochemical reactions of α-Fe2O3−δ are controlled by pseudo-capacitive behavior. The result is in good accordance with the cycling performance. And it can also be employed to explain the reason why α-Fe2O3−δ has a high reversible specific capacity even cycled at 2C.
The EIS of the electrodes were performed to illustrate the effect of oxygen vacancies in sample α-Fe2O3−δ. The Nyquist plots of the electrodes before cycling and after 400 cycles are shown in Additional file 1: Figure S4 with a frequency ranging from 100 to 0.01 Hz. The Nyquist plots are composed of semicircle in the high-to-middle frequency regions and a sloping long line in the low frequency region. The smaller diameter of the semicircle indicates lower contact resistance and charge transfer resistance. The more sloping long line indicates faster kinetics during cycles. Compared with the commercial hematite, Fe2O3−δ delivers a lower contact resistance and charge transfer resistance. This mainly ascribes to the introduction of oxygen vacancies, which could be regarded as electron donor, change the electronic structure, and facilitate the Li+ ion diffusion and electron transportation. After 400 cycles, the diameter of the semicircle became smaller and the long line became more sloping, which indicated the lower resistance and faster ion diffusion rate. This phenomenon may be ascribed to the irreversible reaction during discharge/charge progress, which will lead to the formation of metallic Fe or the activation of the electrode material and the formation of channels for the diffusion of lithium ions [55, 56]. Moreover, the existence of oxygen vacancies in the materials also could suppress the formation of insulated Li2O, which will lower the resistance.
In conclusion, α-Fe2O3−δ nanoparticles with oxygen vacancies were successfully synthesized by a two-step method incorporating a sol–gel process and following calcination of the precursor. The introduction of oxygen vacancies into hematite exerts positive impact on the electrochemical performance of the final product. The as-prepared α-Fe2O3−δ shows enhanced electrochemical performance and cycling stability when being used as anode materials for LIBs. The existence of oxygen vacancies not only provides more space for Li+ storage but also facilitates the transformation of electronic structure. Meanwhile, the introduction of oxygen vacancies could also lower the contact resistance and charge transfer resistance during the discharge/charge process, leading to the enhanced electrochemical performance of the sample.
The financial support of the Natural Science Foundation of China (NSFC 21101091, 21171007, 21671005) and the Programs for Science and Technology Development of Anhui Province (1501021019) is gratefully acknowledged.
PYZ prepared the manuscript and carried out the experiment. YWL, JWL, WWW, and ZF helped in the technical support for the characterizations. ZF designed the experiment. YYZ and XXW participated in the experiment. All the authors discussed the results and approved the final manuscript.
Additional file 1: Fig. S1. Typical XRD pattern of the precursor (a) and the corresponding TEM image (b). Fig. S2 FT-IR spectra of the acrylic acid monomer and as-prepared precursor. Fig. S3 Nitrogen adsorption−desorption isotherm and the corresponding pore size distribution (inset) of the as-prepared Fe2O3−δ. Fig. S4 Nyquist plots of Fe2O3−δ and commerical Fe2O3 before cycling and after 400 cycles at 2 C in the frequency range from 100 kHz to 0.01 Hz. (DOCX 3150 kb)
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1.Key Laboratory of Functional Molecular Solids, Ministry of Education, Center for Nano Science and Technology, College of Chemistry and Materials ScienceAnhui Normal UniversityWuhuPeople's Republic of China
2.School of Chemistry and Chemical EngineeringUniversity of South ChinaHengyangChina
3.WuhuPeople's Republic of China
Zeng, P., Zhao, Y., Lin, Y. et al. Nanoscale Res Lett (2017) 12: 13. https://doi.org/10.1186/s11671-016-1783-0
Received 12 October 2016 | CommonCrawl |
\begin{document}
\title{\bf Improved bound in Roth's theorem on arithmetic progressions}
\author{ By\\ \\{\sc Tomasz Schoen\footnote{The author is partially supported by National Science Centre, Poland grant 2019/35/B/ST1/00264}}}
\date{}
\maketitle
\begin{abstract} We prove that if $A\subseteq \{1,\dots,N\}$ does not contain any non-trivial three-term arithmetic progression, then
$$|A|\ll \frac{(\log\log N)^{3+o(1)}}{\log N^{}}N\,.$$ \end{abstract}
\section{Introduction}\label{s:intro}
In this paper we prove the following bound in Roth's theorem on arithmetic progressions.
\begin{theorem}\label{t:roth} If $A\subseteq \{1,\dots, N\}$ does not contain any non-trivial arithmetic progression of length three then
$$|A|\ll \frac{(\log\log N)^3(\log\log\log N)^5}{\log N^{}}N\,.$$ \end{theorem}
The first non-trivial upper bound concerning the size of progression-free sets was given by Roth \cite{roth} who showed the above inequality with $N/\log\log N$. Then it was subsequently refined by Heath-Brown \cite{heath-brown} and Szemer\'edi \cite{szemeredi-3ap} with a denominator of $(\log N)^c$ for a positive constant $c$,
by Bourgain \cite{bourgain-1/2, bourgain-2/3} and Sanders \cite{sanders-3/4}
by proving that bound with $c=1/2-o(1)$, $c=2/3-o(1)$ and $c=3/4-o(1)$. Sanders \cite{sanders-1} showed a result close to the logarithmic barrier
$$|A|\ll \frac{(\log\log N)^{6}}{\log N^{}}N$$
and Bloom \cite{bloom} further proved that
$$|A|\ll \frac{(\log\log N)^{4}}{\log N^{}}N\,,$$
for set $A\subseteq \{1,\dots, N\}$ avoiding three-term arithmetic progressions. Recently a slightly weaker bound was obtained by a different argument by Bloom and Sisask \cite{bloom-sisask}. Other results related to Roth's theorem can be found in \cite{green-roth}, \cite{helfgott-deroton}, \cite{naslund}, \cite{schoen-shkredov} and \cite{schoen-sisask}.
Let us also comment on the recent progress for the analogous problem in a high-dimensional case. Croot, Lev and Pach \cite{croot-lev-pach} proved, by a polynomial method, an upper estimate $(4-c)^n$ with some constant $c>0$, for the size of progression-free sets in $(\mathbb {Z}/4\mathbb {Z})^n$. Later Ellenberg and Gijswijt \cite{ellenberg-gijswijt} obtained the bound $(3-c)^n$ with a positive constant $c$ for subsets of $\mathbb {F}_3^n$. The latter result significantly improves the previous best bound of Bateman and Katz \cite{bateman-katz}, however this the paper \cite{bateman-katz} contains many deep results and valuable ideas that could potentially be also used in the integer case.
Each of the mentioned papers contains significant novel ideas and methods, any of them are used in our proof of Theorem \ref{t:roth}. We employ the density increment argument obtained via the Fourier analytical method invented by Roth \cite{roth}. We make use of the Bohr set machinery introduced by Bourgain \cite{bourgain-1/2}. We focus on the structure of the large spectrum, explored first by Bourgain \cite{bourgain-2/3} and thenceforth used in all further works. We also take advantage of deep insight into the structure of the large spectrum done by Bateman and Katz in \cite{bateman-katz} and \cite{bateman-katz-nonsmooth}.
Finally, let us mention that as far as the lower bound on the maximal size of progression-free subsets of $\{1,\dots, N\}$ is concerned, the first non-trivial lower estimate $N^{1-c(\log \log N)^{-1}}$ was established by Salem and Spencer \cite{salem-spencer}. Then Behrend \cite{behrend} improved it to $\exp(-c\sqrt{\log N})N.$ Elkin \cite{elkin} refined slightly Behrend's bound by a factor of $(\log N)^{1/2}$ and his argument was simplified in \cite{green-wolf}.
\section{Notation, Bohr sets and standard results} \label{s:notation}
All sets considered in the paper are finite subsets of $\mathbb {Z}$ or $\mathbb {Z}/N\mathbb {Z}$. We write $1_A(x)$ for the indicator function of set $A$. Given functions $f,g:\mathbb {Z}/N\mathbb {Z}\rightarrow\mathbb{C}$, the convolution of $f$ and $g$ is defined by $$ (f*g)(x)=\sum_{t\in\mathbb {Z}/N\mathbb {Z}}f(t)g(x-t). $$ The Fourier coefficients of a function $f:\mathbb {Z}/N\mathbb {Z}\to\mathbb{C}$ are defined by $$ \widehat f(r)=\sum_{x\in\mathbb {Z}/N\mathbb {Z}}f(x)e^{-2\pi ixr/N}, $$ where $r\in\mathbb {Z}/N\mathbb {Z}$, and the above applies to the indicator function of $A\subseteq\mathbb {Z}/N\mathbb {Z}$ as well. Parseval's formula states in particular that
$$\sum_{r=0}^{N-1}|\h {1_A}(r)|^{2}=|A|N\,.$$ We also recall the fact that $$\widehat{(1_A*1_B)}(r)=\h {1_A}(r)\h {1_B}(r)\,.$$ For a real number $\theta\geqslant 0$, the $\theta-$spectrum of a set $A$ is the set $$
\Delta_{\theta}(A)=\big \{r\in\mathbb {Z}/N\mathbb {Z}:|\h {1_A}(r)|\geqslant\theta|A|\big \}. $$ For a specified set $A$ we often write $\Delta_{\theta}$ instead of $\Delta_{\theta}(A).$
For $m\in \mathbb {N}$ by $E_{2m}(A)$ we denote the number of $2m$--tuples $(a_1,\dots,a_m,b_1,\dots,b_m)\in A^{2m}$ such that $$a_1+\dots+a_m=b_1+\dots+b_m.$$ For $m=2$, we simply write $E(A)$ for $E_4(A)$ and we call it the additive energy of set $A.$
We define the span of a finite set $X$ by $${\rm Span\,}(X)=\Big\{\sum_{x\in X}\varepsilon_xx: \varepsilon_x\in \{-1,0,1\} \text{ for all } x\in X\Big\}.$$
The dimension ${\rm dim}(A)$ of set $A$ is the minimal size of set $X$ such that $A\subseteq {\rm Span\,} (X).$ The following theorem proven in \cite{sanders-shkredov} (see also \cite{schoen-shkredov-dim} and \cite{sy}) provides an upper bound on the dimension of a set in terms of its additive doubling $K=|A+A|/|A|.$
\begin{theorem}\label{t:dimension}$\text{\cite{sanders-shkredov}}$ Suppose that $|A+A|= K|A|$. Then ${\rm dim} (A)\ll K\log |A|.$ \end{theorem}
We are going to use a sophisticated concept of Bohr sets-a fundamental tool introduced to modern additive combinatorics by Bourgain \cite{bourgain-1/2}.
Let $G=\mathbb {Z}/N\mathbb {Z}$ be a cyclic group and let us denote the group of its characters by $\widehat G\backsimeq\mathbb {Z}/N\mathbb {Z}$. We define the Bohr set with a generating set $\Gamma\subseteq\widehat G$ and a radius $\gamma\in(0,\frac{1}{2}]$ to be the set $$ B(\Gamma,\gamma)
= \big\{ x\in \mathbb {Z}/N\mathbb {Z}: \ \|{tx}/{N}\|\leqslant\gamma \text { for all } t\in\Gamma\big\}\,.$$
Here $\left\Vert \cdot\right\Vert $
denotes the distance to the integers, i.e. $\left\Vert x\right\Vert =\min_{y\in\mathbb {Z}}|x-y|$ for $x\in\mathbb{R}$. Given $\eta>0$ and a Bohr set $B=B(\Gamma,\gamma)$, by $B_{\eta}$ we mean the Bohr set $B(\Gamma,\eta\gamma).$ The two lemmas below are pretty standard, hence we refer the reader to \cite{tao-vu} for a complete account. The size of $\Gamma$ is called the rank of $B$ and we denote it by ${\rm{ rk}}(B).$ \begin{lemma}\label{l:bohr-size}
For every $\gamma\in(0,\frac{1}{2}]$ we have $$
\gamma^{|\Gamma|}N\leqslant|B(\Gamma,\gamma)|\leqslant 8^{|\Gamma|+1}|B(\Gamma,\gamma/2)|\,. $$ \end{lemma}
Bohr sets do not always behave like convex bodies. The size of Bohr sets can vary significantly even for small changes of the radius which was the motivation behind the following definition.
We call a Bohr set $B(\Gamma,\gamma)$ \emph{regular} if for every $\eta$, with $|\eta|\le1/(100|\Gamma|)$ we have $$
(1-100|\Gamma||\eta|)|B|\leqslant|B_{1+\eta}|\leqslant(1+100|\Gamma||\eta|)|B|. $$ Bourgain \cite{bourgain-1/2} showed that regular Bohr sets are ubiquitous. \begin{lemma}\label{l:bohr-regular} \label{lem:regularity} For every Bohr set $B(\Gamma,\gamma),$ there exists $\gamma'$ such that $\frac{1}{2}\gamma\leqslant\gamma'\leqslant\gamma$ and $B(\Gamma,\gamma')$ is regular. \end{lemma}
The last lemma of this section presents a standard $L^2$ density increment technique introduced by Heath-Brown \cite{heath-brown} and Szemer\'edi \cite{szemeredi-3ap}, see also \cite{pintz-steiger-szemeredi}. A proof of the lemma below can be found in either of the following papers \cite{sanders-3/4}, \cite{sanders-1} and \cite{bloom}.
\begin{lemma} \label{l:l2-increment} Let $A\subseteq \mathbb {Z}/N\mathbb {Z}$ be a set with density $\delta.$ Let $\Gamma\subseteq \Z/N\Z$ and $\nu\geqslant 0$ be such that
$$\sum_{r\in \Gamma\setminus \{0\}}|\h {1_A}(r)|^{2}\geqslant \nu|A|^{2}\,.$$ Then there is a regular Bohr set $B$ with ${\rm{ rk}} ({B})={\rm dim}(\Gamma)$ and radius $\Omega(({\rm dim}(\Gamma))^{-1})$ such that
$$|(A+t)\cap B|\geqslant(1+\Omega(\nu))\delta |B|$$
for some $t$. \end{lemma}
Throughout the paper we assume that set $A$ does not contain any non-trivial arithmetic progression of length three and that $N$ is a large number.
\section {Sketch of the argument}\label{s:sketch}
We apply a widely used density increment argument introduced by Roth \cite{roth}, however we use it in a rather non-standard way. In the first step, we increase the density by a large factor of the form $(\log(1/\delta))^{1-o(1)}$ on some low-rank Bohr set. Then we apply the iterative method of Bloom to our new set with larger density to obtain the desired bound.
The general strategy can be roughly described as follows. Let $A\subseteq [N]$ be a set with density $\delta$ without arithmetic progressions of length three then it is known that $1_A$ must have large Fourier coefficients. To obtain a density increment we would have to find a small set $\Lambda$ such that ${\rm Span\,} (\Lambda)$ has large intersection with the spectrum $\Delta_\delta.$ The size of $\Lambda$ is equal to the rank of a Bohr set, on which we will increase density, and density increment (given by the $L^2$ method) equals
$$(1+\Omega(\delta^2|{\rm Span\,}(\Lambda)\cap \Delta_\delta|))\delta.$$ If we want to obtain the density increment by factor $\Omega(L)$ for some function $L\rightarrow \infty$, we have to locate
$\Lambda$ of size $O(\delta^{-1+c}), c>0$ such that
\begin{equation}\label{span-spec}|{\rm Span\,}(\Lambda)\cap \Delta_\delta| \gg L\delta^{-2}\,. \end{equation} The main problem is that by of Bateman-Katz structural result (see Theorem \ref{t:bateman-katz-structure}) there are sets with spectrum such that the described set $\Lambda$ does not exist. Hence one needs to combine the above method with some new ideas.
In order to obtain the density increment we will consider three separate cases with respect to the size of Fourier coefficients of $1_A$ that in a sense dominate in $\Delta_{\delta^{1+\mu}}$ for a some small constant $\mu>0.$ If the contribution of middle size or small Fourier coefficients is large we follow the method introduced by Bateman and Katz \cite{bateman-katz}. We consider essentially two subcases
according to the additive behavior of the large spectrum $\Delta$. Following \cite{bateman-katz}, we call the cases smoothing and nonsmoothing, respectively. If the higher energy $E_8(\Delta)$ is much bigger than
one can deduce from the H\"older inequality applied to $E(\Delta)$ (smoothing case), then based on the Bateman-Katz argument we can indeed find a small set $\Lambda$ satisfying (\ref{span-spec}). The nonsmoothing case is more delicate.
In that case we use a seminal result of Bateman and Katz \cite{bateman-katz,bateman-katz-nonsmooth} that describes the structure of the spectrum in the nonsmoothing case and it turns out that again we can also find a small set $\Lambda$ satisfying
(\ref{span-spec}), apart from one situation where roughly $\Delta\approx X+H\,,|\Delta|\sim\delta^{-3+O(\mu)}\,, |X|\sim \delta^{-2+O(\mu)}\,, |H|\sim \delta^{-1+O(\mu)}$ and $H$ is a highly structured set. This case is considered separately in Lemma \ref{l:increment-3-hard} which is an important part of our argument. We show that either the density can be increased on a Bohr set generated by $H$ (such a Bohr set has a very low rank) or $X$ contains additive substructure which again leads to a density increment on a low-rank Bohr set. The above argument does not apply when $\Delta_{\delta^{1+\mu}}$ is dominated by large Fourier coefficients. Then assuming that there are very few smaller Fourier coefficients in $\Delta_{\delta^{1+\mu}}$, using a different technique based on Fourier approximation method, we prove that $A$ does indeed have density increment on a low-rank Bohr set.
\section{Middle size Fourier coefficients} \label{s:large-coeff} Assume that $A\subseteq \{1,\dots,N'\}$ does not contain any non-trivial arithmetic progressions of length three. Let $N$ be any prime number satisfying $2N'<N\leqslant 4N'.$ We embed $A$ in $\Z/N\Z$ in a natural way and observe that $A$ also does not contain any non-trivial arithmetic progression of length $3$ in $\Z/N\Z.$ Let us recall a standard argument that shows that $1_A$ must have large Fourier coefficients. The number of three-term arithmetic progressions in $A$ (including trivial ones) is expressed by the sum $\frac1N\sum_{r=0}^{N-1}\h {1_A}(r)^2\h {1_A}(-2r)$, whence we have
$$\frac1N\sum_{r=0}^{N-1}\h {1_A}(r)^2\h {1_A}(-2r)=|A|\,.$$
Clearly, we can assume that $|A|>\sqrt{2 N}$, so by the H\"older inequality
$$\sum_{r\not=0}|\h {1_A}(r)|^3\geqslant |A|^3-N|A|\geqslant \frac12|A|^3\,.$$ Since
$$\sum_{r\not \in \Delta_{\delta/4}(A)}|\h {1_A}(r)|^3\leqslant \frac14\delta |A|\sum_{r=0}^{N-1}|\h {1_A}(r)|^2=\frac14|A|^3$$ it follows that
$$\sum_{r \in \Delta_{\delta/4}\setminus \{0\}}|\h {1_A}(r)|^3\geqslant \frac14|A|^3\,,$$
hence there are non-trivial Fourier coefficients with $|\h {1_A}(r)|\gg \delta|A|.$
However to obtain a large density increment we have to control Fourier coefficients below typical treshold $\delta|A|.$ We will consider three separate cases: \begin{equation}\label{mid}
\sum_{r:\, \delta^{1-\mu} |A|\leqslant |\h {1_A}(r)|\leqslant \delta^{1/10} |A|}|\h {1_A}(r)|^3\geqslant \frac1{10}\delta^{\mu/5}|A|^3\,, \end{equation} \begin{equation}\label{sml}
\sum_{r:\, \delta^{1+\mu} |A|\leqslant |\h {1_A}(r)|\leqslant \delta^{1-\mu} |A|}|\h {1_A}(r)|^3\geqslant \frac1{10}\delta^{\mu/5}|A|^3\,, \end{equation} and the last one if \eqref{mid} and \eqref{sml} do not hold, where $\mu$ is a small positive constant. Throughout the paper we assume that $\delta^{\mu/20}<\log^{-1}(1/\delta)$ since we know that $\delta\rightarrow 0$ as $N\rightarrow\infty.$
By dyadic argument, we obtain \begin{equation}\label{in:l3-fourier}
\sum_{r:\, \theta |A|\leqslant |\h {1_A}(r)|\leqslant 2\theta |A|}|\h {1_A}(r)|^3\geqslant \frac1{10}\delta^{\mu/5}|A|^3\log^{-1}(1/\delta) \end{equation} for some $\delta^{1-\mu}\leqslant \theta \leqslant \delta^{1/10}$, so \begin{equation}\label{in:spectrum-size}
|\Delta_\theta|\gg \theta^{-3}\delta^{\mu/5}\log^{-1}(1/\delta)\geqslant \theta^{-3}\delta^{\mu/4}\,. \end{equation}
In this section we consider the first case \eqref{mid}. We will apply the Bateman-Katz-Bloom lemma, see Lemma 5.3 in \cite{bateman-katz} and Theorem 4.1 in \cite{bloom} (a slightly weaker version of Lemma \ref{l:bloom} can be easily deduced from Lemma \ref{l:bateman-katz-span}).
\begin{lemma}\label{l:bloom}
Let $A\subseteq \mathbb {Z}/N\mathbb {Z}$ be a set with density $\delta,$ and let $\Delta$ be a subset of $\Delta_\theta$. Then there exists a set $\Delta'\subseteq \Delta$ such that $|\Delta'|\gg \theta |\Delta|$ and ${\rm dim}(\Delta')\ll\theta^{-1}\log(1/\delta).$
\end{lemma}
\begin{lemma}\label{l:increment-1}
Let $A\subseteq \mathbb {Z}/N\mathbb {Z}$ be a set with density $\delta,$ and suppose that (\ref{in:spectrum-size}) holds for some
$\delta^{1-\mu}\leqslant \theta \leqslant \delta^{1/10}$. Then there is a regular Bohr set $B$ with ${\rm{ rk}}(B)\ll \delta^{-1+\mu/3}$ and radius $\Omega( \delta^{1-\mu/3})$ such that for some $t$
$$|(A+t)\cap B|\gg \delta^{1-\mu/4}|B|.$$ \end{lemma} \begin{proof} By Lemma \ref{l:bloom} there exists a set
$\Delta_1\subseteq \Delta_\theta$ such that $|\Delta_1|=\Theta (\theta |\Delta_\theta|)$ and $${\rm dim}(\Delta_1)\ll\theta^{-1}\log(1/\delta)\,.$$
By iterative application of Lemma \ref{l:bloom},
we see that there are disjoint sets $\Delta_1,\dots, \Delta_k\subseteq \Delta_\theta$, for
$k=\Theta(\delta^{-\mu/2})$ such that $|\Delta_i|=\Theta(\theta
|\Delta_\theta|)$ and $${\rm dim} (\Delta_i)\ll \theta^{-1}\log(1/\delta)$$
for every $1\leqslant i\leqslant k.$ Put $\Gamma=\bigcup_{i=1}^k \Delta_i\subseteq \Delta_\theta$ then by (\ref{in:spectrum-size}) we have
$$|\Gamma|\gg\delta^{-\mu/2}\theta^{-2}\delta^{\mu/4}\gg \delta^{-\mu/4}\theta^{-2}$$ and $${\rm dim}(\Gamma)\ll\delta^{-\mu/2}\theta^{-1}\log(1/\delta)\ll \delta^{-1+\mu/2}\log(1/\delta)\ll \delta^{-1+\mu/3}\,.$$ Therefore, by Lemma \ref{l:l2-increment} a shift of the set $A$ has density at least
$$(1+\Omega(\theta^2|\Gamma|))\delta\gg \delta^{-1+\mu/3}$$ on a regular Bohr
set with rank $O(\delta^{-1+\mu/3})$ and radius $\Omega( \delta^{1-\mu/3})$.$
\Box$ \end{proof}
\section{Additively smoothing spectrum}\label{s:smoothig}
In sections \ref{s:smoothig} and \ref{s:nonsmoothing} we obtain a density increment provided that \eqref{sml} holds. Hence for some $\delta^{1+\mu}\leqslant \theta\leqslant \delta^{1-\mu}$ we have
$$\sum_{r:\, \theta |A|\leqslant |\h {1_A}(r)|\leqslant 2\theta |A|}|\h {1_A}(r)|^3\gg |A|^3\log^{-1}(1/\delta)\,.$$ Thus \begin{equation}\label{in:spectrum-size-1}
|\Delta_{\theta}|\geqslant \delta^{\mu/5}\theta^{-3}\log^{-1} (1/\delta)\geqslant \delta^{2\mu}\theta^{-2}\delta^{-1}\,, \end{equation} so the size of $\Delta_{\theta}$ is close to the maximal possible value.
A well-known theorem of Shkredov \cite{shkredov-1, shkredov-2} states that for every $\Delta\subseteq \Delta_\theta$ and $m\in \mathbb {N}$ we have
$$E_{2m}(\Delta)\geqslant \theta^{2m}\delta|\Delta|^{2m}.$$
Observe that by the Parseval formula $|\Delta_\theta|\leqslant \theta^{-2}\delta^{-1}$, so if we additionally assume that $|\Delta_\theta|\gg \theta^{-2}\delta^{-1},$ then the H\"older inequality implies that for $m\geqslant 3$
$$E_{2m}(\Delta_\theta)\geqslant E(\Delta_\theta)^{m-1} |\Delta_\theta|^{m-2}\gg_m \theta^{2m}\delta|\Delta_\theta|^{2m}\,,$$ which essentially meets Shkredov's bound. This observation motivates the next definition introduced by Bateman and Katz. We say that a spectrum $\Delta_\theta$ is $\sigma$-additively smoothing (or simply additively smoothing if $\sigma$ is indicated) if
$$E_{8}(\Delta_\theta)\geqslant \delta^{-\sigma}\theta^{8}\delta^{}|\Delta_\theta|^{8}.$$ Otherwise, we say that the spectrum $\Delta_\theta$ is $\sigma$-additively nonsmoothing. In this section, we will obtain a density increment for additively smoothing spectrum.
The following lemma, proven in \cite{schoen-shkredov-dim} (see Corollary 7.5) is an abelian group version of Bateman-Katz Lemma 5.3. The proof of this result requires some modifications, but similarly as in Bloom's Theorem 4.1 in \cite{bloom} it relies on a probabilistic argument of Bateman and Katz.
\begin{lemma}\label{l:bateman-katz-span}
Let $\Delta\subseteq \Z/N\Z$ be a set such that $E_{2s}(\Delta)=\kappa |\Delta|^{2s}\geqslant 10^s s^{2s}|\Delta|^s,$ where $2\leqslant s=\lfloor \log |\Delta|\rfloor$. Then there exists a set $\Lambda\subseteq \Delta$ such that
$|\Lambda|\ll\kappa^{-1/2s}\log^{3/2} |\Delta|$ and
\begin{equation*}\label{f:d_bound1}
|{\rm Span\,} (\Lambda)\cap \Delta|\gg \kappa^{1/2s}|\Delta|\log ^{-3/2}|\Delta|\,.
\end{equation*} \label{l:bateman-katz} \end{lemma}
\begin{lemma}\label{l:increment-2}
Let $A\subseteq \mathbb {Z}/N\mathbb {Z}$, $|A|=\delta N$ and suppose
that for some $\delta^{1+\mu}\leqslant \theta\leqslant \delta^{1-\mu}$ we have $E_8(\Delta_\theta)\geqslant \delta^{-20\mu}\theta^{8}\delta^{}|\Delta_\theta|^{8}.$ Then there is a regular Bohr set $B$ with rank ${\rm{ rk}}(B)\ll \delta^{-1+\mu/2}$ and radius
$\Omega(\delta^{1-\mu/2})$ such that for some $t$
$$|(A+t)\cap B|\gg \delta^{1-\mu/2}|B|.$$ \end{lemma}
\begin{proof} Put $s=\lfloor \log |\Delta_\theta|\rfloor$. Using the H\"older inequality and (\ref{in:spectrum-size}) we have \begin{eqnarray*}
E_{2s}(\Delta_\theta)&\geqslant& E_8(\Delta_\theta)^{\frac{s-1}{3}}|\Delta_\theta|^{-\frac{s-4}{3}}\geqslant \delta^{O(1)}\delta^{\frac13(1-20\mu)s}\theta^{\frac83s}|\Delta_\theta|^{\frac73s}\\
&\geqslant& \delta^{-\frac13(18\mu+o(1))s} \theta^{2s}|\Delta_\theta|^{2s}\geqslant \delta^{-4\mu s} \theta^{2s}|\Delta_\theta|^{2s}\,, \end{eqnarray*} provided that $N$ is large enough. Notice that (\ref{in:spectrum-size-1}) implies that $E_{2s}(\Delta_\theta)\gg 10^{s} s^{2s}
|\Delta_\theta|^s,$ so we can apply
Lemma \ref{l:bateman-katz-span}. Thus, there exists a set $\Lambda$ such that
$$|\Lambda|\ll \delta^{2\mu}\theta^{-1}\log^{3/2}(1/\delta)\ll \delta^{-1+\mu/2}$$ and
$$ |{\rm Span\,} (\Lambda)\cap \Delta_\theta|\gg \delta^{-2\mu}\theta \log^{-3/2}(1/\delta)|\Delta_\theta|\gg \delta^{-2\mu}\theta^{-2}\log^{-5/2}(1/\delta)\gg \delta^{-\mu/2}\theta^{-2}\,.$$ Now it is enough to use Lemma \ref{l:l2-increment} with $\Gamma={\rm Span\,}(\Lambda)$ to get the required result.$
\Box$
\end{proof}
\section{Additively nonsmoothing spectrum} \label{s:nonsmoothing}
In this section, we will obtain a density increment in a more difficult case, when the spectrum $\Delta_\theta$ is an additively nonsmoothing set. Recall that for some $\delta^{1+\mu}\leqslant \theta\leqslant \delta^{1-\mu}$ we have $$
|\Delta_{\theta}|\geqslant \delta^{2\mu}\theta^{-2}\delta^{-1}\,. $$
Bateman and Katz \cite{bateman-katz,bateman-katz-nonsmooth} proved the following fundamental result characterizing the structure of additively nonsmoothing sets.
\begin{theorem}\label{t:bateman-katz-structure}
Let $\tau>0$ be a fixed number. There exists a function $f=f_\tau:(0,1)\rightarrow (0,\infty)$ with $f(x)\rightarrow 0$ as $x\rightarrow 0$ such that the following holds. Let $\Delta$ be a symmetric set of an abelian group and let $\sigma>0$. Assume that $E(\Delta')\gg |\Delta|^{2+\tau}$ for every $\Delta'\subseteq \Delta$ with $|\Delta'|\gg|\Delta|$ and that $E_8(\Delta)\leqslant |\Delta|^{4+3\tau+\sigma}.$ Then there exists $\alpha$, $0\leqslant \alpha\leqslant \frac{1-\tau}{2}$, such that for $i=1,\dots,\lceil |\Delta|^{\alpha-f(\sigma)}\rceil$ there are sets $H_i, X_i$ and $\Delta_i\subseteq \Delta$ such that
\begin{eqnarray}\label{bk1} |H_i|&\ll& |\Delta|^{\tau+\alpha+f(\sigma)},\\
\label{bk2}|X_i|&\ll& |\Delta|^{1-\tau-2\alpha+f(\sigma)},\\
\label{bk3}|H_i+H_i|&\ll& |H_i|^{1+f(\sigma)}, \end{eqnarray} and
\begin{eqnarray}\label{bk4}|(X_i+H_i)\cap \Delta_i|&\gg& |\Delta|^{1-\alpha-f(\sigma)}.\end{eqnarray} Furthermore, the sets $\Delta_i$ are pairwise disjoint. \end{theorem}
We will apply Theorem \ref{t:bateman-katz-structure} to the set $\Delta_\theta.$ By Shkredov's theorem for every $\Delta\subseteq \Delta_\theta$ with $|\Delta|\gg |\Delta_\theta|$ we have
$$E(\Delta)\geqslant \theta^4\delta |\Delta|^4\gg \delta^{10\mu/3}\theta^{2/3}\delta^{-2/3}|\Delta|^{7/3}\geqslant \delta^{4\mu}|\Delta|^{7/3}\gg |\Delta_\theta|^{7/3-2\mu}\,.$$ On the other hand, by Lemma \ref{l:increment-2} we can assume that
$$E_8(\Delta_\theta)\leqslant \delta^{-20\mu}\theta^{8}\delta^{}|\Delta_\theta|^{8}\leqslant |\Delta_\theta|^{5+10\mu}\,.$$ Therefore we can apply Theorem \ref{t:bateman-katz-structure} with $$ \tau=1/3-2\mu \text{~~ and ~~} \sigma=16\mu\,,$$ hence our spectrum $\Delta_\theta$ has structure described in Theorem \ref{t:bateman-katz-structure}.
Throughout the paper assume that $f(\sigma)\geqslant \sigma$ and that $\mu$ and $f=f(16\mu)$ are small constants.
Each of the four inequalities given in Theorem \ref{t:bateman-katz-structure} is crucial in our approach. Note that from \eqref{bk1}, \eqref{bk2} and \eqref{bk4} we can deduce lower bounds for the size of $H_i$ and $X_i.$ In order to apply the last one, we will need the following simple, elementary lemma.
\begin{lemma}\label{l:selection} Let $c,\varepsilon>0$ be such that $|(X+H)\cap \Delta|\geqslant c |X||H|^{1-\varepsilon}$. Then there is a set $X'\subseteq X$ such that $|X'|\geqslant \frac{c}4|X||H|^{-\varepsilon}$ and for every $Y\subseteq X'$ we have
$|(Y+H)\cap \Delta|\geqslant \frac{c^2}{8}|Y||H|^{1-2\varepsilon}.$ \end{lemma} \begin{proof} Put $S=(X+H)\cap \Delta$ and notice that
$$\sum_{t\in X+H}(1_X*1_H)(t)=|X||H|\,.$$
Let us denote by $P$ the set of elements $t$ with $(1_X*1_H)(t)\geqslant \frac2{c}|H|^{\varepsilon}.$ Clearly, $|P|\leqslant \frac{c}2|X||H|^{1-\varepsilon}$ and therefore
$$\sum_{t\in S\setminus P}(1_X*1_H)(t)=\sum_{x\in X}|(x+H)\cap (S\setminus P)|\geqslant |S|-|P|\geqslant \frac{c}2 |X||H|^{1-\varepsilon}\,.$$ Let $X'$ be the set of all $x\in X$ satisfying the inequality
$|(x+H)\cap (S\setminus P)|\geqslant \frac{c}4
|H|^{1-\varepsilon}.$ Observe that
$$\sum_{x\in X\setminus X'}|(x+H)\cap (S\setminus P)|\leqslant \frac{c}4|H|^{1-\varepsilon}|X\setminus X'|\leqslant \frac{c}4|X||H|^{1-\varepsilon}\,,$$ hence
$$|X'||H|\geqslant \sum_{x\in X'}|(x+H)\cap (S\setminus P)|\geqslant \frac{c}4|X||H|^{1-\varepsilon}\,.$$
Thus, $|X'|\geqslant \frac{c}4 |X||H|^{-\varepsilon}$ and if $Y\subseteq X'$, then
$$|(Y+H)\cap \Delta|\geqslant \frac{\frac{c}4|Y||H|^{1-\varepsilon}}{\frac2{c}|H|^\varepsilon}\,,$$ which yields to the required inequality. $
\Box$ \end{proof}
The next lemma provides a density increment in a simpler case-in Bateman-Katz theorem we have $\alpha\geqslant 20f.$
\begin{lemma}\label{l:increment-3-easy}
Let $A\subseteq \mathbb {Z}/N\mathbb {Z}, |A|=\delta N,$ and assume that for every $\Delta'\subseteq \Delta$ with $|\Delta'|\gg|\Delta|$ we have $E(\Delta')\gg |\Delta|^{7/3-2\mu}$ and
$E_8(\Delta)\leqslant |\Delta|^{5+10\mu}.$ Then either there is a regular Bohr set $B$ with ${\rm{ rk}}( B)\ll \delta^{-1+f}$ and radius $\Omega (\delta^{1-f})$ such that
$$|(A+t)\cap B|\gg \delta^{1-f}|B|$$
for some $t$; or there are sets $H$ and $X$ such that $|H|\ll |\Delta_\theta|^{1/3+21f}, |H+H|\ll |H|^{1+f}$, $|X|\ll |\Delta_\theta|^{2/3+2f},$ and
$$|(X+H)\cap \Delta_\theta|\gg |\Delta_\theta|^{1-21f}.$$ \end{lemma} \begin{proof} By Theorem \ref{t:bateman-katz-structure} applied with $\tau=1/3-2\mu$ there exist $0\leqslant \alpha\leqslant 1/3+\mu$ and sets $H_i, X_i$ for
$1\leqslant i\leqslant \lceil |\Delta_\theta|^{\alpha-f}\rceil$ such that
$$|\Delta_\theta|^{1/3-2\mu+\alpha-2f}\ll |H_i|\ll |\Delta_\theta|^{1/3-2\mu+\alpha+f}\,,$$ and
$$|\Delta_\theta|^{2/3+2\mu-2\alpha-2f}\ll |X_i|\ll |\Delta_\theta|^{2/3+2\mu-2\alpha+f}\,,$$ that fulfill inequalities \eqref{bk1}--\eqref{bk4}.
First, we assume that $1/3-20f\leqslant \alpha\leqslant 1/3+\mu$ and put $k=\lceil |\Delta_\theta|^{\alpha-25f}\rceil.$ Then by \eqref{bk4}, \eqref{bk3} and Theorem \ref{t:dimension} we have
$$\big|\bigcup_{i=1}^k(X_i+H_i)\cap \Delta_\theta\big|\gg |\Delta_\theta|^{\alpha-25f}|\Delta_\theta|^{1-\alpha-f}\geqslant |\Delta_\theta|^{1-f}\geqslant \theta^{-2}\delta^{-f}\,,$$ and \begin{eqnarray*}
{\rm dim}\big(\bigcup_{i=1}^k(X_i+H_i)\big)&\ll& \sum_{i=1}^k|X_i|{\rm dim} (H_i)\leqslant |\Delta_\theta|^{\alpha-25f}|X_i||H_i|^f\log |H_i|\\
&\ll& |\Delta_\theta|^{2/3+2\mu-\alpha-22f}\leqslant |\Delta_\theta|^{1/3-f}\ll \delta^{-1+f}\,. \end{eqnarray*}
Next let us assume that $20f\leqslant \alpha\leqslant 1/3-20f$. Observe that by \eqref{bk4} for every $i$ we have \begin{equation}\label{sel}
|(X_i+H_i)\cap \Delta_i|\gg |\Delta_\theta|^{1-\alpha-f}\gg |X_i||H_i||\Delta_\theta|^{-3f}\gg |X_i||H_i|^{1-5f}\,. \end{equation} By Lemma \ref{l:selection} applied with $X_i, H_i$ and $\varepsilon=5f$ there is $X_i'\subseteq X_i$ such that
$$|X_i'|\gg |X_i||H_i|^{-5f}\gg |\Delta_\theta|^{2/3+2\mu-2\alpha-5f}\geqslant |\Delta_\theta|^{1/3-\alpha+15f}\,.$$
Let $Y_i\subseteq X_i'$ be any subset of size $\lceil|\Delta_\theta|^{1/3-\alpha+15f}\rceil.$
By Lemma \ref{l:selection}, we have
$$|(Y_i+H_i)\cap \Delta_i|\gg |\Delta_\theta|^{2/3-2\mu+3f}\geqslant |\Delta_\theta|^{2/3+f}\geqslant \theta^{-2}\delta^{-f}\,.$$ Again Theorem \ref{t:dimension} and \eqref{bk3} imply that \begin{eqnarray*}
{\rm dim}(Y_i+H_i)&\leqslant& {\rm dim}(Y_i){\rm dim}(H_i)\ll |\Delta_\theta|^{1/3-\alpha+15f}|H_i|^{f}\log |H_i|\\
&\leqslant& |\Delta_\theta|^{1/3-\alpha+17f}\leqslant |\Delta_\theta|^{1/3-f}\leqslant \delta^{-1+f}\,. \end{eqnarray*}
In both above considered cases we found a subset of $\Delta_\theta$ of size $\Omega(\theta^{-2}\delta^{-f})$ and dimension $O(\delta^{-1+f})$ hence by Lemma \ref{l:l2-increment} there is a regular Bohr set $B$ with ${\rm{ rk}}( B)\ll \delta^{-1+f}$ and radius $\Omega (\delta^{1-f})$ such that
$$|(A+t)\cap B|\geqslant (1+\Omega(\theta^2\theta^{-2}\delta^{-f}))\delta|B|\gg \delta^{1-f}|B|$$ for some $t.$
Finally, if $\alpha\leqslant 20f$ then for every $i$ we have $
|H_i|\ll
|\Delta_\theta|^{1/3-\mu/3+\alpha+f}\leqslant |\Delta_\theta|^{1/3+21f},
|H_i+H_i|\ll
|H_i|^{1+f}, |X_i|\ll |\Delta_\theta|^{2/3+2\mu/3-2\alpha+f}\leqslant
|\Delta_\theta|^{2/3+2f} $ and
$$|(X_i+H_i)\cap \Delta_\theta|\gg |\Delta_\theta|^{1-f}\,,$$ which completes the proof. $
\Box$ \end{proof}
Finally, we arrived at a more difficult case, where $\Delta\approx X+H, \, |X|\sim\delta^{-2+O(\mu)}, |H|\sim\delta^{-1+O(\mu)}$ and the set $H$ is highly structured.
\begin{lemma}\label{l:increment-3-hard} Let $A\subseteq \mathbb {Z}/N\mathbb {Z}, |A|=\delta N,$ and assume that there are sets $H$ and $X$ such that $|H|\ll |\Delta_\theta|^{1/3+21f}, |H+H|\ll |H|^{1+f}$, $|X|\ll |\Delta_\theta|^{2/3+2f},$ and
$$|(X+H)\cap \Delta_\theta|\gg |\Delta_\theta|^{1-21f}.$$ Then there is a regular Bohr set $B$ with ${\rm{ rk}}( B)\leqslant \delta^{-1+f}$ and radius $\Omega (\delta^{1-f})$ such that
$$|(A+t)\cap B|\gg \delta^{1-f}|B|$$
for some $t$. \end{lemma}
\begin{proof} Since ${\rm dim}(H)\ll \delta^{-2f}$ then there is a set $\Lambda$ such that $|\Lambda|\ll \delta^{-2f}$ and $H\subseteq {\rm Span\,}(\Lambda)$. Let $B=B(\Lambda,\gamma)$ be a regular Bohr set with radius
$1/(6|\Lambda|)\leqslant \gamma\leqslant 1/(3|\Lambda|)$ (the existence of such $\gamma$ is guaranteed by Lemma \ref{l:bohr-regular}). Then clearly $B\subseteq B(H,1/3)$. Furthermore, for $h\in H$ and $b\in B$ we have
$$\|hb/N\|\leqslant \sum_{\lambda\in \Lambda} \|\lambda b/N\|\leqslant 1/3,$$ so
$$|\h {1_B}(h)|\geqslant \sum_{b\in B}\Re \,e^{-2\pi ihb/N}\geqslant \frac12|B|\,.$$
Put $A_t=(A+t)\cap B$ and let us assume that for each $t$ we have \begin{equation}\label{in:density-at}
|A_t|\ll \delta^{1-f} |B|\,, \end{equation} as otherwise we would obtain the required density increment on a Bohr set with the rank $O(\delta^{-2f})$ and radius $\Omega(\delta^{2f})$. For every $x\in \Z/N\Z$ we have $$\widehat {1_{A_t}}(x)=\frac1N\sum_h\h {1_B}(h)\h {1_A}(x-h)e^{2\pi i t(x-h)/N}\,,$$ hence by the Parseval formula
\begin{equation}\label{for}
\sum_t|\widehat {1_{A_t}}(x)|^2=\frac1N\sum_h |\h {1_B}(h)\h {1_A}(x+h)|^2\geqslant \frac1N\sum_{h\in H} |\h {1_B}(h)\h {1_A}(x+h)|^2\,. \end{equation}
Let $Y\subseteq X$ be a set given by Lemma \ref{l:selection} when applied to $X, H$ and $\varepsilon=5f$. Bounding similarly as in \eqref{sel} we have
$$|Y|\gg |X||H|^{-5f}\gg |\Delta_\theta|^{2/3-O(f)}\,.$$ Therefore, summing \eqref{for} over $Y$ we get
\begin{eqnarray*}\sum_{t}\sum_{x\in Y}|\widehat {1_{A_t}}(x)|^2&\geqslant& \frac1N\sum_{h\in H}\sum_{x\in Y} |\h {1_B}(h)\h {1_A}(x+h)|^2 \gg \frac1N |(Y+H)\cap \Delta_\theta||B|^2 \delta^{2-2\mu}|A|^2\\
&\gg &\delta^3 |\Delta_\theta|^{1-O(f)}|B|^2|A|\gg \delta^{O(f)}|B|^2|A|\,. \end{eqnarray*} Using averaging argument and (\ref{in:density-at}) we see that there is a $t$ such that \begin{equation}\label{sum}
\sum_{x\in Y}|\widehat {1_{A_t}}(x)|^2\gg \delta^{1+O(f)}|B|^2\gg \delta^{-1+O(f)}|A_t|^2\,. \end{equation} We can ignore all small terms in \eqref{sum} that satisfy
$$|\widehat {1_{A_t}}(x)|\leqslant c\frac{\delta^{-1/2+O(f)}}{|X|^{1/2}}|A_t|=\delta^{1/2+O(f)}|A_t| \,,$$ where $c>0$ is a sufficiently small constant, so by dyadic argument there is $\eta\gg \delta^{1/2+O(f)}$ such that
$$\sum_{x\in Y: \,\eta |A_t|\leqslant |\widehat {1_{A_t}}(x)|\leqslant 2\eta |A_t|}|\widehat {1_{A_t}}(x)|^2\gg \delta^{-1+O(f)}\log^{-1}(1/\delta)|A_t|^2\gg \delta^{-1+O(f)}|A_t|^2\,.$$ Put
$$S=\big\{x\in Y: \eta |A_t|\leqslant |\widehat {1_{A_t}}(x)|\leqslant 2\eta |A_t|\big\}$$ then by the above inequality it follows that
$$|S|\gg \eta^{-2} \delta^{-1+O(f)}\,.$$
By Lemma \ref{l:bloom}
there is a set $Z\subseteq S\subseteq Y$ such that $|Z|\geqslant \eta |S|\gg \eta^{-1} \delta^{-1+O(f)}$ and ${\rm dim}(Z)\ll \eta^{-1}\log(N/|A_t|).$
From \eqref{sum} one can deduce that $|A_t|\gg \delta^2|B|$ hence by Lemma \ref{l:bohr-size} it follows that
$$|A_t|\gg \delta^2|B|\geqslant \delta^2 \gamma^{\delta^{-2f}}N\geqslant (\delta/8)^{2\delta^{-2f}}N\,,$$ so $${\rm dim}(Z)\ll \eta^{-1}\delta^{-3f}\ll \delta^{-1/2-O(f)}\,.$$
Put $\eta_1=\eta$ and let $Z_1\subseteq Y$ be any set of size $\Theta(\eta_1^{-1}\delta^{-1+O(f)})$ such that ${\rm dim}(Z)\ll \eta_1^{-1}\delta^{-3f}\ll \delta^{-1/2-O(f)}$. Then we apply the above argument to the set $Y\setminus Z_1$ to find $Z_2\subseteq Y\setminus Z_1$ and $\eta_2\gg \delta^{1/2+O(f)}$ with the same properties (observe that the whole argument can be applied for any set $Y'\subseteq Y$ giving essentially the same conclusion, as long as
$|Y'|\gg |Y|$). Applying this procedure $k$ times we obtain disjoint sets $Z_1,\dots,Z_k\subseteq Y$ such that $|Z_i|=\Theta(\eta_i^{-1}\delta^{-1+O(f)})$ and ${\rm dim}(Z_i)\ll
\eta_i^{-1}\delta^{-2f}\ll \delta^{1-O(f)}|Z_i|$ for some $\eta_i\gg \delta^{1/2+O(f)},$ where $k$ is the smallest integer such that
$$|Z_1|+\dots+|Z_k|\geqslant \delta^{-3/2}\,.$$
Since for each $i$ we have $|Z_i|\leqslant \delta^{-3/2}$ it follows that
$$|Z_1|+\dots+|Z_k|\leqslant 2\delta^{-3/2}\,.$$
Put $U=\bigcup_{i=1}^k Z_i\subseteq X$ then $|U|\geqslant \delta^{-3/2}$ and
$${\rm dim}(U+H)\leqslant{\rm dim}(H)\sum_{i=1}^k{\rm dim}(Z_i)\ll \delta^{1-O(f)}\sum_{i=1}^k |Z_i| \ll \delta^{-1/2-O(f)}\,.$$
Again, by Lemma \ref{l:selection} we have
$$|U+H|\gg |U||H|^{1-10f}\gg \delta^{-5/2+O(f)}\,.$$ Lemma \ref{l:l2-increment} implies that there exists a Bohr set $B'$ with ${\rm{ rk}}(B')\ll \delta^{-1/2-O(f)}$ and radius $\Omega(\delta^{1/2+O(f)})$ such that \begin{equation}\label{f}
|(A+t)\cap B'|\gg (1+\Omega(\delta^{2+2\mu} \delta^{-5/2+O(f)}))\delta |B'|\gg \delta^{1/2+O(f)}|B'|\gg \delta^{1-f}|B'| \end{equation} for some $t$ which is a contradiction. $
\Box$ \end{proof}
\section{Large Fourier coefficients} \label{s:large-coeff} In this section we obtain the density increment if \eqref{mid} and \eqref{sml} do not hold, so there is a kind of spectral gap in terms of $L^3$-norm. We will use the well-known Chang's Spectral Lemma \cite{chang}, which states that for every $\theta$ we have $${\rm dim}(\Delta_\theta)\ll \theta^{-2}\log (1/\delta)\,.$$ For any function $f: \mathbb {Z}_N\rightarrow \mathbb R$ define $$T(f)=\sum_{x+y=2z}f(x)f(y)f(2z)\,.$$ We also make use of the following lower bound on the number of $3$-term arithmetic progressions in a set $S\subseteq \Z/N\Z$ with density $\gamma$ proven by Bloom \cite{bloom} $$T(1_A)\geqslant \gamma^{O(\gamma^{-1}\log^4(1/\gamma))}N^2.$$
\begin{lemma}\label{l:increment-4} Let $A\subseteq \mathbb {Z}/N\mathbb {Z},$ be a set with density $\delta$ such that \eqref{mid} and \eqref{sml} do not hold. Then there is a regular Bohr set $B$ with ${\rm{ rk}}( B)\leqslant \delta^{-2/5}$ and radius $\Omega (\delta^{4})$ such that
$$|(A+t)\cap B|\gg \mu\frac{\log(1/\delta)}{\log\log^5(1/\delta)}\delta|B|$$
for some $t$. \end{lemma} \begin{proof} By Chang's lemma $${\rm dim}(\Delta_{\delta^{1/10}})\ll \delta^{-1/5}\log (1/\delta)\leqslant \delta^{-2/5}$$
hence there is a set $\Lambda$ such that $|\Lambda|\ll \delta^{-2/5}$ and
$\Delta_{\delta^{1/10}}\subseteq {\rm Span\,}(\Lambda)$. Let $B=B(\Lambda,\gamma)$ be a regular Bohr set with radius $\gamma\gg \delta^3$. Let $\beta=\frac1{|B|}1_B$ then for every $r\in \Delta_{\delta^{1/10}}$ we have \begin{equation}\label{beta}
\big|\widehat \beta(r)-1\big|\leqslant \frac1{|B|}\sum_{b\in B} |e^{-2\pi i\lambda b/N}-1|\leqslant \frac{2\pi}{|B|}\sum_{b\in B} \sum_{\lambda\in \Lambda}\|rb/N\|\leqslant 2\pi \delta^2\,, \end{equation}
and similarly $|\widehat \beta(2r)-1| \ll \delta^2.$ Let $f: \mathbb {Z}_N\rightarrow [0,1]$ be a function defined by $$f(t)=\beta*1_A(t)\,.$$
We may assume that $f(t)=\frac1{|B|}|(A+t)\cap B|\leqslant L\delta,$ where $$L=c\mu\frac{\log(1/\delta)}{\log\log^5(1/\delta)}$$ and $c>0$ is a small constant. Put $$S=\big\{ t: f(t)\geqslant \delta/2\big \}$$
then by $\sum_tf(t)=|A|$ it follows that $|S|\geqslant N/L$ hence by Bloom's Theorem we have \begin{equation}\label{tf} T(f)\geqslant \frac18\delta^3T(S)\gg \delta^3 \exp({-O(L\log^5 L)})N^2\gg \delta^{3+\mu/10}N^2\,. \end{equation} Our next step is to compare $T(f)$ and $T(1_A).$ By \eqref{beta}, \eqref{mid}, \eqref{sml}, Parseval's formula and H\"older's inequality we have \begin{eqnarray*}
\big|T(1_A)-T(f)\big|&=&\frac1{N}\big|\sum_{r=0}^{N-1} \widehat {1_A}(r)^2\h {1_A}(-2r)-\sum_{r=0}^{N-1} \widehat f(r)^2\widehat f(-2r)\big|\\
&\leqslant&\frac1{N}\sum_{r\in \Delta_{\delta^{1/10}}} |\h {1_A}(r)^2\h {1_A}(-2r)(1-\widehat \beta(r)^2\widehat \beta (-2r))|+\frac2{N}\sum_{r\not\in \Delta_{\delta^{1/10}}}|\h {1_A}(r)|^3\\
&\ll & \delta^2 \frac1{N}\sum_{r\in \Delta_{\delta^{1/10}}}|\h {1_A}(r)|^3+\frac2{N}\sum_{r\in \Delta_{\delta^{1+\mu}}\setminus \Delta_{\delta^{1/10}} }|\h {1_A}(r)|^3+\frac2{N}\sum_{r\not\in \Delta_{\delta^{1+\mu}}}|\h {1_A}(r)|^3\\
&\ll& \delta^2|A|^2+\delta^{1+\mu/5}|A|^2+2\delta^{1+\mu}|A|^2\ll \delta^{3+\mu/5}N^2\,. \end{eqnarray*} Thus, by \eqref{tf} $$T(1_A)\gg \delta^{3+\mu/10}N^2\,,$$ which is a contradiction.$
\Box$ \end{proof}
\section{Proof of Theorem 1}\label{s:iteration}
Summarizing all considered cases, we can state the following result.
\begin{theorem}\label{t:increment} There exists an absolute constant $c>0$ such that the following holds.
Let $A\subseteq \mathbb {Z}/N\mathbb {Z}$ be a set without any non-trivial arithmetic progressions of length three and let $|A|=\delta N$. Then there is a regular Bohr set $B$ with ${\rm{ rk}}( B)\ll \delta^{-1+c}$ and radius $\Omega( \delta^{4})$ such that for some $t$
$$|(A+t)\cap B|\gg \frac{\log(1/\delta)}{\log\log^5(1/\delta)}\delta|B|.$$
\end{theorem} \begin{proof} Let us first make a suitable choice of parameters . Let $\mu>0$ be a constant the such that for every $\sigma\leqslant \mu,$ \eqref{f} holds with $f=f(16\sigma)$. Since we assumed that $f(\mu)\geqslant \mu,$ we see that in all considered cases in Lemma \ref{l:increment-1}, Lemma \ref{l:increment-2}, Lemma \ref{l:increment-3-easy}, Lemma \ref{l:increment-3-hard} and Lemma \ref{l:increment-4} we obtain density increment at least by factor of $\Omega(\mu\log(1/\delta)\log\log^{-5}(1/\delta))$ on a Bohr set with ${\rm{ rk}}( B)\ll \delta^{-1+\mu/3}$ and
radius $\Omega( \delta^{4})$. Thus, it is enough to take $c=\mu/3$. $
\Box$ \end{proof}
After the first step of our iterative procedure we obtain a larger density increment on a low-rank Bohr set and then we apply less effective method of Bloom (Theorem 7.1 \cite{bloom}).
\begin{lemma}\label{l:bloom-iteration} {\rm\cite{bloom}} There exists an absolute constant $c_1 > 0$ such that the following holds. Let $B\subseteq \mathbb {Z}/N\mathbb {Z}$ be a regular Bohr set of rank $d$. Let $A_1 \subseteq B$ and $A_2 \subseteq B_\varepsilon,$ each with relative densities $\alpha_i$. Let $\alpha = \min(c_1, \alpha_1, \alpha_2)$ and assume that $d \leqslant \exp(c_1(\log^2(1/\alpha)).$ Suppose that $B_\varepsilon$ is also regular and $c_1\alpha/(4d) \leqslant \varepsilon \leqslant c_1\alpha/d.$ Then either \begin{itemize} \item[(i)] there is a regular Bohr set $B'$ of rank ${\rm{ rk}}(B') \leqslant d + O(\alpha^{-1}\log(1/\alpha))$ and size
$$ |B'| \geqslant \exp\big(-O(\log^2(1/\alpha)(d + \alpha^{-1}\log(1/\alpha)))\big)|B|$$ such that
$$|(A_1+t)\cap B'|\gg (1+c_1)\alpha_1|B'|$$ for some $t\in \Z/N\Z$;
\item[(ii)] or there are $\Omega(\alpha_1^2\alpha_2 |B| |B_\varepsilon|)$ three-term arithmetic progressions $x+y=2z$ with $x,y\in A_1, z\in A_2$; \end{itemize} \end{lemma}
Now we are in position to finish the proof of our main result. We will not give detailed proof of the iteration procedure as it is very standard and the reader can find details on it in the literature (see \cite{bloom}, \cite{sanders-3/4}). In the first step we apply Theorem \ref{t:increment} to obtain a regular Bohr set $B^0$ with ${\rm{ rk}}( B^0)\ll \delta^{-1+c}$, radius $\Omega( \delta^{4})$ and a progression-free set $A_0\subseteq A+t$ for some $t$ such that
$$|A_0\cap B^0|\gg \alpha|B^0|\,,$$ where $$\alpha\gg \frac{\log(1/\delta)}{\log\log^5(1/\delta)}\delta\,.$$ By Lemma \ref{l:bohr-size} we have
$$|B^0|\geqslant \exp\big (-O(\delta^{-1+c}\log(1/\delta))\big)N\,.$$ Next we iteratively apply Lemma \ref{l:bloom-iteration} and let $B^i$ be Bohr sets obtained in the iterative procedure. Observe that after $k\ll \log (1/\alpha)$ steps we will be in the case $(ii)$ of Lemma \ref{l:bloom-iteration} and that ${\rm{ rk}}(B^i)\ll \alpha^{-1}\log^2(1/\alpha)$ for every $i\leqslant k$. Thus, there are
$$\Omega( \alpha^3 |B^k||B^k_\varepsilon|)$$ three-term arithmetic progressions in $A,$ where $\varepsilon\geqslant c_1\alpha/(4{\rm{ rk}}(B^k))\gg \alpha^2\log^2(1/\alpha).$
Hence by Lemma \ref{l:bloom-iteration} we have
$$|B^k|\geqslant \exp\big(-O(\alpha^{-1}\log^4(1/\alpha))\big)N\geqslant \exp\big(-O(\delta^{-1}\log^3(1/\delta))\log\log^5(1/\delta)\big)N\,,$$ and by Lemma \ref{l:bohr-size} \begin{eqnarray*}
|B^k_\varepsilon|&\geqslant& \exp\big(-O(\alpha^{-1}\log^3(1/\alpha))\big)\exp\big(-O(\alpha^{-1}\log^4(1/\alpha))\big)N\\ &\geqslant& \exp\big(-O(\delta^{-1}\log^3(1/\delta)\log\log^5(1/\delta))\big)N\,. \end{eqnarray*} Therefore $A$ contains
$$\alpha^3\exp\big(-O(\delta^{-1}\log^3(1/\delta)\log\log^5(1/\delta))\big)N^2$$
arithmetic progressions of length three. Since there are only $|A|$ trivial progressions it follows that
$$|A|\geqslant \alpha^3 \exp\big(-O(\delta^{-1}\log^3(1/\delta)\log\log^5(1/\delta))\big)N^2\,,$$ which completes the proof of Theorem \ref{t:roth}.
\section{Concluding remarks}
In Lemma \ref{l:increment-1}, Lemma \ref{l:increment-2}, Lemma \ref{l:increment-3-easy} and Lemma \ref{l:increment-3-hard} we obtained a density increment by factor of $\delta^{-c}$ on a low-rank Bohr set, where $c$ is a positive constant. Such density increment even in the first step of an iterative method would lead to the upper bound $O((\log N)^{-1-c})$ in Theorem \ref{t:roth}. However, in Lemma \ref{l:increment-4} we only were able to prove an increment by factor $(\log(1/\delta))^{1-o(1)}$. Any refinement of Lemma \ref{l:increment-4} will directly imply an improvement of Theorem \ref{t:roth}.
{}
\noindent{Faculty of Mathematics and Computer Science,\\ Adam Mickiewicz University,\\ Umul\-towska 87, 61-614 Pozna\'n, Poland\\} {\tt [email protected]}
\end{document} | arXiv |
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Find the value of k (if any), which makes the function f(x).x=1. f(x)=\begin{cases} { k^2 }...
Find the value of k (if any), which makes the function {eq}f(x) {/eq} continuous at {eq}x=1 {/eq} {eq}\displaystyle f(x)=\begin{cases} { k^2 } x-\frac{5}{4}; & x\leq 1\\ \frac { \sqrt {x^2+3}-2}{x^{2}-1};& x >1 \end{cases} {/eq}
Continuous Function:
A function {eq}f(x) {/eq} is said to be continuous at a given value {eq}x = a {/eq}, if the left-hand limit, the right-hand limit and the value of the function at {eq}x= a {/eq} are all same. i.e., the limit of the function at that value should exist and should be equal to the value of the function at the given value.
$$\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=f(a) $$
The given function is:
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Influence of Ambiguity on Success of Public Infrastructural Megaprojects in Kenya
Austin Baraza Omonyo,
Austin Baraza Omonyo
Centre for Finance & Project Management™ College House, 2nd Floor, Koinange Street, Nairobi, Kenya
Prof. Roselyn Gakure,
Prof. Roselyn Gakure
Jomo Kenyatta University of Agriculture and Technology School of Entrepreneurship & Management, Main Campus, Thika, Kenya
Prof. Romanus Odhiambo,
Prof. Romanus Odhiambo
Jomo Kenyatta University of Agriculture & Technology Office of The Deputy Vice Chancellor, Academics, Main Campus, Thika, Kenya
Article Date Published : 27 March 2018 | Page No.: EM-2018-207-221 | Google Scholar
DOI https://doi.org/10.18535/ijsrm/v6i3.em05
The objective of this study was to investigate the influence of ambiguity on success of public infrastructural megaprojects in Kenya. The need for this study arose from the thesis that ambiguity is a key cause of complexity that results in infrastructural megaprojects being delivered over budget, behind schedule, with benefit shortfalls, over and over again. The study was designed as multiple-method research based on virtual constructionist ontology recognizing that complexity is the mid-point between order and disorder. A cross-sectional census survey of completed public infrastructural megaprojects was conducted using two interlinked questionnaires. Quantitative data was analyzed using descriptive and inferential statistics while qualitative data was analyzed using expert judgment, scenario mapping and retrospective sense-making. The projects surveyed majorly utilized fixed price contracts with the outcome of increased delivery within budget than within schedule. The results showed that ambiguity had significant negative influence on process and overall success of public infrastructural megaprojects but had no significant relationship with product and organizational success. Projects in which the client assumed responsibility for cost and schedule risk had higher chances of meeting both cost and schedule objectives. In order to manage the negative effects of ambiguity, we recommend a new perspective to contract design of public infrastructural megaprojects based on complexity science, blending both outcome and behavior-based contracts. Such contracts should ensure that, in the face of ambiguity, the contractors are able to act in the best interest of their clients and that the clients have access to quality Project Management Information Systems.
Public infrastructural megaprojects are large-scale, complex ventures that cost billions of money, take many years to develop and build, involve multiple public and private stakeholders, are transformational, and impact millions of people (Flyvbjerg, 2014) . These projects are generally "greenfield" in nature as they often create new assets and utilize a variety of delivery models depending on their inherent complexity. They are often trait-making since they are designed to change the structure of society. This is in contrast with smaller and more conventional projects that are trait taking, often fitting into preexisting structures without modification (Hirschman, l995). Brady and Davies (2014) note that megaprojects are among the most complex category of project.
In Kenya, the growth in the use of infrastructural megaprojects to deliver goods and services has been phenomenal over the past few years and there appears to be no end in sight for
their use. This is despite the fact that megaprojects are always delivered over budget, with schedule delays, with benefit shortfalls, over and over again (Flyvbjerg, 2014) . The complexity inherent in the megaproject environment is often cited as the main cause of this poor performance (Cooke-Davies, Crawford & Stephens, 2011) . Without a coherent research agenda to understand both its causes and navigation strategies, complexity continues to result in problems, waste and socio-economic failure (Remington & Zolin, 2011) . As a step in taking this agenda forward, and building on the work of earlier researchers such as Maylor, Vidgen and Carver (2008), the Project Management Institute (PMI) published a Practice Guide on Navigating Complexity in 2014. This Guide describes the causes of complexity in three categories namely; human behavior, ambiguity and system behavior.
Whereas a number of studies have been conducted to explain the centrality of complexity in determining success of infrastructural megaprojects, most of these studies conclude that human behavior is the main cause of poor performance in these projects (Collyer, 2016) (Olaniran, Love, Edwards, Olatunji & Mathews, 2015;) (Meyer, 2014;) (Shore, 2008;) (Flyvbjerg, Holm & Buhl, 2004;) (Lovallo & Kahneman, 2003;) (Bruzelius, Flyvbjerg & Rothengatter, 2002;) (Mackie & Preston, 1998;) (Kahneman & Tversky, 1979) . This is despite existence of the thesis that there is a significant positive relationship between ambiguity and both customer and organizational outcomes (Hargen & Park, 2013) . Ambiguity arises from unclear or misleading events, cause and effect confusion, emergent issues or situations open to more than one interpretation in programs and projects (PMI, 2014). As a result of this ambiguity, the whole is always greater than the sum of the parts leading to uncertainty in performance of the project metrics. As Brady and Davis (2014) demonstrate, handling this inherent ambiguity to ensure stability can indeed enhance success of megaprojects. Once stability is achieved, the project schedule is able to absorb emerging disruption arising from dependencies and connections among its component parts thus ensuring that the dynamics of the system as a whole are kept under control. It is argued that achieving stability is as important as the more common measures of project performance (Swartz, 2008) . It is on the basis of this proposition that this multiple-method research based on virtual constructionist ontology was designed. The study surveyed managers, team members, sponsors and key stakeholders of completed public infrastructural megaprojects to investigate the influence of ambiguity on success of public infrastructural megaprojects with a view to making recommendations on how to manage its effects. Ambiguity was operationalized through its components namely; context, emergence and uncertainty (PMI, 2014). The results of the study showed that each of these components of ambiguity were related to success and that the overall ambiguity had significant negative influence on success of public infrastructural megaprojects.
The remainder of this article is organized as follows: the relevant theoretical and empirical literature are reviewed leading to the formulation of the study hypothesis and a presentation of the conceptual framework. The next section describes the methodology followed by the study and this is followed by a section on results of the study and a discussion of those results. The last section presents the conclusions and references.
2. Theoretical and Conceptual Framework
2.1 Theoretical Framework
This study was operationalized through two theories namely; Complex Adaptive Systems and Project Success theories. Each of these theories is briefly discusses below:
2.1.1 Complex Adaptive Systems Theory
Complex Adaptive Systems (CAS) theory also referred to as Complexity theory, states that critically interacting components self-organize to form potentially evolving structures exhibiting a hierarchy of emergent system properties (Lucas, 2009) . The rise of CAS as a school of thought is usually attributed to the mid-1980's formation of the Santa Fe Institute, a New Mexico think tank formed in part by the former members of the nearby Los Alamos National Laboratory. The scientists here claimed that through the study of complexity theory, one can see both laws of chaos and that of order; through which an explanation for how any collection of components will organize itself can be generated (Waldrop, 1992) .
Complexity theory is concerned with the study of how order, structure, pattern, and novelty arise from extremely complicated, apparently chaotic systems and conversely, how complex behavior and structure emerges from simple underlying rules. The theory attempts to discover how the many disparate elements of a system work with each other to shape the system and its outcomes, as well as how each component changes over time (PMI, 2014). Insights from the study of complexity in the life sciences suggests that there is a natural tendency for all organisms (including human kind and social organisms such as project teams) to evolve complex responses to challenges that they encounter in their environment.
Another important concept in complexity theory is that there is no master controller of any system. Rather, coherent system behavior is generated by the competition and cooperation between actors that is always present. The components of a system have different levels of organization-made up of divisions, which contain different departments, which in turn comprise different workers. But the important differentiation from this organization is that complex adaptive systems are constantly revising and rearranging their building blocks as they gain experience (Caldart & Joan, 2004) .
The CAS theory is reinforced by the Chaos theory which studies the behavior of dynamical systems that are highly sensitive to initial conditions. The theory is attributed to Edward Lorenz who while using a computer to simulate weather systems in 1960 at Massachusetts Institute of Technology, discovered one important aspect of how non-linearity affects the weather-the principle of sensitive dependence on initial conditions (Lorenz, 1963) . Lorenz's discovery of how minute changes can have major and unpredictable consequences in nonlinear systems became known as the "butterfly effect". According to this theory, small differences in initial conditions yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general (Kellert, 1993) . Thus, a small initial schedule delay in delivering one component of a megaproject can lead to more than proportionate delay in the entire project.
The CAS and Chaos theory are helpful in defining the main aspects of ambiguity namely; context, emergence and uncertainty. Emergence is an unanticipated change, spontaneous or gradual, that occurs within the context of a program or project. Uncertainty is a state of imperfect knowledge about future occurrences on the project while context is the environmental set-up within which the project is implemented.
2.1.2 Project Success Theory
There have been various attempts over the history of project management to define suitable criteria against which to anchor and measure project success (McLeod, Doolin & MacDonell, 2012) . The most recognized of these measures is the long established and widely used "iron triangle" of time, cost and quality (Atkinson, 1999; Cooke-Davies, 2002; de Wit, 1988, Ika, 2009; Jugdev, Thomas, & Delisle, 2001) . However, the "iron triangle" dimensions are inherently limited in scope (Atkinson, 1999; Ika, 2009; Wateridge, 1998) . A project that satisfies these criteria may still be considered a failure; conversely a project that does not satisfy them may be considered successful (Baccarini, 1999; de Wit, 1988, Ika, 2009) . The "iron triangle" only focuses on the project management process and does not incorporate the views and objectives of all stakeholders (Atkinson, 1999; Baccarini, 1999; Bannerman, 2008; de Wit, 1988; Jugdev & Muller, 2005; Wateridge, 1998) .
Researchers have progressively widened the scope and constituency of what is meant by project success, recognizing that project success is more than project management success and that it needs to be measured against overall objectives of the project thus reflecting a distinction between the success of a project's process and that of its product (Baccarini, 1999; Markus & Mao, 2004; Wateridge, 1998) . Product success involves such criteria as product use, client satisfaction and client benefits (McLeod et al., 2012) .
Researchers are also increasingly advocating for project success criteria that incorporates achievement of broader set of organizational objectives involving benefits to the wider stakeholder base (see Shenhar, Dvir, & Levy, 1997; Shenhar, Dvir, Levy & Maltz, 2001; Shenhar & Dvir, 2007; Hoegl & Gemuenden, 2001) . This is plausible given that projects are a means of delivering the organization's strategic objectives. Proponents of this school of thought advocate for inclusion of success criteria such as business and strategic benefits. It is this broader context of success that appeals to infrastructural megaprojects.
2.2 Conceptual Framework
Even though there are no studies that have directly assessed the influence of ambiguity on success of public infrastructural megaprojects, studies have been done that show the importance and usefulness of ambiguity in explaining project outcomes. For instance, in a study on managing structural and dynamic complexity, Brady and Davis (2014) used a comparative study of two successful megaprojects to illustrate the importance of handling ambiguity in order to deliver such projects successfully. The study underscored the need for: integrated project teams expected to come up with innovative EM-2018-209 solutions in the face of uncertainty and emergence; prototyping and testing new technology offsite prior to introduction on site; and an integrated change control system to deal with progressive elaboration in scope and its consequences.
To determine the relationship between ambiguity acceptance and project outcomes, Hargen and Park (2013) conducted an online survey of 2 Fortune 100 and 2 Fortune 500 companies, all of which had implemented Six Sigma and used teams as the core deployment tool for improvement projects. Using a combination of Principal Component Analysis, Correlation Analysis and Regression Analysis, the study established a significant positive relationship between ambiguity and both customer and organizational outcomes.
In a study of the antecedents and impacts of ambidexterity in project teams, Liu and Leitner (2012) used an in-depth case study of a complex infrastructure project facing unique challenges and tight budget and schedule to demonstrate the need for simultaneous pursuit of innovation and efficiency in complex engineering projects. They argue that both exploration and exploitation are likely to be needed for complex engineering projects to succeed. The study found that ambidexterity at the project team level is a significant contributor to project performance; the effects of temporal separation and project context on project performance are mediated by the project team's degree of ambidexterity. A key contribution of this study is the characterization of contextual ambidexterity as that which utilizes behavioral and social means of integrating exploration and exploitation. Contextual ambidexterity is achieved through empowering individuals to decide on the time spent on exploration activities or exploitation activities. To achieve contextual ambidexterity, alignment and adaptability of organizational activities and capabilities must be ensured so as to meet changing demands (Birkinshaw, 2004).
In a study to investigate the importance and usefulness of stability (i.e., ability of schedules to absorb emerging disruption) to project outcomes, Swartz (2008) conducted a survey with managers involved in aviation systems development. The study established that stability was perceived to be as important as the more common measures of project performance (time, cost, quality) and that perceptions differed depending on program size, scope (both could be used to explain complexity) and stage of completion and between managers based on their levels of experience and training.
The relationship is shown below:
In a study to examine the influence of
product requirements ambiguity on new product development task structures, Duimering, Ran, Derbentseva and Poile (2006) used interview data from new product development project managers in a large telecom firm to show that knowledge of how the task structures evolve (emergence) can lead to improved strategies for managing projects with ambiguous requirements. These strategies include decomposition of project tasks to reduce interdependence among tasks and flexible adaptation of the task structures. The study also underscores the role of communication, coordination, knowledge and problem solving in resolving ambiguity.
From the foregoing, it is clear that ambiguity plays a major role in success of projects. The direction of the relationship of ambiguity and success is not apparent as there are instances when it leads to negative consequences and others when it leads to positive consequences, depending on how it is handled. It is on this basis that this study sought to test a non-directional hypothesis that: HA: Ambiguity has significant influence on success of public infrastructural megaprojects.
3.1 Research Design
This study was operationalized through exploratory, descriptive and explanatory research goals based on Neuman (2003) classification of research goals. To achieve these goals, a post-positivist philosophy emphasizing virtual constructionist ontology (Gauthier & Ika, 2012) was assumed. The choice of this philosophical perspective was guided by the social world of complex megaprojects. In this social world, megaproject management is neither a practice nor a tool (as is the case with projects implemented in the modern social world) but a EM-2018-210 rallying rhetoric in a context of power play, domination and control (Gauthier & Ika, 2012) .
The study was designed to be mixed-method research combining both quantitative and qualitative strategies (Burch & Carolyn, 2016) . The mixedmethod research provides an epistemological paradigm that occupies the conceptual space between positivism and interpretivism (Tashakkori & Creswell, 2007) , the main epistemologies on which the virtual constructionist ontology thrives. To generate data for this study, a cross-sectional survey design was used. This design entails the collection of data (predominantly by questionnaire or structured interview) on usually quite a lot more than one case and at a single point in time in order to collect a body of quantitative or quantifiable data in connection with two or more variables, which are then examined to detect patterns of association (Bryman & Bell, 2007) .
3.2 Target Population
This study had as its primary population public sector infrastructural megaprojects implemented by the government of Kenya since 2005. Given the continual reorganizations within government and the several projects implemented by the government, it was unlikely that project records and managers for earlier projects could be traced with ease. Following Flyvbjerg (2014), the minimum budget for megaprojects included in this study was approximately Ksh. 1 billion. Managers, team members, sponsors and key stakeholders of these projects constituted the population of respondents from whom data was collected.
The rationale for selecting infrastructure among other foundations of national transformation was based on its huge actual and projected expenditure in comparison to other sectors. Specifically, in the Government of Kenya (2013) Second Medium Term Plan, infrastructure was allocated over Ksh. 7.5 trillion with the second highest allocation of Ksh. 2.5 trillion going to the Information, Communication and Technology sector.
3.3 Sampling Frame and Sample
The sampling frame of this study comprised a listing of completed public sector infrastructural megaprojects implemented in Kenya since 2005 with a minimum budget of approximately Ksh. 1 billion. The list of these projects was obtained from the Vision 2030 Secretariat and counterchecked with key informants from government parastatals. A total of 31 such projects was identified. Given the number of completed infrastructural megaprojects for the period under study as described by the sample frame, a census survey was found to be appropriate. Generally, when a sample frame is known, it can also be construed to mean that the population is known. In this case, collecting data on each member of the population becomes possible.
3.4 Instruments
The fieldwork for this study utilized two interlinked questionnaires namely, the Complexity Assessment Questionnaire, (CAQ) and the Project Success Questionnaire (PSQ). The CAQ was constructed based on the Practice Standard for Navigating Complexity (PMI, 2014) while the PSQ was developed based on Shenhar and Dvir (2004) and McLeod et al. (2012). Questionnaire survey is hailed to be an efficient data collection mechanism when the researcher knows exactly what is required and how to measure the variables of interest (Neuman, 2003) . Both questionnaires utilized a mixture of Likert scale, open ended questions, checklists and probing questions including those soliciting for specific project metrics.
Ambiguity was measured on a 19item scale comprising three constructs namely; context, emergence and uncertainty. The PSQ scale comprised 18 items blending open and closed ended questions on one part and Likert-type questions on the other part. This scale measured success along three constructs namely; process, product and organizational success. The first part involving closed and open ended questions was meant to assess process success while the Likert type questions assessed product and organizational success on a scale of 1 (strongly agree), 2 (agree), 3 (Neither agree nor disagree), 4 (disagree) and 5 (strongly disagree).
3.5 Pilot Test and Reliability
The pilot test involved validating data collection instruments and testing the feasibility of the data collection schedule. Through this study, the reliability and dimensionality of the measurement scales were tested to ensure that the items in the scales actually and reliably measured the intended variables. A total of four public infrastructural megaprojects and 16 respondents were surveyed as part of the pilot study. This was well above the "10% of the sample projected for the larger parent study" rule (Connelly, 2008) . Reliability is concerned with the question of whether the results EM-2018-211 of a study are repeatable. It is concerned with whether or not the measures that are devised for concepts are consistent (Bryman & Bell, 2007) . Cronbach's (1951) coefficient alpha (α) is commonly used to measure the reliability of the scales for Likert-scaled sub-items (Spector, 1992) . This is because the underlying assumption of the Likert scale is that it represents an underlying continuous latent scale, although the observations are ordinal (Likert, 1931) , and a high score of Cronbach's coefficient alpha means high reliability, stability and accuracy (Papadopoulos, Ojiako, Chipulu & Lee, 2012) . If the sub-items have high agreement and are highly correlated then α will be close to 1. Hair, Babin, Money and Samouel (2003) asserted that an alpha coefficient between 0.8 and 0.9 shows very good strength of association. When α is ≥ 0.7, the scale is generally reliable (Nunnally, 1978). Following this rule, both instruments were found to be reliable with the Ambiguity scale recording a Cronbach's alpha of 0.837 while the overall internal reliability of the success scale was 0.889.
To aid data processing and analysis, this study utilized the Statistical Package for Social Sciences (SPSS) version 20, Microsoft Access 2010 and Microsoft Excel 2010. The original database was created in MS Access (due to its versatility) and then transferred to SPSS for ease of analysis. MS Excel was used to complement SPSS in navigating through data sets during analysis. The SPSS software was chosen for its analytical superiority, availability and ability to handle large amounts of data.
Descriptive statistics for quantitative data
analysis were derived using frequencies and percentages, measures of central tendency (mainly the mean), measures of dispersion (mainly standard deviation coefficient of variation, skewness and kurtosis) and Earned Value measures (mainly Cost Performance Index, CPI and Schedule Performance Index, SPI). Inferential statistics were derived using regression and correlation analysis. The overall model for this study assumed the form of a simple linear regression model and was specified as: PSi 1 2 AM i (1) where i the stochastic term and β2 is the slope of the regression-change in the coefficient of project success as a result of a unit change in ambiguity score. β1 is the intercept-the coefficient of project success when there is no ambiguity. The regression coefficients were extracted using Ordinary Least Squares (OLS) method and tested for significance at the 95 percent confidence level using two-tailed ttest based on the hypothesis that: H 0 : bi 0 ; H A : bi 0 ; where bi are the values of individual betas in the estimated regression equation. The significance of the overall model was tested using multiple coefficient of determination (R2) and the Ftest. Qualitative data was analyzed using a combination of expert judgment, scenario mapping, retrospective sense-making and critical thinking
4. Research Findings and Discussion
A total of 27 completed infrastructural megaprojects were studied as part of this research. This represented a return and response rate of 87.1%. Of these projects, 2 (7.4%) were from Kenya Ports Authority, 2 (7.4%) were from Kenya Pipeline Company, 6 (22.2%) were from Kenya Airports Authority, 3 (11.1%) were from Kenya Power and Lighting Company, 1 (3.7%) was from Kenya Electricity Generating Company, 5 (18.5%) were from Kenya Urban Roads Authority, 1 (3.7%) was from Kenya Civil Aviation Authority, 1 (3.7%) was from Geothermal Development Company, with the remaining 6 (22.2%) coming from Kenya National Highways Authority.
The projects surveyed had a budget at appraisal ranging from approximately Ksh. 1 Billion to Ksh. 327 Billion with 8 of these projects (29.6%) having a budget at appraisal of over Ksh. 10 Billion. The scheduled duration for these projects ranged from 4 months to 72 months with most projects having a scheduled duration of above 20 months. The project locations were spread across several counties in Kenya. All the projects were turnkey, involving a variation of Engineer-Procure-Construct (EPC) and Design-Build-Transfer (DBT) delivery arrangements.
4.2 Findings
4.2.1 Project Context
Ambiguity due to project context was assessed using a checklist that contained several statements regarding various project contractual arrangements and risk handling. The respondents were required to choose all the statements in the checklist that applied to their individual projects. The results showed that of the 27 projects surveyed, 20 projects (74.1%) utilized Fixed Price (FP) Contracts with EM-2018-212 CPI 1
Percentage of Projects Delivered With:
SPI 1 CPI ˂1 54.5% 44.4% 66.7% 9(45%) of these utilizing a Firm Fixed Price/Lump sum (FFP) contract and 11 projects (55%) utilizing Fixed Price with Economic Price Adjustment (FPEA) contract. One project (3.7%) utilized a Cost Plus Fixed/Percentage Fee (CPFF/PF)contract while 6 projects (22.2%) utilized some form of Cost Contracts that involved re-measurement and admeasurement based on initial estimates and Bill of Quantities respectively. Table 1 presents the mean performance statistics of the megaprojects studied based on their contract types. 0.13 1 5.4 -
The results showed that with the exception of the 1 project that utilized a CPF/PF contract, the projects that utilized cost contracts involving re-/admeasurement had the highest mean success score at 5.11 points out of the possible 6.00. These projects also recorded the lowest relative variability (CV=0.1174) in the individual mean success scores. Using the CV values as measures of riskiness in project success, projects using re-/ad-measurable contracts had less risk followed by those utilizing FP-EPA. The projects utilizing FFP recorded the highest risk in mean success.
Since contractual arrangements are usually a means of allocating cost and schedule risks, this study established that of the 11 megaprojects that utilized FPEA contracts, 6 (54.5%) recorded cost overrun while 8 (72.7%) experienced schedule slippage. Of the 9 projects that utilized FFP contracts, 4 (44.4%) experience cost overruns while 8 (88.9%) experienced schedule delay. Of the 6 projects that utilized re-/ad-measurable contracts, 4 (66.7%) were delivered over budget with 5 (83.3%) being delivered behind schedule. This study noted that for projects utilizing FFP contracts, project scope ended up being narrowed to fit into the budget. The CPF/PF contract project was delivered on budget but behind schedule. Table 2 summarizes this information. EM-2018-213 FP-EPA FFP Remeasurable CPF/PF
Barring the results of the project utilizing
CPF/PF, the results showed that projects utilizing FFP contracts recorded the highest cost performance (55.6%) but also recorded the lowest schedule performance (88.9%). Projects that utilized FP-EPA contracts exhibited the second highest cost performance (45.5%) but also the third lowest schedule performance (after 83.3% from re-/admeasurable contract projects).
In 13 (48.1%) of the surveyed projects the risk of schedule delay and cost overrun was contractually shared between the client and contractor. Of these projects, 3 (11.1%) were delivered both on budget and on schedule, 5 (18.5%) were delivered on budget but behind schedule with the remaining 5 (18.5%) delivered both with cost overrun and schedule delay. In 3 projects (11.1%) the client assumed full responsibility for all the risks and insured against schedule delay and cost overrun. Of these, 1 (3.7%) was delivered within budget and ahead of schedule, 1 (3.7%) was delivered within budget but behind schedule with the remaining 1 (3.7%) project delivered over budget but within schedule.
In 8 projects (29.6%) the contractor assumed full responsibility for all the risks and provided guarantees. Of these projects, 3 (11.1%) were delivered on budget, while none was delivered on schedule. In 3 projects (11.1%) FIDIC conditions were used and schedule and cost risks were handled as they occurred. Of these projects, 1 (3.7%) was delivered within budget while none was delivered within schedule. In summary, these results show that of the 24 megaprojects that either transferred risk to the contractor or shared it between the client and the contractor, 13 megaprojects (54.2%) were delivered on budget while only 3 megaprojects (12.5%) were delivered within schedule.
Of the 3 megaprojects where the client assumed full responsibility for cost and schedule risk, 2 (66.7%) of these met both cost and schedule objectives. Generally, the results show that megaprojects that either transferred risk to the contractor or shared risk between the contractor and client had better cost performance but poor schedule performance.
4.2.2 Emergence
Project ambiguity arising from emergence was measured using a 6-item Likert type scale largely centred on assessing project stability. The responses on each item were rated on a 5-point mutually exclusive scale where a rating of 1 denoted a "strongly agree" response, 2 denoted "agree" response, 3 denoted "somewhat agree" response, 4 denoted "disagree" response, while 5 denoted a "strongly disagree" response. A choice of either 1 (strongly agree) or 2 (agree) implied low ambiguity while a choice of either 4 (disagree) or 5 (strongly disagree) implied high ambiguity. A choice of 3 (somewhat disagree) implied a neutral and borderline response which did not communicate much on the complexity of projects studied. As such, this neutral response was dropped from further analysis.
The results indicated that a total of 48.1% of the respondents agreed that assumptions, metrics and constraints remained stable throughout the life of their projects while 22.2% of the respondents disagreed with this statement. A total of 40.7% of the respondents agreed that the stakeholder requirements remained stable throughout their project life while 29.6% disagreed. On average, 55.5% of the respondents agreed that their projects were implemented in a politically and environmentally stable context with 18.5% of the respondents disagreeing. An aggregate of 81.5% of the respondents confirmed that the actual rate and type or propensity for change within their projects was manageable while 3.7% disagreed. In 70.3% of the projects surveyed, the respondents agreed that there was a documented change control system with identifiable change authority. However, in 14.8% of the projects, this was not the case. The results further indicated that contractual arrangements included incentives for the parties to assume responsibility for emerging project risks in only 7.4% of the projects. Table 4.20 summarizes these responses.
The cost and schedule performance results were mapped onto the responses with one cluster containing projects that either strongly agreed or agreed with the statements and the other containing those that disagreed or strongly disagreed. The results showed that for the projects in which assumptions, metrics and constraints remained stable through the life of the project, 53.8% were delivered on budget while 15.4% of those projects were delivered on schedule. Where assumptions, metrics and constraints did not remain stable throughout the life of the project, the results indicate that 66.7% of such projects were delivered within budget with only 16.7% of them being delivered within schedule.
Of the 11 projects in which stakeholder requirements remained stable throughout their life cycle, 54.5% were delivered within budget while 18.2% of those projects were delivered within schedule. For the projects in which stakeholder requirements did not remain stable throughout their life cycle, 50% were delivered within budget with only 12.5% of those projects being delivered within schedule. Projects which were conducted in a politically and environmentally stable context recorded a 46.7% chance of delivery within budget but only 20% of those projects were delivered within schedule.
Of the projects which were conducted in a politically and environmentally unstable context, 40% were delivered within budget with none of those projects being delivered within schedule. For the projects in which the actual rate and type or propensity for change was not manageable, 45.5% were delivered on budget while only 13.6% were delivered on schedule. Where the actual rate and type or propensity for change was not manageable, the project was delivered with both cost overrun and schedule delay. For the projects that had a documented change control system with identifiable change authority, 57.9% were delivered within budget while 21.1% were delivered within schedule.
The results further indicated that of the projects in which there was no documented change control system with identifiable change authority, 50% were delivered within budget but none of them was delivered within schedule. None of the projects surveyed utilized contractual arrangements that included incentives for the parties to assume responsibility for emerging project risks. However, for the projects in which contractual arrangements did not include incentives for the parties to assume responsibility for emerging project risks, 48.2% were delivered within budget with 18.5% being delivered within schedule.
4.2.3 Uncertainty
Ambiguity arising from uncertainty was measured using an 11-item Likert-type scale. The responses on each item were rated on a 5-point mutually exclusive scale where a rating of 1 denoted a "strongly agree" response, 2 denoted "agree" response, 3 denoted "somewhat agree" response, 4 EM-2018-214 denoted "disagree" response, while 5 denoted a "strongly disagree" response. A choice of either 1 (strongly agree) or 2 (agree) implied low ambiguity while a choice of either 4 (disagree) or 5 (strongly disagree) implied high ambiguity. A choice of 3 (somewhat disagree) implied a neutral and borderline response which did not communicate much on the complexity of projects studied. As such, this neutral response was dropped from further analysis.
The results showed that 40.7% of the respondents agreed that their projects were conducted over a relatively short period of time with a manageable number of stakeholder changes, while an equal proportion of respondents disagreed. A total of 77.7% of the projects surveyed indicated that the requirements, scope and objectives were clearly developed while 7.4% disagreed. An aggregate 74% of the respondents confirmed that the success criteria for their projects were defined, documented and agreed upon by the stakeholders while 7.4% of the respondents disagreed.
Funding in 63% of the projects surveyed came from a single source or sponsor. In 33.3% of the projects, funding came from multiple sources. A total of 66.7% of the respondents indicated that their organizations had implemented similar projects before while 29.6% had not. In 81.5% of the projects, there were a manageable number of issues, risks and uncertainties but this was not the case in 11.1% of the projects. It was indicated in 66.6% of the projects that suppliers were able to meet their commitments to the projects while in 7.4% of the projects, suppliers did not meet their commitments.
In 71.3% of the projects, the client was prepared in advance to accept and sign off deliverables but this was not the case in 11.1% of the projects. In slightly over 85% of the projects surveyed, project documents and files were kept current in an accessible location by the team but this was not the case in 3.7% of the projects. Finally, 74% of the respondents indicated that their projects had a manageable number of critical paths while 3.7% did not agree. Table 4.22 summarizes these responses. To check the relationship between uncertainty and project success, the cost and schedule performance results for the projects were mapped onto the responses. The results indicated that 54.5% of the projects in which respondents agreed that the project was conducted over a relatively short period of time with a manageable number of stakeholder changes were delivered within budget with 27.3% of those projects being delivered within schedule. Of the 11 projects in which the respondents disagreed with this statement, 63.6% were delivered within budget while only 9.1% of those projects were delivered within schedule.
Of the projects in which requirements, scope and objectives were clearly developed, 54.5% were delivered within budget with only 22.7% of those projects being delivered within schedule. Both the two projects in which the project requirements, scope and objectives were not clearly defined, were delivered within budget but only 1 was delivered on schedule. For the projects in which the success criteria was defined, documented and agreed upon by the stakeholders, 55% were delivered within budget while 25% were delivered within schedule. Both the projects in which the success criteria was not defined, documented and agreed upon by the stakeholders, were delivered on budget with none being delivered on schedule.
An aggregate 46.7% of the projects whose funding was obtained from a single source or sponsor were delivered within budget with 26.7% of those projects being delivered within schedule. Where project funding came from multiple sources or sponsors, 55.6% were delivered within budget with none being delivered on schedule. For projects which had not been undertaken by the organization before, the chance of delivery within budget was 44.4% while that of delivery within schedule was 22.2%. Where that type of project was being delivered by the organization for the first time, the chance of delivery within budget went down to 37.5% while that of delivery on schedule dropped to 12.5%. In circumstances where the project had a manageable number of issues, risks and uncertainties, delivery within budget was recorded in 45.5% of the projects while delivery within schedule was recorded in only 13.6% of the projects. Where there were unmanageable number of issues, risks and uncertainties, cost delivery went down to 33.3% while schedule delivery slipped to zero. In projects where suppliers were able to meet commitments, 44.4% were delivered within budget and 16.7% were delivered within schedule. For the 2 projects in which suppliers could not meet their commitments, one was delivered on budget while both were delivered behind schedule. Of the 7 projects that were delivered to the committed deadlines, only 1 was delivered within budget, none was delivered within schedule. Where the projects did not deliver to the committed deadlines, 25% were delivered within budget but none was delivered within schedule.
In circumstances where the client was EM-2018-215 Ambigui ty score -.687** prepared in advance to accept and sign off deliverables, the chance of delivery within schedule was 52.6% and that of delivery within budget was 26.3%. These went down to 33.3% and 0% in circumstances where the client was not prepared in advance to accept and sign off deliverables. The results also indicate that of the 26 projects in which project documents and files were kept current in an accessible location by the team, 43.5% of those
Project Success Emergen ce
Process -.495** -.706** Success Product -.139 -.236 -.214 Success Organizati .133 -.101 .035 onal Success Composite -.353 -.641** -.561** Success *. Correlation is significant at the 0.05 level (2-tailed). **. Correlation is significant at the 0.01 level (2-tailed). projects were delivered within budget with 17.4% of those projects being delivered within schedule. The project in which this was not the case was delivered within budget but behind schedule. Of the 20 projects in which there were a manageable number of critical paths in the project, delivery within budget was recorded in 40% of the projects while delivery within schedule was recorded in 25% of the projects. The project in which there were many critical paths was delivered within budget but behind schedule.
4.2.4 Hypothesis Testing
In order to test the hypothesis that ambiguity has a significant influence on success of public infrastructural megaprojects, the constructs of Causal relationship between ambiguity and megaproject success was tested using OLS linear regression at the 95% confidence level. The results indicated that there was no serial correlation in the data used to conduct regression analysis given a Durbin-Watson statistic value less than 2. Data was also checked for collinearity using the Tolerance and VIF statistics. The results indicated that the VIF was much lower than the cut-off value of 4 which is used as the threshold to indicate multicollinearity ambiguity were scored to determine their complexity scores. The context construct was used to explain the constructs of emergence and uncertainty and was therefore not scored to determine the overall ambiguity score. The results show that the emergence construct had a mean score of 2.49 with a standard deviation of 0.66 while the uncertainty construct recorded a mean score of 2.17 with a standard deviation of 0.52. The weighted ambiguity score had a mean of 2.33 with a standard deviation of 0.52. The results indicated that the emergence scores had the highest relative variability (CV=0.26) compared to that of uncertainty (CV=0.24) and the weighted score (CV=0.22). With the coefficients of skewness and kurtosis being within the acceptable range of -1 to +1 for skewness and -2.2 to +2.2 for kurtosis (Sposito, Hand, & Skarpness, 1983) , the data was approximately normal and could therefore be used to conduct parametric tests such as correlation and regression.
Ambiguity scores were first correlated with those of project success to determine if they had any association. The results showed that at 99% confidence level, emergence had a significant negative correlation with process success (r=-0.495) and uncertainty had a strong significant negative correlation with process success (r=-0.706). Overall, the results showed that ambiguity had significant negative correlation with process success (CV=0.687). The results indicated that both emergence and uncertainty had no relationship with product and organizational success, but uncertainty had a 0.641 correlation with the overall project success. The correlations are shown in Table 3. particularly in small samples (O'Brien, 2007). The problem of heteroscedasticity was checked using residual statistics plotted on a scatter diagram. The results indicated that almost all the residuals had a mean of 0.000 and approximately equally spread around their mean implying that the data was roughly homoscedastic and was therefore good for OLS regression analysis. The results indicated that the overall model had a 31.5% predictive power (R2=0.315). ANOVA results showed that the overall model was significant with F(1,25)=11.501 and PEM-2018-216 Value< 2 . The regression equation is presented below: bi s( bi ) ˂ 2 for both the intercept and the slope of the equation. Thus, following Koutsoyiannis (1992), the null hypothesis that b0 b1 0 was rejected and a conclusion made that the betas were significant. The results showed that the slope of Ambiguity was significant, implying that a one unit increase in ambiguity reduced project success score by 0.743. Thus, the research hypothesis that ambiguity has a significant influence on success of public infrastructural megaprojects was accepted.
It is generally recognized in normative literature that the FFP is the most commonly used contract type (PMI, 2013c). However, this study established that most infrastructural megaprojects utilized FPEPA and the FFP was utilized by just one third of the projects surveyed. Given the sample size of this study, it may be difficult to draw a conclusion against the postulation of normative literature. The use of FP-EPA contractual arrangements is backed by the long term nature of the projects studied with the implementation of some spanning up to 6 years. With such longer implementation periods, it is likely that factors outside the control of the client or contractor, such as inflation and currency fluctuation, may adversely affect cost performance.
The use of Fixed Price contracts is usually a tactic of transferring the risk of cost overrun to the contractor. In the case of FFP contracts, the entire risk of cost overrun is actually transferred from the client to the contractor. In such cases, the contractors are usually careful not to eat into their profit margins. The results of this study agree with this practice given that a larger proportion of projects that utilized FFP contracts recorded the highest cost performance. This was followed by projects that utilized FP-EPA, which is a variation of FP contracts. Despite having recorded superior cost performance, projects that utilized FFP recorded the highest schedule slippage. This could mean that utilizing FFP contracts could be a zero sum game-since project management success must take into account both cost and schedule performance. It is noted that projects whose contracts included late delivery penalties actually delivered on schedule. This means that the use of FFP should be adjusted to include late delivery penalties if the objectives of both cost and schedule are to be achieved simultaneously.
The results indicated that none of the projects utilized contractual arrangements with incentives for accelerated cost or schedule delivery. Such pain/gain contracts would include Fixed Price Incentive Fee, Cost-Plus Incentive Fee or Cost-Plus Award Fee. The use of these types of contracts incentivizes the contractor for superior delivery of the pre-agreed performance metrics such as schedule and cost performance (PMI, 2013). As such, using these types of contracts is strongly associated with superior project performance (Brady & Davis, 2014) since the parties involved in the project may prefer different actions because of their different risk preferences (Eisenhardt, 1989). Thus, the pain/gain contractual arrangements can be critical in solving the agency problem that characterizes most employer-contractor relationships (Jensen & Meckling, 1976) on projects.
It is argued that transferring risk to the contractor (as in the use of Fixed Price contracts) offers no real protection for the client because the client is always accountable for cost, time, quality and safety (Brady & Davis, 2014) . The results of this study agree with this argument given that 2 out of the 3 projects in which the client assumed full responsibility for risk of cost overrun and schedule delay met their cost and schedule objectives. Thus, even though numbers are still small, the findings of this study could be pointing to the fact that behavior-oriented contracts lead to better results than outcome-based contracts.
The results of this study seem to agree with the postulation of normative literature that emergence can have both positive and negative effect on project outcomes. (PMI, 2014). Emergence may enhance the ability of a project to innovate which in turn could improve project delivery capability. As an illustration, projects in which assumptions, metrics and constraints did not remain stable throughout their life recorded much better cost and schedule performance compared to those in which assumptions, metrics and constraints EM-2018-217 remained stable. The implication of this is that contractual arrangements in projects should ensure collaborative efforts with appropriate incentives to encourage the parties to solve problems that emerge as a result of progress elaboration and during execution (Brady & Davis, 2014) .
Stability of stakeholder requirements is a key aspect of complexity that affects the delivery capability of a project since emergence in stakeholder requirements implies emergence in scope. As the findings of this study attest, emergence in stakeholder requirements throughout the project's life cycle could lead to reduced cost and schedule performance thereby dimming chances of project management success.
Likewise, stability in the political and physical environment of a project could affect project delivery capability. In this study, instability in the political and physical environment adversely affected chances of delivery within budget and schedule. This is plausible particularly given that public infrastructural projects are implemented in delivery of a "political manifesto" and so instability in the political system inevitably affects project management success.
As noted by Swartz (2008), emergence can disrupt a project schedule, and the ability of the schedule to absorb that disruption is critical for the delivery of project outcomes. In situations where the actual rate and type or propensity for change is not manageable, as is the case in complex infrastructural megaprojects, the apparent schedule disruption may also disrupt cost performance. Indeed, the results of this study attest to this-where the actual rate and type or propensity for change was not manageable, cost and schedule delivery slipped to zero. In most cases where the project context was not stable, schedule delivery was adversely affected.
Change management is known to create a superior culture that supports open communication, trust and cooperation (Kerzner, 2009) among various stakeholders on the project. It allows for documented changes within the project to be considered in an integrated fashion while reducing project risk (PMI, 2013c). A well-documented change control system helps to identify, assess and control any potential and approved changes to the project baselines (Axelos, 2017) in order to avoid scope creep. In support of this, the results of this study show that projects that had a documented change control system with identifiable change authority returned a higher probability of delivery within budget and schedule.
Whereas none of the projects surveyed utilized contractual arrangements that included incentives for the parties to assume responsibility for emerging project risks, positive literature suggests that behavior-based contracts could return better project success results than outcome-based contracts. Such contracts enhance ambidexterity (simultaneous pursuit of efficiency and innovation) on projects, which in effect has significant effect on project performance (Liu & Leitner, 2012) . This is supported in a study to compare the complexity of two successful projects by Brady and Davis (2014) who showed that the use of collaborative pain/gain contracts can indeed enhance project success.
Empirical research shows that the longer the duration of the project the larger its cost overrun (Flyvbjerg et al., 2004) . However, based on the findings of this study, longer schedule duration is associated with increased delivery of projects under budget. This finding appears to be in line with the thinking of economic theory that uncertainties are necessary conditions for the existence of opportunities and that without uncertainty entrepreneurial profits would be impossible (Knight, 1948) . What is apparent from the findings of this study is that longer schedule duration is associated with reduced delivery within schedule.
Clear requirements documentation and scope definition are critical prerequisites for designing quality scope baselines. This also involves developing and documenting clear success criteria that is agreed upon by the stakeholders. Complexity arising from uncertainty affects the stability of project baselines and of the decisions made. The results of this study showed that high uncertainty has more negative effect on schedule performance than on cost performance. Chances of delivery over budget and behind schedule were high where the client was not prepared in advance to accept and sign off deliverables, implying that benefits realization planning is important for project success.
The results also showed that delivery over budget and behind schedule was equally high where the project had a large number of issues, risks and uncertainties. The implication of this is that change and risk management planning are critical to avoid firefighting whenever the vagaries of uncertainty hit a project. The Practice Guide on Navigating Complexity (PMI, 2014) recognizes that adequate risk and change management procedures should be in place to enable proper actions during the times of uncertainty, and risk sharing and collaboration are key strategies to handle uncertainty. This study EM-2018-218 found that chances of delivery over budget and behind schedule are also high in circumstances where that type of project had not been undertaken by the organization before. This confirms that the principle of learning from experience could lead to better project outcomes (Axelos, 2017) .
The results of this study indicate that ambiguity has only significant relationship with process success. The implication of this is that accepting and dealing with ambiguity should be a key aspect of project management. Hagen and Park (2013) assert that acceptance of ambiguity is a key trait that differentiates between effective and ineffective leaders. The results of this study also indicate a significant positive relationship between emergence and uncertainty (r=.454). The implication of this is that there is an overlap between these constructs and distinguishing their independent effects on project outcomes can be difficult.
It is recognized in normative literature that ambiguity can have either positive or negative effect on project outcomes (PMI, 2014). In a study on the effects of ambiguity on project task structure in new product development, Duimering, Ran, Derbentseva and Poile (2006) concluded that knowledge of how the task structures evolve (emergence) can lead to improved strategies for managing projects with ambiguous requirements. However, the findings of this study only established that ambiguity leads to negative project outcomes.
6. Conclusions and Recommendation
Ambiguity had significant negative influence on process and overall project success but had no significant relationship with product or organizational success. Thus, as the infrastructural megaproject's context becomes ambiguous, its success is negatively impacted. Both uncertainty and emergence appeared to negatively affect schedule delivery more than cost delivery. This is explained by the contractual context in which these projects were implemented. Infrastructural megaprojects majorly utilized FFP and FP-EPA contracts. Even though usage of these contract types led to increased chances of delivery within budget, they carried with them an inherent risk of delivery behind schedule. However, inclusion of late delivery penalties in these contracts helped remedy the risk of schedule slippage. Projects in which the client assumed responsibility for cost and schedule risk had higher chances of meeting both cost and schedule objectives when compared to those in which risk was either transferred to a third party or shared contractually. Transferring or sharing project risk increased chances of achieving the cost objective but greatly reduced chances of meeting the schedule objective. The results of this study point to the fact that the effects of ambiguitywhether from the project's context, emergence or uncertainty, can be managed contractually.
Neither the usage of outcome-based nor behavior-based contracts in isolation can solve the negative effect of ambiguity on success of public infrastructural megaprojects. We recommend that the design of these projects utilizes hybrid contracts based on complexity science and blending both outcome and behavior based contracts. Such contracts should ensure that the contractors are able to act in the best interest of their clients and that the clients have access to Project Management Information Systems (PMIS) capable of supplying them with information required to verify the behavior of contractors. Such PMIS could include Earned Value Management Systems (EVMS) or Risk Management System (RMS) to enable identification of management of Early Warning Signs (EWS).
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DOI: https://doi.org/10.18535/ijsrm/v6i3.em05
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\begin{document}
\title{On Some Sufficient Conditions for Strong Ellipticity}
\begin{abstract}
We establish several sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor in this paper.
The first presented sufficient condition is an extension of positive definite matrices, which states that the strong ellipticity holds if the unfolding matrix of this fourth-order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries.
An alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not.
Conditions for some special cases beyond the first sufficient condition are further investigated, which includes some important cases for the isotropic and some particular anisotropic linearly elastic materials.
\vskip 12pt
\noindent {\bf Key words.} { Elasticity tensor, isotropic material, anisotropic material, strong ellipticity, M-positive definite, S-positive definite, alternating projection, bi-quadratic form. }
\vskip 12pt
\noindent {\bf AMS subject classifications. }{74B20, 74B10, 15A18, 15A69, 15A99.}
\end{abstract}
\section{Introduction}
The strong ellipticity condition is known for a long time as one of the basic constitutive ingredients in the theory of elasticity, which guarantees the existence of solutions of basic boundary-value problems of elastostatics and thus ensures an elastic material to satisfy some mechanical properties, see eg. \cite{Aron83,ChiritaGhiba10,GourgiotisBigoni16,Gurtin73,KnowlesSternberg75,KnowlesSternberg76,Rosakis90} and works cited therein. Thereby, it is essential to identify whether the strong ellipticity holds or not in subdomains of the domain of the strain energy function.
The strong ellipticity of isotropic materials is relatively easier to handle, thus the early related works focus on isotropic materials. In particular cases, it is well-known that the strong ellipticity of an isotropic linear elastic material is equivalent to two simple inequalities about the Lam\'{e} moduli \cite{Gurtin73}. Knowles and Sternberg \cite{KnowlesSternberg75,KnowlesSternberg76} established necessary and sufficient conditions for both ordinary and strong ellipticity of the equations governing finite plane equilibrium deformations of a compressible hyperelastic solid. Simpson and Spector \cite{SimpsonSpector} extended their works to the spacial case using the representation theorem for copositive matrices. Some reformulations were also given in Rosakis \cite{Rosakis90} and Wang and Aron \cite{WangAron96}. One can refer to \cite{Dacorogna01} as a review of necessary and sufficient conditions for strong ellipticity for isotropic cases.
As to the strong ellipticity of anisotropic materials, Walton and Wilber \cite{WaltonWilber} provided sufficient conditions for strong ellipticity of a general class of anisotropic hyperelastic materials, which require the first partial derivatives of the reduced-stored energy function to satisfy several simple inequalities and the second partial derivatives to satisfy a convexity condition. Chiri\c t\u a, Danescu, and Ciarletta\cite{ChiritaDanescuCiarletta07} and Zubov and Rudev \cite{ZubovRudev16} proposed sufficient and necessary conditions for the strong ellipticity of certain classes of anisotropic linearly elastic materials. Gourgiotis and Bigoni \cite{GourgiotisBigoni16} investigated the strong ellipticity of materials with extreme mechanical anisotropy.
Qi, Dai, and Han \cite{QiDaiHan09} introduced M-eigenvalues for ellipticity tensors and proved that the strong ellipticity holds if and only if all the M-eigenvalues of the ellipticity tensor is positive. Wang, Qi, and Zhang \cite{WangQiZhang09} proposed a practical power method for computing the largest M-eigenvalue of any ellipticity tensor, which can also be employed to verify the strong ellipticity. Very recently, Huang and Qi \cite{HuangQi17} generalized the M-eigenvalues of fourth-order ellipticity tensors and related algorithms to higher order cases. Chang, Qi, and Zhou \cite{ChangQiZhou10} defined another type of ``eigenvalues'' for ellipticity tensors named as singular values, and the positivity of all the singular values of the ellipticity tensor is also a necessary and sufficient condition for the strong ellipticity. Han, Dai, and Qi \cite{HanDaiQi09} linked the strong ellipticity condition to the rank-one positive definiteness of three second-order tensors, three fourth-order tensors, and a sixth-order tensor.
The present paper is built up as follows. We briefly introduce the strong ellipticity for linearly elastic materials and some related concepts and notations in Section \ref{sec_se}. Then in Section \ref{sec_convex}, we establish a sufficient condition for strong ellipticity by identifying whether the intersection of two closed convex sets are nonempty, and propose an alternating projection verification algorithm. In Section \ref{sec_special}, we focus on special cases beyond the first sufficient condition, which includes important cases for the isotropic and particular anisotropic linearly elastic materials. Finally, we draw concluding remarks and propose open questions in Section \ref{sec_conclusion}.
\section{Strong ellipticity for linearly elastic materials}\label{sec_se}
For a linearly elastic material, the tensor of elastic moduli in a Cartesian coordinate system is a fourth-order three-dimensional tensor $\mathscr{A} = (a_{ijkl}) \in \mathbb{R}^{3 \times 3 \times 3 \times 3}$ which is invariant under the following permutations of indices \begin{equation}\label{eq_sym}
a_{ijkl} = a_{jikl} = a_{ijlk}. \end{equation} We use $\mathbb{E}$ to denote the set of all fourth-order three-dimensional tensors satisfying \eqref{eq_sym}. The strong ellipticity condition (SE-condition) is stated by \begin{equation}\label{eq_se}
\mathscr{A} {\bf x}^2 {\bf y}^2 := \sum_{i,j,k,l=1}^3 a_{ijkl} x_i x_j y_k y_l > 0 \end{equation} for any nonzero vectors ${\bf x}, {\bf y} \in \mathbb{R}^3$. We also call a tensor satisfying the SE-condition to be M-positive definite (M-PD) \cite{QiLuo17}. Similarly, a tensor $\mathscr{A} \in \mathbb{E}$ is said to be M-positive semidefinite (M-PSD) \cite{QiLuo17} if $\mathscr{A} {\bf x}^2 {\bf y}^2 \geq 0$ for any vectors ${\bf x}, {\bf y} \in \mathbb{R}^3$.
We can introduce another type of positive definiteness for fourth-order tensors, which is often involved in the investigations of linearly elastic materials. Let $\mathscr{A}$ be a fourth-order three-dimensional tensor and ${\bf Z}$ be a three-by-three matrix. Define \begin{equation}
\mathscr{A} {\bf Z}^2 := \sum_{i,j,k,l = 1}^3 a_{ijkl} z_{ik} z_{jl}. \end{equation} If $\mathscr{A} {\bf Z}^2 > 0$ ($\geq 0$) for any nonzero matrix ${\bf Z} \in \mathbb{R}^{3 \times 3}$, then $\mathscr{A}$ is said to be S-positive (semi)definite \cite{QiLuo17}. Denote ${\bf A}_{kl} = (a_{ijkl}) \in \mathbb{R}^{3 \times 3}$ for each $k,l$ and ${\bf z}_k = (z_{ik}) \in \mathbb{R}^3$ for each $k$. Then the tensor $\mathscr{A}$ and the matrix ${\bf Z}$ can be unfolded into a matrix ${\bf A} \in \mathbb{R}^{9 \times 9}$ and a vector ${\bf z} \in \mathbb{R}^9$ as $$ {\bf A} = \begin{bmatrix}
{\bf A}_{11} & {\bf A}_{12} & {\bf A}_{13} \\
{\bf A}_{21} & {\bf A}_{22} & {\bf A}_{23} \\
{\bf A}_{31} & {\bf A}_{32} & {\bf A}_{33} \end{bmatrix} \quad {\rm and} \quad {\bf z} = \begin{bmatrix}
{\bf z}_1 \\ {\bf z}_2 \\ {\bf z}_3 \end{bmatrix}, $$ respectively. Moreover, it can be verified that $\mathscr{A} {\bf Z}^2 = {\bf z}^\top {\bf A} {\bf z}$. Therefore the fourth-order tensor $\mathscr{A}$ is S-PD or S-PSD if and only if the matrix ${\bf A}$ is PD or PSD, respectively.
The S-positive definiteness is a sufficient condition for the M-positive definiteness, i.e., the SE-condition, which can be proved by observing $\mathscr{A} {\bf Z}^2 = \mathscr{A} {\bf x}^2 {\bf y}^2$ when ${\bf Z} = {\bf x} {\bf y}^\top$. Nevertheless, the S-positive definiteness is not a necessary condition for the SE-condition. A counter example is a tensor with $$ a_{1111} = a_{2222} = a_{3333} = 2, \quad a_{1221} = a_{2121} = a_{2112} = a_{1212} = 1, $$ and all other entries equal to zero. Then the bi-quadratic form is $$ \mathscr{A} {\bf x}^2 {\bf y}^2 = 2(x_1 y_1 + x_2 y_2)^2 + 2 x_3^2 y_3^2, $$ thus $\mathscr{A}$ is apparently M-PSD, while the unfolding matrix $$ \left[
\begin{array}{ccc|ccc|ccc}
2 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\hline
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \end{array} \right] $$ is not PSD. Therefore, we desire to explore weaker but still easily checkable sufficient conditions for the SE-condition.
\section{A sufficient condition with a verification algorithm}\label{sec_convex}
For any $\mathscr{A} \in \mathbb{E}$ and ${\bf y} \in \mathbb{R}^3$, we denote a three-by-three matrix $\mathscr{A} {\bf y}^2$ as $$ (\mathscr{A} {\bf y}^2)_{ij} := \sum_{k,l=1}^n a_{ijkl} y_k y_l, \quad i,j = 1,2,3. $$ Since $\mathscr{A}$ admits the symmetries in \eqref{eq_sym}, this matrix $\mathscr{A} {\bf y}^2$ is a symmetric matrix. Noticing that $\mathscr{A} {\bf x}^2 {\bf y}^2 = {\bf x}^\top (\mathscr{A} {\bf y}^2) {\bf x}$ for any vectors ${\bf x}, {\bf y} \in \mathbb{R}^3$, we can prove the following necessary and sufficient condition for the M-positive (semi)definiteness. \begin{proposition}
Let $\mathscr{A} \in \mathbb{E}$.
Then $\mathscr{A}$ is M-PD or M-PSD if and only if the matrix $\mathscr{A} {\bf y}^2$ is PD or PSD for each nonzero ${\bf y} \in \mathbb{R}^3$, respectively. \end{proposition}
Generally speaking, the above necessary and sufficient condition is as hard as the SE-condition to check. However, it motivates some checkable sufficient conditions. Recall that every positive semidefinite matrix can be decomposed into the sum of rank-one positive semidefinite matrices and the minimal number of terms is exactly its rank \cite{HornJohnson13}. Thus, we have the following sufficient condition for a tensor $\mathscr{A}$ to be M-PSD \begin{equation}\label{eq_rank1+}
\mathscr{A} {\bf y}^2 = \sum_{s = 1}^r \alpha_s {\bf f}_s({\bf y}) {\bf f}_s({\bf y})^\top, \quad \alpha_s > 0, \end{equation} where each ${\bf f}_s({\bf y})$ is a homogeneous function of degree one, i.e., ${\bf f}_s({\bf y}) = {\bf U}_s {\bf y}$ for $s = 1,2,\dots,r$. Any matrix $\mathscr{A} {\bf y}^2$ in the above form is PSD and thus $\mathscr{A}$ is M-PSD. Furthermore, if $\mathscr{A}$ is M-PD then the number of terms in the summation should be no less than $3$, i.e., $r \geq 3$. Denote the entries of each ${\bf U}_s$ as $u_{ij}^{(s)}$ ($i,j=1,2,3$). Then \eqref{eq_rank1+} reads \[ \begin{split} \sum_{k,l=1}^n a_{ijkl} y_k y_l &= \sum_{s=1}^r \alpha_s \Big( \sum_{k=1}^n u_{ik}^{(s)} y_k \Big) \Big( \sum_{l=1}^n u_{jl}^{(s)} y_l \Big) \\ &= \sum_{k,l=1}^n \Big( \sum_{s=1}^r \alpha_s u_{ik}^{(s)} u_{jl}^{(s)} \Big) y_k y_l. \end{split} \] Therefore, given ${\bf U}_s$ ($s = 1,2,\dots,r$), the entries of $\mathscr{A}$ are uniquely determined by \begin{equation}\label{eq_con}
a_{ijkl} = \frac{1}{2} \sum_{s=1}^r \alpha_s \Big( u_{ik}^{(s)} u_{jl}^{(s)} + u_{jk}^{(s)} u_{il}^{(s)} \Big), \end{equation} which satisfies the symmetries $a_{ijkl} = a_{jikl} = a_{ijlk}$.
Next, we shall discuss when a tensor in $\mathbb{E}$ can be represented by \eqref{eq_con}. Denote another fourth-order three-dimensional tensor $\mathscr{B}$ by $$ b_{ijkl} = \sum_{s=1}^r \alpha_s u_{ik}^{(s)} u_{jl}^{(s)}. $$ Note that $\mathscr{B}$ may not in the set $\mathbb{E}$, i.e., it is not required to obey \eqref{eq_sym}, but it still satisfies a weaker symmetry that $b_{ijkl} = b_{jilk}$. It can be seen that its unfolding ${\bf B}$ is a PSD matrix from $$ {\bf B} = \sum_{s=1}^r \alpha_s {\bf u}_s {\bf u}_s^\top, $$ where ${\bf u}_s$ is the unfolding (or vectorization) of ${\bf U}_s$ ($s = 1,2,\dots,r$). Hence $\mathscr{B}$ is S-PSD since all the coefficients $\alpha_s$ are positive. Furthermore, comparing the entries of $\mathscr{A}$ and $\mathscr{B}$, we will find that $$ a_{ijkl} = a_{jikl} = \frac{1}{2} (b_{ijkl} + b_{jikl}), \quad i,j,k,l = 1,2,3, $$ and thus $$ \mathscr{A} {\bf x}^2 {\bf y}^2 = \mathscr{B} {\bf x}^2 {\bf y}^2 = \mathscr{B} \big( {\bf x} {\bf y}^\top \big)^2. $$ Therefore, $\mathscr{A}$ is M-PD or M-PSD when $\mathscr{B}$ is S-PD or S-PSD, respectively.
Given a fourth-order tensor $\mathscr{A} \in \mathbb{E}$, we denote $$ \mathbb{T}_{\mathscr{A}} := \{ \mathscr{T}:\, t_{ijkl} = t_{jilk},\, t_{ijkl} + t_{jikl} = 2 a_{ijkl} \}. $$ We also denote the set of all fourth-order S-PSD tensors as $$ \mathbb{S} := \{ \mathscr{T}:\, t_{ijkl} = t_{jilk},\, \mathscr{T} \text{ is S-PSD} \}. $$ Note that both $\mathbb{T}_{\mathscr{A}}$ and $\mathbb{S}$ are closed convex sets, where $\mathbb{T}_{\mathscr{A}}$ is a linear subspace of the whole space of all the fourth-order three-dimensional tensor with $t_{ijkl} = t_{jilk}$ and $\mathbb{S}$ is isomorphic with the nine-by-nine symmetric PSD matrix cone. Furthermore, we have actually proved the following sufficient condition for a tensor to be M-PD or M-PSD. \begin{theorem}
Let $\mathscr{A} \in \mathbb{E}$.
If $\mathbb{T}_{\mathscr{A}} \cap \mathbb{S} \neq \emptyset$, then $\mathscr{A}$ is M-PSD;
If $\mathbb{T}_{\mathscr{A}} \cap (\mathbb{S} \setminus \partial\mathbb{S}) \neq \emptyset$, then $\mathscr{A}$ is M-PD. \end{theorem}
A method called projections onto convex sets (POCS) \cite{BauschkeBorwein96,EscalanteRaydan11} is often employed to check whether the intersection of two closed convex sets is empty or not. POCS is also known as the alternating projection algorithm. Denote ${\cal P}_1$ and ${\cal P}_2$ as the projections onto $\mathbb{T}_{\mathscr{A}}$ and $\mathbb{S}$, respectively. Then POCS is stated by $$ \left\{ \begin{array}{l}
\mathscr{B}^{(t+1)} = {\cal P}_2( \mathscr{A}^{(t)} ), \\
\mathscr{A}^{(t+1)} = {\cal P}_1( \mathscr{B}^{(t+1)} ), \end{array} \right. \quad t = 0,1,2,\dots. $$ The algorithm can be described as the following iterative scheme: \begin{equation}\label{alg_pocs} \left\{ \begin{array}{l}
\text{Eigendecomposition } {\bf A}^{(t)} = {\bf V}^{(t)} {\bf D}^{(t)} ({\bf V}^{(t)})^\top, \\
{\bf B}^{(t+1)} = {\bf V}^{(t)} {\bf D}_+^{(t)} ({\bf V}^{(t)})^\top, \\
a_{iikl}^{(t+1)} = a_{iikl} \text{ for } i,k,l = 1,2,3, \\
a_{ijkk}^{(t+1)} = a_{ijkk} \text{ for } i,j,k = 1,2,3, \\
a_{ijkl}^{(t+1)} = a_{ijkl} + \frac{1}{2} (b_{ijkl}^{(t+1)} - b_{jikl}^{(t+1)}) \text{ for } i \neq j,\ k \neq l, \end{array} \right. \quad t = 0,1,2,\dots. \end{equation} where $\mathscr{A}^{(0)} = \mathscr{A}$, ${\bf A}^{(t)}$ and ${\bf B}^{(t)}$ are the unfolding matrices of $\mathscr{A}^{(t)}$ and $\mathscr{B}^{(t)}$ respectively, and ${\bf D}_+^{(t)} = {\rm diag}\big( \max(d_{ii}^{(t)}, 0) \big)$. The convergence of the alternating projection method between two closed convex sets has been known for a long time \cite{CheneyGoldstein59}.
\begin{theorem}
Let $\mathscr{A} \in \mathbb{E}$. If $\mathbb{T}_{\mathscr{A}} \cap \mathbb{S} \neq \emptyset$, then the sequences $\{ \mathscr{A}^{(t)} \}$ and $\{ \mathscr{B}^{(t)} \}$ produced by Algorithm \eqref{alg_pocs} both converge to a point $\mathscr{A}^\ast \in \mathbb{T}_{\mathscr{A}} \cap \mathbb{S}$. \end{theorem}
Because the convergence of POCS requires the involved convex sets to be closed, Algorithm \eqref{alg_pocs} is only suitable for identifying the M-positive semidefiniteness. If we want to check the M-positive definiteness, then some modifications are needed. Denote $\mathscr{E} \in \mathbb{E}$ with $e_{iikk} = 1$ ($i,k=1,2,3$) and other entries being zero. Note that $\mathscr{E} {\bf x}^2 {\bf y}^2 = ({\bf x}^\top {\bf x}) ({\bf y}^\top {\bf y})$. Hence $\mathscr{E}$ is M-PD, which implies that $\mathscr{A}$ is M-PD if and only if $\mathscr{A} - \epsilon \mathscr{E}$ is M-PSD for some sufficiently small $\epsilon > 0$. From such observation, we can apply POCS to $\mathscr{A} - \epsilon \mathscr{E}$ with a very small $\epsilon$. If the iteration converges and both $\{ \mathscr{A}^{(t)} \}$ and $\{ \mathscr{B}^{(t)} \}$ converge to the same tensor, then we can conclude that $\mathscr{A}$ is M-PD, i.e., the strong ellipticity holds.
\section{Further sufficient conditions in particular cases}\label{sec_special}
One may conjecture that every M-PSD tensor can be modified into an S-PSD by preserving the summations of the corresponding entries. Unfortunately, this can be disproved by the following counter example given by Choi and Lam \cite{Choi75,ChoiLam77}: \begin{equation}\label{eq_choi} \begin{split}
\mathscr{A}_\gamma {\bf x}^2 {\bf y}^2 = &
\ x_1^2 y_1^2 + x_2^2 y_2^2 + x_3^2 y_3^2 \\
& - 2(x_1 x_2 y_1 y_2 + x_2 x_3 y_2 y_3 + x_3 x_1 y_3 y_1) \\
& + \gamma (x_1^2 y_2^2 + x_2^2 y_3^2 + x_3^2 y_1^2), \end{split} \end{equation} where $\gamma \geq 1$. When we apply POCS to tensor $\mathscr{A}_\gamma$, the limitations of $\{ \mathscr{A}^{(t)} \}$ and $\{ \mathscr{B}^{(t)} \}$ differ from each other, which implies that the intersection of $\mathbb{T}_{\mathscr{A}_\gamma}$ and $\mathbb{S}$ is empty. Furthermore, Choi and Lam \cite{ChoiLam77} also proved that $\{ \lambda \mathscr{A}_1:\, \lambda \geq 0 \}$ is an extremal ray of the M-PSD cone. Therefore, we desire to extend our sufficient condition of the SE-condition to include more situations.
Given a fourth-order tensor $\mathscr{A} \in \mathbb{E}$, we can always calculate the eigendecomposition of its unfolding matrix $$ {\bf A} = \sum_{s=1}^r \alpha_s {\bf u}_s {\bf u}_s^\top, $$ thus matrix $\mathscr{A} {\bf y}^2$ can be correspondingly decomposed into \begin{equation}\label{eq_grad}
\mathscr{A} {\bf y}^2 = \sum_{s = 1}^r \alpha_s ({\bf U}_s {\bf y}) ({\bf U}_s {\bf y})^\top, \end{equation} where ${\bf u}_s$ is the vectorization of ${\bf U}_s$ ($s = 1,2,\dots,r$). Note that the coefficients $\alpha_s$ are not necessarily positive, otherwise this is exactly the case discussed in the previous section. The eigendecomposition of the unfolding matrix guarantees the existence of such decompositions as \eqref{eq_grad}. Actually, the orthogonality of the vectorizations of ${\bf U}_s$ is not required in the following discussion, and the number of terms, i.e., $r$, may also be larger than nine. Without loss of generality, we assume that $\alpha_1,\dots,\alpha_q > 0$ and $\alpha_{q+1},\dots,\alpha_r < 0$.
{\bf Case 1:} $q=3$ and ${\bf U}_s$ ($s=1,2,3$) are rank-one, i.e., $$ {\bf U}_s = {\bf v}_s {\bf w}_s^\top, \quad s = 1,2,3, $$ and furthermore ${\bf V} = \big[ {\bf v}_1, {\bf v}_2, {\bf v}_3 \big]$ and ${\bf W} = \big[ {\bf w}_1, {\bf w}_2, {\bf w}_3 \big]$ are nonsingular. Then the summation of the first three PSD terms is written into $$ \sum_{s = 1}^3 \alpha_s ({\bf U}_s {\bf y}) ({\bf U}_s {\bf y})^\top = {\bf V} \begin{bmatrix}
\alpha_1 ({\bf w}_1^\top {\bf y})^2 \\
& \alpha_2 ({\bf w}_2^\top {\bf y})^2 \\
& & \alpha_3 ({\bf w}_3^\top {\bf y})^2 \end{bmatrix} {\bf V}^\top. $$ Denote $\widehat{\bf U}_s := {\bf V}^{-1} {\bf U}_s$ and $\widehat{\bf U}_s = [\widehat{\bf u}_{s1},\widehat{\bf u}_{s2},\widehat{\bf u}_{s3}]^\top$ ($s = 4,5,\dots,r$). If $\mathscr{A} {\bf y}^2$ is PSD for any nonzero ${\bf y} \in \mathbb{R}^3$, then $\sum_{s = 1}^4 \alpha_s ({\bf U}_s {\bf y}) ({\bf U}_s {\bf y})^\top$ is also PSD since $\alpha_5,\dots,\alpha_r < 0$. That is, $$ \begin{bmatrix}
\alpha_1 ({\bf w}_1^\top {\bf y})^2 \\
& \alpha_2 ({\bf w}_2^\top {\bf y})^2 \\
& & \alpha_3 ({\bf w}_3^\top {\bf y})^2 \end{bmatrix} + \alpha_4 \begin{bmatrix}
\widehat{\bf u}_{41}^\top {\bf y} \\ \widehat{\bf u}_{42}^\top {\bf y} \\ \widehat{\bf u}_{43}^\top {\bf y} \end{bmatrix} \begin{bmatrix}
\widehat{\bf u}_{41}^\top {\bf y} & \widehat{\bf u}_{42}^\top {\bf y} & \widehat{\bf u}_{43}^\top {\bf y} \end{bmatrix} \succeq 0. $$ When ${\bf y}$ is selected such that ${\bf w}_l^\top {\bf y} = 0$, it must hold that $\widehat{\bf u}_{sl}^\top {\bf y} = 0$ to guarantee the positive definiteness, which implies that $\widehat{\bf u}_{sl} = \sigma_{sl} {\bf w}_l$ ($l=1,2,3$). Similarly, we can prove that in this case if $\mathscr{A}$ is M-PSD then \begin{equation}\label{eq_case1_cond1}
{\bf U}_s = {\bf V} {\bm \Sigma}_s {\bf W}^\top, \quad s = 4,5,\dots,r, \end{equation} where ${\bm \Sigma}_s = {\rm diag}(\sigma_{s1},\sigma_{s2},\sigma_{s3})$. Denote ${\bf D} = {\rm diag}({\bf w}_1^\top {\bf y},{\bf w}_2^\top {\bf y},{\bf w}_3^\top {\bf y})$ and ${\bm \sigma}_s = [\sigma_{s1},\sigma_{s2},\sigma_{s3}]^\top$ ($s = 4,5,\dots,r$). Then $\mathscr{A} {\bf y}^2 = {\bf D} {\bf C} {\bf D}^\top$, where \begin{equation} {\bf C} = {\rm diag}(\alpha_1, \alpha_2, \alpha_3) + \sum_{s=4}^r \alpha_s {\bm \sigma}_s {\bm \sigma}_s^\top. \end{equation} Therefore, $\mathscr{A}$ in this case is M-PSD if and only if ${\bf U}_s$ ($s = 4,5,\dots,r$) are in the form \eqref{eq_case1_cond1} and the matrix ${\bf C}$ is PSD.
\begin{theorem}
Let $\mathscr{A} \in \mathbb{E}$ be given by \eqref{eq_grad} with (i) $\alpha_1,\alpha_2,\alpha_3 > 0$, $\alpha_4,\dots,\alpha_r < 0$, (ii) ${\bf U}_s = {\bf v}_s {\bf w}_s^\top$ ($s=1,2,3$), and (iii) ${\bf V} = \big[ {\bf v}_1, {\bf v}_2, {\bf v}_3 \big]$, ${\bf W} = \big[ {\bf w}_1, {\bf w}_2, {\bf w}_3 \big]$ are nonsingular. Then $\mathscr{A}$ is M-PSD if and only if (i) ${\bf U}_s = {\bf V} {\bm \Sigma}_s {\bf W}^\top$ with ${\bm \Sigma}_s = {\rm diag}(\sigma_{s1},\sigma_{s2},\sigma_{s3})$ ($s = 4,\dots,r$), and (ii) the matrix ${\rm diag}(\alpha_1, \alpha_2, \alpha_3) + \sum_{s=4}^r \alpha_s {\bm \sigma}_s {\bm \sigma}_s^\top$ is PSD. \end{theorem}
Nevertheless, we can show that the tensors of this type also satisfy the sufficient condition established in Section \ref{sec_convex}. We can calculate the eigendecomposition of ${\bf C} = \sum_{s=1}^{\widetilde{r}} \widetilde{\alpha}_s \widetilde{\bm \sigma}_s \widetilde{\bm \sigma}_s^\top$. Then all the coefficients $\widetilde{\alpha}_s$ are positive since ${\bf C}$ is PSD. Denote $\widetilde{\bf U}_s := {\bf V} \widetilde{\bm \Sigma}_s {\bf W}^\top$, where $\widetilde{\bf \Sigma}_s = {\rm diag}(\widetilde{\sigma}_{s1},\widetilde{\sigma}_{s2},\widetilde{\sigma}_{s3})$ ($s = 1,2,\dots,\widetilde{r}$). We can easily verify that $\mathscr{A} {\bf y}^2 = \sum_{s = 1}^{\widetilde{r}} \widetilde{\alpha}_s (\widetilde{\bf U}_s {\bf y}) (\widetilde{\bf U}_s {\bf y})^\top$ with $\widetilde{\alpha}_s > 0$ ($s = 1,2,\dots,\widetilde{r}$).
{\bf Case 2}: $r=7$, $q=6$, and ${\bf U}_s$ ($s = 1,2,\dots,6$) are rank-one with $$ {\bf U}_s = \left\{ \begin{array}{ll}
{\bf v}_s {\bf w}_s^\top, & s = 1,2,3, \\
{\bf v}_{s-3} {\bf w}_s^\top, & s = 4,5,6, \end{array} \right. $$ and ${\bf V} = [{\bf v}_1, {\bf v}_2, {\bf v}_3]$, ${\bf W} = [{\bf w}_1, {\bf w}_2, {\bf w}_3]$, $\widetilde{\bf W} = [{\bf w}_4, {\bf w}_5, {\bf w}_6]$ are nonsingular. It is reasonable to further assume that ${\bf w}_s$ and ${\bf w}_{s+3}$ are linearly independent ($s=1,2,3$). Otherwise we can simply collect them into one term. Similarly to our analysis of Case 1, we can conclude that $\mathscr{A}$ in Case 2 is M-PSD if and only if $$ {\bf U}_7 = {\bf V} {\bm \Sigma}_7 {\bf W}^\top + {\bf V} \widetilde{\bm \Sigma}_7 \widetilde{\bf W}^\top, $$ where ${\bm \Sigma}_7 = {\rm diag}(\sigma_{1},\sigma_{2},\sigma_{3})$ and $\widetilde{\bm \Sigma}_7 = {\rm diag}(\sigma_{4},\sigma_{5},\sigma_{6})$, and \begin{multline}\label{eq_case2_cond1} \begin{bmatrix}
\alpha_1 ({\bf w}_1^\top {\bf y})^2 + \alpha_4 ({\bf w}_4^\top {\bf y})^2 \\
& \alpha_2 ({\bf w}_2^\top {\bf y})^2 + \alpha_5 ({\bf w}_5^\top {\bf y})^2 \\
& & \alpha_3 ({\bf w}_3^\top {\bf y})^2 + \alpha_6 ({\bf w}_6^\top {\bf y})^2 \end{bmatrix} \\ + \alpha_7 \begin{bmatrix}
(\sigma_{1} {\bf w}_{1} + \sigma_{4} {\bf w}_4)^\top {\bf y} \\ (\sigma_{2} {\bf w}_{2} + \sigma_{5} {\bf w}_5)^\top {\bf y} \\ (\sigma_{3} {\bf w}_{3} + \sigma_{6} {\bf w}_6)^\top {\bf y} \end{bmatrix} \cdot \begin{bmatrix}
(\sigma_{1} {\bf w}_{1} + \sigma_{4} {\bf w}_4)^\top {\bf y} \\ (\sigma_{2} {\bf w}_{2} + \sigma_{5} {\bf w}_5)^\top {\bf y} \\ (\sigma_{3} {\bf w}_{3} + \sigma_{6} {\bf w}_6)^\top {\bf y} \end{bmatrix}^\top \succeq 0. \end{multline} Denote \begin{equation}\label{eq_case2_fun}
\eta({\bf y}) := \sum_{s=1}^3 \frac{(\sigma_s {\bf w}_s^\top {\bf y} + \sigma_{s+3} {\bf w}_{s+3}^\top {\bf y})^2}{\alpha_s ({\bf w}_s^\top {\bf y})^2 + \alpha_{s+3} ({\bf w}_{s+3}^\top {\bf y})^2}. \end{equation} Then the matrix in \eqref{eq_case2_cond1} is PSD for all nonzero ${\bf y}$ if and only if \begin{equation}\label{eq_case2_cond2}
\sup \bigg\{ \eta({\bf y}):\, {\bf y} \notin \bigcup_{s=1,2,3} \big( {\bf w}_s^\bot \cap {\bf w}_{s+3}^\bot \big) \bigg\} \leq \frac{1}{-\alpha_7}, \end{equation} where ${\bf w}_s^\bot$ denotes the orthogonal complement subspace to ${\rm span}({\bf w}_s)$. Furthermore, the elasticity tensor in both Case 1 and Case 2 cannot be M-PD, which can be seen by taking a vector ${\bf y}$ in $\bigcup_{s=1,2,3} \big( {\bf w}_s^\bot \cap {\bf w}_{s+3}^\bot)$ and thus $\mathscr{A} {\bf y}^2$ has a nonempty null space.
\begin{theorem}
Let $\mathscr{A} \in \mathbb{E}$ be given by \eqref{eq_grad} with (i) $\alpha_1,\dots,\alpha_6 > 0$, $\alpha_7 < 0$, (ii) ${\bf U}_s = {\bf v}_s {\bf w}_s^\top$ ($s=1,2,\dots,6$), ${\bf v}_s = {\bf v}_{s+3}$ ($s=1,2,3$), (iii) ${\bf V} = \big[ {\bf v}_1, {\bf v}_2, {\bf v}_3 \big]$, ${\bf W} = \big[ {\bf w}_1, {\bf w}_2, {\bf w}_3 \big]$, $\widetilde{\bf W} = \big[ {\bf w}_4, {\bf w}_5, {\bf w}_6 \big]$ are nonsingular, and (iv) ${\bf w}_s$ and ${\bf w}_{s+3}$ are linearly independent ($s=1,2,3$). Then $\mathscr{A}$ is M-PSD if and only if (i) ${\bf U}_7 = {\bf V} {\bm \Sigma}_7 {\bf W}^\top + {\bf V} \widetilde{\bm \Sigma}_7 \widetilde{\bf W}^\top$ with ${\bm \Sigma}_7 = {\rm diag}(\sigma_{1},\sigma_{2},\sigma_{3})$ and $\widetilde{\bm \Sigma}_7 = {\rm diag}(\sigma_{4},\sigma_{5},\sigma_{6})$, and (ii) $\sup \Big\{ \eta({\bf y}):\, {\bf y} \notin \bigcup_{s=1,2,3} \big( {\bf w}_s^\bot \cap {\bf w}_{s+3}^\bot \big) \Big\} \leq \frac{1}{-\alpha_7}$, where $\eta(\cdot)$ is defined by \eqref{eq_case2_fun}. \end{theorem}
Generally speaking, the supreme in the left-hand side of \eqref{eq_case2_cond2} is not easy to obtain. In some particular case, however, we can calculate the exact supreme of $\eta({\bf y})$ in the domain of definition. For instance, recall the counter example given in \eqref{eq_choi}, where the matrix $\mathscr{A}_1 {\bf y}^2$ can be decomposed into \[ \begin{split} \mathscr{A}_1 {\bf y}^2 &= \begin{bmatrix}
y_1^2 + y_2^2 & -y_1 y_2 & -y_3 y_1 \\
-y_1 y_2 & y_2^2 + y_3^2 & -y_2 y_3 \\
-y_3 y_1 & -y_2 y_3 & y_3^2 + y_1^2 \end{bmatrix} \\ &= \begin{bmatrix}
2 y_1^2 + y_2^2 \\
& 2 y_2^2 + y_3^2 \\
& & 2 y_3^2 + y_1^2 \end{bmatrix} - \begin{bmatrix}
y_1 \\ y_2 \\ y_3 \end{bmatrix} \cdot \begin{bmatrix}
y_1 & y_2 & y_3 \end{bmatrix}. \end{split} \] In this example, the coefficient $\alpha_7 = -1$ and the function $$ \eta({\bf y}) = \frac{y_1^2}{2 y_1^2 + y_2^2} + \frac{y_2^2}{2 y_2^2 + y_3^2} + \frac{y_3^2}{2 y_3^2 + y_1^2}. $$ We can verify that $\eta({\bf y}) \leq 1$ and the equality holds if $y_1 = y_2 = y_3 \neq 0$, which implies $\sup \big\{ \eta({\bf y}):\, (y_1,y_2)\neq(0,0), (y_2,y_3)\neq(0,0), (y_3,y_1)\neq(0,0) \big\} = 1 = -1/\alpha_7$. Therefore $\mathscr{A}_1$ is M-PSD.
{\bf Case 3}: $r=10$, $q=9$, and ${\bf U}_s$ ($s = 1,2,\dots,9$) are rank-one with $$ {\bf U}_s = \left\{ \begin{array}{ll}
{\bf v}_s {\bf w}_s^\top, & s = 1,2,3, \\
{\bf v}_{s-3} {\bf w}_s^\top, & s = 4,5,6, \\
{\bf v}_{s-6} {\bf w}_s^\top, & s = 7,8,9, \end{array} \right. $$ and ${\bf V} = [{\bf v}_1, {\bf v}_2, {\bf v}_3]$, ${\bf W} = [{\bf w}_1, {\bf w}_2, {\bf w}_3]$, $\widetilde{\bf W} = [{\bf w}_4, {\bf w}_5, {\bf w}_6]$, $\widehat{\bf W} = [{\bf w}_7, {\bf w}_8, {\bf w}_9]$ are nonsingular. It is also reasonable to further assume that $\{ {\bf w}_s, {\bf w}_{s+3}, {\bf w}_{s+6} \}$ are linearly independent ($s=1,2,3$). Similarly to Cases 1 and 2, we can conclude that $\mathscr{A}$ in Case 3 is M-PSD if and only if $$ {\bf U}_{10} = {\bf V} {\bm \Sigma}_{10} {\bf W}^\top + {\bf V} \widetilde{\bm \Sigma}_{10} \widetilde{\bf W}^\top + {\bf V} \widehat{\bm \Sigma}_{10} \widehat{\bf W}^\top, $$ where ${\bm \Sigma}_{10} = {\rm diag}(\sigma_{1},\sigma_{2},\sigma_{3})$, $\widetilde{\bm \Sigma}_{10} = {\rm diag}(\sigma_{4},\sigma_{5},\sigma_{6})$, $\widehat{\bm \Sigma}_{10} = {\rm diag}(\sigma_{7},\sigma_{8},\sigma_{9})$, and \begin{multline}\label{eq_case3_cond1} \begin{bmatrix}
\sum\limits_{h=0,1,2} \alpha_{1+3h} ({\bf w}_{1+3h}^\top {\bf y})^2 \\
& \sum\limits_{h=0,1,2} \alpha_{2+3h} ({\bf w}_{2+3h}^\top {\bf y})^2 \\
& & \sum\limits_{h=0,1,2} \alpha_{3+3h} ({\bf w}_{3+3h}^\top {\bf y})^2 \end{bmatrix} \\ + \alpha_{10} \begin{bmatrix}
(\sigma_{1} {\bf w}_1 + \sigma_{4} {\bf w}_4 + \sigma_{7} {\bf w}_7)^\top {\bf y} \\ (\sigma_{2} {\bf w}_2 + \sigma_{5} {\bf w}_5 + \sigma_{8} {\bf w}_8)^\top {\bf y} \\ (\sigma_{3} {\bf w}_3 + \sigma_{6} {\bf w}_6 + \sigma_{9} {\bf w}_9)^\top {\bf y} \end{bmatrix} \cdot \begin{bmatrix}
(\sigma_{1} {\bf w}_1 + \sigma_{4} {\bf w}_4 + \sigma_{7} {\bf w}_7)^\top {\bf y} \\ (\sigma_{2} {\bf w}_2 + \sigma_{5} {\bf w}_5 + \sigma_{8} {\bf w}_8)^\top {\bf y} \\ (\sigma_{3} {\bf w}_3 + \sigma_{6} {\bf w}_6 + \sigma_{9} {\bf w}_9)^\top {\bf y} \end{bmatrix}^\top \succeq 0. \end{multline} Denote \begin{equation}\label{eq_case3_fun}
\eta({\bf y}) := \sum_{s=1}^3 \frac{(\sigma_s {\bf w}_s^\top {\bf y} + \sigma_{s+3} {\bf w}_{s+3}^\top {\bf y} + \sigma_{s+6} {\bf w}_{s+6}^\top {\bf y})^2}{\alpha_s ({\bf w}_s^\top {\bf y})^2 + \alpha_{s+3} ({\bf w}_{s+3}^\top {\bf y})^2 + \alpha_{s+6} ({\bf w}_{s+6}^\top {\bf y})^2}. \end{equation} According to the assumptions, $\eta({\bf y})$ is well-defined for any nonzero ${\bf y}$. Thus the matrix in \eqref{eq_case3_cond1} is PSD for all nonzero ${\bf y}$ if and only if $$ \max \big\{ \eta({\bf y}):\, {\bf y}^\top {\bf y} = 1 \big\} \leq \frac{1}{-\alpha_{10}}. $$ Furthermore, Case 3 is of interest since ${\bf V} = [{\bf e}_1,{\bf e}_2,{\bf e}_3]$, ${\bf W} = [{\bf e}_1,{\bf e}_2,{\bf e}_3]$, $\widetilde{\bf W} = [{\bf e}_2,{\bf e}_3,{\bf e}_1]$, $\widehat{\bf W} = [{\bf e}_3,{\bf e}_1,{\bf e}_2]$ are exactly the case for isotropic and some particular anisotropic linearly elastic materials \cite{ZubovRudev16}.
\begin{theorem}
Let $\mathscr{A} \in \mathbb{E}$ be given by \eqref{eq_grad} with (i) $\alpha_1,\dots,\alpha_9 > 0$, $\alpha_{10} < 0$, (ii) ${\bf U}_s = {\bf v}_s {\bf w}_s^\top$ ($s=1,2,\dots,9$), ${\bf v}_s = {\bf v}_{s+3} = {\bf v}_{s+6}$ ($s=1,2,3$), (iii) ${\bf V} = \big[ {\bf v}_1, {\bf v}_2, {\bf v}_3 \big]$, ${\bf W} = \big[ {\bf w}_1, {\bf w}_2, {\bf w}_3 \big]$, $\widetilde{\bf W} = \big[ {\bf w}_4, {\bf w}_5, {\bf w}_6 \big]$, $\widehat{\bf W} = \big[ {\bf w}_7, {\bf w}_8, {\bf w}_9 \big]$ are nonsingular, and (iv) ${\bf w}_s, {\bf w}_{s+3}, {\bf w}_{s+6}$ are linearly independent ($s=1,2,3$). Then $\mathscr{A}$ is M-PSD if and only if (i) ${\bf U}_{10} = {\bf V} {\bm \Sigma}_{10} {\bf W}^\top + {\bf V} \widetilde{\bm \Sigma}_{10} \widetilde{\bf W}^\top + {\bf V} \widehat{\bm \Sigma}_{10} \widehat{\bf W}^\top$ with ${\bm \Sigma}_{10} = {\rm diag}(\sigma_{1},\sigma_{2},\sigma_{3})$, $\widetilde{\bm \Sigma}_{10} = {\rm diag}(\sigma_{4},\sigma_{5},\sigma_{6})$, $\widehat{\bm \Sigma}_{10} = {\rm diag}(\sigma_{7},\sigma_{8},\sigma_{9})$, and (ii) $\max \big\{ \eta({\bf y}):\, {\bf y}^\top {\bf y} = 1 \big\} \leq \frac{1}{-\alpha_{10}}$, where $\eta(\cdot)$ is defined by \eqref{eq_case3_fun}. Furthermore, $\mathscr{A}$ is M-PD when the strict inequality holds in the last condition. \end{theorem}
\section{Conclusions}\label{sec_conclusion}
We have established several sufficient conditions for the strong ellipticity (M-positive definiteness) of general elasticity tensors. S-positive definiteness has been already known as an easily checkable sufficient conditions for M-positive definiteness. However, the range of S-positive (semi)definiteness is too narrow. Thus our first sufficient condition extends the coverage of S-PSD tensors, which states that $\mathscr{A}$ is M-PSD or M-PD if it can be modified into an S-PSD or S-PD tensor $\mathscr{B}$ respectively by preserving $b_{ijkl} = b_{jilk}$ and $b_{ijkl} + b_{jikl} = a_{ijkl} + a_{jikl}$. To check whether a tensor satisfies this condition, we employ an alternating projection method called POCS and verify its convergence.
Next, we have considered three particular cases and provided the necessary and sufficient conditions of the strong ellipticity. Actually, we can understand these three cases in the following way. Any tensor $\mathscr{A} \in \mathbb{E}$ can be factorized into two parts $\mathscr{A} = \mathscr{A}_1 - \mathscr{A}_2$, where $\mathscr{A}_1$ and $\mathscr{A}_2$ satisfy that $\mathbb{T}_{\mathscr{A}_1} \cap \mathbb{S} \neq \emptyset$ and $\mathbb{T}_{\mathscr{A}_2} \cap \mathbb{S} \neq \emptyset$. We further assume that $\mathscr{A}_1 = \sum_{s=1}^9 \alpha_s {\bf v}_s \circ {\bf v}_s \circ {\bf w}_s \circ {\bf w}_s$ ($\alpha_s \geq 0$), where $\circ$ stands for the outer product of tensors. It is interesting to note that such $\mathscr{A}_1$ is in the dual cone of the M-PSD cone. We have established necessary and sufficient conditions for the strong ellipticity for the cases in which ${\bf v}_s$ and ${\bf w}_s$ satisfy some dependence conditions and the rank of $\mathscr{A}_2 {\bf y}^2$ is no greater than one for all ${\bf y} \in \mathbb{R}^3$.
Our further investigation will be threefold. The first one is to find out whether there are other M-PSD or M-PD elasticity tensors beyond our sufficient conditions. Alternatively, we desire to know whether every M-PSD elasticity tensor can be represented by a convex combination of several tensors satisfying the conditions presented in this paper. The next part is the characterization of the M-PSD cone. We already know that $\{ \alpha \mathscr{A}:\, \alpha \geq 0 \}$ is an extremal ray of the M-PSD cone if there is a tensor in $\mathbb{T}_{\mathscr{A}} \cap \mathbb{S}$ whose unfolding matrix is rank-one. We would like to find more extremal rays of this convex cone. Of course, if all the extremal rays were discovered, then the M-PSD cone could be characterized thoroughly. Our third target in the future is to generalize these results to higher order elasticity tensors.
\end{document} | arXiv |
Title: Constraints on primordial magnetic fields from the optical depth of the cosmic microwave background
Authors: Kerstin E. Kunze, Eiichiro Komatsu
(Submitted on 31 Dec 2014 (v1), last revised 24 May 2015 (this version, v2))
Abstract: Damping of magnetic fields via ambipolar diffusion and decay of magnetohydrodynamical (MHD) turbulence in the post decoupling era heats the intergalactic medium (IGM). Delayed recombination of hydrogen atoms in the IGM yields an optical depth to scattering of the cosmic microwave background (CMB). The optical depth generated at $z\gg 10$ does not affect the "reionization bump" of the CMB polarization power spectrum at low multipoles, but affects the temperature and polarization power spectra at high multipoles. Writing the present-day energy density of fields smoothed over the damping scale at the decoupling epoch as $\rho_{B,0}=B_{0}^2/2$, we constrain $B_0$ as a function of the spectral index, $n_B$. Using the Planck 2013 likelihood code that uses the Planck temperature and lensing data together with the WMAP 9-year polarization data, we find the 95% upper bounds of $B_0<0.63$, 0.39, and 0.18~nG for $n_B=-2.9$, $-2.5$, and $-1.5$, respectively. For these spectral indices, the optical depth is dominated by dissipation of the decaying MHD turbulence that occurs shortly after the decoupling epoch. Our limits are stronger than the previous limits ignoring the effects of the fields on ionization history. Inverse Compton scattering of CMB photons off electrons in the heated IGM distorts the thermal spectrum of CMB. Our limits on $B_0$ imply that the $y$-type distortion from dissipation of fields in the post decoupling era should be smaller than $10^{-9}$, $4\times10^{-9}$, and $10^{-9}$, respectively.
Comments: 14 pages, 30 figures, calculations revised and updated, accepted for publication in JCAP
Subjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO)
From: Kerstin E. Kunze [view email]
[v1] Wed, 31 Dec 2014 14:26:18 GMT (1978kb)
[v2] Sun, 24 May 2015 15:05:21 GMT (1900kb) | CommonCrawl |
Corporate Finance & Accounting Financial Ratios
Net Present Value (NPV)
By Jason Fernando
Reviewed By Julius Mansa
What is Net Present Value (NPV)?
Net present value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time. NPV is used in capital budgeting and investment planning to analyze the profitability of a projected investment or project.
The following formula is used to calculate NPV:
?NPV=∑t=1nRt(1+i)twhere:Rt=Net?cash?inflow-outflows?during?a?single?period?ti=Discount?rate?or?return?that?could?be?earned?inalternative?investmentst=Number?of?timer?periods\begin{aligned} &NPV = \sum_{t = 1}^n \frac { R_t }{ (1 + i)^t } \\ &\textbf{where:} \\ &R_t=\text{Net cash inflow-outflows during a single period }t \\ &i=\text{Discount rate or return that could be earned in} \\ &\text{alternative investments} \\ &t=\text{Number of timer periods} \\ \end{aligned}?NPV=t=1∑n?(1+i)tRt??where:Rt?=Net?cash?inflow-outflows?during?a?single?period?ti=Discount?rate?or?return?that?could?be?earned?inalternative?investmentst=Number?of?timer?periods??
If you are unfamiliar with summation notation – here is an easier way to remember the concept of NPV:
?NPV=TVECF?TVICwhere:TVECF=Today's?value?of?the?expected?cash?flowsTVIC=Today's?value?of?invested?cash\begin{aligned} &\textit{NPV} = \text{TVECF} - \text{TVIC} \\ &\textbf{where:} \\ &\text{TVECF} = \text{Today's value of the expected cash flows} \\ &\text{TVIC} = \text{Today's value of invested cash} \\ \end{aligned}?NPV=TVECF?TVICwhere:TVECF=Today's?value?of?the?expected?cash?flowsTVIC=Today's?value?of?invested?cash??
A positive net present value indicates that the projected earnings generated by a project or investment - in present dollars - exceeds the anticipated costs, also in present dollars. It is assumed that an investment with a positive NPV will be?profitable,?and an investment with a negative NPV will result in a net loss. This concept is the basis for the?Net Present Value Rule, which dictates that only investments with positive NPV values should be considered.
Apart from the formula itself, net present value can be calculated using tables, spreadsheets, or calculators.
Understanding Net Present Value
How to Calculate Net Present Value (NPV)
Money in the present is worth more than the same amount in the future due to inflation and to earnings from alternative investments that could be made during the intervening time. In other words, a dollar earned in the future won't be worth as much as one earned in the present.?The discount rate element of the NPV formula is a way to account for this.
For example, assume that an investor could choose a $100 payment today or in a year. A rational investor would not be willing to postpone payment. However, what if an investor could choose to receive $100 today or $105 in a year? If the payer was reliable, that extra 5% may be worth the wait, but only if there wasn't anything else the investors could do with the $100 that would earn more than 5%.
An investor might be willing to wait a year to earn an extra 5%, but that may not be acceptable for all investors. In this case, the 5% is the discount rate which will vary depending on the investor. If an investor knew they could earn 8% from a relatively safe investment over the next year, they would not be willing to postpone payment for 5%. In this case, the investor's discount rate is 8%.
A company may determine the discount rate using the expected return of other projects with a similar level of risk or the cost of borrowing money needed to finance the project. For example, a company may avoid a project that is expected to return 10% per year if it costs 12% to finance the project or an alternative project is expected to return 14% per year.
Imagine a company can invest in equipment that will cost $1,000,000 and is expected to generate $25,000 a month in revenue for five years. The company has the capital available for the equipment and could alternatively invest it in the stock market for an expected return of 8% per year. The managers feel that buying the equipment or investing in the stock market are similar risks.
Step One: NPV of the Initial Investment
Because the equipment is paid for up front, this is the first cash flow included in the calculation. There is no elapsed time that needs to be accounted for so today's outflow of $1,000,000 doesn't need to be discounted.
Identify the number of periods (t)
The equipment is expected to generate monthly cash flow and last for five years, which means there will be 60 cash flows and 60 periods included in the calculation.
Identify the discount rate (i)
The alternative investment is expected to pay 8% per year. However, because the equipment generates a monthly stream of cash flows, the annual discount rate needs to be turned into a periodic or monthly rate. Using the following formula, we find that the periodic rate is 0.64%.
?Periodic?Rate=((1+0.08)112)?1=0.64%\text{Periodic Rate} = (( 1 + 0.08)^{\frac{1}{12}}) - 1 = 0.64\%Periodic?Rate=((1+0.08)121?)?1=0.64%?
Step Two: NPV of Future Cash Flows
Assume the monthly cash flows are earned at the end of the month, with the first payment arriving exactly one month after the equipment has been purchased. This is a future payment, so it needs to be adjusted for the time value of money. An investor can perform this calculation easily with a spreadsheet or calculator. To illustrate the concept, the first five payments are displayed in the table below.
Image by Sabrina Jiang ? Investopedia?2020
The full calculation of the present value is equal to the present value of all 60 future cash flows, minus the $1,000,000 investment. The calculation could be more complicated if the equipment was expected to have any value left at the end of its life, but, in this example, it is assumed to be worthless.
?NPV=?$1,000,000+∑t=16025,00060(1+0.0064)60NPV = -\$1,000,000 + \sum_{t = 1}^{60} \frac{25,000_{60}}{(1 + 0.0064)^{60}}NPV=?$1,000,000+∑t=160?(1+0.0064)6025,00060???
That formula can be simplified to the following calculation:
?NPV=?$1,000,000+$1,242,322.82=$242,322.82NPV = -\$1,000,000 + \$1,242,322.82 = \$242,322.82NPV=?$1,000,000+$1,242,322.82=$242,322.82?
In this case, the NPV is positive; the equipment should be purchased. If the present value of these cash flows had been negative because the discount rate was larger, or the net cash flows were smaller, the investment should have been avoided.
Net Present Value Drawbacks and Alternatives
Gauging an investment's profitability with NPV relies heavily on assumptions and estimates, so there can be substantial room for error. Estimated factors include investment costs, discount rate, and projected returns. A project may often require unforeseen expenditures to get off the ground or may require additional expenditures at the project's end.
Payback period,?or "payback method," is a simpler alternative to NPV. The payback method calculates how long it will take for the original investment to be repaid. A drawback is that this method?fails to account for the time value of money. For this reason, payback periods calculated for longer investments have a greater potential for inaccuracy.
Moreover, the payback period is strictly limited to the amount of time required to earn back initial investment costs. It is possible that the investment's rate of return could experience sharp movements. Comparisons using payback periods do not account for the long-term profitability of alternative investments.
Net Present Value vs. Internal Rate of Return
Internal rate of return (IRR) is very similar to NPV except that the discount rate is the rate that reduces the NPV of an investment to zero. This method is used to compare projects with different lifespans or amount of required capital.
For example, IRR could be used to compare the anticipated profitability of a three-year project that requires a $50,000 investment with that of a 10-year project that requires a $200,000 investment. Although the IRR is useful, it is usually considered inferior to NPV because it makes too many assumptions about reinvestment risk and capital allocation.
Net present value (NPV) is the calculation used to find today's value of a future stream of payments. It accounts for the time value of money and can be used to compare investment alternatives that are similar. The NPV relies on a discount rate of return that may be derived from the cost of the capital required to make the investment, and any project or investment with a negative NPV should be avoided. An important drawback of using an NPV analysis is that it makes assumptions about future events that may not be reliable.
What does the net present value mean?
Net present value (NPV) is a financial metric that seeks to capture the total value of a potential investment opportunity. The idea behind NPV is to project all of the future cash inflows and outflows associated with an investment, discount all those future cashflows to the present day, and then add them together. The resulting number after adding all the positive and negative cashflows together is the investment's NPV. A positive NPV means that, after accounting for the time value of money, you will make money if you proceed with the investment.
What is the difference between NPV and IRR?
NPV and IRR are closely related concepts, in that the IRR of an investment is the discount rate that would cause that investment to have an NPV of zero. Another way of thinking about this is that NPV and IRR are trying to answer two separate but related questions. For NPV, the question is, "What is the total amount of money I will make if I proceed with this investment, after taking into account the time value of money?" For IRR, the question is, "If I proceed with this investment, what would be the equivalent annual rate of return that I would receive?"
What is a good NPV?
In theory, an NPV is "good" if it is greater than zero. After all, the NPV calculation already takes into account factors such as the investor's cost of capital, opportunity cost, and risk tolerance through the discount rate. And the future cashflows of the project, together with the time value of money, are also captured. Therefore, even an NPV of $1 should theoretically qualify as "good". In practice, however, many investors will insist on certain NPV thresholds, such as $10,000 or greater, in order to provide themselves with an additional margin of safety.
Internal Rate of Return (IRR)
The internal rate of return (IRR) is a metric used in capital budgeting to estimate the return of potential investments.
Discounted cash flow (DCF) is a valuation method used to estimate the attractiveness of an investment opportunity.
Net Present Value Rule Definition
The net present value rule (NPV) states that an investment should be accepted if the NPV is greater than zero, and it should be rejected otherwise.
Rate of Return (RoR)
A rate of return is the gain or loss of an investment over a specified period of time, expressed as a percentage of the investment's cost.
How to Use the Profitability Index (PI) Rule
The profitability index (PI) rule is a calculation of a venture's profit potential, used to decide whether or not to proceed.
What Is Capital Budgeting?
Capital budgeting is a process a business uses to evaluate potential major projects or investments. It allows a comparison of estimated costs versus rewards.
What Is the Formula for Calculating Net Present Value (NPV)?
What is the formula for calculating net present value (NPV) in Excel?
What's the Difference Between ROI and IRR?
An Introduction to Capital Budgeting
Disadvantages of Net Present Value (NPV) for Investments
How do you calculate IRR in Excel? | CommonCrawl |
Central limit theorem for sample medians
If I calculate the median of a sufficiently large number of observations drawn from the same distribution, does the central limit theorem state that the distribution of medians will approximate a normal distribution? My understanding is that this is true with the means of a large number of samples, but is it also true with medians?
If not, what is the underlying distribution of sample medians?
normal-distribution mathematical-statistics sampling median central-limit-theorem
Nick Stauner
$\begingroup$ You need some regularity conditions so that the median will have a normal distribution under rescaling in the limit. To see what can go wrong, consider any distribution over a finite number of points, say, $X$ uniform on $\{-1,0,1\}$. $\endgroup$ – cardinal Dec 4 '12 at 18:47
$\begingroup$ Regarding regularity conditions: If the underlying distribution has a density that is differentiable at the (true) median, then the sample median will have an asymptotic normal distribution with a variance that depends on said derivative. This holds more generally for arbitrary quantiles. $\endgroup$ – cardinal Dec 4 '12 at 18:54
$\begingroup$ @cardinal I believe you need additional conditions: when the density is second differentiable, is equal to zero at the median, and has zero first derivative there, then the asymptotic distribution of the sample median will be bimodal. $\endgroup$ – whuber♦ Dec 4 '12 at 19:24
$\begingroup$ @whuber: Yes, because the density (not its derivative as I inadvertently stated earlier) enters into the variance as a reciprocal, the value of the density at that point must not be zero. Apologies for dropping that condition! $\endgroup$ – cardinal Dec 4 '12 at 19:41
$\begingroup$ Elementary counterexamples can be created using any distribution that assigns probability of $1/2$ to an interval $(-\infty,\mu]$ and probability $1/2$ to $[\mu+\delta,\infty)$ where $\delta\gt 0,$ such as a Bernoulli$(1/2)$ ($\mu=0,\delta=1$). Sample medians will be less than or equal to $\mu$ as often as they are greater than or equal to $\mu+\delta$. The chance that the median is not in $(\mu,\mu+\delta)$ approaches $0$ for large samples, effectively leaving a "gap" in $(\mu,\mu+\delta)$ in the limiting distribution--which obviously then will be non-normal, no matter how it is standardized. $\endgroup$ – whuber♦ Feb 11 '14 at 22:32
If you work in terms of indicator variables (i.e. $Z_i = 1$ if $X_i \leq x$ and $0$ otherwise), you can directly apply the Central limit theorem to a mean of $Z$'s, and by using the Delta method, turn that into an asymptotic normal distribution for $F_X^{-1}(\bar{Z})$, which in turn means that you get asymptotic normality for fixed quantiles of $X$.
So not just the median, but quartiles, 90th percentiles, ... etc.
Loosely, if we're talking about the $q$th sample quantile in sufficiently large samples, we get that it will approximately have a normal distribution with mean the $q$th population quantile $x_q$ and variance $q(1-q)/(nf_X(x_q)^2)$.
Hence for the median ($q = 1/2$), the variance in sufficiently large samples will be approximately $1/(4nf_X(\tilde{\mu})^2)$.
You need all the conditions along the way to hold, of course, so it doesn't work in all situations, but for continuous distributions where the density at the population quantile is positive and differentiable, etc, ...
Further, it doesn't hold for extreme quantiles, because the CLT doesn't kick in there (the average of Z's won't be asymptotically normal). You need different theory for extreme values.
Edit: whuber's critique is correct; this would work if $x$ were a population median rather than a sample median. The argument needs to be modified to actually work properly.
edited Oct 6 '18 at 23:32
Glen_bGlen_b
$\begingroup$ I think one logical piece of this explanation may be missing: how exactly does one use indicators to obtain sample medians? I can see how when $x$ is the underlying median, the indicator $X_i\le x$ will work: but this indicator does not coincide with the sample median or any function of it. $\endgroup$ – whuber♦ Feb 2 '13 at 17:44
$\begingroup$ How do you go from asymptotic normal distributions for $F^{−1}_X (\overline{Z})$ to get asymptotic normality for fixed quantiles of X? Edit: I got it, that $\overline{Z}$ becomes a percent value 0-100% thus quantile values are asymptotically normal $\endgroup$ – adam Feb 18 '14 at 13:33
The key idea is that the sampling distribution of the median is simple to express in terms of the distribution function but more complicated to express in terms of the median value. Once we understand how the distribution function can re-express values as probabilities and back again, it is easy to derive the exact sampling distribution of the median. A little analysis of the behavior of the distribution function near its median is needed to show that this is asymptotically Normal.
(The same analysis works for the sampling distribution of any quantile, not just the median.)
I will make no attempt to be rigorous in this exposition, but I do carry out it out in steps that are readily justified in a rigorous manner if you have a mind to do that.
These are snapshots of a box containing 70 atoms of a hot atomic gas:
In each image I have found a location, shown as a red vertical line, that splits the atoms into two equal groups between the left (drawn as black dots) and right (white dots). This a median of the positions: 35 of the atoms lie to its left and 35 to its right. The medians change because the atoms are moving randomly around the box.
We are interested in the distribution of this middle position. Such a question is answered by reversing my procedure: let's first draw a vertical line somewhere, say at location $x$. What is the chance that half the atoms will be to the left of $x$ and half to its right? The atoms at the left individually had chances of $x$ to be at the left. The atoms at the right individually had chances of $1-x$ to be at the right. Assuming their positions are statistically independent, the chances multiply, giving $x^{35}(1-x)^{35}$ for the chance of this particular configuration. An equivalent configuration could be attained for a different split of the $70$ atoms into two $35$-element pieces. Adding these numbers for all possible such splits gives a chance of
$${\Pr}(x\text{ is a median}) = C x^{n/2} (1-x)^{n/2}$$
where $n$ is the total number of atoms and $C$ is proportional to the number of splits of $n$ atoms into two equal subgroups.
This formula identifies the distribution of the median as a Beta$(n/2+1, n/2+1)$ distribution.
Now consider a box with a more complicated shape:
Once again the medians vary. Because the box is low near the center, there isn't much of its volume there: a small change in the volume occupied by the left half of the atoms (the black ones once again)--or, we might as well admit, the area to the left as shown in these figures--corresponds to a relatively large change in the horizontal position of the median. In fact, because the area subtended by a small horizontal section of the box is proportional to the height there, the changes in the medians are divided by the box's height. This causes the median to be more variable for this box than for the square box, because this one is so much lower in the middle.
In short, when we measure the position of the median in terms of area (to the left and right), the original analysis (for a square box) stands unchanged. The shape of the box only complicates the distribution if we insist on measuring the median in terms of its horizontal position. When we do so, the relationship between the area and position representation is inversely proportional to the height of the box.
There is more to learn from these pictures. It is clear that when few atoms are in (either) box, there is a greater chance that half of them could accidentally wind up clustered far to either side. As the number of atoms grows, the potential for such an extreme imbalance decreases. To track this, I took "movies"--a long series of 5000 frames--for the curved box filled with $3$, then with $15$, then $75$, and finally with $375$ atoms, and noted the medians. Here are histograms of the median positions:
Clearly, for a sufficiently large number of atoms, the distribution of their median position begins to look bell-shaped and grows narrower: that looks like a Central Limit Theorem result, doesn't it?
Quantitative Results
The "box," of course, depicts the probability density of some distribution: its top is the graph of the density function (PDF). Thus areas represent probabilities. Placing $n$ points randomly and independently within a box and observing their horizontal positions is one way to draw a sample from the distribution. (This is the idea behind rejection sampling.)
The next figure connects these ideas.
This looks complicated, but it's really quite simple. There are four related plots here:
The top plot shows the PDF of a distribution along with one random sample of size $n$. Values greater than the median are shown as white dots; values less than the median as black dots. It does not need a vertical scale because we know the total area is unity.
The middle plot is the cumulative distribution function for the same distribution: it uses height to denote probability. It shares its horizontal axis with the first plot. Its vertical axis must go from $0$ to $1$ because it represents probabilities.
The left plot is meant to be read sideways: it is the PDF of the Beta$(n/2+1, n/2+1)$ distribution. It shows how the median in the box will vary, when the median is measured in terms of areas to the left and right of the middle (rather than measured by its horizontal position). I have drawn $16$ random points from this PDF, as shown, and connected them with horizontal dashed lines to the corresponding locations on the original CDF: this is how volumes (measured at the left) are converted to positions (measured across the top, center, and bottom graphics). One of these points actually corresponds to the median shown in the top plot; I have drawn a solid vertical line to show that.
The bottom plot is the sampling density of the median, as measured by its horizontal position. It is obtained by converting area (in the left plot) to position. The conversion formula is given by the inverse of the original CDF: this is simply the definition of the inverse CDF! (In other words, the CDF converts position into area to the left; the inverse CDF converts back from area to position.) I have plotted vertical dashed lines showing how the random points from the left plot are converted into random points within the bottom plot. This process of reading across and then down tells us how to go from area to position.
Let $F$ be the CDF of the original distribution (middle plot) and $G$ the CDF of the Beta distribution. To find the chance that the median lies to the left of some position $x$, first use $F$ to obtain the area to the left of $x$ in the box: this is $F(x)$ itself. The Beta distribution at the left tells us the chance that half the atoms will lie within this volume, giving $G(F(x))$: this is the CDF of the median position. To find its PDF (as shown in the bottom plot), take the derivative:
$$\frac{d}{dx}G(F(x)) = G'(F(x))F'(x) = g(F(x))f(x)$$
where $f$ is the PDF (top plot) and $g$ is the Beta PDF (left plot).
This is an exact formula for the distribution of the median for any continuous distribution. (With some care in interpretation it can be applied to any distribution whatsoever, whether continuous or not.)
Asymptotic Results
When $n$ is very large and $F$ does not have a jump at its median, the sample median must vary closely around the true median $\mu$ of the distribution. Also assuming the PDF $f$ is continuous near $\mu$, $f(x)$ in the preceding formula will not change much from its value at $\mu,$ given by $f(\mu).$ Moreover, $F$ will not change much from its value there either: to first order,
$$F(x) = F\left(\mu + (x-\mu)\right) \approx F(\mu) + F^\prime(\mu)(x-\mu) = 1/2 + f(\mu)(x-\mu).$$
Thus, with an ever-improving approximation as $n$ grows large,
$$g(F(x))f(x) \approx g\left(1/2 + f(\mu)(x-\mu)\right) f(\mu).$$
That is merely a shift of the location and scale of the Beta distribution. The rescaling by $f(\mu)$ will divide its variance by $f(\mu)^2$ (which had better be nonzero!). Incidentally, the variance of Beta$(n/2+1, n/2+1)$ is very close to $n/4$.
This analysis can be viewed as an application of the Delta Method.
Finally, Beta$(n/2+1, n/2+1)$ is approximately Normal for large $n$. There are many ways to see this; perhaps the simplest is to look at the logarithm of its PDF near $1/2$:
$$\log\left(C(1/2 + x)^{n/2}(1/2-x)^{n/2}\right) = \frac{n}{2}\log\left(1-4x^2\right) + C' = C'-2nx^2 +O(x^4).$$
(The constants $C$ and $C'$ merely normalize the total area to unity.) Through third order in $x,$ then, this is the same as the log of the Normal PDF with variance $1/(4n).$ (This argument is made rigorous by using characteristic or cumulant generating functions instead of the log of the PDF.)
Putting this altogether, we conclude that
The distribution of the sample median has variance approximately $1/(4 n f(\mu)^2)$,
and it is approximately Normal for large $n$,
all provided the PDF $f$ is continuous and nonzero at the median $\mu.$
whuber♦whuber
$\begingroup$ I like that 4th figure. Did you make it using R? $\endgroup$ – EngrStudent Mar 24 '14 at 19:29
$\begingroup$ @Engr I probably could have made one like it in R, perhaps using layout, but in fact it was done with Mathematica 9. $\endgroup$ – whuber♦ Mar 24 '14 at 19:36
$\begingroup$ 'Tis a thing of beauty. $\endgroup$ – EngrStudent Mar 24 '14 at 23:50
$\begingroup$ @Tim I do not understand the relevance of the reference to a prior, but I do appreciate you pointing out that the correct name of the Beta distribution identified in the "Intuition" section is Beta$(n/2+1,n/2+1)$. I'll fix that wherever it occurs (which is in several places in the discussion). $\endgroup$ – whuber♦ Feb 4 '16 at 14:57
$\begingroup$ @Engr And behold--you had commented on it over six years ago! After so much time and so many posts, we easily forget so much... . $\endgroup$ – whuber♦ Sep 23 '20 at 19:33
@EngrStudent illuminating answer tells us that we should expect different results when the distribution is continuous, and when it is discrete (the "red" graphs, where the asymptotic distribution of the sample median fails spectacularly to look like normal, correspond to the distributions Binomial(3), Geometric(11), Hypergeometric(12), Negative Binomial(14), Poisson(18), Discrete Uniform(22).
And indeed this is the case. When the distribution is discrete, things get complicated. I will provide the proof for the Absolutely Continuous Case, essentially doing no more than detailing the answer already given by @Glen_b, and then I will discuss a bit what happens when the distribution is discrete, providing also a recent reference for anyone interested in diving in.
ABSOLUTELY CONTINUOUS DISTRIBUTION
Consider a collection of i.i.d. absolutely continuous random variables $\{X_1,...X_n\}$ with distribution function (cdf) $F_X(x) = P(X_i\le x)$ and density function $F'_X(x)=f_X(x)$. Define $Z_i\equiv I\{X_i\le x\}$ where $I\{\}$ is the indicator function. Therefore $Z_i$ is a Bernoulli r.v., with $$E(Z_i) = E\left(I\{X_i\le x\}\right) = P(X_i\le x)=F_X(x),\;\; \text{Var}(Z_i) = F_X(x)[1-F_X(x)],\;\; \forall i$$
Let $Y_n(x)$ be the sample mean of these i.i.d. Bernoullis, defined for fixed $x$ as $$Y_n(x) = \frac 1n\sum_{i=1}^nZ_i$$ which means that $$E[Y_n(x)] = F_X(x),\;\; \text{Var}(Y_n(x)) = (1/n)F_X(x)[1-F_X(x)]$$ The Central Limit Theorem applies and we have
$$\sqrt n\Big(Y_n(x) - F_X(x)\Big) \rightarrow_d \mathbb N\left(0,F_X(x)[1-F_X(x)]\right) $$
Note that $Y_n(x) = \hat F_n(x)$ i.e. non else than the empirical distribution function. By applying the "Delta Method" we have that for a continuous and differentiable function $g(t)$ with non-zero derivative $g'(t)$ at the point of interest, we obtain
$$\sqrt n\Big(g[\hat F_n(x)] - g[F_X(x)]\Big) \rightarrow_d \mathbb N\left(0,F_X(x)[1-F_X(x)]\cdot\left(g'[F_X(x)]\right)^2\right) $$
Now, choose $g(t) \equiv F^{-1}_X(t),\;\; t\in (0,1)$ where $^{-1}$ denotes the inverse function. This is a continuous and differentiable function (since $F_X(x)$ is), and by the Inverse Function Theorem we have
$$g'(t)=\frac {d}{dt}F^{-1}_X(t) = \frac 1{f_x\left(F^{-1}_X(t)\right)}$$
Inserting these results on $g$ in the delta-method derived asymptotic result we have
$$\sqrt n\Big(F^{-1}_X(\hat F_n(x)) - F^{-1}_X(F_X(x))\Big) \rightarrow_d \mathbb N\left(0,\frac {F_X(x)[1-F_X(x)]}{\left[f_x\left(F^{-1}_X(F_X(x))\right)\right]^2} \right) $$
and simplifying,
$$\sqrt n\Big(F^{-1}_X(\hat F_n(x)) - x\Big) \rightarrow_d \mathbb N\left(0,\frac {F_X(x)[1-F_X(x)]}{\left[f_x(x)\right]^2} \right) $$
.. for any fixed $x$. Now set $x=m$, the (true) median of the population. Then we have $F_X(m) = 1/2$ and the above general result becomes, for our case of interest,
$$\sqrt n\Big(F^{-1}_X(\hat F_n(m)) - m\Big) \rightarrow_d \mathbb N\left(0,\frac {1}{\left[2f_x(m)\right]^2} \right) $$
But $F^{-1}_X(\hat F_n(m))$ converges to the sample median $\hat m$. This is because
$$F^{-1}_X(\hat F_n(m)) = \inf\{x : F_X(x) \geq \hat F_n(m)\} = \inf\{x : F_X(x) \geq \frac 1n \sum_{i=1}^n I\{X_i\leq m\}\}$$
The right-hand side of the inequality converges to $1/2$ and the smallest $x$ for which eventually $F_X \geq 1/2$, is the sample median.
So we obtain
$$\sqrt n\Big(\hat m - m\Big) \rightarrow_d \mathbb N\left(0,\frac {1}{\left[2f_x(m)\right]^2} \right) $$ which is the Central Limit Theorem for the sample median for absolutely continuous distributions.
DISCRETE DISTRIBUTIONS
When the distribution is discrete (or when the sample contains ties) it has been argued that the "classical" definition of sample quantiles, and hence of the median also, may be misleading in the first place, as the theoretical concept to be used in order to measure what one attempts to measure by quantiles.
In any case it has been simulated that under this classical definition (the one we all know), the asymptotic distribution of the sample median is non-normal and a discrete distribution.
An alternative definition of sample quantiles is by using the concept of the "mid-distribution" function, which is defined as $$F_{mid}(x) = P(X\le x) - \frac 12P(X=x)$$
The definition of sample quantiles through the concept of mid-distribution function can be seen as a generalization that can cover as special cases the continuous distributions, but also, the not-so-continuous ones too.
For the case of discrete distributions, among other results, it has been found that the sample median as defined through this concept has an asymptotically normal distribution with an ...elaborate looking variance.
Most of these are recent results. The reference is Ma, Y., Genton, M. G., & Parzen, E. (2011). Asymptotic properties of sample quantiles of discrete distributions. Annals of the Institute of Statistical Mathematics, 63(2), 227-243., where one can find a discussion and links to the older relevant literature.
Alecos PapadopoulosAlecos Papadopoulos
$\begingroup$ (+1) For the article. This is an excellent answer. $\endgroup$ – ahwillia Feb 18 '14 at 15:23
$\begingroup$ Can you please explain why $F^{-1}_X(\hat F_n(m))$ converges to the sample median $\hat m$? $\endgroup$ – honeybadger Oct 24 '18 at 15:59
$\begingroup$ I know that $\hat F_n(m) \to F_X(m) $ in distribution, but I cannot see how sample median $\hat m$ is equal to $F^{-1}_X(\hat F_n(m))$ $\endgroup$ – honeybadger Oct 24 '18 at 16:13
$\begingroup$ @kasa I elaborated a bit on the matter. $\endgroup$ – Alecos Papadopoulos Oct 25 '18 at 23:55
$\begingroup$ I am sorry to keep bringing this up again: But the smallest $x$ for which eventually $F_X(x) ≥ 1/2 $, is the population median, not the sample median, isn't it? $\endgroup$ – honeybadger Oct 26 '18 at 8:20
Yes it is, and not just for the median, but for any sample quantile. Copying from this paper, written by T.S. Ferguson, a professor at UCLA (his page is here), which interestingly deals with the joint distribution of sample mean and sample quantiles, we have:
Let $X_1, . . . ,X_n$ be i.i.d. with distribution function $F(x)$, density $f(x)$, mean $\mu$ and finite variance $\sigma^2$. Let $0 < p < 1$ and let $x_p$ denote the $p$-th quantile of $F$, so that $F(x_p) = p$. Assume that the density $f(x)$ is continuous and positive at $x_p$. Let $Y_n = X_{(n:\lceil np\rceil)}$ denote the sample $p$-th quantile. Then
$$\sqrt n(Y_n − x_p) \xrightarrow{d} N(0, p(1 − p)/(f(x_p))^2)$$
For $p=1/2 \Rightarrow x_p=m$ (median), and you have the CLT for medians,
$$\sqrt n(Y_n − m) \xrightarrow{d} N\left(0, [2f(m)]^{-2}\right)$$
COOLSerdash
$\begingroup$ Nice. It is worth mentioning that the variance of the sample median is not as easy to estimate as the one for the sample mean. $\endgroup$ – Michael M Nov 10 '13 at 7:30
$\begingroup$ @Alecos - how did you get two answers for this question? $\endgroup$ – EngrStudent Apr 19 '17 at 15:28
$\begingroup$ @EngrStudent The system allows it, it just asks you to verify that you want indeed to add a second answer. $\endgroup$ – Alecos Papadopoulos Apr 19 '17 at 15:31
I like the analytic answer given by Glen_b. It is a good answer.
It needs a picture. I like pictures.
Here are areas of elasticity in an answer to the question:
There are lots of distributions in the world. Mileage is likely to vary.
Sufficient has different meanings. For a counter-example to a theory, sometimes a single counter-example is required for "sufficient" to be met. For demonstration of low defect rates using binomial uncertainty hundreds or thousands of samples may be required.
For a standard normal I used the following MatLab code:
mysamples=1000;
loops=10000;
y1=median(normrnd(0,1,mysamples,loops));
cdfplot(y1)
and I got the following plot as an output:
So why not do this for the other 22 or so "built-in" distributions, except using prob-plots (where straight line means very normal-like)?
And here is the source code for it:
loops=600;
y=zeros(loops,23);
y(:,1)=median(random('Normal', 0,1,mysamples,loops));
y(:,2)=median(random('beta', 5,0.2,mysamples,loops));
y(:,3)=median(random('bino', 10,0.5,mysamples,loops));
y(:,4)=median(random('chi2', 10,mysamples,loops));
y(:,5)=median(random('exp', 700,mysamples,loops));
y(:,6)=median(random('ev', 700,mysamples,loops));
y(:,7)=median(random('f', 5,3,mysamples,loops));
y(:,8)=median(random('gam', 10,5,mysamples,loops));
y(:,9)=median(random('gev', 0.24, 1.17, 5.8,mysamples,loops));
y(:,10)=median(random('gp', 0.12, 0.81,mysamples,loops));
y(:,11)=median(random('geo', 0.03,mysamples,loops));
y(:,12)=median(random('hyge', 1000,50,20,mysamples,loops));
y(:,13)=median(random('logn', log(20000),1.0,mysamples,loops));
y(:,14)=median(random('nbin', 2,0.11,mysamples,loops));
y(:,15)=median(random('ncf', 5,20,10,mysamples,loops));
y(:,16)=median(random('nct', 10,1,mysamples,loops));
y(:,17)=median(random('ncx2', 4,2,mysamples,loops));
y(:,18)=median(random('poiss', 5,mysamples,loops));
y(:,19)=median(random('rayl', 0.5,mysamples,loops));
y(:,20)=median(random('t', 5,mysamples,loops));
y(:,21)=median(random('unif',0,1,mysamples,loops));
y(:,22)=median(random('unid', 5,mysamples,loops));
y(:,23)=median(random('wbl', 0.5,2,mysamples,loops));
figure(1); clf
for i=2:23
subplot(4,6,i-1)
probplot(y(:,i))
title(['Probplot of ' num2str(i)])
axis tight
if not(isempty(find(i==[3,11,12,14,18,22])))
set(gca,'Color','r')
When I see the analytic proof I might think "in theory they all might fit" but when I try it out then I can temper that with "there are a number of ways this doesn't work so well, often involving discrete or highly constrained values" and this might make me want to be more careful about applying the theory to anything that costs money.
EngrStudentEngrStudent
$\begingroup$ Am I wrong or the distribution for which the median is not normally distributed are discrete? $\endgroup$ – SeF Mar 19 '19 at 19:00
$\begingroup$ @SeF - some of the other answers are very illuminating here. $\endgroup$ – EngrStudent Sep 15 '20 at 14:24
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\begin{document}
\title{Spectral Methods for Partial Differential \\
Equations in Irregular Domains: \\
The Spectral Smoothed Boundary Method \thanks{This work was supported by
grants BFM2003-02832 (Ministerio de Ciencia y Tecnolog\'{\i}
\setcounter{page}{1}
\begin{abstract} In this paper, we propose a numerical method to approximate the solution of partial differential equations in irregular domains with no-flux boundary conditions by means of spectral methods. The main features of this method are its capability to deal with domains of arbitrary shape and its easy implementation via Fast Fourier Transform routines. We discuss several examples of practical interest and test the results against exact solutions and standard numerical methods. \end{abstract}
\begin{keywords} Spectral methods, Irregular domains, Phase Field methods, Reaction-diffusion equations \end{keywords}
\begin{AMS} 65M70, 65T50, 65M60. \end{AMS}
\pagestyle{myheadings} \thispagestyle{plain} \markboth{A. BUENO-OROVIO, V. M. P\'EREZ-GARC\'IA AND F. H. FENTON}{SPECTRAL METHODS IN IRREGULAR DOMAINS}
\section{Introduction}
Spectral methods \cite{Fornberg,Trefethen,Sanzserna} are among the most extensively used methods for the discretization of spatial variables in partial differential equations and have been shown to provide very accurate approximations of sufficiently smooth solutions. Because of their high-order accuracy, the use of spectral methods has become widespread over the years in various fields, including fluid dynamics, quantum mechanics, heat conduction and weather prediction \cite{Canuto,Gottlieb,Roger,solitones,BunYuGuo}. However, these methods have some limitations which have prevented them from being extended to many problems where finite-difference and finite-element methods continue to be used predominantly. One limitation is that the discretization of partial differential equations by spectral methods leads to the solution of large systems of linear or nonlinear equations involving \emph{full} matrices. Finite-difference and finite-element methods, on the other hand, lead to systems involving sparse matrices that can be handled by appropriate methods to reduce the complexity of the calculations substantially. Another drawback of spectral methods is that the geometry of the problem domain must be simple enough to allow the use of an appropriate orthonormal basis to expand the full set of possible solutions to the problem. This inability to handle irregularly shaped domains is one reason why these methods have had limited use in many engineering problems, where finite-element methods are preferred because of their flexibility to describe complex geometries despite the computational costs associated with constructing an appropriate solution grid. Although there have been attempts to use spectral methods in irregular domains \cite{Orszag,Korczak}, these approaches usually involve either incorporating finite-element preconditioning or the use of so-called spectral elements which are similar to finite elements. We are not aware of any previous study where purely spectral methods, particularly those involving Fast Fourier Transforms (FFTs), have been used to obtain solutions in complex irregular geometries.
In this paper we present an accurate and easy-to-use method for approximating the solution of partial differential equations in irregular domains with no-flux boundary conditions using spectral methods. The idea is based on what in dendritic solidification is known as the phase-field method \cite{KarRap98}. This method is used to locate and track the interface between the solid and liquid states and has been applied to a wide variety of problems including viscous fingering \cite{Foletal99a,Foletal99b}, crack propagation \cite{Karetal01} and the tumbling of vesicles \cite{BibMis03}. For a comprehensive review see \cite{Casademunt}.
In what follows we use the idea behind phase-field methods to illustrate how the solution of several partial differential equations can be obtained in various irregular and complex domains using spectral methods. Throughout the manuscript, for simplicity, we will refer to the combination of the phase-field and spectral methods as the spectral smoothed boundary (SSB) method. Our approach consists of two steps. First, the idea of the phase-field method is formalized and its convergence analyzed for the case of homogeneous Neumann boundary conditions. Then we discuss how the new formulation is useful for the direct use of spectral methods, specifically those based on trigonometric polynomials. This formulation makes the problem suitable for efficient solution using FFTs \cite{FFTW}. Since it is our intention that the resulting methodology be used in a variety of problems in engineering and applied science, we have concentrated on the important underlying concepts, reserving some of the more formal questions related to these methods for a subsequent analysis.
\section{The phase-field (smoothed boundary) method} \label{ppi} In this work we focus on applying the phase-field method to partial differential equations of the form \begin{subequations} \begin{equation}\label{equa} \nabla (\boldsymbol{D}^{(j)}\nabla u_j) + f(u_1,...,u_N,t) = \partial_t u_j \end{equation} for $N$ unknown real functions $u_j$ defined on an irregular domain $\Omega \subset \mathbb{R}^n$, where $n=1,2,3$ is the spatial dimensionality of the problem, together with appropriate initial conditions $u_j(x,0) = u_{j0}(x)$ and subject to Neumann boundary conditions
\begin{equation} \label{boundary}
(\boldsymbol {\Vec{n}} \cdot \boldsymbol{D}^{(j)}\nabla u_j) = 0
\end{equation}
\end{subequations} on $\partial \Omega$, where $\boldsymbol{D}^{(j)}(x)$ is a family of $n\times n$ matrices that may depend on the spatial variables.
Equations (\ref{equa}) and (\ref{boundary}) include many reaction-diffusion models, such as those describing population dynamics or cardiac electrical activity. In here we will restrict the analysis to equations of the form \eqref{equa} although we believe that the idea behind the method can be extended to many other problems involving complex boundaries and different types of partial differential equations.
Instead of discretizing Eq. (\ref{equa}) the smoothed boundary method relies on considering the auxiliary problem \begin{equation}\label{equa2} \nabla (\phi^{(\xi)} \boldsymbol{D}^{(j)}\nabla u^{(\xi)}_j) + \phi^{(\xi)}f(u^{(\xi)}_1,...,u^{(\xi)}_N,t) = \partial_t (\phi^{(\xi)}u^{(\xi)}_j), \end{equation} for the unknown functions $u^{(\xi)}_j$ on an enlarged domain
$\Omega'$ satisfying the following conditions: (i) $\Omega \subset \Omega'$ and (ii) $\partial \Omega \cup \partial \Omega' = \emptyset$. The function $\phi^{(\xi)}$ is continuous in $\Omega'$ and has the value one inside $\Omega$ and smoothly decays to zero outside $\Omega$, with $\xi$ identifying the width of the decay. That is, if $\chi_{\Omega}$ is the characteristic function of the set $\Omega$ defined as \begin{equation}\label{chieq} \chi_{\Omega}=\left\{\begin{array}{cl}
1, & x \in \Omega \\
0, & x \in \Omega'-\Omega
\end{array}\right. \end{equation} then $\phi^{(\xi)}: \Omega' \rightarrow \mathbb{R}$ is a regularized approximation to $\chi_{\Omega}$ such that $\lim_{\xi\rightarrow 0} \phi_{\xi} = \chi_{\Omega}$.
The key idea of the Smoothed boundary method (SBM) is that when $\xi \rightarrow 0$ the solutions $u_j^{(\xi)}$ of Eqs. (\ref{equa2}) on any domain $\Omega'$ with arbitrary boundary conditions on $\partial \Omega'$ satisfy
$u^{(\xi)}_j \rightarrow u_j$, that means, they tend to the solution of Eqs. (\ref{equa}), automatically incorporating the boundary conditions (\ref{boundary}). To see why, let us first realize that inside $\Omega$ the statement is inmediate since
$\phi^{(\xi)}\rightarrow 1$ in $\Omega$ as $\xi \rightarrow 0$ and Eq. \eqref{equa2} becomes Eq. \eqref{equa}. At the boundary we consider for simplicity the situation with $n=2$ (the extension to $n=3$ is immediate). Assuming smoothness of $\partial \Omega$ (which in this case will be a curve) and choosing any connected curve $\Gamma \subset \partial \Omega$, we define two families of differentiable curves $\Gamma_{\delta^{(+)}} \subset \Omega'/\Omega$ and
$\Gamma_{\delta^{(-)}} \subset \Omega$ whose ends coincide with those of $\Gamma$
and which tend uniformly to $\Gamma$ following the parameter $\delta$.
The curves $\Gamma_{\delta^{(+)}}$ and
$\Gamma_{\delta^{(-)}}$ are then the boundaries of a region
$A_{\delta}$ whose boundary $\partial A_{\delta}=\Gamma_{\delta^{(+)}} \cup \Gamma_{\delta^{(-)}}$ (see Fig. \ref{prima}).
We now integrate Eq. (\ref{equa2}) over $A_{\delta}$: \begin{equation}\label{integr} \int \int_{A_{\delta}} \left[ \nabla (\phi^{(\xi)} \boldsymbol{D}^{(j)}\nabla u^{(\xi)}_j) + \phi^{(\xi)}f(u,t) \right] dx = \int \int_{A_{\delta}} \partial_t (\phi^{(\xi)}u^{(\xi)}_j) dx. \end{equation} Using the Gauss (or Green) theorem for the first term of Eq. (\ref{integr}) we obtain \begin{equation}\label{intro}
\oint_{\partial A_{\delta}} \boldsymbol{n} \cdot \phi^{(\xi)} \boldsymbol{D}^{(j)}\nabla u^{(\xi)}_j dx + \int \int_{A_{\delta}}\phi^{(\xi)}f(u,t) dx = \int \int_{A_{\delta}} \partial_t (\phi^{(\xi)}u^{(\xi)}_j) dx, \end{equation} where $\oint$ denotes a line integral over $\partial A_{\delta}$.
Now we take the limit $\xi \rightarrow 0$ in (\ref{intro})
to obtain \begin{eqnarray} \lim_{\xi \rightarrow 0} \oint_{\partial A_{\delta}} \boldsymbol{n} \cdot \phi^{(\xi)} \boldsymbol{D}^{(j)}\nabla u^{(\xi)}_j dx & = & - \lim _{\xi \rightarrow 0} \int \int_{A_{\delta}}\phi^{(\xi)}f(u,t) dx \nonumber \\ & & + \lim_{\xi \rightarrow 0} \int \int_{A_{\delta}} \partial_t (\phi^{(\xi)}u^{(\xi)}_j) dx \nonumber \\ & & = m(A_{\delta}) \left[ -\phi^{(\xi)}f(u,t) + \partial_t (\phi^{(\xi)}u^{(\xi)}_j) \right]_{x=\zeta}, \label{finita} \end{eqnarray} where the last equality comes from the mean value theorem for integrals and $m(A_{\delta})$ is the measure of the set $A_{\delta}$. Here we assume that the solutions to Eq. \eqref{equa2} and its time derivatives are bounded so that the right-hand side of Eq. (\ref{finita}) is finite. On the left-hand side we decompose $\oint_{\partial A_{\delta}}$ as $\int_{\Gamma_{\delta^{(+)}}} + \int_{\Gamma_{\delta^{(-)}}}$. It is evident that $\lim_{\xi \rightarrow 0} \int_{\Gamma_{\delta^{(+)}}} \boldsymbol{n} \cdot \phi^{(\xi)} \boldsymbol{D}^{(j)}\nabla u^{(\xi)}_j dx = 0$, since in this limit $\phi^{(\xi)}=0$ over all $\Gamma_{\delta^{(+)}}$ and $\phi^{(\xi)}=1$ on $\Gamma_{\delta^{(-)}}$. As we were interested in proving that the boundary conditions are satisfied, we now make the width of the integration region $A_\delta$ tend to zero. Since $\Gamma_{\delta(-)} \rightarrow \Gamma$ as $\delta \rightarrow 0$, and $\lim_{\delta \rightarrow 0} m(A_{\delta}) = 0$, we obtain \begin{eqnarray}\label{final} \lim_{\delta\rightarrow 0} \lim_{\xi\rightarrow 0} \oint_{\partial A_{\delta}} \boldsymbol{n} \cdot \phi^{(\xi)} \boldsymbol{D}^{(j)}\nabla u^{(\xi)}_j dx & = & \lim_{\delta\rightarrow 0} \int_{\Gamma_{\delta(-)}} \boldsymbol{n} \cdot \boldsymbol{D}^{(j)}\nabla u^{(\xi)}_j dx \nonumber \\ & = & \int_{\Gamma} \boldsymbol{n} \cdot \boldsymbol{D}^{(j)}\nabla u^{(\xi)}_j dx = 0. \end{eqnarray} Since Eq. (\ref{final}) is true for any boundary segment $\Gamma$, we obtain the final result that in the limit $\xi \rightarrow 0$ Eq. \eqref{equa2} satisfies $\boldsymbol{n} \cdot \boldsymbol{D}^{(j)}\nabla u^{(\xi)}_j = 0$ for $j=1,...,N$, i.e., the boundary conditions.
\begin{figure}
\caption{Illustration of an irregular domain $\Omega$ with an example of the connected curve $\Gamma$ and the domain $A_\delta$ used in the proof of convergence of the method. }
\label{prima}
\end{figure}
From what we have shown it is clear that $\Omega'$ can be any closed region containing $\Omega$. The idea of the smoothed boundary method is then to consider Eq. \eqref{equa2} for a small but finite $\xi$ and to discretize this problem instead of Eqs. (\ref{equa}) and (\ref{boundary}). The main advantage is that one can search for the approximation $u^{(\xi)}$ on any enlarged domain $\Omega'$ such that $\Omega \subset \Omega'$ (for instance, a rectangular region). The enlarged discrete problem can then be solved with any proper boundary conditions on $\partial \Omega'$, since the fulfilment of the boundary conditions for $u$ on $\partial \Omega$ is guaranteed in the limit $\xi \rightarrow 0$. In our case, we will use the basis of trigonometric polynomials $e^{ikx}$ to approximate the solutions; thus, we will seek an extension of the solution $u$ of Eqs. \eqref{equa} and \eqref{boundary} that is periodic on the enlarged region $\Omega'$.
\section{The Spectral Smoothed Boundary Method}
We want to discretize Eq. (\ref{equa2}) on an enlarged region. As discussed earlier, we choose $\Omega'$ to be a rectangular region containing $\Omega$ and we will expand $u^{(\xi)}$ in the basis of Cartesian products of trigonometric polynomials $e^{ik_xx}e^{ik_yy}$.
Let us rewrite Eq. (\ref{equa2}) without subscripts and superscripts as \begin{equation} \label{equa3} \nabla \phi \cdot \boldsymbol{D}\nabla u + \phi \nabla (\boldsymbol{D}\nabla u) + \phi f(u,t) = \partial_t (\phi u). \end{equation} Note that since $\phi$ is located inside of the time derivative of the right term of Eq. (\ref{equa2}), it is possible for the integration domain itself to evolve in time, and thus this method could be used to solve moving boundary problems once a coupling equation is added for the movement of $\phi$. However, in this manuscript we will only deal with stationary integration domains; thus, $\partial_t \phi = 0$ and the right side of Eq. (\ref{equa3}) can be simplified as $\phi \partial_t u$. Dividing Eq. (\ref{equa3}) by $\phi$, we get \begin{equation} \label{equa4} \nabla \log \phi \cdot \boldsymbol{D}\nabla u + \nabla (\boldsymbol{D}\nabla u) + f(u,t) = \partial_t u, \end{equation} which is the equation of the smoothed boundary method that we will use to perform numerical simulations.
To implement numerically any solution method for Eq. \eqref{equa4}, we need to make a specific choice for $\phi^{(\xi)}$. In practice, any method that produces a smooth characteristic function can be used. In the context of phase-field methods, the standard procedure for obtaining the values of $\phi^{(\xi)}$ (which is called the ``phase-field") is to integrate an auxiliary diffusion equation of the form $\partial_t \phi = \xi^2 \nabla^2 \phi +(2 \phi-1)/2 - (2 \phi-1)^3/2, \label{evalphi}$ with initial conditions $\phi^{(\xi)}(t=0)$ = $\chi_{\Omega}$, until a steady state is reached \cite{Casademunt,Fenton}. Alternatively, since we only seek a smoothed boundary we choose to obtain $\phi^{(\xi)}$ from $\chi_{\Omega}$ using a convolution of the form \begin{equation}\label{convolution} \phi^{(\xi)} = \chi_{\Omega} \ast G^{({\xi})}, \end{equation} where $G^{(\xi )}$ is any family of functions such that $\lim_{\xi \rightarrow 0} G^{(\xi)}(x) = \delta(x)$, where $\delta$ is the Dirac delta function. In particular Gaussian functions of the form \begin{equation} \label{gaussian} G^{(\xi)}(x) = \prod_{k=1}^n \exp(-x_k^2/\xi^2) \end{equation} can be chosen. In this paper, all the functions $\phi^{(\xi)}$ have been obtained using this $n$-dimensional discrete convolution of $\chi_{\Omega}$ with a Gaussian function of the form given by Eq. (\ref{gaussian}). An example of the creation of $\phi^{(\xi)}$ is shown in Fig. \ref{fig:phi}, where it can be seen that the width of the interface in which $\phi^{(\xi)}$ changes from zero to one depends on the value used for $\xi$ (in fact it is of order $\xi$).
To avoid computational difficulties for very small values of $\phi$ we approximate
$\log \phi \approx \log (\phi + \epsilon)$, where $\epsilon$ is the machine precision. Numerically, $\phi$ and $(\phi + \epsilon)$ are equal up to roundoff errors, but this correction bounds the value of $\log \phi$ as $\phi \rightarrow 0$. Choosing $\phi$ to be a periodic function to avoid Gibbs phenomenom when computing its derivatives forces us to choose a computational domain in which this function becomes small enough near $\partial \Omega'$. In practice, this restriction requires us to leave a reasonable margin between the boundaries of the physical and the enlarged domains. We found that a margin of value $M = 10\xi$ is sufficient for all the simulations
to be stable.
All the spatial derivatives in Eq. (\ref{equa4}) are computed in Cartesian coordinates with spectral accuracy in Fourier space. If $g(x)$ is a periodic and sufficiently smooth function, then its $n^{th}$ derivative is given by \begin{equation} \frac{\partial^n g}{\partial x^n} = \mathcal{F}^{-1}\left\{ (ik_x)^n \mathcal{F} \{g\} \right\}, \end{equation} where $\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the direct and inverse Fourier Transform respectively, $k_x$ are the wave numbers associated with each Fourier mode, and $i$ is the imaginary unit. As mentioned previously, the use of this representation for $u$ implicitly assumes
periodic boundary conditions on $\partial \Omega'$. It is significant that only Fourier Transforms are used for these calculations instead of differentiation matrices, thereby avoiding the generation and storage of these matrices and yielding more efficient codes and shorter execution times, especially when Fast Fourier Transforms routines are used.
In this paper we are not concerned with designing the most efficient SSBM, but only to prove that such a method can be used to integrate PDEs in irregular domains. Thus, for time integration we use a simple second-order explicit method. In the particular case where all the coefficients of the diffusion tensor $\boldsymbol{D}$ are constants, we can write Eq. (\ref{equa4}) in the form \begin{equation} \mathcal{L}u + \mathcal{N}(u,t) = \partial_t u, \end{equation} where $\mathcal{L}u = \nabla (\boldsymbol{D}\nabla u)$ is the linear term and $\mathcal{N}(u,t) = \nabla \log \phi \cdot \boldsymbol{D}\nabla u + f(u,t)$ is the nonlinear part of the equation. Then a second-order in time operator splitting scheme of the form \begin{equation} U(t + \Delta t) = e^{\mathcal{L} \Delta t/2} e^{\mathcal{N} \Delta t} e^{\mathcal{L} \Delta t/2} U(t) \end{equation} can be used to solve the equation in time \cite{Strang}. For the examples to be presented later, we solve the nonlinear term by a second-order (half-step) explicit method and integrate the linear part exactly in Fourier space by exponential differentiation, which reduces the stiffness of the problem considerably and allows the use of larger time steps. The operator splitting scheme can then be written as \begin{subequations} \begin{eqnarray} U^{\ast} & = & \mathcal{F}^{-1} \{e^{\mathcal{L}\Delta t/2} \mathcal{F} \{ {U}^k \} \} \\ U^{\ast \ast} & = & U^{\ast} + \mathcal{N}(U^{\ast} + \mathcal{N}(U^{\ast},t_k + \Delta t/2) \cdot \Delta t/2, t_k + \Delta t) \cdot \Delta t \\ U^{k+1} & = & \mathcal{F}^{-1} \{e^{\mathcal{L}\Delta t/2} \mathcal{F} \{ {U}^{\ast \ast} \} \}. \end{eqnarray} \end{subequations}
\begin{figure}
\caption{Left: Example of an irregular domain $\Omega$ defined on a Cartesian grid and an enlarged domain $\Omega'$. Right: Smoothing of the irregular boundary in a one-dimensional section of the domain. The solid line shows a small section of the characteristic function $\chi_{\Omega}$ (with value 0 or 1) corresponding to part of the thicker line shown in the left part of the figure. Phase-field functions $\phi^{(\xi)}$ obtained from $\chi_{\Omega}$ for $\xi = 0.10$, $0.05$ and $0.025$ are labeled by diamonds, circles and stars respectively.}
\label{fig:phi}
\end{figure}
\section{Examples of the methodology}
\label{applications}
\subsection{The heat equation}
As a first example, we will consider a simple linear heat equation. This first case will allow us to make a quantitative study of the errors of the SSB method. Specifically, we are interested in solving the following heat equation with sources: \begin{equation} \label{eqheat} \partial_t u = D \Delta u - r \cos(2\theta) \end{equation} in the annulus $\Omega$ defined by $1 \leq r \leq 2$ with homogeneous Neumann boundary conditions on $\partial
\Omega$, $\partial_r u |_{r = 1} = \partial_r u |_{r = 2} = 0$, and initial data $u(r,\theta,0) = 0$. The diffusion coefficient is taken to be constant with value $D = 1$. Figure \ref{fig:phi2d} shows one example of the generation of the smoothed boundary for this domain. \begin{figure}
\caption{Left: The rectangular domain $\Omega'$ in which Eq. (\ref{eqheat}) is solved using the SSB method, with the irregular domain $\Omega$ (an annulus) shown in gray. Right: Smoothed boundary function $\phi^{(\xi)}$ given by Eq. \eqref{convolution} for $\xi = 0.10$.}
\label{fig:phi2d}
\end{figure}
Equation (\ref{eqheat}) in this geometry has an explicit steady state-solution of the form \begin{equation} \label{eqsteady} u_{st}(r,\theta) = \left( \frac{1}{5}r^3 - \frac{31}{50}r^2 - \frac{8}{25} \frac{1}{r^2} \right) \cos(2\theta), \end{equation} which can be compared with the numerical steady solution of Eq. \eqref{eqheat} (in practice, we stop simulations at $t = 6$ since by this time the numerical solutions have approximately reached the steady state) and obtain error estimates. Figure \ref{fig:errorcircle}
shows the maximum absolute error $E = \| u - U^{(\xi)}\|_{\infty}$ and relative error $e = \| u - U^{(\xi)} \|_{\infty}/\|u\|_{\infty}$ of several simulations for different values of $\xi$ and grid resolutions, where the relative error is defined with respect to the maximum value of the analytical solution. Note that in general these maximum errors decrease as the thickness of the interface is reduced ($\xi \rightarrow 0$), and that in most of the simulations the relative error is less than $1\%$. \begin{figure}
\caption{ Maximum absolute (left) and relative (right) errors of the numerical solution of Eq. (\ref{eqheat}) at time $t = 6$ in the annulus compared to the steady solution
as a function of the parameter
$\eta=\xi/\Delta x$. }
\label{fig:errorcircle}
\end{figure}
Although $\phi^{(\xi)}$ is a continuous function, it is necessary to have a grid fine enough to resolve properly the boundary layers in which it quickly changes from zero to one. For this reason, errors are represented in Fig. \ref{fig:errorcircle} not as a function of the number of grid points but as a function of the parameter $\eta = \xi/\Delta x$, which gives an idea of the number of points that lie in the interface. As mentioned before, it is also necessary to use a margin between the integration domain $\Omega$ and the computational domain $\Omega'$ for $\phi$ to become sufficiently small on $\partial\Omega'$. In this particular case, \begin{equation} \eta = \frac{\xi}{\Delta x} = \xi \frac{N}{2(R+M)}, \end{equation} where $R$ is the outer radius of the annulus, $N$ is the grid resolution, and $M = 10\xi$ is the margin used. This expression implies that for solving Eq. (\ref{eqheat}) in the range $\eta \in [1.5,5]$, the grid resolution varies from 90 to 300 points if $\xi = 0.10$, from 150 to 500 points if $\xi = 0.05$, and equivalently from 270 to 900 points when $\xi = 0.025$. The last important point concerning Fig. \ref{fig:errorcircle} is that, once the interface is properly solved ($\eta \sim 3-4$), the error converges to an approximately constant value that depends only on $\xi$, so there is no reason for using excessively clustered grids. To ensure that this error is produced only by the spatial discretization and is not due to the order of the method chosen to perform the time integration, we have also run the simulations with a first-order explicit (Euler) time-integration method and obtained errors of the same order of magnitude. Figure \ref{fig:heatcircle} shows the solution to Eq. (\ref{eqheat}) obtained at time $t = 6$ with the SSB method. Contour lines are also included to illustrate that the no-flux boundary conditions at $r = 1$ and $r = 2$ are satisfied.
\begin{figure}\label{fig:heatcircle}
\end{figure}
The analytical solution of Eq. (\ref{eqheat}) also satisfies homogeneous Neumann boundary conditions on the quarter-annulus delimited by $1 \leq r \leq 2$, $0 \leq \theta \leq \pi/2$ (see Fig. \ref{fig:heatcircle}), which allows us to use this related geometry to show how the SSB method performs when sharp corners are present in a given geometry. Maximum absolute and relative errors of the simulations for the quarter-annulus are shown in Fig.
\ref{fig:errorquarter}. Both geometries show similar good convergence properties. However, errors are slightly larger but of the same order of magnitude than for the full annulus due to the presence of the sharp corners, which become slightly blunted. This can be seen in Fig. \ref{fig:heatquarter}, which shows the error distribution $E = \| u - U^{(\xi)}\|_{\infty}$ over the domain $\Omega$, along with the corresponding solution.
In Figs. \ref{fig:heatcircle} and \ref{fig:heatquarter} we have shown the solutions of Eq. (\ref{eqheat}) within the irregular geometries $\Omega$. However, the solutions $U^{(\xi)}$ are calculated over the entire domain $\Omega'$. Figure \ref{fig:periodic} shows the solutions over $\Omega'$ in the full and quarter-annulus examples. While no-flux boundary conditions are implemented along $\partial \Omega$, the overall solution has periodic boundary conditions. Note that as the solution $U^{(\xi)}$ is not discontinuous on $\partial \Omega$, our solutions never present Gibbs phenomena due to the irregular boundaries.
\begin{figure}
\caption{Maximum absolute (left) and relative (right) errors of the numerical solution of Eq. (\ref{eqheat}) in the quarter-annulus $1 \leq r \leq 2$, $0 \leq \theta \leq \pi/2$ at time $t = 6$ compared to the steady solution of the problem as a function of the parameter
$\eta=\xi/\Delta x$. }
\label{fig:errorquarter}
\end{figure}
\begin{figure}
\caption{(Left) Solution to Eq. (\ref{eqheat}) in the quarter annulus $1 \leq r \leq 2$, $0 \leq \theta \leq \pi/2$ at $t = 6$ using the SSB method. Grid resolution is $500 \times 500$ and $\xi = 0.025$. (Right)
Spatial distribution of absolute error $E = \| u - U^{(\xi)}\|_{\infty}$ over the quarter-annulus for the simulation solution compared to the analytical steady solution. }
\label{fig:heatquarter}
\end{figure}
\begin{figure}
\caption{Solution to Eq. (\ref{eqheat}) using the SSB method over the entire domain of integration $\Omega'$ for the annulus shown in Fig.\ref{fig:heatcircle} (left) and the quarter-annulus shown in Fig.\ref{fig:heatquarter} (right). Note that the periodicity along $\partial \Omega'$ imposed by the FFT does not alter the solution in $\Omega$ or at the boundary $\partial\Omega$ (shown in white) where the zero-flux boundary conditions are satisfied. }
\label{fig:periodic}
\end{figure}
An important advantage of the SSB method is that when the new formulation given by Eq. (\ref{equa4}) is used, separate equations are not written for the boundaries, as the solution automatically adapts to satisfy the boundary conditions on $\partial \Omega$, which results in a very simple computational implementation. Alterations to the domain geometry therefore can be handled straightforwardly, without generating and implementing additional boundary condition equations. As an example, we present in Fig. \ref{fig:heathole} the solution to Eq. (\ref{eqheat}) in a more complicated geometry that combines both polar and Cartesian coordinates. As we cannot obtain an analytic steady solution to the equation in this domain, we evolve it until time $t = 5$ and compare our solution to one obtained using the \emph{finite elements} toolbox of {\sc Matlab}. Since it is well known that spectral methods have higher accuracy than second-order finite elements, we obtained the finite-element solution using the finest possible grid that our machine could handle (31152 nodes and 61440 triangles). At this resolution, as shown in Fig. \ref{fig:heathole}, the maximum difference between the solutions obtained by both
methods is very small ($\approx 10^{-3}$). We expect that as the grid is refined for the finite elements method the solution will converge to the spectral solution, resulting in an even smaller difference error.
\begin{figure}
\caption{ Solution to Eq. (\ref{eqheat}) at time $t = 5$ in a more complicated domain (left) and spatial distribution of the difference between the solutions obtained by the SSB method and using finite elements (right). Grid resolution is $400 \times 400$ and $\xi = 0.05$.}
\label{fig:heathole}
\end{figure}
\subsection{The Allen-Cahn equation}
An important partial differential equation which arises in the modeling of the formation and motion of phase boundaries is the Allen-Cahn \cite{Allen} equation: \begin{equation} \label{eq:allencahn} \begin{array}{ccl} \partial_t u = \epsilon^2 \Delta u - f(u), & & x \in \Omega \\ \partial_n u = 0, & & x \in \partial \Omega, \end{array} \end{equation} where $\epsilon$ is a small positive constant and $f(u)$ is the derivative of a potential function $W(u)$ that has two wells of equal depth. For simplicity, we will assume that $W(u) = (u^2-1)^2/4$, which makes $f(u) = u^3 - u$. In this manner, the Allen-Cahn equation may be seen as a simple example of a nonlinear reaction-diffusion equation. As explained in Ref. \cite{Trefethen}, this equation has three fixed-point solutions, $u=-1$, $u=0$ and $u = 1$. The middle state is unstable, but the states $u = \pm 1$ are attracting, and solutions tend to exhibit flat areas close to these values separated by interfaces that may coalesce or vanish on long time scales, a phenomenon known as metastability. Figure \ref{fig:allencahn} shows the solution of the Allen-Cahn equation solved with Neumann boundary conditions on an annulus with a $z$-shaped hole using the SSB method, with $\epsilon = 0.01$. The annulus structure is given by $1 \le r \le 5$ and the z-hole is formed using radii at 2, 3, and 4 and angles in steps of $15^\circ$ degrees (15, 30, 60 and $75^\circ$). For initial conditions we have chosen two positive and two negative Gaussian functions located in different parts of the domain: \begin{equation} \label{eq:eqgaussian} u=\sum_{i=1}^{n=4} (-1)^{n+1} \exp(-20 ((x-x_i)^2 + (y-y_i)^2)), \end{equation} where \begin{equation} \begin{tabular}{ll} $x_1 = 1.5\cos(\pi/4)$, &$y_1 = 1.5\sin(\pi/4)$, \\ $x_2 = 4\cos(\pi/12)$, &$y_2 = 4\sin(\pi/12)$, \\ $x_3 = 4.5\cos(\pi/4)$, &$y_3 = 4.5\sin(\pi/4)$, \\ $x_4 = 4\cos(11\pi/24)$, &$y_4 = 4\sin(11\pi/24)$. \\ \end{tabular} \nonumber \end{equation}
For comparison we also solved the equation using a second order in space finite difference method in polar coordinates, since despite the complexity of this geometry all the boundaries are parallel to the axes in polar coordinates and so implementing no-flux boundary conditions is straightforward. As shown in Fig. \ref{fig:allencahn}, both the SSB and polar finite-difference solutions are in very good agreement, except in the interface separating the two phases that lies just in one of the corners. These larger but still overall small differences between both schemes are due to the sharp transition of the solution between -1 and 1 and the fact that the finite difference scheme approximate spatial derivatives using lower-order accuracy than the spectral method. \begin{figure}
\caption{ Solution of the Allen-Cahn equation (\ref{eq:allencahn}) at time $t = 65$ with $\epsilon = 10^{-2}$ using the SSB method (left) and errors when compared to the solution obtained by a second-order finite-difference scheme in polar coordinates (right). Grid size is $360 \times 360$ with $\xi = 0.05$ for the SSB method and grid resolution is $\Delta r=0.025$ and $\Delta \theta$=0.3 degrees for the polar finite-difference solution.}
\label{fig:allencahn}
\end{figure}
\subsection{Reaction-diffusion equations and excitable media}
Models of excitable media form another significant class of nonlinear parabolic partial differential equations and describe systems as diverse as chemical reactions \cite{bz,bar}, aggregation of amoebae in the cellular slime mold Dictyostelium \cite{Foerster},
calcium waves \cite{Marni}, and the electrical properties of neural \cite{HH} and cardiac cells \cite{BR,LR}, among others. The equations of excitable systems extend the Allen-Cahn equation by including one or more additional variables that govern growth and decay of the waves. Solutions of excitable media consist of excursions in state space from a stable rest state and a return to rest, with the equations describing the additional variables determining the time courses of excitation and recovery. In spatially extended systems, diffusive coupling allows excitation to propagate as nonlinear waves, and in multiple dimensions complex patterns can be formed, including two-dimensional spiral waves \cite{bz,bar,Foerster,Marni,Davidenko} and their three-dimensional analogs, scroll waves \cite{gray,Fenton2}. Well-known examples of excitable media equations include the Hodgkin-Huxley \cite{HH} model of neural cells and its generalized simplification, the FitzHugh-Nagumo \cite{FHN} model.
The dynamics of wave propagation in excitable media has been studied extensively in regular domains. However, the complex geometry inherent to some systems, such as the heart, often can have an essential influence on
wave stability and dynamics \cite{Fenton3}. This fact, combined with the need for high-order accuracy to resolve the sharp wave fronts characteristic of cardiac models, should make the SSB method a useful tool for studying electrical waves in realistic heart geometries. Figure \ref{fig:heart} shows an example of a propagating wave of action potential in both an idealized (left) and a realistic (right) slice \cite{Andrew} of ventricular tissue using the SSB method and a phenomenological ionic cell model \cite{Fenton2,Fenton3} with equations of the form \begin{subequations} \label{eq:hearteqs} \begin{eqnarray} & & \partial_t u({\bf x},t) = \nabla \cdot (\boldsymbol{D}\nabla u)-J_{fi}(u,v)-J_{so}(u)-J_{si}(u,w) \\ & &\partial_t v({\bf x},t) = \Theta(u_c-u)(1-v)/\tau_v^{-}(u) - \Theta(u-u_c)v/\tau_v^{+} \\ & &\partial_t w({\bf x},t) = \Theta(u_c-u)(1-w)/\tau_w^{-} - \Theta(u-u_c)w/\tau_w^{+} \\ & &J_{fi}(u,v) = -\frac{v}{\tau_d} \Theta(u-u_c) (1-u)(u-u_c) \\ & &J_{so}(u) = \frac{u}{\tau_0} \Theta(u_c-u) + \frac{1}{\tau_r} \Theta(u-u_c) \\ & &J_{si}(u,w) = -\frac{w}{2\tau_{\mathrm{si}}}(1+\tanh[k(u-u_c^{\mathrm{si}})]) \\ & &\tau_v^{-}(u) = \Theta(u-u_v)\tau_{v1}^{-} + \Theta(u_v-u)\tau_{v2}^{-} \end{eqnarray} \end{subequations}
where u is the membrane potential; $J_{fi}, J_{so}$ and $J_{si}$ are phenomenological currents; $v$ and $w$ are ionic gate variables; and $\boldsymbol{D}$ is the diffusion tensor (isotropic for these simulations, with value $D=1$ $\mathrm{cm}^2/\mathrm{s}$). In all these formulas, $\Theta(x)$ is the standard Heaviside step function defined by $\Theta(x)=1$ for $x \geq 0$ and $\Theta(x) = 0$ for $x < 0$, and the set of parameters of the model are chosen to reproduce different cellular dynamics measured experimentally. \begin{figure}\label{fig:heart}
\end{figure}
\section{Discussion and future work}
In this paper we have presented a new method for implementing homogeneous Neumann boundary conditions using spectral methods for several problems of general interest. The spectral smoothed boundary method offers several advantages over finite-difference and finite-element alternatives. Because ghost cells are not needed, the implementation of boundary conditions requires less coding than finite-difference stencils, and in addition the spatial derivatives are represented with higher accuracy. The use of simple Cartesian grids makes the SSB method easier to use with multiple domain shapes than finite elements, since \emph{grid generation is not necessary}. Furthermore, the use of FFT routines in the SSB method ensures efficiency and also makes extension of the method to three dimensions straightforward using well-established routines. Since the method is directly based on the FFT it is very simple to implement on high performance computers by using
native parallel or vector FFT libraries.
The most significant limitation of the SSB method is that the error of the method depends directly on the ratio of the width of the smoothed boundary $\xi$ to the spatial step $\Delta x$, what implies that using uniform grids the number of points in the discretization needs to be large. This limitation is perhaps not important for certain classes of problems in which the solution contains steep wavefronts or other sharp features that require a fine spatial resolution to correctly reproduce the dynamics of the system, such as electrical waves in cardiac tissue or shock waves in fluid mechanics. However, the adequate reduction of error in domains with irregular boundaries using this method may require an increase in spatial resolution of a factor of 10 or 20 in each direction of the mesh for problems with smooth behavior like the heat equation with slowly varying sources compared to what is typically needed to obtain the same accuracy without complex boundaries.
Thus, an important future extension of this work is to improve the performance of the SSB method for problems that do not track features with sharp spatial gradients. For instance, if the boundary is stationary, it could be useful to use a non-uniform grid with extra resolution along the boundaries combined with a non-uniform fast Fourier transform (NFFT) to calculate the spatial derivatives. However, if the boundary moves over time, it might be more efficient to use a fine spatial discretization than to keep track of the boundary for such problems. Other planned future work includes properly handling complex anisotropies in the diffusion matrices such as those found in cardiac muscle, examining whether the method can be used to satisfy other types of boundary conditions including Dirichlet and Robin, and implementing non-stationary boundaries.
In conclusion, we have presented and analyzed a new numerical method which imposes homogeneous Neumann boundary conditions in complex geometries using spectral methods. We have used this method to solve different partial differential equations in domains with irregular boundaries and have found
good agreement with the exact analytical solutions when such solutions can be obtained. Along with the overall advantage of allowing domains of different shapes to be considered with spectral methods in a very simple way, this method also offers highly accurate discretizations of spatial derivatives, ease of implementation, straightforward extension to 3D, and applicability to a wide variety of equations. Moreover, SSB codes need not change to implement different geometries since all the information on the geometry is contained in the function $\phi^{(\xi)}$, with the additional advantage that
this function is easy to generate and, unlike finite element methods, does not require the use of special software for grid generation.
\hbox{}
\textbf{Acknowledgments.} We would like to thank Elizabeth M. Cherry for useful discussions and valuable comments on the manuscript. A. Bueno-Orovio also would like to thank the Physics Department at Hofstra University for their hospitality during his visit there (July - September, 2003). We also acknowledge the National Biomedical Computation Resource (NIH Grant P41RR08605, USA).
\end{document} | arXiv |
Viechtbauer (2007) is a general article about meta-analysis focusing in particular on random- and mixed-effects (meta-regression) models. An example dataset, based on a meta-analysis by Linde et al. (2005) examining the effectiveness of Hypericum perforatum extracts (St. John's wort) for treating depression, is used in the paper to illustrate the various methods. The data are given in the paper in Table 1 (p. 105).
Variables ai and ci indicate the number of participants with significant improvements between baseline and the follow-up assessment in the treatment and the placebo group, respectively, variables n1i and n2i are the corresponding group sizes, variable yi is the log of the relative improvement rate (i.e., the improvement rate in the treatment group divided by the improvement rate in the placebo group), vi is the corresponding sampling variance, dosage is the weekly dosage (in grams) of the Hypericum extract used in each study, major indicates whether a study was restricted to participants with major depression or not (1 or 0, respectively), baseline denotes the average score on the Hamilton Rating Scale for Depression (HRSD) at baseline (i.e., before treatment begin), and duration indicates the number of treatment weeks before response assessment. Variables yi and vi are not actually included in the original dataset and were added by means of the escalc() function.
Note that, for illustration purposes, only a subset of the data from the Linde et al. (2005) meta-analysis are actually included in this example. Therefore, no substantive interpretations should be attached to the results of the analyses given below.
With transf=exp, the values of the outcome measure (i.e., the log relative improvement rates) and corresponding confidence interval bounds are exponentiated and hence transformed back from the log scale. Therefore, variable yi now indicates the relative improvement rate, and ci.lb and ci.ub are the bounds of an approximate 95% confidence interval for the true relative improvement rate in the individual studies (note that this is not a permanent change – object dat still contains the log transformed values, which we need for the analyses below).
Therefore, the estimated relative rate is 1.38 with an approximate 95% CI of 1.26 to 1.52. However, the Q-test suggests that the true (log) relative rates are not homogeneous.
We can interpret the model estimate obtained above as an estimate of the (weighted) average of the true log relative rates for these 17 studies. This is the so-called fixed-effects model, which allows us to make a conditional inference (about the average effect) that only pertains to this set of studies.
We can model the heterogeneity in the true log relative rates and apply a random-effects model. This allows us to make an unconditional inference about a larger population of studies from which the included set of studies are assumed to be a random selection.
The baseline HRSD score will be used to reflect the severity of the depression in the patients. Since these two variables may interact, their product will also be included in the model. Finally, for easier interpretation, we will also center the variables at (roughly) their means when including them in the model.
I(baseline - 20) -0.0672 0.0352 -1.9086 0.0563 -0.1363 0.0018 .
These are the same results as given in Table 2 on page 113. Therefore, it appears that St. John's wort is more effective for lower baseline HRSD scores (the coefficient is negative, but just misses being significant at $\alpha = .05$ with $p = .06$). On the other hand, the total dosage of St. John's wort administered during the course of a study does not appear to be related to the treatment effectiveness ($p = .56$) and there does not appear to be an interaction between the two moderators ($p = .65$).
So, for a low baseline HRSD score (i.e., mildly depressed patients), the estimated average relative improvement rate is quite high (2.67 with 95% CI: 1.46 to 4.88), but at a high baseline HRSD score (i.e., more severely depressed patients), the estimated average relative improvement rate is low (1.26 with 95% CI: 0.99 to 1.61) and in fact not significantly different from 1.
Linde, K., Berner, M., Egger, M., & Mulrow, C. (2005). St John's wort for depression: Meta-analysis of randomised controlled trials. British Journal of Psychiatry, 186, 99–107.
Viechtbauer, W. (2007). Accounting for heterogeneity via random-effects models and moderator analyses in meta-analysis. Zeitschrift für Psychologie / Journal of Psychology, 215(2), 104–121.
Note that the equation used to compute these bounds is slightly different from the equation given in footnote 4 in the article. The bounds given above do take the uncertainty in the estimate of $\mu$ into consideration and are therefore a bit wider than the ones reported in the article. | CommonCrawl |
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\title{On equivariant Dirac operators for $\pmb{SU_q(2)}$}
\markboth{Partha Sarathi Chakraborty and Arupkumar Pal}{On equivariant Dirac operators for $SU_q(2)$}
\author{PARTHA SARATHI CHAKRABORTY and ARUPKUMAR PAL$^{*}$}
\address{Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai~600~113, India\\ \noindent $^{*}$Indian Statistical Institute, 7, SJSS Marg, New Delhi~110~016, India\\ \noindent E-mail: [email protected]; [email protected]\\[1.2pc] \noindent {\it Dedicated to Prof.~Kalyan Sinha on his
sixtieth birthday}}
\volume{116}
\mon{November}
\parts{4}
\pubyear{2006}
\Date{}
\begin{abstract} We explain the notion of minimality for an equivariant spectral triple and show that the triple for the quantum $SU(2)$ group constructed by Chakraborty and Pal in~\cite{c-p1} is minimal. We also give a decomposition of the spectral triple constructed by Dabrowski {\it et~al} \cite{dlssv} in terms of the minimal triple constructed in~\cite{c-p1}. \end{abstract}
\keyword{Spectral triple; quantum group.}
\maketitle
\section{Introduction}
The interaction between noncommutative geometry and quantum groups, in particular the (noncommutative) geometry of quantum groups, had been one of the less understood and less explored areas of both the theories for a while. In the last few years, however, there has been some progress in this direction. The first important step was taken by the authors in \cite{c-p1} where they found an optimal family of Dirac operators for the quantum $SU(2)$ group acting on $L_2(h)$, the $L_2$ space of the Haar state $h$, and equivariant with respect to the (co-)action of the group itself. This family has quite a few remarkable features. They are:
\begin{enumerate} \renewcommand\labelenumi{\arabic{enumi}.} \leftskip -.25pc \item Any element of the $K$-homology group can be realized by a member from this family, which means that all elements of the $K$-homology group are realizable through some Dirac operator acting on the single Hilbert space $L_2(h)$ in a natural manner.
\item The sign of any equivariant Dirac operator on $L_2(h)$ is a compact perturbation of the sign of a Dirac operator from this family,
\item Given any equivariant Dirac operator $\tilde{D}$ acting on $L_2(h)$, and any Dirac operator $D$ from this family, there exist
two positive reals $k_1$ and $k_2$ such that \begin{equation*}
|\widetilde{D}| \leq k_1 + k_2|D|. \end{equation*} \item They exhibit features that are unique to the quantum case ($q\neq 1$). It was proved in~\cite{c-p1} that for classical $SU(2)$, there does not exist any Dirac operator acting on (one copy of) the $L_2$ space that is both equivariant as well as 3-summable. \end{enumerate} These triples were later analysed by Connes \cite{co4} in great detail, where the general theory of Connes--Moscovici was applied to obtain a beautiful local index formula for $SU_q(2)$.
Recently, Dabrowski {\it et~al} \cite{dlssv} have constructed another family of Dirac operators that act on two copies of the $L_2$ space, has the right summability property, is equivariant in a sense described in \cite{dlssv}, and is isospectral to the classical Dirac operator. In this note, we will give a decomposition of this Dirac operator in terms of the Dirac operators constructed in \cite{c-p1}.
\section{Equivariance and minimality}
In this section, we will formulate the notion of an equivariant spectral triple for a compact quantum group and what one means by its minimality, or irreducibility. For basic notions on compact quantum groups, we refer the reader to~\cite{w3}. To fix the notation, let us recall a few things briefly here. Let $G=(C(G),\Delta)$ be a compact quantum group, where $C(G)$ is the unital $C^*$-algebra of `continuous functions on $G$' and $\Delta$ the comultiplication map. The symbols $\kappa$ and $h$ will denote the antipode map and the Haar state for $G$. For two functionals $\rho$ and $\sigma$ on $C(G)$, the convolution product $\rho\ast\sigma$ is the functional $a\mapsto (\rho\otimes\sigma)\Delta(a)$. For $\rho$ as above and $a\in C(G)$, we will denote by $a\ast\rho$ the element $(\mbox{id}\otimes\rho)\Delta(a)$ and by $\rho\ast a$ the element $(\rho\otimes \mbox{id})\Delta(a)$. A unitary representation $u$ of $G$ acting on a Hilbert space $\mathcal{H}$ is a unitary element of the multiplier algebra $M(\mathcal{K}(\mathcal{H})\otimes C(G))$, where $\mathcal{K}(\mathcal{H})$ denotes the space of compact operators on $\mathcal{H}$, that satisfies the condition $(\mbox{id}\otimes\Delta)u=u_{12}u_{13}$. For a unitary representation $u$ and a continuous linear functional $\rho$ on $C(G)$, we will denote by $u_\rho$ the operator $(\mbox{id}\otimes\rho)u$ on $\mathcal{H}$. The GNS space associated with the state $h$ will be denoted by $L_2(h)$ and the cyclic vector will be denoted by $\Omega$. While using the comultiplication $\Delta$, we will often use the Sweedler notation (i.e.\ $\Delta(a)=a_{(1)}\otimes a_{(2)}$).
Let $\mathcal{A}$ be a unital $C^*$-algebra, $G$ be a compact quantum group and let $\tau$ be an action of $G$ on $\mathcal{A}$, i.e.\ $\tau$ is a unital $C^*$-homomorphism from $\mathcal{A}$ to $\mathcal{A}\otimes C(G)$ satisfying the condition $(\mbox{id}\otimes\Delta)\tau = (\tau\otimes\mbox{id})\tau$. In other words, let $(\mathcal{A}, G, \tau)$ be a $C^*$-dynamical system. Recall \cite{b-s} that a covariant representation of $(\mathcal{A}, G,\tau)$ on a Hilbert space $\mathcal{H}$ is a pair $(\pi,u)$ where $\pi$ is a unital *-representation of $\mathcal{A}$ on $\mathcal{H}$, $u$ is a unitary representation of $G$ on $\mathcal{H}$ and they obey the condition \begin{equation}\label{cov_1} u(\pi(a)\otimes I)u^* =(\pi\otimes\mbox{id})\tau(a),\quad a\in\mathcal{A}. \end{equation} By an {\it odd $G$-equivariant spectral data for $\mathcal{A}$}, we mean a quadruple $(\pi,u,\mathcal{H},D)$ where \begin{enumerate} \renewcommand\labelenumi{\arabic{enumi}.} \leftskip -.25pc
\item $(\pi,u)$ is a covariant representation of $(\mathcal{A}, G,\tau)$ on the Hilbert space $\mathcal{H}$,
\item $\pi$ is faithful,
\item $u(D\otimes I)u^* =D\otimes I$,
\item $(\pi,\mathcal{H},D)$ is an odd spectral triple. \end{enumerate} We will often be sloppy and just say $(\pi,\mathcal{H},D)$ is an odd $G$-equivariant spectral triple for $\mathcal{A}$, omitting $u$. We say that an operator $D$ on a Hilbert space $\mathcal{H}$ is an {\it odd $G$-equivariant Dirac operator} for $\mathcal{A}$ if there exists a unitary representation $u$ of $G$ on $\mathcal{H}$ such that $(\pi,u,\mathcal{H},D)$ gives a $G$-equivariant spectral data for $\mathcal{A}$.
Similarly, an even $G$-equivariant spectral data for $\mathcal{A}$ consists of an even spectral data $(\pi,u,\mathcal{H}, D,\gamma)$ where $(\pi,u,\mathcal{H},D)$ obeys conditions~1, 2 and~3 above, and moreover $(\pi,\mathcal{H},D,\gamma)$ is an even spectral `triple', and one has $u(\gamma\otimes I)u^* =\gamma \otimes I$. An even $G$-equivariant Dirac operator is also defined similarly.
We say that an equivariant odd spectral data $(\pi,u,\mathcal{H},D)$ is {\it minimal} if the covariant representation $(\pi,u)$ is irreducible.
Note that if we take $\mathcal{A}=C(G)$, then the groups $G$ and $G_{\rm op}$ have natural actions $\Delta$ and $\Delta_{\rm op}$ on $\mathcal{A}$. In what follows, we will mainly be concerned about these two systems $(\mathcal{A}=C(G), G, \Delta)$ and $(\mathcal{A}=C(G), G_{\rm op}, \Delta_{\rm op})$. A $G$-equivariant spectral triple for $C(G)$ will be called a {\it right equivariant spectral triple} for $C(G)$. A~ {\it right equivariant Dirac operator} for $C(G)$ will mean a $G$-equivariant Dirac operator for $C(G)$. Similarly, a $G_{\rm op}$-equivariant spectral triple for $C(G)$ will be called a {\it left equivariant spectral triple} for $C(G)$ and a $G_{\rm op}$-equivariant Dirac operator for $C(G)$ will be called a {\it left equivariant Dirac operator} for $C(G)$.
We will next study covariant representations of the right $G$-action on $C(G)$, i.e. representations of the system $(C(G), G,\Delta)$.
\begin{lemma}\label{tech_0} Let $(\pi, u)$ be a covariant representation of $(C(G), G, \Delta)$. If the Haar state $h$ of $G$ is faithful, then $\pi$ is faithful. \end{lemma}
\begin{proof} Assume $\pi(a)=0$. Then $\pi(a^*a)=0$ and hence $(\pi\otimes\mbox{id})\Delta(a^*a)=u(\pi(a^*a)\otimes I)u^*=0$. Applying $(\mbox{id}\otimes h)$ on both sides, we get $h(a^*a)I=0$. Since $h$ is faithful,\break $a=0$. \qed \end{proof}
\begin{remark}{\rm The above lemma helps ensure that if we have a compact quantum group with a faithful Haar state, take a covariant representation $(\pi,u)$ of the system $(C(G), G,\Delta)$ on a Hilbert space $\mathcal{H}$, and look at a Dirac operator $D$ on $\mathcal{H}$, then we really get a spectral triple for the space $G$ rather than that of some subspace (i.e.\ quotient $C^*$-algebra of $C(G)$)\break of it.} \end{remark}
\begin{lemma}\label{tech_1} Let $(\pi, u)$ be a covariant representation of $(C(G), G, \Delta)$. Then the operator $u_h$ is a projection and for any continuous linear functional $\rho$ on $\mathcal{A},$ one has $u_h u_\rho=u_\rho u_h=\rho(1)u_h$. \end{lemma}
\begin{proof} Using Peter--Weyl decomposition for $u$, one can assume without loss in generality that $u$ is finite dimensional. Take two vectors $w$ and $w'$ in $\mathcal{H}$. Then \begin{align*} \langle w, u_h w'\rangle
&= \langle w\otimes\Omega , u ( w'\otimes\Omega)\rangle\\[.25pc]
&= \langle u^*(w\otimes\Omega) , w'\otimes\Omega\rangle\\[.25pc]
&= \langle ((\mbox{id}\otimes\kappa)u)(w\otimes\Omega) ,
w'\otimes\Omega\rangle\\[.25pc]
&= \overline{\langle w'\otimes\Omega,
((\mbox{id}\otimes\kappa)u)(w\otimes\Omega)\rangle}\\[.25pc]
&= \overline{\langle w', ((\mbox{id}\otimes h\kappa)u) w\rangle}\\[.25pc]
&= \overline{\langle w', ((\mbox{id}\otimes h)u) w\rangle}\\[.25pc]
&= \langle u_h w, w'\rangle. \end{align*} Thus $u_h$ is self-adjoint.
Next, for any continuous linear functional $\rho$, \begin{align*} u_\rho u_h &= (\mbox{id}\otimes\rho)u (\mbox{id}\otimes h)u\\[.25pc]
&= (\mbox{id}\otimes\rho\otimes h)(u_{12}u_{13})\\[.25pc]
&= (\mbox{id}\otimes\rho\otimes h)(\mbox{id}\otimes\Delta)u\\[.25pc]
&= (\mbox{id}\otimes \rho\ast h)u\\[.25pc]
&= \rho(1)u_h. \end{align*} Similary one has $u_h u_\rho = \rho(1)u_h$. In particular, $u_h^2=u_h$, so that $u_h$ is a projection. \qed \end{proof}
\begin{lemma}\label{tech_2} Let $A\equiv A(G)$ be the *-subalgebra of $C(G)$ generated by matrix entries of all finite dimensional unitary representations of $G$. Let $(A,\mathcal{U})$ be a dual pair of Hopf$^{\,\,*}$-algebras {\rm (}see {\rm \cite{v})}. Then \begin{equation}\label{dlssv_eq1} u_\rho \pi(a)=\pi(a\ast \rho_{(1)})u_{\rho_{(2)}}\quad \mbox{for all }\ \rho\in\mathcal{U} \mbox{ and } \ a\in A(G). \end{equation} \end{lemma}
\begin{proof} Apply $(\mbox{id}\otimes\rho)$ on both sides in the equality \begin{equation*} u(\pi(a)\otimes I)=((\pi\otimes\mbox{id})\Delta(a))u \end{equation*} and use the fact that $\rho(ab)=\rho_{(1)}(a)\rho_{(2)}(b)$.\qed \end{proof}
\begin{lemma}\label{tech_3} Let $w\in\mathcal{H}$ be a vector in the range of $u_h$. Then for any $a\in A(G)$ and $\rho\in\mathcal{U}${\rm ,} one has $u_\rho\pi(a)w=\pi(a\ast\rho)w$. In particular{\rm ,} one has $u_h \pi(a) w= h(a)w$. \end{lemma}
\begin{proof} Use Lemma~\ref{tech_2} to get \begin{align*} u_\rho\pi(a)w &= \pi(a\ast \rho_{(1)})u_{\rho_{(2)}}w\\[.2pc]
&= \pi(a\ast \rho_{(1)})u_{\rho_{(2)}} u_h w\\[.2pc]
&= \rho_{(2)}(1)\pi(a\ast \rho_{(1)}) u_h w\\[.2pc]
&= \pi(\rho_{(2)}(1)a\ast \rho_{(1)}) w\\[.2pc]
&= \pi(a_{(1)}\rho_{(1)}(a_{(2)})\rho_{(2)}(1))w \\[.2pc]
&= \pi(a_{(1)}\rho(a_{(2)}))w\\[.2pc]
&= \pi(a\ast \rho)w, \end{align*} for $a\in A(G)$. \qed \end{proof}
\begin{lemma}\label{tech_4} The linear span of $\{\pi(a)u_h w\hbox{\rm :}~a\in A(G), w\in\mathcal{H}\}$ is dense in $\mathcal{H}$. In particular{\rm ,} $u_h$ is nonzero. \end{lemma}
\begin{proof} Using Peter--Weyl decomposition of $u$ and the observation that $h(\kappa(a))=h(a)$ for all $a\in A$, it follows that $u_h=(u^*)_h$. Now take a vector $w'$ in $\mathcal{H}$ such that $\langle w', \pi(a)u_h w\rangle=0$ for all $w\in\mathcal{H}$ and $a\in A$. But then $\langle w', \pi(a)(u^*)_h w\rangle=0$, i.e.\ $\langle w'\otimes \Omega, (\pi(a)\otimes I)u^* (w\otimes \Omega)\rangle=0$. The covariance condition~(\ref{cov_1}) now gives $\langle u (w'\otimes\Omega), (\pi\otimes\mbox{id})\Delta(a) (w\otimes\Omega)\rangle=0$ for all $w\in\mathcal{H}$ and $a\in A$. In particular, one has $\langle u (w'\otimes\Omega), (\pi\otimes\mbox{id})\Delta(a) (\pi(b)w\otimes\Omega)\rangle=0$ for all $w\in\mathcal{H}$, and $a,b\in A$. Since $(\pi\otimes\mbox{id})\Delta(a)(\pi(b)\otimes I)=(\pi\otimes\mbox{id})(\Delta(a)(b\otimes I))$ and $\{\Delta(a)(b\otimes I)\hbox{:}~a,b\in A\}$ is total in $\mathcal{A}\otimes\mathcal{A}$, we get $u(w'\otimes\Omega)=0$ and consequently $w'=0$.\qed \end{proof}
For $w\in\mathcal{H}$, denote by $P_w$ the projection onto the closed linear span of $\{\pi(a)w\hbox{:}~a\in A\}$.
\begin{lemma}\label{tech_5} Let $w\in u_h\mathcal{H}$. Then $(P_w\otimes I)u(P_w\otimes I)=u(P_w\otimes I)$. If $w'$ is another vector in $u_h\mathcal{H}$ such that $\langle w,w'\rangle=0${\rm ,} then the projections $P_w$ and $P_{w'}$ are orthogonal.
\end{lemma}
\begin{proof} For the first part, it is enough to show that $P_w u_\rho P_w =u_\rho P_w$ for all $\rho\in\mathcal{U}$. But this is clear because from Lemma~\ref{tech_3}, we have $u_\rho \pi(a) w = \pi(a\ast \rho) w$.
For the second part, take $a,a'\in A$. Then using Lemma~\ref{tech_4} one gets \begin{align*} \langle \pi(a)w,\pi(a')w'\rangle &= \langle w, \pi(a^*a')w'\rangle\\[.2pc]
&= \langle u_h w, \pi(a^*a')w'\rangle\\[.2pc]
&= \langle w, u_h \pi(a^*a')w'\rangle\\[.2pc]
&= \langle w, h(a^*a')w'\rangle\\[.2pc]
&= 0. \end{align*} Thus $P_w$ and $P_{w'}$ are orthogonal. \qed \end{proof}
\begin{proposition}$\label{tech_6}\left.\right.$
\noindent Let $\{w_1,w_2,\ldots\}$ be an orthonormal basis for $u_h\mathcal{H}$. Write $P_n$ for $P_{w_n}${\rm ,} and let $\pi_n(\cdot):=P_n \pi(\cdot)P_n${\rm ,} $u_n := (P_n\otimes I)u(P_n\otimes I)$. Then \begin{enumerate} \renewcommand\labelenumi{{\rm \arabic{enumi}.}} \leftskip -.25pc
\item For each $n${\rm ,} $(\pi_n, u_n)$ is a covariant representation of the system $(\mathcal{A}, G,\Delta)$ on $P_n\mathcal{H}${\rm ,}
\item $\pi=\oplus \pi_n${\rm ,} $u=\oplus u_n${\rm ,}
\item $(\pi_n,u_n)$ is unitarily equivalent to the pair $(\pi_{\rm L}, u_{\rm R})$ where $\pi_{\rm L}$ is the representation of $\mathcal{A}$ on $L_2(G)$ by left multiplications and $u_{\rm R}$ is the right regular representation of $G$. \end{enumerate} \end{proposition}
\begin{proof} It follows from Lemmas~\ref{tech_5} and \ref{tech_4} that $P_n$'s are orthogonal, $\sum P_n=I$ and consequently $\pi=\oplus \pi_n$ and $u=\oplus u_n$.
Define $V_n\hbox{:}~P_n\mathcal{H}\rightarrow L_2(G)$ by \begin{equation*} V_n \pi(a)w_n = \pi_L(a)\Omega,\quad a\in A. \end{equation*} Since $\langle \pi_{\rm L}(a)\Omega,\pi_{\rm L}(b)\Omega\rangle =h(a^*b)=\langle \pi(a)w_n,\pi(b) w_n\rangle$, $\{\pi(a)w_n\hbox{:}~a\in A\}$ is total in $P_n\mathcal{H}$ and $\{\pi_{\rm L}(a)\Omega\hbox{:}~a\in A\}$ is total in $L_2(G)$, $V_n$ extends to a unitary from $P_n\mathcal{H}$ onto $L_2(G)$. Next, for $a,b\in A$, one has $V_n\pi(a)\pi(b)w_n=V_n\pi(ab)w_n=\pi_{\rm L}(ab)\Omega =\pi_{\rm L}(a)\pi_{\rm L}(b)\Omega =\pi_{\rm L}(a)V_n \pi(b)w_n$. So $V_n \pi(a)=\pi_{\rm L}(a)V_n$ for all $a\in A$ and hence for all $a\in\mathcal{A}$.
Finally, we will show that $(V_n\otimes I)u(V_n^*\otimes I) = u_{\rm R}$. Write $\tilde{u}_n:=(V_n\otimes I)u(V_n^*\otimes I)$. Then for any $\rho\in \mathcal{U}$, one has \begin{align*} (\mbox{id}\otimes \rho)\tilde{u}_n \pi_{\rm L}(a)\Omega
&= V_n u_\rho V_n^* \pi_{\rm L}(a)\Omega\\[.25pc]
&= V_n u_\rho \pi(a) w_n \\[.25pc]
&= V_n u_\rho \pi(a) u_h w_n \\[.25pc]
&= V_n \pi(a\ast \rho) w_n\\[.25pc]
&= V_n \pi(a\ast \rho) V_n^* V_n w_n\\[.25pc]
&= \pi_{\rm L}(a\ast\rho) \Omega. \end{align*}
By \cite{w3}, $\tilde{u}_n$ must be the right regular representation $u$ on $L_2(G)$. \qed \end{proof}
\begin{remark} {\rm The above proposition leads to an alternative proof of the Takesaki--Takai duality for compact quantum groups (Theorem~7.5 of \cite{b-s}).} \end{remark}
\begin{theorem}[\!] The covariant representation $(\pi,u)$ is irreducible if and only if the operator $u_h$ is a rank one projection. \end{theorem}\pagebreak
\begin{proof} Immediate corollary of Proposition~\ref{tech_6}. \qed \end{proof}
\begin{remark} {\rm In particular, it follows from the above theorem that the covariant representation $(\pi_{\rm L}, u_{\rm R})$ on $L_2(G)$ is irreducible. Thus the equivariant Dirac operator constructed in~\cite{c-p1} is {\it minimal}.} \end{remark}
\section{The decomposition}
\subsection*{\it Canonical triples for $SU_q(2)$}
Let $q$ be a real number in the interval $(0,1)$. Let $\mathcal{A}$ denote the $C^*$-algebra of continuous functions on $SU_q(2)$, which is the universal $C^*$-algebra generated by two elements $\alpha$ and $\beta$ subject to the relations \begin{align*} \alpha^*\alpha+\beta^*\beta&=I=\alpha\alpha^*+q^2\beta\beta^*, \quad \alpha\beta-q\beta\alpha=0=\alpha\beta^*-q\beta^*\alpha,\\ \beta^*\beta&=\beta\beta^* \end{align*} as in \cite{c-p1}. Let $\pi\hbox{:}~\mathcal{A}\rightarrow\mathcal{L}(L_2(h))$ be the representation given by left multiplication by elements in $\mathcal{A}$. Let $u$ denote the right regular representation of $SU_q(2)$. Recall~\cite{w3} that $u$ is the unique representation acting on $L_2(h)$ that obeys the condition \begin{equation} ((\mbox{id}\otimes\rho)u)\pi(a)\Omega
= \pi((\mbox{id}\otimes\rho)\Delta(a))\Omega \end{equation} for all $a\in\mathcal{A}$ and for all continuous linear functionals $\rho$ on $\mathcal{A}$. In \cite{c-p1}, the authors studied right equivariant Dirac operators, those Dirac operators that commute with the right regular representation, i.e.\ $D$ acting on $L_2(h)$ for which \begin{equation*} (D\otimes I)u=u(D\otimes I). \end{equation*} In particular, an optimal family of equivariant Dirac operators were found. A generic member of this family is of the form \begin{equation*} e^{(n)}_{ij}\mapsto\begin{cases} (an+b)e^{(n)}_{ij}, &\hbox{if} -n\leq i< n-k,\\[.6pc] (cn+d)e^{(n)}_{ij}, &\hbox{if} \ i=n-k,n-k+1,\ldots,n,\end{cases} \end{equation*} where $k$ is a fixed nonnegative integer and $a$, $b$, $c$, $d$ are reals with $ac<0$. If one looks at left equivariant Dirac operators, the same arguments would then lead to the following theorem.
\setcounter{defin}{0} \begin{theorem}[\!] Let $v$ be the left regular representation of $SU_q(2)$. Let $k$ be a nonnegative integer and let $a, b, c$, $d$ be real numbers with $ac<0$. Then the operator $D\equiv D(k,a,b,c,d)$ on $L_2(h)$ given by \begin{equation*} e^{(n)}_{ij}\mapsto \begin{cases} (an+b)e^{(n)}_{ij}, &\hbox{if} \ -n\leq j< n-k,\\[.6pc] (cn+d)e^{(n)}_{ij}, &\hbox{if}\ j=n-k,n-k+1,\ldots,n, \end{cases} \end{equation*} gives a spectral triple $(\pi,L_2(h),D)$ having nontrivial Chern character and obeys \begin{equation}\label{equiv} (D\otimes I)v=v(D\otimes I). \end{equation}
Conversely{\rm ,} given any spectral triple $(\pi,L_2(h), \tilde{D})$ with nontrivial Chern character such that $(\tilde{D}\otimes I)v=v(\tilde{D}\otimes I)${\rm ,} there exist a nonnegative integer $k$ and reals $a, b, c$, $d$ with $ac<0$ such that \begin{enumerate} \renewcommand\labelenumi{{\rm \arabic{enumi}.}} \leftskip -.25pc \item $\hbox{\rm sign}\,\tilde{D}$ is a compact perturbation of the sign of $D\equiv D(k,a,b,c,d)$, and
\item there exist constants $k_1$ and $k_2$ such that \begin{equation*}
|\tilde{D}|\leq k_1+k_2|D|. \end{equation*} \end{enumerate} \end{theorem}
\begin{proof} The key point is to note that the characterizing property of the left regular representation $v$ is \begin{equation} ((\mbox{id}\otimes\rho)v^*)\pi(a)\Omega
= \pi((\rho\otimes\mbox{id})\Delta(a))\Omega. \end{equation} Thus on the right-hand side, one now has left convolution of $a$ by $\rho$ instead of right convolution by $\rho$. Therefore any self-adjoint operator on $L_2(h)$ with discrete spectrum that obeys $(D\otimes I)v=v(D\otimes I)$ will be of the form \begin{equation*} e^{(n)}_{ij}\mapsto \lambda(n,j)e^{(n)}_{ij}. \end{equation*} Hence if one now proceeds exactly along the same lines as in \cite{c-p1}, one gets all the desired conclusions. \qed \end{proof}
Observe at this point that the whole analysis carried out in~\cite{co4} will go through for this Dirac operator as well. Let us now take two such Dirac operators $D_1$ and $D_2$ on $L_2(h)$ given by \begin{align} D_1 e^{(n)}_{ij} &=\begin{cases}-2n e^{(n)}_{ij}, &\hbox{if}\ j\neq n\\[.6pc] (2n+1) e^{(n)}_{ij}, &\hbox{if}\ j=n\end{cases},\nonumber\\[.5pc] D_2 e^{(n)}_{ij}&=\begin{cases}(-2n-1) e^{(n)}_{ij}, &\hbox{if}\ j\neq n\\[.6pc]
(2n+1) e^{(n)}_{ij}, &\hbox{if}\
j=n\end{cases}. \end{align} Now look at the triple \begin{equation*}
(\pi\oplus\pi, L_2(h)\oplus L_2(h), D_1\oplus |D_2|). \end{equation*} It is easy to see that this is a spectral triple. Nontriviality of its Chern character is a direct consequence of that of $D_1$. We will show in the next paragraph that in a certain sense, the spectral triple constructed in \cite{dlssv} is equivalent to this above triple.
\subsection*{\it The decomposition}
Let us briefly recall the Dirac operator constructed in~\cite{dlssv}. The carrier Hilbert space $\mathscr{H}$ is a direct sum of two copies of $L_2(h)$ that decomposes as \begin{equation*} \mathscr{H}=W_0^\uparrow\oplus\left(\bigoplus_{n\in\half\mathbb{Z}_+}
( W_n^\uparrow\oplus W_n^\downarrow )\right), \end{equation*} where \begin{align*} W_n^\uparrow &=\mbox{span}\bigg\{ u^n_{ij}\hbox{:}~i=-n,-n+1,\ldots,n,\,\\ &\hskip 1.1cm\, j=-n-\half,-n+\half,\ldots,n+\half\bigg\},\\[.4pc] W_n^\downarrow &=\mbox{span}\bigg\{ d^n_{ij}\hbox{:}~i=-n,-n+1,\ldots,n,\,\\ &\hskip 1.1cm\, j=-n+\half,-n+\frac{3}{2},\ldots,n-\half\bigg\}. \end{align*} ($u^n_{ij}$ and $d^n_{ij}$ correspond to the basis elements
$|nij\!\uparrow\rangle$ and $|nij\!\downarrow\rangle$ respectively in the notation of~\cite{dlssv}.) Now write \begin{equation*} v^n_{ij}= \begin{pmatrix}u^n_{ij}\\[.5pc] d^n_{ij}\end{pmatrix} \end{equation*} with the convention that $d^n_{ij}=0$ for $j=\pm\big(n+\half\big)$. Then the representation $\pi'$ of $\mathcal{A}$ on $\mathscr{H}$ is given by \begin{align*} \pi'(\alpha^{*}) \,v^n_{ij} &= a^+_{nij} v^{n+\half}_{ i+\half, j+\half}
+ a^-_{nij} v^{n-\half}_{i+\half, j+\half}, \nonumber \\[.5pc] \pi'(-\beta) \,v^n_{ij} &= b^+_{nij} v^{n+\half}_{ i+\half, j-\half}
+ b^-_{nij} v^{n-\half}_{ i+\half, j-\half}, \nonumber \\[.5pc] \pi'(\alpha) \,v^n_{ij} &= \tilde a^+_{nij} v^{n+\half}_{ i-\half, j-\half}
+ \tilde a^-_{nij} v^{n-\half}_{ i-\half, j-\half},
\\[.5pc]
\pi'(-\beta^*) \, v^n_{ij} &= \tilde b^+_{nij} v^{n+\half}_{ i-\half, j+\half}
+ \tilde b^-_{nij} v^{n-\half}_{ i-\half, j+\half}, \nonumber \end{align*} where $a^\pm_{nij}$ and $ b^\pm_{nij}$ are the following $2 \times 2$ matrices: \begin{align*} a^+_{nij} &= q^{\big(i+j-\half\big)/2} [n + i + 1]^\half \begin{pmatrix}{q^{-n-\half}} \, \frac{[n+j+\frac{3}{2}]^{1/2}}{[2n+2]} & 0\\[.6pc] q^\half \,\frac{[n-j+\half]^{1/2}}{[2n+1]\,[2n+2]} & q^{-n} \, \frac{[n+j+\half]^{1/2}}{[2n+1]}\end{pmatrix}, \nonumber \\[.7pc] a^-_{nij} &= q^{\big(i+j-\half\big)/2} [n - i]^\half \begin{pmatrix} {q^{n+1}} \, \frac{[n-j+\half]^{1/2}}{[2n+1]} & - q^\half \,\frac{[n+j+\half]^{1/2}}{[2n]\,[2n+1]}\\[.6pc] 0 & q^{n+\half} \, \frac{[n-j-\half]^{1/2}}{[2n]}\end{pmatrix}, \nonumber \\[.7pc]
b^+_{nij} &= q^{\big(i+j-\half\big)/2} [n + i + 1]^\half \begin{pmatrix}\frac{[n-j+\frac{3}{2}]^{1/2}}{[2n+2]} & 0\\[.6pc] - q^{-n-1} \,\frac{[n+j+\half]^{1/2}}{[2n+1]\,[2n+2]} & q^{-\half} \, \frac{[n-j+\half]^{1/2}}{[2n+1]}\end{pmatrix}, \\[.7pc] b^-_{nij} &=q^{\big(i+j-\half\big)/2} [n - i]^\half\begin{pmatrix} - q^{-\half} \, \frac{[n+j+\half]^{1/2}}{[2n+1]} & - q^n \,\frac{[n-j+\half]^{1/2}}{[2n]\,[2n+1]}\\[.6pc] 0 & - \frac{[n+j-\half]^{1/2}}{[2n]}\end{pmatrix}, \nonumber \end{align*} ($[m]$ being the $q$-number $\frac{q^m-q^{-m}}{q-q^{-1}}$) and $\tilde a^\pm_{nij}$ and $\tilde b^\pm_{nij}$ are the hermitian conjugates of the above ones: \begin{equation*} \tilde a^\pm_{nij} = (a^\mp_{n\pm\half, i-\half, j-\half})^*, \qquad \tilde b^\pm_{nij} = (b^\mp_{n\pm\half, i-\half, j+\half})^*. \end{equation*} The operator $D$ is given by \begin{equation*} D u^n_{ij}=(2n+1) u^n_{ij}, \qquad D d^n_{ij}= -2n d^n_{ij}. \end{equation*} The triple $(\pi', \mathscr{H}, D)$ is precisely the triple constructed in~\cite{dlssv}.
\begin{theorem}[\!] Let $\mathcal{K}_q$ be the two-sided ideal of $\mathcal{L}(\mathscr{H})$ generated by the operator \begin{equation*} d^n_{ij}\mapsto q^n d^n_{ij}, \quad u^n_{ij}\mapsto q^n u^n_{ij}, \end{equation*} and let $\mathcal{A}_f$ denote the *-subalgebra of $\mathcal{A}$ generated by $\alpha$ and $\beta$. Then there is a unitary $U\hbox{\rm :}~L_2(h)\oplus L_2(h)\rightarrow \mathscr{H}$ such that \begin{align}
U(D_1\oplus |D_2|)U^{*} &= D,\\ U(\pi(a)\oplus\pi(a))U^* -\pi'(a)
&\in \mathcal{K}_q \quad \mbox{for all} \ a\in\mathcal{A}_f. \end{align} \end{theorem}
\begin{proof} Define $U\hbox{:}~L_2(h)\oplus L_2(h)\rightarrow \mathscr{H}$ as follows: \begin{align*} U(e^{(n)}_{ij}\oplus 0) &= d^n_{i,j+\half}, \quad i=-n,-n+1,\ldots,n,\;\\[.4pc] &\qquad\,j=-n,-n+1,\ldots,n-1,\\[.4pc] U(e^{(n)}_{in}\oplus 0) &= u^n_{i,n+\half}, \quad i=-n,-n+1,\ldots,n,\\[.4pc] U(0\oplus e^{(n)}_{ij}) &= u^n_{i,j-\half}, \quad i=-n,-n+1,\ldots,n,\; j=-n,-n+1,\ldots,n. \end{align*}
It is immediate that $U(D_1\oplus |D_2|)U^{*} = D$. Therefore all that we need to prove now is that $U(\pi(a)\oplus\pi(a))U^{*} -\pi'(a)\in\mathcal{K}_q$ for all $a\in \mathcal{A}_f$. For this, let us introduce the representation $\hat{\pi}\hbox{:}~\mathcal{A}\rightarrow\mathcal{L}(L_2(h))$ given by \begin{equation*} \hat{\pi}(\alpha)=\hat{\alpha}, \qquad \hat{\pi}(\beta)=\hat{\beta}, \end{equation*} where $\hat{\alpha}$ and $\hat{\beta}$ are the following operators on $L_2(h)$ (see Lemma~2.2 of \cite{c-p3}): \begin{align} \hat{\alpha}\hbox{:}~e^{(n)}_{ij} &\mapsto q^{2n+i+j+1} e^{\big(n+\half\big)}_{i-\half ,j-\half }\nonumber\\[.4pc] &\quad\, + (1-q^{2n+2i})^\half(1-q^{2n+2j})^\half
e^{\big(n-\half\big)}_{i-\half ,j-\half },\label{halpha}\\[.4pc] \hat{\beta}\hbox{:}~e^{(n)}_{ij} &\mapsto
- q^{n+j}(1-q^{2n+2i+2})^\half
e^{\big(n+\half\big)}_{i+\half ,j-\half }\nonumber\\[.4pc] &\quad\, + q^{n+i}(1-q^{2n+2j})^\half
e^{\big(n-\half\big)}_{i+\half ,j-\half }.\label{hbeta} \end{align} It is easy to see that \begin{equation*} \pi(a)\oplus\pi(a)-\hat{\pi}(a)\oplus\hat{\pi}(a) \in U^*\mathcal{K}_q U \end{equation*} for $a=\alpha^*$ and $a=\beta$. Therefore it is enough to verify that \begin{equation*} U(\hat{\pi}(a)\oplus\hat{\pi}(a))U^*-\pi'(a)\in\mathcal{K}_q \end{equation*} for $a=\alpha^*$ and for $a=\beta$.
Next observe that \begin{align*}
a^+_{nij} &= (1-q^{2n+2i+2})^\half \begin{pmatrix}
(1-q^{2n+2j+3})^\half & 0\\[.5pc] 0 &
(1-q^{2n+2j+1})^\half \end{pmatrix} + O(q^{2n}),
\\[.65pc]
a^-_{nij} &= q^{2n+i+j+\half}(1-q^{2n-2i})^\half \begin{pmatrix} q(1-q^{2n-2j+1})^\half & 0\\[.5pc] 0 & (1-q^{2n-2j-1})^\half \end{pmatrix}\\[.5pc] &\quad\,+O(q^{2n}),
\\[.65pc]
b^+_{nij} &= q^{n+j-\half}(1-q^{2n+2i+2})^\half \begin{pmatrix} q(1-q^{2n-2j+3})^\half & 0\\[.5pc] 0 & (1-q^{2n-2j+1})^\half \end{pmatrix}\\[.5pc] &\quad\,+O(q^{2n}), \\[.65pc] &= q^{n+j-\half}(1-q^{2n+2i+2})^\half \begin{pmatrix} q & 0\\[.2pc] 0 & 1 \end{pmatrix}+O(q^{2n}), \\[.65pc] b^-_{nij} &=- q^{n+i}(1-q^{2n-2i})^\half \begin{pmatrix} (1-q^{2n+2j+1})^\half & 0 \\[.5pc] 0 &(1-q^{2n+2j-1})^\half \end{pmatrix}\\[.5pc] &\quad\,+O(q^{2n})\\[.65pc] &= - q^{n+i} \begin{pmatrix}(1-q^{2n+2j+1})^\half & 0 \\[.5pc] 0 &(1-q^{2n+2j-1})^\half \end{pmatrix}+O(q^{2n}). \end{align*} The required result now follows from this easily. \qed \end{proof}
\begin{remark} {\rm The above decomposition in particular tells us that the spectral triples $(\pi\oplus\pi, L_2(h)\oplus L_2(h), D_1\oplus
|D_2|)$ and $(\pi', \mathscr{H}, D)$ are essentially unitarily equivalent at the Fredholm module level. Therefore by Proposition~8.3.14 of \cite{h-r}, they give rise to the same element in $K$-homology.} \end{remark}
\begin{remark} {\rm In the spectral triple in~\cite{dlssv}, the Hilbert space can be decomposed as a direct sum of two isomorphic copies in such a manner that in each half Dirac operator has constant sign. So positive and negative signs come with equal frequency. However this symmetry is only superficial, as the decomposition above illustrates. This asymmetry might be a reflection of the inherent asymmetry in the {\it growth graph} associated with quantum $SU(2)$ (see \cite{c-p}). For classical $SU(2)$ the graph is symmetric whereas in the quantum case it is not.
It should also be pointed out here that, at least as far as classical odd dimensional spaces are concerned, this kind of sign symmetry is always superficial. They are always inherent in the even cases, but not in the odd ones.} \end{remark}
\end{document} | arXiv |
JUMP Math | Eighth Grade
Home Reports Center Math Eighth Grade JUMP Math
JUMP Math - Eighth Grade
The instructional materials reviewed for JUMP Math Grade 8 meet expectations for not assessing topics before the grade level in which the topic should be introduced.
The instructional materials reviewed for JUMP Math Grade 8 meet expectations for assessing grade-level content. The Sample Unit Quizzes and Tests along with Scoring Guides and Rubrics were reviewed for this indicator. Examples of grade-level assessment items include:
Teacher Resource, Part 1, Sample Quizzes and Tests, Unit 2 Test, Item 6, "John thinks 6 + 6 + 6 + 6 = 64 . Is he correct? Explain." Students apply the properties of exponents to solve the problem. (8.EE.1)
Teacher Resource, Part 1, Sample Quizzes and Tests, Unit 7 Test, Item 4, "a. Draw a scatter plot for data. b. Circle the cluster. c. Describe the association in as much detail as possible. d. Identify the outlier(s). Explain." Students construct and interpret scatter plots and describe the patterns and outliers. (8.SP.1)
Teacher Resource, Part 2, Sample Quizzes and Tests, Unit 1 Test, Item 6, "The graph shows the cost of renting an e-bike per hour. a. What is the y-intercept? What is the flat rate? b. Find the slope and write an equation for the line. Slope = riserun = ____= y=_______ c. Jack has $22. For how many hours can he rent an e-bike?" Students find the slope and y-intercept of a graph and write an equation. (8.F.6)
Teacher Resource, Part 2, Sample Quizzes and Tests, Unit 3 Quiz, Item 4, "A cube has volume 0.512 m$$^3$$. What is the length of one side of the cube?" Students find the cube root of a number. (8.EE.2)
Teacher Resource, Part 2, Sample Quizzes and Tests, Unit 4 Test, Item 6, "An engineer stands 200 m away from the base of a building. The distance to the top of the building is 350 m. a. What is the height of the building to nearest tenth of a meter? b. If the engineer stands 100 m from the base of the building, what will be the distance to the top of the building to the nearest tenth of a meter?" Students apply the Pythagorean Theorem to find a missing side length in a right triangle. (8.G.7)
The materials for Grade 8 include 14 units. In the materials, there are 156 lessons, and of those, 29 are Bridging lessons. According to the materials, Bridging lessons should not be "counted as part of the work of the year" (page A-56), so the number of lessons examined for this indicator is 127 lessons. The supporting clusters were also reviewed to determine if they could be factored in due to how strongly they support major work of the grade. There were connections found between supporting clusters and major clusters, and due to the strength of the connections found, the number of lessons addressing major work was increased from the approximately 101 lessons addressing major work as indicated by the materials themselves to 103.5 lessons.
Lessons – Approximately 81 percent, 103.5 out of 127; and
Teacher Resource, Part 2, Unit 4, Lessons G8-46, G8-47, and G8-48 connect 8.NS.2 with 8.G.7,8 as students are expected to use rational approximations of irrational numbers in order to apply the Pythagorean Theorem to real-world and mathematical problems or find the distance between two points in the coordinate plane.
Teacher Resource, Part 2, Unit 6, Lessons G8-53 and G8-54 connect 8.G.7 with 8.G.9 as students use the Pythagorean Theorem in real-world and mathematical problems in order to find the volumes of cones and spheres.
Teacher Resource, Part 2, Unit 7, Lessons SP8-7 and SP8-8 connect 8.F.4 with 8.SP.2 as students construct a linear model for a relationship between two quantities because a scatterplot of the two quantities suggests a linear association between them.
The instructional materials reviewed for JUMP Math Grade 8 partially meet the expectation for being consistent with the progressions in the Standards. Overall, the materials address the standards for this grade level and provide all students with extensive work on grade-level problems. The materials make connections to content in future grades, but they do not explicitly relate grade-level concepts to prior knowledge from earlier grades.
At the beginning of each unit, "This Unit in Context" provides a description of prior concepts and standards students have encountered during the grade levels before this one. The end of this section also makes connections to concepts that will occur in future grade levels. For example, "This Unit in Context" from Teacher Resource, Part 1, Unit 6, Functions: Defining, Evaluating, and Comparing Functions describes the topics from Operations and Algebraic Thinking that students encountered in Grades 4 and 5, specifically generating patterns given rules, and from Equations and Expressions in Grade 6, specifically analyzing the relationship between dependent and independent variables. The description then includes topics from Functions, specifically the graph of a function along with its rate of change and initial value, and it concludes with how the work of this unit builds to the study of functions in high school.
In the Advanced Lessons, students get the opportunity to engage with more difficult problems, but the problems are still aligned to grade-level standards. For example, the problems in Teacher Resource, Part 1, Unit 2, Lesson EE8-23 engage students in estimation with numbers written in scientific notation and the four operations, and these problems still align to 8.EE.3,4. Also, in Teacher Resource, Part 2, Unit 5, Lesson EE8-55, the problems align to 8.EE.8.
Every lesson identifies "Prior Knowledge Required" even though the prior knowledge identified is not aligned to any grade-level standards. For example, Teacher Resource, Part 2, Unit 1, Lesson F8-15 states that its goals are to introduce the y-intercept and find the y-intercept by graphing and draw a line using the slope and y-intercept. The prior knowledge required is adding, subtracting, and dividing integers and plotting points on a grid.
There are 29 lessons identified as Bridging lessons, but these lessons are not explicitly aligned to standards from prior grades even though they do state for which grade-level standards they are preparation. For example, in Teacher Resource, Part , Unit 1, all fourteen lessons are Bridging Lessons and are labeled as "preparation for" various standards in 8.EE and 8.F.3. However, none of these fourteen Bridging lessons are explicitly aligned to standards prior to Grade 8. Also, Teacher Resource, Part 2, Unit 1, Lessons F8-13 and F8-14 are Bridging lessons labeled as "preparation for 8.G.1, 8.G.3, and 8.F.4" that have students plotting points in coordinate grids and finding the lengths of horizontal and vertical line segments, but the lessons are not explicitly aligned to standards prior to Grade 8.
In the materials, the units are organized by domains and are clearly labeled. For example, Teacher Resource, Part 1, Unit 7 is titled Statistics and Probability: Patterns in Scatter Plots, and Teacher Resource, Part 2, Unit 4 is titled Geometry: Pythagorean Theorem. Within the units, there are goals for each lesson, and the language of the goals is visibly shaped by the CCSSM cluster headings. For example, in Teacher Resource, Part 1, Unit 5, one of the goals for Lesson EE8-44 states "Students will review the concept of unit rate and find the unit rate in proportional relationships represented in different ways, including on the line of a graph." The language of this goal is visibly shaped by 8.EE.B, "Understand the connections between proportional relationships, lines, and linear equations."
In Teacher Resource, Part 2, Unit 1, Lessons F8-15 through F8-19 and F8-23, the materials connect 8.F.B with 8.F.A as students are expected to be able to interpret the equation y=mx+b as defining a linear function and construct a function to model a linear relationship between two quantities.
In Teacher Resource, Part 2, Unit 2, Lessons G8-37 and G8-38, the materials connect 8.EE.B with 8.G.A as students use an understanding of similar triangles in order to explain why the slope is the same between any two points on a line and derive the equation of a line through the origin or any other point on the vertical axis.
In Teacher Resource, Part 2, Unit 4, Lessons G8-42 and G8-43, the materials connect 8.EE.A with 8.G.B as students work with radicals, specifically square root and cube root symbols, in order to represent solutions of equations that arise from developing an understanding of and being able to apply the Pythagorean Theorem.
Teacher Resource, Part 2, Unit 2, Lesson G8-31 focuses on similarity; the student proves figures are similar by showing that the ratios of the side lengths are proportional. Similarity is introduced before students learn dilations. The two topics are connected in Teacher Resource, Part 2, Unit 2, Lesson G8-35.
Teacher Resource, Part 1, Unit 3, Lesson G8-9, Exercises, "The triangles are congruent. a. Sketch the triangles. Mark the equal sides with hash marks (p. D-58)." Transformations are mentioned briefly when teachers are directed to say, "I need to turn the second triangle 90 clockwise to get it to the same position as the first triangle."
Teacher Resource, Part 2, Unit 1, Lesson F8-16, "The y-intercept of lines that go through the origin is zero. Remind students that lines that go through the origin represent a proportional relationship between x and y. For example, in y = 3x. SAY: the coordinates of the origin is (0,0), so one row in the table of values is (0,0). ASK: If a line goes through the origin, what is the y-value when x is equal to zero? (0) What is the y-intercept for the line that goes through the origin? (0) Ask a volunteer to circle the y intercept in the equations y = x + 2 and y = 2x - 3. (+2, -3) Explain that in the equation y = 1.5x, the y-intercept is 0 because you can write the equation as y = 1.5x + 0."
Standard 8.EE.7 expects students to develop procedural skills when solving linear equations in one variable. Lessons that include on-grade-level practice to develop fluency with linear equations with one variable include:
Teacher Resource, Part 1, Unit 4, Lesson EE8-35, Exercises, "Solve the equation. a. 3x + 3 − x = 5 b. 7x + 2 = 4x + 11." Students solve equations with the distributive property and combine like terms.
Teacher Resource, Part 1, Unit 4, Lesson EE8-37, Exercises, "Solve the equation. If there is no solution, write 'no solution'. a. x + 15 = 33 b. 9x = 18 c. 0x =7." Students must know if the equations have one solution, no solution, or infinitely many solutions.
Teacher Resource, Part 2, Unit 3, Lesson NS8-1, "√x = 4 SAY: To isolate x, we need to undo the square root. ASK: What operation does this? (squaring) SAY: To keep our equation balanced, we need to square both sides. Continue writing on the board: (√x)$$^2$$ = 4$$^2$$." Students are reminded how to undo operations when solving equations and relate this idea to equations with radicals and exponents and practice similar equations.
Standard 8.G.9 expects students to know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems with procedural skill. Examples include:
Teacher Resource, Part 2, Unit 6, Lesson G8-50, students find the volume of a cylinder given the radius or diameter. For example, students are given exercises with pictures of cylinders that have various measures and are asked to calculate the volume.
Book 2 Unit 6 Lesson G8-53, Exercises, Item 1, "A paper cup in the shape of a cone has a radius of 1.5 inches and a height of 4 inches. What is the volume of water that the cup can hold? (p. Q-42)." Students solve problems with the volume of cones, cylinders, and spheres.
Teacher Resource, Part 2, Unit 1, Lesson F8-23, Exercises, "There is some water in the bathtub. Sam pulls out the plug from the bathtub to let the water drain. After 10 seconds, there are 14 gallons of water left in the tub. After 30 seconds, there are 10 gallons of water left. a. Use two points, A (10, 14) and B (30, 10), and find the slope of the line. b. When Sam pulls out the plug, how much water was there? c). Write an equation for the number of gallons (g) left in the tub after s seconds. d. How many gallons are there after 20 seconds? e. How long does it take for all the water to drain from the tub? f. Find the y-intercept and the x-intercept of the equation and compare them to the answers to parts a and d. What did you notice? g. What does the slope represent?" (8.F.B) This problem misses the opportunity to have students apply the mathematics of using functions to model relationships between quantities. The questions include multiple prompts, guiding students along step by step for each question, rather than allowing students to attempt to apply the math and solve the problems independently.
Student Resource, Assessment & Practice Book, Part 2, Lesson EE8-52, Item 4, "Write the equations for the word problem. Then solve by graphing. The intersection point may have fractions or decimals. a. Two trains left Union Station at different times. Train A is 12 km from the station and is traveling 60 km/h. Train B is 27 km from the station and is traveling 50 km/h. When will Train A catch up to Train B? How far will they be from the station?" (8.EE.8c) Students solve real-world problems by writing two linear equations for the word problems and finding the solution by graphing them.
Teacher Resource, Part 2, Unit 5, Lesson EE8-52, Exercises, "Write a formula for the word problem. a. A gravel company charges $25 per cubic yard and a delivery charge of $75. b. A yearbook company charges $500 plus $15 per yearbook." (8.EE.8c) All questions in the lesson are structured as a "flat fee" plus a rate. The application questions follow given examples closely.
Non-routine problems are occasionally found in the materials. For example,
Teacher Resource, Part 2, Unit 5, Lesson EE8-55, Extensions, Item 1, "Alex is four times as old as Clara. In 5 years, Alex will only be three times as old as Clara. How old are Alex and Clara today?" Students are solving a non-routine problem leading to two linear equations in two variables. (8.EE.8c)
Conceptual Understanding: Teacher Resource, Part 2, Unit 2, Lesson G8-37, Extensions, Item 1, "Use the fact that dilations produce similar triangles to explain why a line AB and its dilation A*B*have the same slope. Hint: Draw an example. Construct a triangle ABC used to find the slope of AB. Use similarity rules and cross multiplication instead of counting grid squares." Students develop conceptual understanding of slope.
Application: Student Resource, Assessment and Practice, Part 2, Lesson G8-53, Item 11, "The Mayon volcano in the Philippines is cone shaped. The diameter of its base is 20km and the distance up the curved side, from the base to the apex , is 10.3km. Find the height of the volcano." Students engage in application when solving the word problem.
Procedural Skills and Fluency: Teacher Resource, Part 1, Unit 4, Lesson EE8-35, Exercises, "Solve the equation. a. 3x + 3 − x = 5 b. 7x + 2 = 4x + 11." Students use the distributive property to solve equations.
Teacher Resource, Part 1, Unit 2, Lesson EE8-16, Exercises, "Simplify the expression to as few powers as possible by multiplying powers with the same base. a. 4$$^2$$x 7$$^2$$x 7$$^4$$x 4$$^3$$." Conceptual understanding is developed while also practicing procedural skill.
Teacher Resource, Part 2, Unit 3, Lesson NS8-6, Exercises, Item 2, "Alex plays baseball. Last month, he was at bat 33 times and got 19 hits. How many hits did Alex get as a percentage of the number of times he was at bat?" Students develop both procedural skill and application.
"Mathematical Practices in this Unit" gives suggestions on how students can show they have met a Mathematical Practice. For example, in Teacher Resource, Part 2, Unit 5, Mathematical Practices in this Unit, "MP.4: EE8-49 Extension 6, EE8-50 Extension 5, EE8-53 Extension 1, EE8-54 Extensions 2-3, EE8-55 Extensions 1-4 In EE8-53 Extension 1, students make sense of and persevere in solving a non-routine problem when they find the area of a triangle given the equations of the lines that form the triangle by finding the base and height. STudents need to recognize the horizontal side as the base and the vertical distance to the intersection of the other two lines as the height."
"Mathematical Practices in this Unit" gives the Mathematical Practices that can be assessed in the unit. For example, in Teacher Resources, Part 1, Unit 7, Mathematical Practices in this Unit, "In this unit, you will have the opportunity to assess MP.2, MP.3, MP.5, and MP.8."
The instructional materials reviewed for JUMP Math Grade 8 partially meet expectations for carefully attending to the full meaning of each practice standard. The materials do not attend to the full meaning of MP4.
MP1: Teacher Resource, Part 1, Unit 3, Lesson G8-15, Extensions, Item 3, "Which is larger, 2$$^{75}$$ or 3$$^{50}$$? Do not use a calculator. Explain how you know your answer is correct." Students would have to persevere when solving the problem as they would likely need to try multiple strategies to determine which is larger."
MP2: Teacher Resource, Part 1, Unit 5, Lesson EE8-44, Extensions, Item 7, "When Amy cut $$\frac{3}{4}$$ of a foot off the bottom of her curtain, the curtain became $$\frac{3}{4}$$ of the length it originally was. How long was the curtain originally? Write your answer as a full sentence." Students decontextualize the problem to solve the equation they write, and they put their answer back in context at the end when they write the answer as a full sentence.
MP5: Teacher Resource, Part 2, Unit 3, Lesson NS8-1, Extensions, Item 2, "Alice buys a house for $200,000. The price of her house goes up by $8,000 every 3 years. Ken buys a house for $100,000. The price of his house doubles every 10 years. When will the two houses be worth the same amount of money? Use any tool you think will help." Students can use any tool which will help solve the problem.
MP6: Teacher Resource, Part 1, Unit 2, Lesson EE8-23, Extensions, Item 2, "a. If a year is 365.2422 days, and the universe is about 13.8 billion years old, about how many seconds old is the universe? Write your answer using scientific notation. b. What decimal hundredths might have been rounded to 13.8? c. What range of values might the number of seconds actually be, given the range of decimal hundredths that round to 13.8? d. What place value does it make sense to round your answer from part a to? Explain." Students attend to precision as they work with scientific notation to solve the problem.
MP7: Teacher Resource, Part 1, Unit 2, Lesson EE8-17, Extensions, Item 3, "Give students an easier problem: write 10$$^3$$ x 8 + 10$$^3$$ x 2 as a single power of 10. Encourage students to compute it first, and then to look back and explain why the result happened. Then encourage students to use the same technique for the harder problem." By redirecting students in this way, students make use of the structure.
MP8: Teacher Resource, Part 2, Unit 1, Lesson F8-19, Extensions, Item 1, "b. Describe what you are always doing the same in part a. c. Find a formula for finding the x-intercept of the line y = mx + b." Students are given four equations in slope intercept form and calculate the x-intercept of the line. Students use repeated reasoning to find the formula as a generalization.
Examples of the materials not carefully attending to the meaning of MP4 include:
Teacher Resource, Part 2, Unit 7, Lesson SP8-11, Extensions, Item 1, students are given a set of data about late phone charges and create a two-way table for the data. Students answer follow-up questions, "b. How many landline customers did not pay on time this year? Make a row two-way relative frequency table for the data. c. Based on the data, is there an association between the type of phone and repeated lateness? Explain." By scaffolding the questions into a step-by-step process, students do not model with mathematics.
Teacher Resource, Part 2, Unit 4, Performance Task, Fire Department Ladder Problems, Item 5, "Two firefighters need to reach the top of a building. The building is 20 feet high and has a stone wall around it. The stone wall is: 8 feet high, 4 feet out from the building, 1 foot thick. The firefighters need to bring a ladder (not one mounted on the truck). The firefighters have two options." Students are shown pictures of the two options, one with the ladder between the wall and the building, and the other with the ladder outside of the wall. Students complete the following: "a. Label each picture with the distances given. b. What is the slope of the ladder in each option? c. To be safe to climb, the absolute value of the slope of the ladder has to be less than 4. Circle the option(s) that are safe." By asking each question separately, students do not model with mathematics independently.
Teacher Resource, Part 1, Unit 4, Lesson EE8-34, Extensions, Item 5, "a. Glen says that the expression -5x - 3 is always negative because both the coefficient and the constant terms are negative. Do you agree with Glen? Why or why not? b. In pairs, discuss your answers to part a). Do you agree with each other? Discuss why or why not."
In Teacher Resource, Part 1, Unit 6, Lesson F8-15, Extensions, Item 1, students make an argument about where a y-intercept would be given a point and a slope. "A line passes through A (1, 2) with a negative slope. Can the y-intercept be negative? Why? Hint: Draw lines with negative slope from point A."
Teacher Resource, Part 2, Unit 2, Lesson G8-26, Extensions, Item 4, "A Transformation takes the point (x, y) to the point (2x, y+3). Does the transformation take the line y = 2x + 1 to another line? Explain how you know."
Teacher Resource, Part 1, Unit 6, Lesson F8-10, Extensions, Item 2, "Tessa says that when you find the rate of change from point A to point B, you always get the same answer as when you find the rate of change from point B to point A. a. Do you agree with Tessa? Why or why not? b. In pairs, explain your answers to part a. Do you agree with each other? Discuss why or why not."
Teacher Resource, Part 1, Unit 4, Lesson EE8-32, Extensions, Item 5, "a. Find a positive number x that makes the equation true: 125$$^8$$ = x$$^6$$. Explain how you got your answer? b). In pairs, compare your answers. Do you agree with each other? Discuss why or why not. c. Is your answer to part a) greater or less than 125? Explain why this makes sense."
Teacher Resource, Part 1, Unit 4, Lesson EE8-37, Extensions, Item 1, "Simplify the equation. Does the equation have one unique solution, no solution, or infinitely many solutions? a. 6x + 3 = 6x + 6 b. 7x + 1 = 43 c. 9(x+1) = 9x + 9"
Many MP3 problems in the extension sections follow a similar structure. Students are given a problem and "explain." Then, students compare their answers with a partner and discuss if they agree or not. This one dimensional approach does not offer guidance to students on how to construct an argument or critique the reasoning of others. For example, Teacher Resource, Part 2, Unit 3, Lesson NS8-6, Extensions, Item 6, "b. Explain why, in a right triangle, the side opposite the right angle is always the longest side. Use any tool you think would help. c. In pairs, explain your answers to part b. Do you agree with each other? Discuss why or why not."
Students are given extension questions when they are asked to analyze the math completed by a fictional person. For example, Teacher Resource, Part 1, Unit 4, Lesson EE8-33, Extensions, Item 4, "Marta says that the expression 5x + 20 is always a multiple of 5 because 5 and 20 are both multiples of 5. Do you agree with Marta? Why or why not?" These problems begin to develop students' ability to analyze the mathematical reasoning of others but do not fully develop this skill. Students analyze an answer given by another, but do not develop an argument or present counterexamples.
A rubric for the Mathematical Practices is provided for teachers on page K-91. For MP3, a Level 3 is stated as, "Is able to use objects, drawings, diagrams, and actions to construct an argument" and "Justifies conclusions, communicates them to others, and responds to the arguments of others." This rubric would provide some guidance to teachers about what to look for in student answers but no further direction is provided about how to use it to coach students to improve their arguments or critiques.
In the Math Practices in this Unit Sections, MP3 is listed multiple times. The explanation of MP3 in the unit often consists of a general statement. For example, in Teacher Resource, Part 1, Unit 4, the MP3 portion of the section states, "In EE8-37 Extension 4, students critique an argument when they explain where the mistake is in a fictional student's argument. Students construct an argument when they explain how to correct the mistake." These explanations do not provide guidance to teachers in how to get students to construct arguments or critique the reasoning of others.
Some guidance is provided to teachers for constructing a viable argument when teachers are provided solutions to questions labeled as MP3 in the extension questions. Some of these questions include wording that could be used as an exemplar response about what a viable argument is. For example, in Teacher Resource, Part 1, Unit 5, Lesson EE8-46, Extensions, Item 5, students are given the work of another student and asked if they agree. Teachers are provided with a sample solution, "a. I do not agree with Kyle. I solved the equation and got x = −13. In this case, 2x + 6 = − 20 and 2x + 1 = −25. It is true that 5 > 4 and −20 > −25, but it is not true that when you multiply two greater numbers, you always get a greater answer. Indeed, 5 × (−20) = −100 and 4 × (−25) = −100. Kyle would be correct if 2x + 6 and 2x + 1 were positive numbers; then it would be true that 5(2x + 6) > 5(2x + 1) > 4(2x + 1), but since 2x + 1 is negative and 5 > 4, you get 5(2x + 1) < 4(2x + 1), so the inequalities become 5(2x + 6) > 5(2x + 1) < 4(2x + 1)." In addition to the sample solution that could be used as an exemplar, teachers are also given the note, "Encourage students to not only explain why Kyle's answer is incorrect, but why his reasoning is incorrect (when you multiply both sides of the inequality 5 > 4 by the same negative number, the inequality changes direction)." This guidance would help teachers develop students' arguments and emphasizes the importance of not just explaining an answer but looking specifically at the mathematical reasoning.
In Teacher Resource, Part 2, Unit 7, Lesson SP8-6, Extensions, Item 3, students construct an argument to explain why a sphere with a given volume will or will not fit inside a box with a given, larger, volume. Students have the opportunity to critique a partner's argument. For example, if one student argues incorrectly that the ball will fit into the cube because 11.3 < 20, their partner will need to explain that they need to compare the diameter, not the radius, to the width of the box.
In Teacher Resource, Part 2, Unit 3, Lesson NS8-6, Extension, a sample answer is provided but no support on engaging students in how to analyze the reasoning of others: "b. Explain why, in a right triangle, the side opposite the right angle is always the longest side. Use any tool you think will help. c. In pairs, explain your reasoning from part b. Do you agree with each other? Discuss why or why not."
In Teacher Resource, Part 1, Unit 4, Lesson EE8-31, Extensions, Item 4, students are given the question, "Without using a calculator, show that 2$$^{100}$$ has at least 31 digits." Teachers are provided with the answer, "2$$^{10}$$ = 1,024 > 10$$^3$$, so 2$$^{100}$$ = (2$$^{10}$$)$$^{10}$$ > (10$$^3$$)$$^{10}$$= 10$$^{30}$$ , which is the smallest number with 31 digits, so 2$$^100$$ has at least 31 digits." This sample answer does not provide any assistance for developing students ability to construct viable arguments.
In Teacher Resource, Part 2, Unit 1, Lesson 8F-15, MP3 is identified in the section titled, "Drawing a line using the y-intercept and the slope (page L-18)." In this section, teachers are told to ask students a series of questions about drawing lines. "ASK: But how can we use the slope to find another point? SAY: As an example, let's draw a line with y-intercept = 2 and slope = $$\frac{1}{3}$$. First, mark the y-intercept on the y-axis. SAY: The y-intercept is 2, so we mark (0, 2) on the grid. The slope is $$\frac{1}{3}$$. ASK: What does the slope fraction stand for? ($$\frac{rise}{run}$$) SAY: The fraction of rise over run is 1 over 3, so I can say the rise is 1 and the run is 3. ASK: Do the rise and run have to be 1 and 3? (no) What other numbers would work? (2 and 6, or 3 and 9)." These questions are not developing students' abilities to construct arguments or to critique the reasoning of others.
Accurate mathematics vocabulary is present in the materials, however, while vocabulary is identified throughout the materials, there is no explicit directions for instruction of the vocabulary in the teacher materials of the lesson. Examples include, but are not limited to: | CommonCrawl |
Model selection and averaging of nonlinear mixed-effect models for robust phase III dose selection
Yasunori Aoki ORCID: orcid.org/0000-0002-5881-20231 nAff2,
Daniel Röshammar3 nAff4,
Bengt Hamrén3 &
Andrew C. Hooker1
Journal of Pharmacokinetics and Pharmacodynamics volume 44, pages581–597(2017)Cite this article
Population model-based (pharmacometric) approaches are widely used for the analyses of phase IIb clinical trial data to increase the accuracy of the dose selection for phase III clinical trials. On the other hand, if the analysis is based on one selected model, model selection bias can potentially spoil the accuracy of the dose selection process. In this paper, four methods that assume a number of pre-defined model structure candidates, for example a set of dose–response shape functions, and then combine or select those candidate models are introduced. The key hypothesis is that by combining both model structure uncertainty and model parameter uncertainty using these methodologies, we can make a more robust model based dose selection decision at the end of a phase IIb clinical trial. These methods are investigated using realistic simulation studies based on the study protocol of an actual phase IIb trial for an oral asthma drug candidate (AZD1981). Based on the simulation study, it is demonstrated that a bootstrap model selection method properly avoids model selection bias and in most cases increases the accuracy of the end of phase IIb decision. Thus, we recommend using this bootstrap model selection method when conducting population model-based decision-making at the end of phase IIb clinical trials.
Introduction and background
Quantifying the probability of achieving the targeted efficacy and safety response is crucial for go/no-go investment decision-making in a drug development program. This is particularly crucial when analyzing phase IIb (PhIIb) dose-finding studies to select the phase III dose(s) given the costs of phase III studies.
It has previously been shown that population model-based (pharmacometric) approaches can drastically increase the power to identify drug effects in clinical trial data analysis compared to conventional statistical analysis (e.g., [1]). On the other hand, the model-based approach can be hindered by model selection bias if a single model structure is assumed and used for the analysis (e.g., [2, 3]). There have been several attempts through model averaging and model selection to weaken the model structure assumptions by considering multiple possible model candidates in the analysis [4,5,6,7,8,9].
In this paper, we introduce four methods that assume a number of pre-defined model candidates and then combine or select those candidate models in different ways to make predictions and to account for uncertainty in those predictions. The first method is "simple" model selection where a set of model structures are pre-specified and a model is chosen according to a statistical criterion. Uncertainty in model prediction is then derived from parameter uncertainty, based on a bootstrap procedure using the selected model. The second method is a bootstrapped model selection procedure, where, for each bootstrap dataset, the best-fit of the candidate models is chosen according to a statistical criterion. Simulation from each bootstrap selected model with its optimal parameter will then generate a distribution of the quantities of interest, accounting for both model and parameter uncertainty (similar methods can be found in the literature, e.g., [11, 12]). The third method is a conventional model averaging procedure where each candidate model is fit to the data and uncertainty is quantified via bootstrap. Simulations (including parameter uncertainty) from each candidate model of the distributions of the quantities of interest are then combined as a weighted average depending on model fit to the original data. The fourth method is a bootstrapped model averaging procedure, where the weighting for the weighted average calculations are based on model fit to each bootstrapped dataset (as opposed to the model-fit to the original data).
Comparison of these methods and a standard statistical method (pair-wise ANOVA and the groupwise estimate of an average change from baseline) are done using clinical trial simulations of dose-finding studies. To make the simulations as realistic as possible, we have based them on the protocol of an actual PhIIb trial for an oral asthma drug candidate (AZD1981) as well as the data from the placebo arm of that trial. Drug effects using various model structures were simulated for five different dose arms (placebo plus four active arms). The different analysis methods were then used to calculate the probability of achieving target endpoint and then choose the minimum effective dose (MED).
Phase IIb dose-finding case study
Part of the PhIIb clinical trial data and the study protocol of the asthma drug candidate AZD1981 (ClinicalTrials.gov/NCT01197794) was utilized in this work. One endpoint goal of the study was to demonstrate that the drug improved the forced expiratory volume in 1 s (FEV1) of asthma patients by, on average, at least 0.1 L (placebo and baseline adjusted). This clinical trial was chosen as a case study since FEV1 is a highly variable endpoint (standard deviation of 0.3 L in the placebo effect) relative to the expected effect magnitude; hence it is hard to characterize the dose–effect relationship from PhIIb clinical trials.
This study was conducted for 12 weeks and FEV1 was measured every 2 weeks (for a total of 7 measurements, or visits). The first measurement was a screening visit and the second measurement was used as a baseline measurement after which either placebo, AZD1981 10, 20, 100 or 400 mg was administered twice daily (bid).
The data from the placebo group and the lowest dose group of the PhIIb clinical trial for AZD1981 was provided for this analysis. Dosing information was not provided; however, as there were no statistically significant differences between the placebo group and the lowest dose group as described in [13], in this paper we refer this dataset as a "placebo" dataset. This dataset comprises 324 patients with a total of 1803 FEV1 measurements.
Placebo model
The following placebo model was developed using the placebo dataset from the PhIIb clinical trial for AZD1981:
$$\begin{aligned} {\text{FEV}}1 & = \left( {{\text{FEV}}1_{\text{Baseline}} + \left\{ {\begin{array}{*{20}l} 0 \hfill & {{\text{if visit}} = 1,2} \hfill \\ {{\text{FEV}}1_{\text{Placebo}} } \hfill & {{\text{if visit}} = 3,4,5,6,7} \hfill \\ \end{array} } \right\}} \right) \cdot (1 + \epsilon_{1} ) + \epsilon_{2} \\ {\text{FEV}}1_{\text{Placebo}} & = \theta_{1} + \eta_{1} \\ {\text{FEV}}1_{\text{Baseline}} & = \theta_{2} e^{{\eta_{2} }} \\& \quad \times (1 + \theta_{3} ({\text{FEV}}1_{{{\text{\% of normal}}}} - \overline{{{\text{FEV}}1}}_{\text{\% of normal}})) \\& \quad \times (1 + \theta_{4} ({\text{Age}} - \overline{\text{Age}} )) \\& \quad \times \left\{ {\begin{array}{*{20}l} 1 \hfill & {\text{if Male}} \hfill \\ {\theta_{5} } \hfill & {\text{if Female}} \hfill \\ \end{array} } \right. \\ \eta_1 & \sim {\mathcal{N}}(0,\omega_{1}^{2} ) \\ \eta_2 & \sim {\mathcal{N}}(0,\omega_{2}^{2} ) \\ \epsilon_1 & \sim {\mathcal{N}}(0,\sigma_{1}^{2} ) \\ \epsilon_2 & \sim {\mathcal{N}}(0,\sigma_{2}^{2} ) \\ \end{aligned}$$
where \({\text{FEV}}1_{{{\text{\% of normal}}}}\) is the percentage of FEV1 at visit 2 compared to the predicted normal and \(\overline{{{\text{FEV}}1}}_{\text{\% of normal}}\) is its population mean, \(\overline{\text{Age}}\) is the mean of the age of the patients. All the estimated model parameters can be found in Table 1.
Table 1 Estimated Parameters of the placebo model of FEV1 of Asthma patients based on the placebo and lowest dose group of the PhIIb clinical trial for AZD 1981
Previously Wang et al. [14] have modelled a placebo effect on the FEV1 measurement. The model presented here differs slightly from Wang et al. in that this model employs a step function for the placebo effect model with respect to visit, while Wang et al. have used exponential models with time as the independent variable. Wang et al. state that the placebo effect plateaus at 0.806 \({\text{week}}^{ - 1}\) while the current dataset contains FEV1 measurements only every 2 weeks; hence the rate constant of the exponential model was not estimable from this dataset.
Drug effect models
In this work, we simulate and estimate using a number of different dose–effect relationships DEj:
$$\begin{aligned} {\text{DE}}_{0} &= 0\; ( {\text{no treatment effect)}} \hfill \\ {\text{DE}}_{1} ({\text{dose}};p_{1} ) &= p_{1} \cdot {\text{dose}}\; ( {\text{linear model)}} \hfill \\ {\text{DE}}_{2} ({\text{dose}};p_{1} ,p_{2} ) &= p_{1} \cdot { \log }(1 + p_{2} {\text{dose}})\; ( {\text{log-linear model)}} \hfill \\ {\text{DE}}_{3} ({\text{dose}};{\text{EMAX}},{\text{EC}}50) &= {\text{EMAX}} \cdot \frac{\text{dose}}{{{\text{EC}}50 + {\text{dose}}}}\; ( {\text{Emax model)}} \hfill \\ {\text{DE}}_{4} ({\text{dose}};{\text{EMAX}},{\text{EC}}50,\gamma ) &= {\text{EMAX}} \cdot \frac{{{\text{dose}}^{\gamma } }}{{{\text{EC}}50^{\gamma } + {\text{dose}}^{\gamma } }}\; ( {\text{sigmoid Emax model)}} \hfill \\ \end{aligned}$$
To create simulated datasets, we add different simulated drug effects, with different parameters, using the above models, to the FEV1 measurements of the placebo data (more detail below). For estimation using the model-based analysis methods described below, we embed these dose–effect relationships into the placebo model as follows:
$${\text{FEV}}1_{j} = \left( {{\text{Baseline}} + {\text{PlaceboEffect}} + {\text{DE}}_{j} } \right) \cdot (1 + \epsilon_{1} ) + \epsilon_{2}$$
Analysis methods
Statistical analysis used for the PhIIb clinical trial for AZD1981
The primary statistical analysis of the data from the PhIIb clinical trial for AZD1981 to determine the MED was performed using a pair-wise ANOVA and a group wise estimate of treatment effect. Briefly, the treatment effect was measured as the change from baseline (average of all available data from visits 3–7 minus baseline) per dose group. The MED was identified via a two-stage step-down procedure to select either 400, 100, 40, 10 mg or "stop" (do not proceed to phase III). The procedure was as follows: (1) starting with the highest (400 mg) dose-arm conduct a one-sided ANOVA comparison with the placebo-arm. (2) If the difference is significant (significance level of 0.05) check that the average treatment effect in the arm is greater than the primary efficacy variable (0.1 L). (3) If both steps 1 and 2 are satisfied then proceed to the next dose dose-arm (100 mg) and repeat, otherwise move to step 4. (4) Choose the lowest dose arm where both steps 1 and 2 are satisfied (Note that if 100 mg satisfies steps 1 and 2 but 40 mg does not then the MED will be 100 mg in this process, even if 10 mg might also satisfy steps 1 and 2).
Model selection and averaging analysis methods
Below is an overview of four methods that assume a number of pre-defined model candidates and then combine or select those candidate models in different ways to make predictions and to account for uncertainty in those predictions. For the given example, the methods were meant to compare with the standard determination of the MED, identified in the original study using the two-stage step-down procedure described earlier in this section. Thus, in the following methods, there should be a test for drug effect as well as a determination if that effect is greater than a given minimum effect size. In all methods, the test for drug effect is done using a likelihood ratio test (LRT) against the placebo model (5% significance level) [8]. Determination of effect sizes at specific doses is done by first computing the change from baseline average (population mean) effect size, and uncertainty around that effect size, predicted by the different methods described below, for a given dose. MED is then chosen as the lowest studied dose with a predefined probability to be above a target effect. For a full technical description of the methods, we refer the readers to the Appendix. The methods below are previously presented by the authors as a conference contribution [9], and Method 3 was presented at an earlier conference [10].
Method 1: model selection
Fit each candidate model structure to the original data and estimate the model parameters and maximum likelihood (see Fig. 1).
Method 1 model selection
For each candidate model, perform a numerical identifiability test (see appendix for detail) and LRT against the placebo model and reject any model structure that fails either of these tests.
Select one model structure among the remaining model candidates using a statistical criterion based on the maximum likelihood (e.g., the Akaike information criterion, AIC, the Bayesian information criterion, BIC, etc.).
Quantify parameter uncertainty using case sampling bootstrap and the selected model structure.
Simulate the quantities of interest (with uncertainty); in this case, the dose-endpoint (change from baseline population mean effect size) relationships using the selected model structure and model parameters obtained from the bootstrap procedure.
Make a decision; in this case, choose the lowest dose (given allowed dose levels) that has a probability of achieving target endpoint greater than a predefined limit.
Method 2: bootstrap model selection
Create bootstrap datasets based on the original data using a case sampling bootstrap procedure (see Fig. 2).
Method 2 bootstrap model selection
For each bootstrap dataset estimate parameters and the maximum likelihood for each candidate model structure.
For each bootstrap dataset, and for each candidate model, perform a numerical identifiability test and LRT against the placebo model and reject any model structure that fails either of these tests.
For each bootstrap dataset, select one model structure among the remaining model candidates using a statistical criterion based on the maximum likelihood (e.g., AIC, BIC, etc.).
For each bootstrap dataset, simulate the quantities of interest; in this case, the dose-endpoint (change from baseline population mean effect size) relationships using the selected model structure and model parameters obtained from that bootstrap dataset.
Summarize the simulations; in this case, compute the probability of achieving the target endpoint at each dose of interest using the simulated dose-endpoint relationships.
Make a decision; in this case, choose the lowest dose (given allowed dose levels) that has a probability of achieving the target endpoint greater than a predefined limit.
Method 3: model averaging
Fit each candidate model structure to the original data and estimate model parameters and maximum likelihood (see Fig. 3).
Method 3 model averaging
For each candidate model, perform a numerical identifiability test and LRT against the placebo model and reject any model structure that fails either of these tests.
For each model structure, quantify parameter uncertainty using case sampling bootstrap methodology to obtain the distribution of the model parameters.
For each model structure, simulate the quantities of interest; in this case, the dose-endpoint relationships using the model parameters obtained from the bootstrap method.
For each model structure, summarize the simulations; in this case, compute the probability of achieving the target endpoint at each dose of interest using the simulated dose-endpoint relationships.
Compute the weighted average of the summary variables obtained in step 5; in this case, the probability of achieving the target endpoint at each dose over the model structures, where the weights are derived from the maximum likelihood obtained in step 1 (e.g., AIC, BIC, etc.).
Method 4: bootstrap model averaging
Method 4 bootstrap model averaging
For each bootstrap dataset and each model structure, simulate the quantities of interest; in this case, the dose-endpoint (change from baseline population mean effect size) relationships using the selected model structure and model parameters obtained from that bootstrap dataset.
Summarize the simulations; in this case, compute the weighted average of the probability of achieving the target endpoint at each dose using the dose-endpoint relationships for all the model structures and all the bootstrap datasets (except the ones that failed the LRT or identifiability test). The weights are derived from the maximum likelihood obtained in step 2 (using AIC, BIC, etc.).
Single model based approach
To compare the proposed methods against the idealized situation where the underlining true model structure is known before the analysis, we compare with a single model based approach where the model used to analyze the dataset is the same as the model used to simulate that dataset. Note that this single model based analysis using the simulation model is an idealistic scenario. In a real PhIIb dataset analysis (i.e., when analyzing data that was not simulated) it is not realistic to assume the exact underlying model structure is known a priori. The method has the following steps:
Perform LRT between the model with and without drug effect. If the model does not pass the LRT, make a "stop" decision.
If the model with drug effect passes the LRT, estimate the parameter uncertainty using a case sampling bootstrap.
Simulate the quantities of interest (with uncertainty); in this case, the dose-endpoint (change from baseline population mean effect size) relationships using the model parameter distribution obtained from the bootstrap procedure.
Make a decision; in this case, select the dose based on the required probability of achieving the target endpoint.
Numerical experiments
To test the proposed model averaging and selection methodologies, we have simulated dose-finding studies under various designs and experimental scenarios. All numerical computations were done using NONMEM [15] version 7.3, PsN [16] version 4.6 on a Linux Cluster, with Intel Xeon E5645 2.4 GHz processors, 90 GB of memory, Scientific Linux release 6.5, GCC 4.4.7 and Perl version 5.10.1. To assure reproducibility of the numerical experiments we had a fixed random seed when the bootstrap method was performed using PsN. All computation outside of NONMEM and PsN was done using R version 3.2 [17] and all plots are made using ggplot2 [18].
Simulation studies based on placebo data
To create simulated datasets, we have simply added different simulated drug effects to the FEV1 measurements of the placebo data. We have randomly generated the artificial drug effect so that the theoretical minimum effective dose (tMED, i.e., the exact dose that achieves a drug effect of 0.1 L) is uniformly distributed in the ranges shown in Table 2.
Table 2 Various scenarios of the simulation studies
For each Simulation Study 1–5, we have constructed 300 PhIIb clinical trial simulation datasets (1500 datasets in total). Simulation Studies 1–4 are constructed to test each analysis method for the accuracy of finding tMED, while Simulation Study 5 is constructed to test each method for the accuracy of Type-1 error control.
In each Simulation study, log-linear, emax and sigmoidal models (described above) were used to simulate the drug effects (DEj) (100 datasets each for each of three model structures, hence 300 total simulated datasets in one simulation study). For each data set, we first randomly choose tMED in the range shown in Table 2. Then the model parameters are chosen randomly as follows:
For the log-linear model, \(p_{1}\) and \(p_{2}\) are chosen so that
$$\begin{aligned} {\text{DE}}_{2} (1000,p_{1} ,p_{2} )&\sim {\text{unif}}(0.2,0.3) \hfill \\ {\text{DE}}_{2} ({\text{tMED}};p_{1} ,p_{2} ) &= 0.1 \hfill \\ \end{aligned}$$
For the emax model, \({\text{EMAX}}\) and \({\text{EC}}50\) are chosen so that
$$\begin{aligned} {\text{EMAX}}&\sim {\text{unif}}(0.2,0.3) \hfill \\ {\text{DE}}_{3} ({\text{tMED}};{\text{EMAX}},{\text{EC}}50) &= 0.1 \hfill \\ \end{aligned}$$
For the sigmoidal model, \({\text{EMAX}}\), \({\text{EC}}50\) and \(\gamma\) are chosen so that
$$\begin{aligned} {\text{EMAX}}&\sim {\text{unif}}(0.2,0.3) \hfill \\ \gamma &\sim {\text{unif}}(0.5,4) \hfill \\ {\text{DE}}_{4} ({\text{tMED}};{\text{EMAX}},{\text{EC}}50,N) &= 0.1 \hfill \\ \end{aligned}$$
Note that to determine the parameters \(p_{1}\) and \(p_{2}\) for the log-linear model, we need to solve a nonlinear equation numerically and we do so by using the uniroot function in R. As can be seen in Fig. 5, we can create diverse realistic simulated drug effects by the above choice of model parameters while the range of tMED is constrained.
Plot of (some of) the simulated drug effect for Simulation Study 3. The theoretical minimum effective dose (the exact dose that achieves the target endpoint of 0.1 L) ranges between 40 and 100 mg hence the 100 mg dose is the correct dose selection for this simulation study
Numerical experiment 1: dose finding accuracy
The simulated data from Simulation Studies 1–4 (when a drug effect is present) was analyzed using the methods presented above to determine the dose finding accuracy of the methods. Each method was used to find the MED for each trial simulation dataset and the probability of finding the correct dose was calculated (see Table 2).
For the model-based approaches, the MED dose was chosen as the minimum dose arm (of the investigated doses) with more than a 50% probability of achieving the target endpoint. 50% was chosen to match the statistical analysis used for the PhIIb clinical trial for AZD198, which evaluates if the average treatment effect in a dose arm is greater than the primary efficacy variable.
Numerical experiment 2: type-1 error control accuracy
All methods presented above were used to determine the MED based on the data from Simulation Study 5 (the simulation study without simulated drug effect) to test the type-I error rate of the proposed methods. That is, the probability of choosing the MED to be either 10, 40, 100, or 400 mg while there is no simulated drug effect. The MED selection using the model-based approaches were determined at a 50% confidence level to fairly compare the method with the pairwise ANOVA method.
Numerical experiment 3: decision-making accuracy
In the previous two numerical experiments the MED using the model-based approaches are determined at a 50% confidence level to fairly compare the method with the pairwise ANOVA method. However, in reality, more than 50% certainty may be desired when making a decision about which dose to use in a phase III trial [20]. For example, from an investment perspective, it may be more crucial to reduce the risk of proceeding to a phase III trial with insufficient effect than to determine the exact MED of a drug.
For this experiment, we define the "correct" decision to be when any dose higher than the theoretical MED is selected. For example, for Simulation Study 3 (\( 40\;{\text{mg}} < {\text{tMED}} \le 100\;{\text{mg}} \)), if either 100 or 400 mg is chosen then the correct decision was made; while if dose 10 or 40 mg is chosen, or a "stop" decision is made, then the incorrect decision was made. Each method was then used to find the MED (70% confidence level for the model-based approaches) for each simulated dataset from Simulation Studies 1–4 (when a drug effect is present). The probability of each method making the correct decision was then calculated.
Numerical experiment 4: probability of achieving target endpoint estimation accuracy
In the model averaging and selection methods investigated here, the dose selection is based on the probability of achieving the target endpoint, hence, accurate estimation of this probability is crucial. In this experiment, we investigate this probability estimation for each simulated dataset from Simulation Studies 1–4 (when a drug effect is present) in the following manner:
Select a predefined limit, p, for the probability of achieving the target effect.
Allow any dose (any positive real number) to be selected (not just the investigated dose levels) and choose the dose that is estimated to achieve the target endpoint with probability \(p\) using the proposed model-based methods.
Repeat steps 1 and 2 for all 1200 simulated phase IIb datasets and count the number of times a dose above the theoretical minimum effective dose (tMED) is selected, from which the empirical probability of achieving the target effect is calculated.
Repeat steps 1–3 for \(p = 0.01, 0.02, \ldots , 0.99\).
Note that if a method estimates the probability of achieving the target endpoint without bias, then the selected doses should be above tMED with probability \(p\).
To concisely present the results for each of Numerical Experiments 1, 3, and 4, we has combined the results of Simulation Studies 1–4. Hence, for those experiments, the results are based on 1200 PhIIb clinical trial simulations. We refer the readers to the Appendix for a detailed discussion of the result for each simulation study. Further, the uncertainty of the numerical experiments has been quantified by randomly sampling trial simulations with replacement (1200 trial simulations for Numerical Experiments 1, 3, and 4, and 300 trial simulations for Numerical Experiment 2) and repeated the numerical experiments. For example, for Numerical Experiments 1, 3, and 4, 1200 trial simulations were sampled with replacement 100 times to produce 100 sets of the 1200 trial simulations. For each set of trial simulations, the numerical experiments were performed.
The dose finding accuracy of the various investigated methods is presented in Fig. 6. As can be seen, all the model based methods could find the correct dose more often than the statistical method used in the PhIIb AZD1981 study protocol. In addition, we can see that Methods 2 and 4 outperform Methods 1 and 3 and the Single Model Based approach (using the simulation model).
Probability of finding the correct dose. The edges of the boxes are 75th and 25th percentiles. The line in the box is the median and the whiskers extend to the largest and the smallest value within 1.5*inter-quartile range. Dots are the outliers outside of the whiskers
The Type-I error control of the various investigated methods is presented in Fig. 7. As can be seen, Methods 1–4 control the type-I error accurately. Furthermore, we can see that the LRT is necessary for Methods 1, 2, and 4 to properly control the Type-1 error. Lastly, we see that the type-I error is lower than expected for the standard statistical test and Single Model Based method (using the Simulation Model).
Type-1 error rate, the probability of choosing either 10 mg, 40 mg, 100 mg, or 400 mg while there is no simulated drug effect. The significance level of all the methods was set to 0.05 hence if the Type-1 error is correctly controlled the Type-1 error rate should be at 5% (indicated by the horizontal line)
The decision-making accuracy of the various investigated methods is presented in Fig. 8. As can be seen, all model based method (Methods 1–4 and the Single Model Based method) makes the correct decision more often than the Statistical method employed in the AZD 1981 study protocol. Also, we can see that Method 4 performs relatively poorly compared to Methods 1–3.
Probability of making the correct decision (the correct decision is defined as choosing the dose that is above tMED)
The Probability of achieving target endpoint estimation accuracy of the various investigated model-based methods is presented in Fig. 9. Note that if the investigated method estimates the probability of achieving the target endpoint without bias then the QQplot in Fig. 9 should follow the line of unity.
The accuracy of the calculated probability of achieving a target endpoint. The x-axis is the predefined limit for the probability of achieving target endpoint where the dose was chosen. The y-axis is the probability that the chosen dose by the various methods is above tMED. If the probability of achieving the target endpoint is estimated without bias, the plot should lie on the line of identity (red straight line). Grey shaded areas are 95% confidence intervals calculated by the random sampling with replacement of the 1200 trial simulation datasets
As can be seen in Fig. 9, Methods 2 and 4, using AIC as the statistical criteria in the methods, can calculate the probability of achieving target endpoint accurately. The bias on the calculated probability of achieving the target endpoint of the conventional model selection method (Method 1) is clearly observed. As discussed in literature (e.g., [2, 3]), if model selection is made based on one dataset the bias in the model selection procedure will be carried forward to subsequent analyses and any resulting quantity may be biased. Although the regular model averaging method (Method 3) should significantly decrease the effect of model selection bias, we still observe the presence of bias. Lastly, we observed that AIC is a more suitable statistical criterion than BIC for the proposed model averaging and selection methods.
This work presents model averaging and selection methods that incorporate both model structure and parameter estimation. We have tested the proposed methods through realistic PhIIb dose finding and decision-making scenarios and demonstrated that the proposed methods could help increase the overall probability of making the correct decision at the end of PhIIb studies.
Through all the numerical experiments, the model based approaches (Methods 1–4 and Single Model Based method) outperformed the pairwise ANOVA based method used in the AZD1981 study protocol. Numerical Experiments 1 and 4 have shown that Methods 2 and 4 perform better than other methods for finding MED and estimating probability of achieving endpoint. Numerical Experiment 3 has shown that Method 2 can be used to make the investment decision more accurately than Method 4. Experiment 2 has shown that Type-1 error can be appropriately controlled using the LRT and the Type-1 error control of Method 2 is marginally better than the other methods (Method 1, 3 and 4). Thus, within the scope of our numerical experiment, Method 2 was the most accurate and precise compared to the other tested methods.
The numerical experiments indicated that AIC is a more suitable statistical criterion than BIC for the model averaging and selection methods we have tested. BIC takes the number of observations into account when weighing the penalty for the extra degrees of freedom. For nonlinear mixed effect models, the informativeness of the dataset not only depends on the number of observations but also a number of individuals. Hence, we conjecture that, by naively using the number of observations, BIC does not properly weigh the penalty term and some other way of quantifying the 'informativeness of observations' is necessary.
Although we have conducted a wide-range of numerical experiments within the scope of this project, we believe the accumulation of more experiences of these and other methods through applying them to more scenarios would be desirable. For example, it would be interesting to compare and/or integrate the methods presented here with the MCP-Mod approach [21, 22]. The MCP-Mod methodology allows model averaging and selection methods for the "Mod" portion of that framework, but entails an initial multiple comparison procedure (the "MCP" portion) that may be redundant with the LRT used here. To promote the application and further development of the proposed methodologies, we have made the methodologies investigated here available as a GUI based open source software (available at www.bluetree.me and the Mac App Store, app name: modelAverage) as well as an R script supplied as the supplementary material of this paper.
We recommend the use of the bootstrap model selection method (Method 2) presented in this paper when conducting model-based decision-making at the end of phase IIb study. The studies here indicate the proposed method reduces the analysis bias originating from model selection bias of single model structure based analyses. As a consequence of including model structure uncertainty, the quantified uncertainty may appear to be larger than single model based uncertainty; however, the method appears to more accurately reflect the true uncertainty of the investigated models and estimated parameters. The proposed method increases the probability of making the correct decisions at the end of phase IIb trial compared to conventional ANOVA-based Study Protocols.
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Authors would like to thank Professor Mats O. Karlsson for useful discussions, the Methodology group of the Uppsala University Pharmacometrics Group for discussions and Ms. Ida Laaksonen for refining the diagrams in this work.
Yasunori Aoki
Present address: National Institute of Informatics, Tokyo, Japan
Daniel Röshammar
Present address: SGS Exprimo, Mechelen, Belgium
Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Yasunori Aoki & Andrew C. Hooker
Quantitative Clinical Pharmacology, Innovative Medicines and Early Development, IMED Biotech Unit, AstraZeneca, Gothenburg, Sweden
Daniel Röshammar & Bengt Hamrén
Bengt Hamrén
Andrew C. Hooker
Correspondence to Yasunori Aoki.
Below is the link to the electronic supplementary material. the following two supplementary materials are missing from the list below: AOKI_etal_ModelAveraging_RScript.docx , AOKI_etal_ModelAveraging_RScript.rmd
Supplementary material 1 (CSV 2 kb)
Supplementary material 10 (CSV 4 kb)
Supplementary material 11 (DOCX 62 kb)
Supplementary material 12 (RMD 23 kb)
Appendix: Detailed description of the method
In this section, we present step by step explanations of our model selection and averaging methodologies. We have implemented this methodology in the C++ language, and an open source software with a graphical user-interface is available at www.bluetree.me and the Mac App Store (app name: modelAverage). Also, easy to read (computationally not optimized) R script used for the numerical experiments presented in this is available as the supplementary material.
Prior to the analysis
We assume that prior to the analysis there are multiple candidate models pre-specified before the collection of the data, and subsequently collected study data is available.
We denote \(X_{0}\) to be the independent variables (e.g., patient number, dosage, observation time, covariates) of the original dataset, and \({\varvec{y}}_{0}\) be the dependent variable (e.g., observed biomarker values). For simplicity, we consider \(X_{0}\) to be \(N_{\text{records}} \times N_{{{\text{ind}}\_{\text{variables}}}}\) matrix and \({\varvec{y}}_{0}\) to be a vector of \(N_{\text{records}}\) elements. We denote candidate model structures as \({\text{model}}_{i}\; ({\varvec{x}};{\varvec{\theta}},{\varvec{\eta}},\varvec{ \epsilon })\) where \({\varvec{x}}\) is a vector of independent variables, \({\varvec{\theta}}\) is a vector of fixed effect parameters, \({\varvec{\eta}}\) is a vector of random effect variables related to individual variabilities, and \(\varvec{ \epsilon }\) is a vector of random variable related to residual (unexplainable) variabilities. We denote the dose–effect relationship embedded in model structure \({\text{model}}_{i}\) by \({\text{DE}}_{i} \;({\text{dose}};{\varvec{\theta}},{\varvec{\eta}})\). We assume we have \(N_{\text{model}}\) candidate models and we denote the placebo model by \({\text{model}}_{0}\) hence we have \(N_{\text{model}} + 1\) models in total.
Create bootstrap datasets based on the original data and estimate parameters for each bootstrap dataset
Construct bootstrap datasets based on \((X_{0} ,{\varvec{y}}_{0} )\) and we denote them as \(\{ (X_{i} ,{\varvec{y}}_{i} )\}_{i = 1}^{{N_{\text{bootstrap}} }}\). We have used case sampling bootstrap in our numerical experiment; however, it can be extended to other types of bootstrap methods.
Estimate parameters and the maximum likelihood from each bootstrap dataset and estimate maximum likelihood parameters for each model for each bootstrap dataset and denote them as \(({\widehat{\varvec{\theta }}}_{ij} ,{\widehat{\Omega}}_{ij} ,\widehat{\Sigma }_{ij} )\) for \(i = 0, \ldots ,N_{\text{bootstrap}}\) and \(j = 0, \ldots ,N_{\text{model}}\), i.e.,
$$({\widehat{\varvec{\theta }}}_{ij} ,\widehat{\Omega }_{ij} ,\widehat{\Sigma }_{ij} ) = {\text{argmax}}({\varvec{\theta}},\Omega ,\Sigma) l({\text{model}}_{j} ( \cdot ;{\varvec{\theta}}, \cdot , \cdot );\Omega ,\Sigma ;X_{i} ,{\varvec{y}}_{i} ),$$
where \(l\) is a likelihood function for the nonlinear mixed-effect model (we refer the readers to [23] for more detailed discussion and approximation methods for this likelihood function), and we denote the maximum likelihood as a \(\widehat{l_{ij}}\), i.e.,
$$\begin{array}{*{20}c} {\widehat{l}_{ij} } = {l({\text{model}}_{j} ( \cdot ;\widehat{\varvec{\theta}}_{ij} , \cdot , \cdot );\widehat{\Omega }_{ij} ,\widehat{\Sigma }_{ij} ;X_{i} ,{\varvec{y}}_{i} ).} \\ \end{array}$$
Conduct numerical identifiability test and LRT
In order to have a rigorous Type-I error control in our model selection and averaging methods, each model that we use is subject to the LRT against the placebo model. That is to say, we have imposed the following to the estimated likelihood:
$$\begin{array}{*{20}c} {\widehat{l}_{ij} = \left\{ {\begin{array}{*{20}l} 0 \hfill & { {\text{if}}\, |\widehat{l}_{ij} - \widehat{l}_{i0} | < \chi_{0.05}^{2} ({\text{df}}),} \hfill \\ {\widehat{l}_{ij} } \hfill & {{\text{otherwise}}.} \hfill \\ \end{array} } \right.} \\ \end{array}$$
where \(\widehat{l}_{i0}\) is the estimated likelihood of the placebo model, \({\text{df}}\) is the degree of freedom of the Chi square distribution that is calculated as the number of the dose–effect relationship related parameters (i.e., linear: \({\text{df}} = 1\), logLinear: \({\text{df}} = 1\), emax: \({\text{df}} = 2\), sigmoidal \({\text{df}} = 3\)). Also, to reduce the chance of contaminating the model averaging and selection by non-identifiable models, we conduct numerical identifiability test to remove the models that are locally-practically non-estimable from a bootstrap dataset. We do so by re-estimating the model parameters using preconditioning [19]. We denote the estimated parameter and maximum likelihood using preconditioning by \((\widetilde{{\varvec{\theta}}}_{ij} ,\widetilde{\Omega }_{ij} ,\widetilde{\Sigma }_{ij} )\) and \(\widetilde{l}_{ij}\), respectively. We reject the model by setting the likelihood to be zero if \(\widehat{\varvec{\theta }}_{ij}\) and \(\widetilde{{\varvec{\theta}}}_{ij}\) are significantly different while \(\widehat{l}_{ij}\) and \(\widetilde{l}_{ij}\) are similar. In particular, for our numerical experiment, we have imposed the following:
$$\begin{array}{*{20}c} {\widehat{l}_{ij} = \left\{ {\begin{array}{*{20}l} 0 \hfill& { {\text{if }}|\widehat{l}_{ij} - \widetilde{l}_{ij} | < 0.1 {\text{ and }}\left| {\frac{{{\widehat{\varvec{\theta }}}_{ij} - \widetilde{{\varvec{\theta}}}_{ij} }}{{{\widehat{\varvec{\theta }}}_{ij} + \widetilde{{\varvec{\theta}}}_{ij} }}} \right|_{ \inf } > 0.10,} \hfill \\ {\widehat{l}_{ij} } \hfill & { {\text{otherwise}}.} \hfill \\ \end{array} } \right.} \\ \end{array}$$
where the division of \(\displaystyle{\frac{{{\widehat{\varvec{\theta }}}_{ij} - \widetilde{{\varvec{\theta}}}_{ij} }}{{{\widehat{\varvec{\theta }}}_{ij} + \widetilde{{\varvec{\theta}}}_{ij} }}}\) is the elementwise division of the vectors.
We acknowledge that the presented numerical identifiability test can only provide the evidence of non-estimability and does not necessarily prove the estimability of the model parameters; however, we have observed that this simple identifiability test has successfully reduced the number of non-estimable models included in the model averaging and selection schemes.
Simulate the quantities of interest (e.g., dose-endpoint relationships)
In this case, we construct the dose-endpoint relationships based on the estimated parameters in Step 2 and the definition of the model based clinical trial endpoint. Construct the estimated dose-endpoint relationships for each bootstrap dataset for each model and denote them as \(h_{ij} ({\text{dose}})\). For example, if the endpoint is defined as the average effect like this FEV1 case study
$$h_{ij} ({\text{dose}}) = E_{{\varvec{\eta}}} ({\text{DE}}_{j} ({\text{dose}};{\varvec{\theta}}_{ij} ,{\varvec{\eta}});\Omega_{ij} ),$$
where \(E_{{\varvec{\eta}}} ( \cdot ;\Omega )\) denotes expectation over \({\varvec{\eta}}\) with \({\varvec{\eta}} \sim {\mathcal{N}}(\Omega )\). Other choices of the endpoint definition would be the median or percentile of \(({\text{DE}}_{j} ({\text{dose}};{\varvec{\theta}}_{ij} ,{\varvec{\eta}});\Omega_{ij} )\).
Depending on the definition of the endpoint and the structure of the dose–effect relationship with respect to \({\varvec{\eta}}\), a stochastic simulation may be required to compute \(h_{ij} ( \cdot )\). The candidate drug effect for this case study is linear with respect to \({\varvec{\eta}}\) and the end point defined by the study protocol is an average over the population, which we can analytically determine, \(h_{ij} ({\text{dose}})\).
Summarize the simulations
Based on the computed likelihood \(\widehat{l}_{ij}\) and the dose-endpoint relationship \(h_{ij} ({\text{dose}})\), we compute the probability of achieving target endpoint versus dose relationship. In this step, we need to choose a weighting scheme where models are selected or averaged. We denote this weight function as \(w_{j}\) and it will depend on the likelihood \(\widehat{l}_{ij}\) and the structure of the model (i.e., the number of model parameters). We denote the weight of the \(i\)th bootstrap sample with Model \(j\) as \(w_{ij}\).
For the weights calculated based on AIC, we let \(w_{ij}\) to be the following:
$$w_{ij} = w_{j} (\widehat{l}_{ij} ) = \exp \left( {\ln (\widehat{l}_{ij} ) - N_{{{\text{para }}j}} } \right)$$
where \(N_{{{\text{para}}\,j}}\) is the number of parameters of Model \(j\).
For the weights calculated based on BIC, we let \(w_{ij}\) to be the following:
$$w_{ij} = w_{j} (\widehat{l}_{ij} ) = { \exp }\left( {{ \ln }(\widehat{l}_{ij} ) - N_{{{\text{para}}\,j}} { \ln }(N_{\text{obs}} )/2} \right).$$
where \(N_{\text{obs}}\) is the number of observations (total number of FEV1 measurements in a dataset).
Using this weight function, we can define the probability of achieving the target endpoint \(p({\text{dose}})\) as follows:
$$p({\text{dose}}) = \mathop \sum \limits_{{i = 0, \ldots ,N_{\text{bootstrap}} }} \left\{ {\begin{array}{*{20}l} {1/(1 + N_{\text{bootstrap}} )} \hfill & {{\text{if }}h_{ik} ({\text{dose}}) > {\text{TV}}} \hfill \\ 0 \hfill & {{\text{otherwise}} .} \hfill \\ \end{array} } \right.$$
where \(k = {\text{argmax}}_{j} (w_{0j} ).\)
Method 2: model selection using bootstrap maximum likelihood
$$p({\text{dose}}) = \mathop \sum \limits_{{i = 0, \ldots ,N_{\text{bootstrap}} }} \left\{ {\begin{array}{*{20}l} {1/(1 + N_{\text{bootstrap}} )} \hfill & { {\text{if }}h_{{ik_{i} }} ({\text{dose}}) > {\text{TV}}} \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right.$$
where \(k_{i} = {\text{argmax}}_{j} (w_{ij} )\).
$$p({\text{dose}}) = \mathop \sum \limits_{{i = 0, \ldots ,N_{\text{bootstrap}} \,j = 0, \ldots ,N_{\text{model}} }} \left\{ {\begin{array}{*{20}l} {\frac{{w_{0j} }}{{\mathop \sum \nolimits_{{j = 0, \ldots ,N{\text{model}}}} w_{0j} }}} \hfill & {{\text{if }}h_{ij} ({\text{dose}}) > {\text{TV}}} \hfill \\ 0 \hfill & {{\text{otherwise}}.} \hfill \\ \end{array} } \right.$$
Method 4: model averaging using bootstrap maximum likelihood
$$p({\text{dose}}) = \mathop \sum \limits_{{i = 0, \ldots ,N_{\text{bootstrap}} \,j = 0, \ldots ,N_{\text{model}} }} \left\{ {\begin{array}{*{20}l} {\frac{{w_{ij} }}{{\mathop \sum \nolimits_{{j = 0, \ldots ,N_{\text{model}} }} w_{ij} }}} \hfill & {{\text{if }}h_{ij} ({\text{dose}}) > {\text{TV}}} \hfill \\ 0 \hfill & {{\text{otherwise}}.} \hfill \\ \end{array} } \right.$$
Detailed analyses of numerical experiments
In this section, we investigate the numerical computational results presented in the "Results" Section more in detail.
Effect of excluding the simulation model from the set of candidate models
All the numerical experiments presented so far has the simulation model (the model that was used to create a trial simulation dataset) included as one of the candidate models. It is natural to suspect superior performance of the model averaging methods compared to the study protocol can be spoiled if the simulation model is not included in the set of candidate models. Numerical Experiments 1 and 4 were re-run with candidate models excluding the simulation models.
As can be seen in Fig. 10, Method 2 and 4 can still be used to accurately estimate the probability of achieving target endpoints. As can be seen in Fig. 11, if the simulation model is excluded from the set of candidate models, the probability of finding correct dose decreases; however, it still performs superior to the ANOVA based statistical method used in the Study Protocol.
The accuracy of calculated probability of achieving target endpoint. The Methods 1–4 used in this example did not include the simulation model
Probability of finding the correct dose
Accuracy of the probability of achieving target endpoint estimation for each simulation study
Q–Q plots similar to Fig. 9 are plotted for each Simulation Study in Fig. 12. Surprisingly, the calculated probability of achieving the target endpoint based on the model-based approach using the simulation model (i.e., using the model structure that was used to simulate the drug effect) was not very accurate especially in Simulation Studies 1 and 4. Further investigation on this simulation study has shown that the values of ED50 used for Emax or Sigmoidal models in Simulation Study 1 were below 10 mg and for the Simulation Study 4 were near 400 mg hence the design of the experiment was poor for these simulation studies. Due to the uninformative design of the study, the model-based analysis of data with simpler models provided more accurate predictions than with the model used to simulate the drug effect.
The accuracy of calculated probability of achieving target endpoint. The x-axis is the predefined limit for the probability of achieving target endpoint where the dose was chosen. The y-axis is the probability that the chosen dose by the various methods is above tMED. If the probability of achieving the target endpoint is estimated without bias, the plot should lie on the red straight line
We can observe that Methods 2 and 4 slightly underestimate the probability of achieving target endpoint for Simulation Study 3; however, the inaccuracy of these methods are significantly less than that of other methods. Methods 2 and 4 are consistently more accurate than the other methods, hence, they can help reduce the risk of inaccurate estimation of the probability of achieving target endpoint by properly averaging over multiple possible model structures.
Precision of the estimation of the MED
To quantify the precision of the proposed methods, for each method and simulation, we have calculated the difference between the estimated dose that achieves a target effect with 70% probability and the estimated dose that achieves a target effect with 50% probability (we refer to this as the 'estimated MED range'). For comparison, the estimated MED range obtained using Methods 1, 3, 4 and Single Model Based method are compared against Method 2. The differences of the estimated MED range of various methods and Method 2 are depicted in Fig. 13. As can be seen, the estimated MED range is usually wider when estimated using Method 4 when comparing with Method 2. That is to say, Method 2 usually estimates the MED more precisely than Method 4. Although Methods 2 and 4 are similarly accurate, as demonstrated in Figs. 9, 10, 11 and 12, since Method 4 is less precise than Method 2, we have observed worse performance in Experiment 3.
Precision of the estimation of the MED compared to Method 2. The various methods were compared against Method 2 for the estimated MED range. A positive difference indicates the method has a larger estimated MED range, hence poorer precision
Method 1 is typically more precise than Method 2 and the Single Model Based method is often more precise than Method 2. Both Method 1 and the Single Model Based method only used one model to simulate the endpoint hence more precision; however, as can be seen in Figs. 9, 10, 11 and 12 these methods are not accurate and, hence, not desirable methods.
Dose finding accuracy for each simulation study
In Table 3, the probability of choosing the correct dose was tabulated. By using Method 2, the probability of choosing the correct dose has increased from 39.92 to 51.67% compared to the study protocol. What is particularly noteworthy is that the dose finding accuracy has increased from 49 to 65.7% for Simulation Study 4 where the highest tested dose was the correct dose choice.
Table 3 Probability of selecting the correct dose (either 10, 40, 100, or 400 mg)
For all simulation studies except for simulation study 1, the bootstrap model selection and averaging methods (Methods 2 and 4) outperformed simple model selection and averaging methods (Methods 1 and 3). For all simulation studies except for simulation study 3, the model selection and averaging methods outperformed the study protocol even if the simulation model is not included in the candidate models. The overall performance of Methods 2 and 4 are similar to the case where idealized single model based analysis was done using the simulation model.
Decision-making accuracy for each simulation study
As can be seen in Table 4, Method 2 (bootstrap model selection) consistently outperforms Method 4. As discussed in the previously, Method 4 is generally less precise than 2. As a result, Method 4 does not perform as well as Method 2 when a dose is selected not based on the median.
Table 4 Number of times a "correct" decision was made (dose selected above the theoretical MED). For the model-based methods the dose was chosen with at 70% of confidence level
Effect of the identifiability test
We have repeated all of the numerical experiments without identifiability tests. No significant difference in the results was observed. In order to correctly count the degree of freedom for AIC, we need to reject the models that are not identifiable; however, in practice, the inclusion of non-identifiable models did not influence the analysis results within the scope of this investigation. Figure 14 shows the probability of finding the correct dose for Numerical Experiment 1 both with and without the identifiability test. As can be seen, the identifiability test does not significantly influence the dose finding accuracy.
Probability of finding correct dose with and without numerical identifiability test
Aoki, Y., Röshammar, D., Hamrén, B. et al. Model selection and averaging of nonlinear mixed-effect models for robust phase III dose selection. J Pharmacokinet Pharmacodyn 44, 581–597 (2017). https://doi.org/10.1007/s10928-017-9550-0
Phase IIb clinical trial
Dose finding study
Dose–effect relationship | CommonCrawl |
The (homological) persistence of gerrymandering
An extension of the angular synchronization problem to the heterogeneous setting
Constrained Ensemble Langevin Monte Carlo
Zhiyan Ding and Qin Li ,
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53705 USA
* Corresponding author: Qin Li
Received September 2021 Revised October 2021 Early access December 2021
Fund Project: Q.L. acknowledges support from Vilas Early Career award. The research of Z.D., and Q.L is supported in part by NSF via grant DMS-1750488, DMS-2023239 and Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin Madison with funding from the Wisconsin Alumni Research Foundation
The classical Langevin Monte Carlo method looks for samples from a target distribution by descending the samples along the gradient of the target distribution. The method enjoys a fast convergence rate. However, the numerical cost is sometimes high because each iteration requires the computation of a gradient. One approach to eliminate the gradient computation is to employ the concept of "ensemble." A large number of particles are evolved together so the neighboring particles provide gradient information to each other. In this article, we discuss two algorithms that integrate the ensemble feature into LMC, and the associated properties.
In particular, we find that if one directly surrogates the gradient using the ensemble approximation, the algorithm, termed Ensemble Langevin Monte Carlo, is unstable due to a high variance term. If the gradients are replaced by the ensemble approximations only in a constrained manner, to protect from the unstable points, the algorithm, termed Constrained Ensemble Langevin Monte Carlo, resembles the classical LMC up to an ensemble error but removes most of the gradient computation.
Keywords: Langevin Monte Carlo, ensemble methods, variance, gradient free.
Mathematics Subject Classification: Primary: 62D05; Secondary: 82C31, 65C05.
Citation: Zhiyan Ding, Qin Li. Constrained Ensemble Langevin Monte Carlo. Foundations of Data Science, doi: 10.3934/fods.2021034
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Figure 1. Example 1: Evolution of samples using CEnLMC. $ N = 10^4 $
Figure 2. Example 1: Evolution of samples using LMC and MALA. $ N = 10^4 $
Figure 3. Example 1: Evolution of $ \mathcal{R}_m $ when $ N = 2\times10^3, 6\times10^3 $ or $ 10^4 $
Figure 4. Example 2: Evolution of samples using CEnLMC when $ N = 10^4 $
Figure 5. Example 2: Evolution of samples using LMC and MALA when $ N = 10^4 $
Figure 6. Example 2: Evolution of $ \mathcal{R}_m $ with $ m $ when $ N = 2\times10^3, 6\times10^3, 10^4 $
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Zhiyan Ding Qin Li | CommonCrawl |
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DCDS-B Home
A two-group age of infection epidemic model with periodic behavioral changes
doi: 10.3934/dcdsb.2020020
Long term behavior of random Navier-Stokes equations driven by colored noise
Anhui Gu 1,, , Boling Guo 2, and Bixiang Wang 3,
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA
* Corresponding author: Anhui Gu
Received June 2019 Published December 2019
Fund Project: This work is supported by NSF of Chongqing grant cstc2018jcyjA0897
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This paper is devoted to the study of long term behavior of the two-dimensional random Navier-Stokes equations driven by colored noise defined in bounded and unbounded domains. We prove the existence and uniqueness of pullback random attractors for the equations with Lipschitz diffusion terms. In the case of additive noise, we show the upper semi-continuity of these attractors when the correlation time of the colored noise approaches zero. When the equations are defined on unbounded domains, we establish the pullback asymptotic compactness of the solutions by Ball's idea of energy equations in order to overcome the difficulty introduced by the noncompactness of Sobolev embeddings.
Keywords: Random attractor, colored noise, unbounded domain, Navier-Stokes equations, energy equations.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L30.
Citation: Anhui Gu, Boling Guo, Bixiang Wang. Long term behavior of random Navier-Stokes equations driven by colored noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020020
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# Notes on fermionic Fock space for number theorists
$$
\text { Greg W. Anderson }
$$
Author address: This is the Mar. 8, 2000 version of my "notebook". It is a compilation of exercises, worked examples and key references (along with provocative remarks) that I have compiled in order to help myself and (fingers crossed) others learn their way around fermionic Fock space. Eventually (again, fingers crossed) the notebook will become a monograph suitable for use by, say, second year graduate students with an interest in number theory.
## CHAPTER 1
## Essential tools
Throughout this chapter we fix a commutative artinian local ring $k$ with maximal ideal $m$. We call elements of $k$ scalars and elements of $k$-modules vectors. We denote the set of rational integers by $\mathbb{Z}$ and the set of positive integers by $\mathbb{N}$. We denote the cardinality of a set $S$ by $|S|$. The principal results of this chapter are Theorems 15.10 and 17.1.
## Linear algebra over an artinian local ring
ExAmple 1.1. Since $k$ is artinian and local, the following also hold:
- $k$ is noetherian.
- $m$ is the set of zero divisors of $k$.
- $m$ is nilpotent.
We refer the reader to [Matsumura CRT] for background in commutative algebra.
ExAmPle 1.2. For any power $q$ of a prime number the quotient $\mathbb{Z} / q \mathbb{Z}$ is an artinian local ring.
EXAMPLE 1.3. Let $t_{1}, \ldots, t_{n}$ be independent variables. Let $k_{0}$ be a field. Let $I$ be an ideal of the power series ring $k_{0}\left[\left[t_{1}, \ldots, t_{n}\right]\right]$ contained in the maximal ideal $\left(t_{1}, \ldots, t_{n}\right)$ and containing the ideal $\left(t_{1}^{N}, \ldots, t_{n}^{N}\right)$ for some positive integer $N$. The quotient $k_{0}\left[\left[t_{1}, \ldots, t_{n}\right]\right] / I$ is an artinian local ring.
EXAMPle 1.4. Let $X$ be an $n$ by $n$ matrix with entries in the maximal ideal $m$. There exist unique matrices $Y$ and $Z$ with entries in $m$ such that $Y$ is upper triangular $\left(Y_{i j} \neq 0 \Rightarrow i \leq j\right), Z$ is strictly lower triangular $\left(Z_{i j} \neq 0 \Rightarrow i>j\right)$, and $(1+X)=(1+Y)(1+Z)$. The idea developed in this example is often exploited in the sequel.
ExAmple 1.5. Let $n$ be a positive integer. Put $\mathcal{G}:=\operatorname{GL}_{n}(k)$. Let $\mathcal{B} \subseteq \mathcal{G}$ be the subgroup consisting of upper triangular matrices. Let $\mathcal{U} \subseteq \mathcal{G}$ be the subgroup consisting of matrices differing from the identity matrix by a strictly lower triangular matrix with entries in $m$. Let $\mathcal{W} \subseteq \mathcal{G}$ be the subgroup consisting of permutation matrices. One has a disjoint union decomposition $\mathcal{G}=\coprod_{W \in \mathcal{W}} \mathcal{U} \mathcal{B} W$. EXAMPLE 1.6. Let $n$ be a positive integer. If there exists $t \in k^{\times}$ such that $1-t \in k^{\times}$, then $\operatorname{SL}_{n}(k)$ is the commutator subgroup of $\mathrm{GL}_{n}(k)$. Hint:
$$
\begin{aligned}
{\left[\begin{array}{ll}
1 & x \\
0 & 1
\end{array}\right] } & =\left[\begin{array}{ll}
t & 0 \\
0 & 1
\end{array}\right]\left[\begin{array}{cc}
1 & \frac{x}{t-1} \\
0 & 1
\end{array}\right]\left[\begin{array}{ll}
t & 0 \\
0 & 1
\end{array}\right]^{-1}\left[\begin{array}{cc}
1 & \frac{x}{t-1} \\
0 & 1
\end{array}\right]^{-1} \quad(x \in k) \\
{\left[\begin{array}{ll}
0 & 1 \\
-1 & 0
\end{array}\right] } & =\left[\begin{array}{ll}
1 & 1 \\
0 & 1
\end{array}\right]\left[\begin{array}{ll}
1 & 0 \\
-1 & 1
\end{array}\right]\left[\begin{array}{ll}
1 & 1 \\
0 & 1
\end{array}\right] \\
{\left[\begin{array}{ll}
y & 0 \\
0 & y^{-1}
\end{array}\right] } & =\left[\begin{array}{ll}
y & 0 \\
0 & 1
\end{array}\right]\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]\left[\begin{array}{ll}
y & 0 \\
0 & 1
\end{array}\right]^{-1}\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right] \quad\left(y \in k^{\times}\right)
\end{aligned}
$$
DEFINITION 1.7. Let $n$ be a positive integer. An $2 n$ by $2 n$ matrix $A$ with scalar entries is said to be split orthogonal if $A \in \mathrm{GL}_{2 n}(k)$ and $A$ preserves the quadratic form $h \mapsto \sum_{i=1}^{n} h_{i} h_{2 n+1-i}$ on the space of column vectors of length $2 n$ with scalar entries.
DEFINITION 1.8. Let $n$ be a positive integer. Given an $n$ by $n$ matrix $A$ with scalar entries, let $A^{\dagger}$ be the matrix given by the rule
$$
\left(A^{\dagger}\right)_{i j}:=A_{n+1-j, n+1-i}
$$
for $i, j=1, \ldots, n$. In other words, $A^{\dagger}$ is obtained from $A$ by reflecting in the anti-diagonal $\{i+j=n+1\}$. We say that an $n$ by $n$ matrix $A$ with scalar entries is dagger-alternating if $A^{\dagger}+A=0$ and $A$ vanishes on the anti-diagonal.
EXAMPle 1.9. Let $n$ be a positive integer. Fix a $2 n$ by $2 n$ matrix $A$ with scalar entries. Let
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right], \quad A^{\dagger}=\left[\begin{array}{ll}
d^{\dagger} & b^{\dagger} \\
c^{\dagger} & a^{\dagger}
\end{array}\right]
$$
be the decomposition of $A$ and corresponding decomposition of $A^{\dagger}$ into $n$ by $n$ blocks. The matrix $A$ is split orthogonal if and only if
$$
a d^{\dagger}+b c^{\dagger}=1, \quad d^{\dagger} a+b^{\dagger} c=1
$$
and the matrices
$$
c^{\dagger} a, \quad d^{\dagger} b, \quad a b^{\dagger}, \quad c d^{\dagger}
$$
are dagger-alternating. An $2 n$ by $2 n$ permutation matrix $W$ is split orthogonal if and only if the $2 n$ by $2 n$ permutation matrix
commutes with $W$.
EXAMPLE 1.10. Let $n$ be a positive integer and let $\mathcal{G} \subseteq \mathrm{GL}_{2 n}(k)$ be the group of split orthogonal matrices. Let $\mathcal{B} \subseteq \mathcal{G}$ be the subgroup consisting of upper triangular matrices. Let $\mathcal{U} \subseteq \mathcal{G}$ be the subgroup consisting of matrices differing from the identity matrix by a strictly lower triangular matrix with entries in $m$. Let $\mathcal{W} \subseteq \mathcal{G}$ be the subgroup consisting of permutation matrices. One has a disjoint union decomposition $\mathcal{G}=\coprod_{W \in \mathcal{W}} \mathcal{U} \mathcal{B} W \mathcal{B}$. Let $\mathcal{P} \subseteq \mathcal{G}$ be the subgroup consisting of matrices vanishing in the lower left $n$ by $n$ block. Let $R \in \mathcal{W}$ be the permutation matrix representing the transposition exchanging $n$ and $n+1$. The group $\mathcal{G}$ is generated by the subgroup $\mathcal{P}$ and the matrix $R$.
ExAmple 1.11. Let $E$ be a $k$-module and let $V \subseteq E$ be a $k$ submodule.
- If $V$ is finitely generated over $k$ and $E$ is free over $k$, then $V$ is contained in a finitely generated free $k$-submodule of $E$.
- If $E$ is free and $x V=0$ for some $0 \neq x \in k$, then $V \subseteq m E$.
- If $V$ is free over $k$, then $V \cap m E=m V$.
- If $V+m E=E$, then $E=V$.
The last assertion is a version of Nakayama's Lemma; note that it is not necessary to assume that $E / V$ is finitely generated.
Definition 1.12. We say that a $k$-module $E$ is flat if for every positive integer $n$, column vector $e$ of length $n$ with entries in $E$, and row vector $x$ of length $n$ with scalar entries such that $x e=0$, there exists some positive integer $N$, matrix $A$ of $n$ rows and $N$ columns with scalar entries, and column vector $f$ of length $N$ with entries in $E$ such that $x A=0$ and $e=A f$.
Proposition 1.13. Let $E$ be a flat (e. g. free) k-module. For any family $\left\{e_{i}\right\}_{i \in S}$ of vectors of $E$, the following assertions are equivalent:
1. The family $\left\{e_{i}+m E\right\}_{i \in S}$ is linearly independent over $k / m$.
2. The family $\left\{e_{i}\right\}_{i \in S}$ is linearly independent over $k$.
3. The family $\left\{e_{i}\right\}_{i \in S}$ can be extended to a $k$-basis of $E$. (In particular, the $k$-module $E$ is free.) Proof. $(1 \Rightarrow 2)$ Without loss of generality we may assume that $S=\{1, \ldots, n\}$ for some positive integer $n$. Arrange the family $\left\{e_{i}\right\}_{i=1}^{n}$ into a column vector $e$ of length $n$. Let $x$ be any row vector of length $n$ with scalar entries such that $x e=0$. By hypothesis there exists a positive integer $N$ and a matrix $A$ of $n$ rows and $N$ columns, and a column vector $f$ of length $N$ with entries in $E$ such that $x A=$ 0 and $e=A f$. The matrix $A$ reduced modulo $m$ must have rank $n$, for otherwise we arrive at a contradiction to our assumption that $e_{1}+m E, \ldots, e_{n}+m E$ are $(k / m)$-linearly independent. It follows that some $n$ by $n$ minor of $A$ is an invertible scalar, and hence that $x=0$ by Cramer's rule.
$(1 \Rightarrow 3)$ Taking this implication for granted if $m=0$, i. e., if $k$ is a field, we may assume without loss of generality that the family $\left\{e_{i}+m E\right\}_{i \in S}$ is a $(k / m)$-basis for $E / m E$. Then the family $\left\{e_{i}\right\}_{i \in S}$ spans $E$ over $k$ by Example 1.11, is $k$-linearly independent by what we have already proved, and hence is already a $k$-basis for $E$.
$(3 \Rightarrow 1,2)$ Trivial.
EXAMPle 1.14. The following hold:
- Each idempotent $k$-linear endomorphism $\alpha$ of a free $k$-module $E$ gives rise to a $k$-linear direct sum decomposition
$$
E=\alpha E \oplus(1-\alpha) E
$$
both summands of which are free $k$-modules.
- A $k$-linear map $A \stackrel{\phi}{\rightarrow} B$ of free $k$-modules is injective (resp. surjective) if and only if the induced $\operatorname{map} A / m A \stackrel{\phi \bmod m}{\rightarrow} B / m B$ is injective (resp. surjective).
- Given a short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ of $k$ modules, if two of the $k$-modules $A, B, C$ are free, then so is the third and the sequence splits.
ExAmple 1.15. Let $S$ be a set and let $2^{S}$ be the family of all subsets of $S$. A subfamily $\Phi \subseteq 2^{S}$ is called a boolean ideal under the following conditions:
- $\emptyset \in \Phi$.
- For all $I, J \in \Phi$ one has $I \cup J \in \Phi$.
- For all $I \in \Phi$ and $J \subseteq S$ one has $I \cap J \in \Phi$.
Given a $k$-module $E$ and a boolean ideal $\Phi \subseteq 2^{S}$, put
$$
E(S, \Phi):=\{e: S \rightarrow E \mid\{s \in S \mid e(s) \neq 0\} \in \Phi\} .
$$
The $k$-module-valued functor $E \mapsto E(S, \Phi)$ of $k$-modules is exact. If $E$ is a free $k$-module, the $k$-module $E(S, \Phi)$ is again free.
## Algebras and ideals
Definition 2.1. A $k$-algebra is a $k$-module equipped with a $k$ bilinear associative (but possibly noncommutative) product with respect to which there exists a (necessarily unique) two-sided identity. The identity element of a $k$-algebra $\mathcal{A}$ is assumed to act as the identity on each left (resp. right) $\mathcal{A}$-module. We say that a $k$-algebra $\mathcal{A}$ is flat if flat (and hence free) as a $k$-module. The group consisting of the elements of a $k$-algebra $\mathcal{A}$ possessing two-sided inverses is denoted by $\mathcal{A}^{\times}$. A homomorphism $\phi: \mathcal{A} \rightarrow \mathcal{B}$ of $k$-algebras is a $k$-linear map such that $\phi(1)=1$ and $\phi\left(a a^{\prime}\right)=\phi(a) \phi\left(a^{\prime}\right)$ for all $a, a^{\prime} \in \mathcal{A}$.
Definition 2.2. For any positive integer $n$ and $k$-algebra $\mathcal{A}$, we denote the $k$-algebra of $n$ by $n$ matrices with entries in $\mathcal{A}$ by $\operatorname{Mat}_{n}(\mathcal{A})$, and we put $\operatorname{GL}_{n}(\mathcal{A}):=\operatorname{Mat}_{n}(\mathcal{A})^{\times}$. Given positive integers $p$ and $q$, we denote by $\operatorname{Mat}_{p \times q}(\mathcal{A})$ the set of $p$ by $q$ matrices with entries in $\mathcal{A}$.
Definition 2.3. Fix a $k$-algebra $\mathcal{A}$. A $k$-submodule $\mathcal{A}_{0} \subseteq \mathcal{A}$ is said to be a $k$-subalgebra if $1 \in \mathcal{A}_{0}$ and $a b \in \mathcal{A}_{0}$ for all $a, b \in \mathcal{A}_{0}$. A $k$-submodule $\mathcal{I} \subseteq \mathcal{A}$ is said to be a left ideal of $\mathcal{A}$ if $a b \in \mathcal{I}$ for all $a \in \mathcal{A}$ and $b \in \mathcal{I}$. A $k$-submodule $\mathcal{I}^{\star} \subseteq \mathcal{A}$ is said to be a right ideal of $\mathcal{A}$ if $b a \in \mathcal{I}^{\star}$ for all $a \in \mathcal{A}$ and $b \in \mathcal{I}^{\star}$. Symbols bearing the superscript $\star$ are reserved for use in denoting right ideals. A $k$-submodule of $\mathcal{A}$ is said to be a two-sided ideal if both a left and a right ideal. We say that a left ideal of $\mathcal{A}$ is flat if free as a $k$-module; similarly we speak of flat right ideals and flat $k$-subalgebras of $\mathcal{A}$.
ExAmple 2.4. Let a $k$-algebra $\mathcal{A}$ and left ideals $\mathcal{I}, \mathcal{J} \subseteq \mathcal{A}$ be given. Put
$$
(\mathcal{I}: \mathcal{J}):=\{a \in \mathcal{A} \mid \mathcal{J} a \subseteq \mathcal{I}\}
$$
The sequence
$$
0 \rightarrow \mathcal{I} \subset(\mathcal{I}: \mathcal{J}) \stackrel{a \mapsto(x+\mathcal{J} \mapsto x a+\mathcal{I})}{\rightarrow} \operatorname{Hom}_{\mathcal{A}}(\mathcal{A} / \mathcal{J}, \mathcal{A} / \mathcal{I}) \rightarrow 0
$$
is exact. Example 2.5. Put $\mathcal{A}:=\operatorname{Mat}_{n}(k)$. Fix $x \in \mathcal{A}$. Let $\mathcal{J} \subset \mathcal{A}$ be the flat left ideal consisting of matrices with vanishing first column. Let $\mathcal{I}^{\star} \subset \mathcal{A}$ be the flat right ideal consisting matrices with vanishing first row. The following assertions hold:
- $x-c \in \mathcal{I}^{\star}+\mathcal{J}$ for a unique scalar $c$.
- If $\mathcal{A} x \mathcal{A} \subseteq \mathcal{I}^{\star}+\mathcal{J}$, then $x=0$.
- If $x \mathcal{A} \subseteq \mathcal{J}$, then $x=0$.
- If $\mathcal{A} x \subseteq \mathcal{I}^{\star}$, then $x=0$.
- If $x \mathcal{A} \subseteq \mathcal{I}^{\star}+\mathcal{J}$, then $x \in \mathcal{I}^{\star}$.
- If $\mathcal{A} x \subseteq \mathcal{I}^{\star}+\mathcal{J}$, then $x \in \mathcal{J}$.
- If $\mathcal{J} x \subseteq \mathcal{J}$, then $x-c \in \mathcal{J}$ for a unique scalar $c$.
- If $x \mathcal{I}^{\star} \subseteq \mathcal{I}^{\star}$, then $x-c \in \mathcal{I}^{\star}$ for a unique scalar $c$.
- If $x$ is central in $\mathcal{A}$, then $x$ is a scalar.
- $(x+\mathcal{J}) \cap \mathcal{A}^{\times} \neq \emptyset$ if and only if $x \notin \mathcal{J}+m \mathcal{A}$.
- $\left(x+\mathcal{I}^{\star}\right) \cap \mathcal{A}^{\times} \neq \emptyset$ if and only if $x \notin \mathcal{I}^{\star}+m \mathcal{A}$.
There exist triples $\left(\mathcal{A}, \mathcal{I}^{\star}, \mathcal{J}\right)$ consisting of an infinite rank flat $k$-algebra $\mathcal{A}$, a flat right ideal $\mathcal{I}^{\star}$ of $\mathcal{A}$ and a flat left ideal $\mathcal{J}$ of $\mathcal{A}$ such that for every $x \in \mathcal{A}$ all the statements above make sense and remain true. The theory of Clifford algebras provides natural examples of such triples.
Example 2.6. Let $E$ be a free $k$-module. Put
$$
\mathcal{T}(E):=\bigoplus_{n=0}^{\infty} \underbrace{E \otimes_{k} \cdots \otimes_{k} E}_{n} .
$$
The $k$-module $\mathcal{T}(E)$ contains $E$ as a $k$-submodule, forms a $k$-algebra with unit under the tensor product operation, and has the following universal property in the category of $k$-algebras:
- For any $k$-algebra $\mathcal{A}$ and $k$-linear map $\phi: E \rightarrow \mathcal{A}$ there exists a unique extension of $\phi$ to a $k$-algebra homomorphism $\mathcal{T}(E) \rightarrow \mathcal{A}$.
One calls $\mathcal{T}(E)$ the tensor algebra of $E$ over $k$. Now let $\left\{e_{i}\right\}_{i \in S}$ be a $k$-basis for $E$. We declare a word $W$ in $S$ to be an element of the disjoint union $\coprod_{n=0}^{\infty} S^{n}$. Given a word $W=\left(i_{1}, \ldots, i_{n}\right) \in S^{n}$, put
$$
e_{W}:=e_{i_{1}} \otimes \cdots \otimes e_{i_{n}} \in \mathcal{T}(E) .
$$
The family $\left\{e_{W}\right\}$ indexed by words in $S$ is a $k$-basis for $\mathcal{T}(E)$. One has
$$
e_{W} \otimes e_{W^{\prime}}=e_{W W^{\prime}}
$$
for all words $W$ and $W^{\prime}$ in $S$, where $W W^{\prime}$ denotes the concatenation of $W$ and $W^{\prime}$. Every $k$-algebra is of the form $\mathcal{T}(E) / \mathcal{I}$ for some free $k$-module $E$ and two-sided ideal $\mathcal{I} \subseteq \mathcal{T}(E)$. EXAmple 2.7. Let $E$ be a free $k$-module. Let $\mathcal{I}$ be the two-sided ideal of $\mathcal{T}(E)$ generated by the set $\{e \otimes e \mid e \in E\}$. The quotient $\bigwedge(E):=\mathcal{T}(E) / \mathcal{I}$ is called the exterior algebra of $E$ over $k$. The product in $\Lambda(E)$ is traditionally denoted by the wedge symbol $\wedge$. It is well known that the natural map $(e \mapsto e+\mathcal{I}): E \rightarrow \bigwedge(E)$ is injective; $E$ is traditionally identified via this map with a $k$-submodule of $\bigwedge(E)$. The exterior algebra has the following universal property:
- For all $k$-algebras $\mathcal{A}$ and $k$-linear maps $\phi: E \rightarrow \mathcal{A}$ such that
$$
\phi(e)^{2}=0
$$
for all $e \in E$, there exists a unique $k$-algebra homomorphism $\bigwedge(E) \rightarrow \mathcal{A}$ extending $\phi$
Let $\left\{e_{i}\right\}_{i \in S}$ be a $k$-basis for $E$ indexed by a linearly ordered set $S$. Put
$$
e_{I}:=e_{i_{1}} \wedge \cdots \wedge e_{i_{r}} \in \bigwedge(E)
$$
for each finite subset $I=\left\{i_{1}<\cdots<i_{r}\right\} \subseteq S$. The family $\left\{e_{I}\right\}$ indexed by finite subsets of $S$ is a $k$-basis for $\bigwedge(E)$. One has
$$
e_{I} \wedge e_{J}= \begin{cases}(-1)^{|\{(i, j) \in I \times J \mid i>j\}|} e_{I \cup J} & \text { if } I \cap J=\emptyset \\ 0 & \text { if } I \cap J \neq \emptyset\end{cases}
$$
for all finite subsets $I, J \subseteq S$.
Example 2.8. Suppose $E$ has a $k$-basis $e_{1}, \ldots, e_{n}$. Let $A \in \operatorname{Mat}_{n}(k)$ be given. Put
$$
f_{j}:=\sum_{i=1}^{n} A_{i j} e_{i}
$$
for $j=1, \ldots, n$. Then
$$
f_{1} \wedge \cdots \wedge f_{n}=(\operatorname{det} A) e_{1} \wedge \cdots \wedge e_{n}
$$
Thus the theory of determinants is linked to the theory of exterior algebras.
## Laurent series
Definition 3.1. Let $t$ be a variable. A series $f=\sum_{i \in \mathbb{Z}} a_{i} t^{i}$ with scalar coefficients $a_{i}$ vanishing for $i \ll 0$ is called a Laurent series in $t$; if the coefficients $a_{i}$ vanish for $i<0$ we call $f$ a power series in $t$. We denote the $k$-module of Laurent series in $t$ with scalar coefficients by $k((t))$ and the $k$-submodule of power series in $t$ by $k[[t]]$. The $k$ modules $k((t))$ and $k[[t]]$ are free. Under the standard rule for series multiplication the $k$-module $k((t))$ becomes a commutative ring with unit and $k[[t]]$ a subring. The ring $k((t))$ is artinian and local. The unique maximal ideal of $k((t))$ is generated by $m$ and consists of all Laurent series in $t$ with coefficients in $m$. The ring $k[[t]]$ is noetherian and local but not artinian.
Definition 3.2. Given a Laurent series $f=\sum_{i} a_{i} t^{i} \in k((t))$, put
$$
w(f):=\min \left(\left\{i \mid a_{i} \not \equiv 0 \bmod m\right\} \cup\{+\infty\}\right),
$$
thereby defining the winding number of $f$. One has
$$
w(f)<+\infty \Leftrightarrow f \in k((t))^{\times}
$$
for all $f \in k((t))$. One has
$$
w(f g)=w(f)+w(g), \quad w(f+g) \geq \min (w(f), w(g))
$$
for all $f, g \in k((t))$.
ExAmPle 3.3. Fix a positive integer $n$ and $f \in k[[t]]$ of winding number $n$. The Weierstrass Division Theorem says that for each power series $g \in k[[t]]$ there exist unique $q, r \in k[[t]]$, the latter a polynomial in $t$ of degree $<n$, such that $f=g q+r$. The Weierstrass Preparation Theorem says that there exist unique $u, r \in k[[t]]$, the former a power series in $t$ with invertible constant term and the latter a polynomial in $t$ of degree $<n$ all coefficients of which belong to $m$, such that $f=\left(t^{n}+r\right) u$.
DeFinition 3.4. For each $f \in k((t))^{\times}$there exists a unique family $\left\{a_{i}\right\}_{i \in \mathbb{Z}}$ of scalars with the following properties:
- $a_{i}=0$ for $i \ll 0$.
- $a_{i} \equiv 0 \bmod m$ for $i<0$.
- $a_{0} \not \equiv 0 \bmod m$.
- $f^{-1} t^{w(f)} \prod_{i=-\infty}^{N}\left\{\begin{array}{ll}\left(1-a_{i} t^{i}\right) & \text { if } i \neq 0 \\ a_{0} & \text { if } i=0\end{array} \in 1+t^{N+1} k[[t]]\right.$ for $N \geq 0$
We call $\left\{a_{i}\right\}_{i \in \mathbb{Z}}$ the family of Witt parameters of $f$.
EXAMPle 3.5. Let $n$ be a positive integer. Put $\mathcal{G}:=\mathrm{GL}_{n}(k((t)))$. Given $A \in \mathcal{G}$, write $A=\sum_{i \in \mathbb{Z}} A^{(i)} t^{i}$, where the coefficients $A^{(i)}$ are $n$ by $n$ matrices with scalar coefficients vanishing for $i \ll 0$. Let $\mathcal{B} \subseteq \mathcal{G}$ be the subgroup consisting of matrices $A$ such that $A^{(i)}=0$ for $i<0$ and $A^{(0)}$ is upper triangular. Let $\mathcal{U} \subseteq \mathcal{G}$ be the subgroup consisting of matrices $A$ such that $A^{(i)}=0$ for $i>0, A^{(0)}$ differs from the identity matrix by a strictly lower triangular matrix with entries in $m$, and $A^{(i)}$ has all entries in $m$ for $i<0$. Let $\mathcal{W} \subseteq \mathcal{G}$ be the subgroup consisting of matrices factoring as a permutation matrix times a diagonal matrix with power of $t$ on the diagonal. One has a disjoint union decomposition $\mathcal{G}=\coprod_{W \in \mathcal{W}} \mathcal{U} \mathcal{B} W \mathcal{B}$. EXAMPLE 3.6. For each $f \in k((t))$ of winding number 0 there exists a positive integer $N$ such that $t^{N} f^{i} \in k[[t]]$ for all $i \in \mathbb{Z}$.
## Almost upper triangular matrices
Definition 4.1. Let $I$ and $J$ be subsets of $\mathbb{Z}$ and let $A$ be an $I$ by $J$ matrix with scalar entries. Put
$$
\operatorname{supp} A:=\left\{(i, j) \in I \times J \quad \mid A_{i j} \neq 0\right\},
$$
thereby defining the support of $A$. We say that $A$ is finitely supported if
$$
|\operatorname{supp} A|<\infty \text {. }
$$
We say that $A$ is upper triangular if
$$
\{(i, j) \in \operatorname{supp} A \mid i>j\}=\emptyset
$$
strictly upper triangular if
$$
\{(i, j) \in \operatorname{supp} A \mid i \geq j\}=\emptyset,
$$
and almost upper triangular if
$$
\forall n \in \mathbb{Z}|\{(i, j) \in \operatorname{supp} A \mid i \geq n \geq j\}|<\infty .
$$
The "lower" analogues of the preceding "upper" notions are defined in the obvious way. We denote the set of $I$ by $J$ almost upper triangular matrices with scalar entries by $\mathcal{Q}(I, J)$. If $I=J$ we write $\mathcal{Q}(I)$, and if $I=J=\mathbb{Z}$ we write $\mathcal{Q}$. We permit one or both of $I$ or $J$ to be the empty set, in which case we set $\mathcal{Q}(I, J)=\{0\}$.
Definition 4.2 . Let $\mathcal{H}$ be the $k$-module consisting of column vectors $h$ with scalar entries indexed by $\mathbb{Z}$ such that $h_{i}=0$ for all $i \gg 0$. For each subset $I \subseteq \mathbb{Z}$, put
$$
\mathcal{H}(I):=\{h \in \mathcal{H} \mid \operatorname{supp} h \subset I\}
$$
where
$$
\operatorname{supp} h:=\left\{i \in \mathbb{Z} \mid h_{i} \neq 0\right\}
$$
for each $h \in \mathcal{H}$.
ExAmple 4.3. For all subsets $I, J \subseteq \mathbb{Z}$, the corresponding $k$-module $\mathcal{Q}(I, J)$ is free. For all subsets $I \subseteq \mathbb{Z}$, the corresponding $k$-module $\mathcal{H}(I)$ is free.
ExAmple 4.4. Let $L \subseteq \mathcal{H}$ be a flat $k$-submodule such that the $k$-modules $L \cap \mathcal{H}(\mathbb{Z} \backslash \mathbb{N})$ and $\mathcal{H} /(L+\mathcal{H}(\mathbb{Z} \backslash \mathbb{N}))$ are finitely generated. Then there exists a subset $I \subset \mathbb{Z}$ such that $\sup I<+\infty, \inf \mathbb{Z} \backslash I>-\infty$ and $\mathcal{H}=\mathcal{H}(I) \oplus L$. Definition 4.5. A gauge is a function $\sigma: \mathbb{Z} \rightarrow \mathbb{Z}$ with the following properties:
- $\sigma(n) \leq \sigma(n+1)$ for all $n \in \mathbb{Z}$.
- $\lim _{n \rightarrow-\infty} \sigma(n)=-\infty$.
- $\lim _{n \rightarrow+\infty} \sigma(n)=+\infty$.
Given subsets $I, J \subseteq \mathbb{Z}$, an $I$ by $J$ matrix $A$ with scalar entries and a gauge $\sigma$, we say that $\sigma$ dominates the matrix $A$ if
$$
A_{i j} \neq 0 \Rightarrow i \leq \sigma(j)
$$
for all $(i, j) \in I \times J$.
EXAMPle 4.6. For any gauges $\sigma$ and $\tau$, the functions
$$
\left(n \mapsto\left\{\begin{array}{l}
\sigma(\tau(n)) \\
\max (\sigma(n), \tau(n)) \\
\min (\sigma(n), \tau(n)) \\
\min \{j \in \mathbb{Z} \mid n \leq \sigma(j)\} \\
\max \{j \in \mathbb{Z} \mid \sigma(j) \leq n\}
\end{array}\right): \mathbb{Z} \rightarrow \mathbb{Z}\right.
$$
are again gauges.
LemMA 4.7. Let subsets $I, J \subseteq \mathbb{Z}$ and an $I$ by $J$ matrix $A$ with scalar entries be given. The matrix $A$ is almost upper triangular if and only if there exists a gauge $\sigma$ dominating $A$.
Proof. $(\Rightarrow)$ For each $n \in \mathbb{Z}$, put
$$
S_{n}:=\{\ell \in \mathbb{Z} \mid \ell \leq n\} \cup \bigcup_{\substack{j \in J \\ n \geq j}}\left\{i \in I \mid A_{i j} \neq 0\right\} \subseteq \mathbb{Z} .
$$
Clearly, one has $S_{n} \neq \emptyset, S_{n} \subseteq S_{n+1}$ and $\bigcup S_{n}=\mathbb{Z}$. By hypothesis, $S_{n}$ is bounded above and one has $\bigcap S_{n}=\emptyset$. The function
$$
\left(n \mapsto \max S_{n}\right): \mathbb{Z} \rightarrow \mathbb{Z}
$$
is therefore a gauge dominating $A$.
$(\Leftarrow)$ Let $\sigma$ be a gauge dominating $A$. For all $n \in \mathbb{Z}$ one has
$$
\left\{\begin{array}{l|l|l}
(i, j) \in I \times J & \begin{array}{l}
i \geq n \geq j \\
A_{i j} \neq 0
\end{array}
\end{array}\right\} \subseteq\left\{\begin{array}{l}
(i, j) \in I \times J \mid \begin{array}{l}
i \geq n \geq j \\
i \leq \sigma(n) \\
n \leq \sigma(j)
\end{array}
\end{array}\right\},
$$
and the set on the right is finite. Definition 4.8. Let subsets $I_{0}, I_{1}, I_{2} \subseteq \mathbb{Z}$ be given. For $\nu=1,2$, let a matrix $A^{(\nu)} \in \mathcal{Q}\left(I_{\nu-1}, I_{\nu}\right)$ and a gauge $\sigma_{\nu}$ dominating $A^{(\nu)}$ be given. One has
$$
A_{i_{0} i_{1}}^{(1)} A_{i_{1} i_{2}}^{(2)} \neq 0 \Rightarrow\left\{\begin{array}{l}
i_{1} \geq \min \left\{j \in \mathbb{Z} \mid i_{0} \leq \sigma_{1}(j)\right\}>-\infty \\
i_{1} \leq \sigma_{2}\left(i_{2}\right) \\
i_{0} \leq \sigma_{1}\left(\sigma_{2}\left(i_{2}\right)\right)
\end{array}\right.
$$
for all $\left(i_{0}, i_{1}, i_{2}\right) \in I_{0} \times I_{1} \times I_{2}$. We refer to this estimate as the gauge trick. It follows that the product $A^{(1)} A^{(2)}$ is a well defined $I_{0}$ by $I_{2}$ matrix with scalar entries dominated by the gauge $\sigma_{1} \circ \sigma_{2}$. Thus a natural product
$$
\mathcal{Q}\left(I_{0}, I_{1}\right) \times \mathcal{Q}\left(I_{1}, I_{2}\right) \rightarrow \mathcal{Q}\left(I_{0}, I_{2}\right)
$$
is defined.
Example 4.9. Consider the $\mathbb{N}$ by $\mathbb{N}$ matrices
$$
A=\left[\begin{array}{rrrrr}
1 & -1 & 0 & 0 & \ldots \\
0 & 1 & -1 & 0 & \ldots \\
0 & 0 & 1 & -1 & \\
\vdots & \vdots & & \ddots & \ddots
\end{array}\right], \quad B:=\left[\begin{array}{rrrr}
1 & 0 & 0 & \ldots \\
1 & 0 & 0 & \ldots \\
1 & 0 & 0 & \ldots \\
\vdots & \vdots & \vdots &
\end{array}\right]
$$
One has
$$
\left(B^{T} A\right) B=\left[\begin{array}{ccc}
1 & 0 & \ldots \\
0 & 0 & \ldots \\
\vdots & \vdots &
\end{array}\right], \quad B^{T}(A B)=\left[\begin{array}{ccc}
0 & 0 & \ldots \\
0 & 0 & \ldots \\
\vdots & \vdots &
\end{array}\right] .
$$
Moral: Associativity of products of infinite matrices cannot be taken for granted.
Definition 4.10. Let subsets $I, J \subseteq \mathbb{Z}$ be given. Let an $I$ by $J$ matrix $A$ with scalar entries and a sequence $\left\{A^{(n)}\right\}_{n=1}^{\infty}$ of $I$ by $J$ matrices with scalar entries be given. The sequence $\left\{A^{(n)}\right\}$ is said to be uniformly dominated if there exists a gauge dominating all of the matrices $A^{(n)}$. We say that the sequence $\left\{A^{(n)}\right\}$ converges entrywise to $A$ or that the entrywise limit of the sequence $\left\{A^{(n)}\right\}$ is $A$ if for every pair $(i, j) \in I \times J$ there exist only finitely many $n$ such that $A_{i j}^{(n)} \neq A_{i j}$. We say that the sequence $\left\{A^{(n)}\right\}$ is entrywise convergent if there exists some (necessarily unique) $I$ by $J$ matrix with scalar entries to which the sequence $\left\{A^{(n)}\right\}$ converges entrywise. EXAMPle 4.11. A sequence $\left\{A^{(n)}\right\}_{n=1}^{\infty}$ in $\mathcal{Q}$ is uniformly dominated if and only if for all $\left(i_{0}, j_{0}\right) \in \mathbb{Z}^{2}$ the set
$$
\left(\bigcup_{n=1}^{\infty} \operatorname{supp} A^{(n)}\right) \bigcap\left\{(i, j) \in \mathbb{Z} \times \mathbb{Z} \mid \begin{array}{l}
i \geq i_{0} \\
j \leq j_{0}
\end{array}\right\}
$$
is finite.
Lemma 4.12. Let matrices $A, B \in \mathcal{Q}$ and uniformly dominated sequences $\left\{A^{(n)}\right\}_{n=1}^{\infty}$ and $\left\{B^{(n)}\right\}_{n=1}^{\infty}$ in $\mathcal{Q}$ be given. Assume that $\left\{A^{(n)}\right\}$ converges entrywise to $A$ and that $\left\{B^{(n)}\right\}$ converges entrywise to $B$. Then the sequence $\left\{A^{(n)} B^{(n)}\right\}_{n=1}^{\infty}$ is uniformly dominated and converges entrywise to $A B$.
Proof. Let $\sigma$ be a gauge dominating all the matrices $A^{(n)}$ and hence also the matrix $A$. Let $\tau$ be a gauge dominating all the matrices $B^{(n)}$ and hence also the matrix $B$. By the gauge trick, the gauge $\sigma \circ \tau$ dominates all the matrices $A^{(n)} B^{(n)}$ and hence the sequence $\left\{A^{(n)} B^{(n)}\right\}$ is uniformly dominated. Now fix $\left(i_{0}, j_{0}\right) \in \mathbb{Z} \times \mathbb{Z}$ and let $S$ be the finite set consisting of all $\ell \in \mathbb{Z}$ such that $\ell \leq \tau\left(j_{0}\right)$ and $i_{0} \leq \sigma(\ell)$. By hypothesis there exists $n_{0} \in \mathbb{N}$ such that $A_{i_{0} \ell}^{(n)}=A_{i_{0} \ell}$ and $B_{\ell j_{0}}^{(n)}=B_{\ell j_{0}}$ for all $\ell \in S$ and all $n \geq n_{0}$. By the gauge trick one has
$$
\left(A^{(n)} B^{(n)}\right)_{i_{0} j_{0}}=\sum_{\ell \in S} A_{i_{0} \ell}^{(n)} B_{\ell j_{0}}^{(n)}=\sum_{\ell \in S} A_{i_{0} \ell} B_{\ell_{j}}=(A B)_{i_{0} j_{0}}
$$
for all $n \geq n_{0}$, and hence $\left\{A^{(n)} B^{(n)}\right\}$ converges entrywise to $A B$ as claimed.
Definition 4.13. One can deduce the associativity of multiplication of almost upper triangular matrices from the associativity of multiplication of finitely supported matrices via Lemma 4.12. Thus, in particular, $\mathcal{Q}$ becomes a ring with unit and $\mathcal{H}$ a left $\mathcal{Q}$-module under the standard rule for matrix multiplication. We sometimes refer to $\mathcal{Q}^{\times}$as the Japanese group since in various versions this group figures prominently in the works of the Kyoto school of soliton theory.
EXAmPle 4.14. One has
$$
A \mathcal{H}\left(\left\{n \in \mathbb{Z} \mid n \leq n_{0}\right\}\right) \subseteq \mathcal{H}\left(\left\{n \in \mathbb{Z} \mid n \leq \sigma\left(n_{0}\right)\right\}\right)
$$
for all $A \in \mathcal{Q}$, gauges $\sigma$ dominating $A$, and $n_{0} \in \mathbb{Z}$. EXAMPLE 4.15. Let $\left\{X^{(n)}\right\}_{n=1}^{\infty}$ be a sequence of strictly upper triangular elements of $\mathcal{Q}$ with pairwise disjoint supports. Define a sequence $\left\{A^{(n)}\right\}_{n=1}^{\infty}$ inductively by the rules
$$
A^{(1)}:=X^{(1)}, \quad A^{(n+1)}:=\left(1+X^{(n)}\right) A^{(n)} .
$$
The sequence $\left\{A^{(n)}\right\}$ is uniformly dominated and entrywise convergent. The entrywise limit of the sequence $\left\{A^{(n)}\right\}$ differs from the identity by a strictly upper triangular matrix.
Definition 4.16. Given subsets $I, J \subseteq \mathbb{Z}$, we denote by $\mathcal{Q}(I, J)^{\times}$ the subset of $\mathcal{Q}(I, J)$ consisting of matrices $A$ such that for some (necessarily unique) $B \in \mathcal{Q}(J, I)$ one has $A B=1 \in \mathcal{Q}(I)$ and $B A=1 \in$ $\mathcal{Q}(J)$. If $I=J$, we write $\mathcal{Q}(I)^{\times}$, and if $I=J=\mathbb{Z}$, we write $\mathcal{Q}^{\times}$.
ExAmple 4.17. Let gauges $\sigma$ and $\tau$ dominating matrices $A, B \in$ $\mathcal{Q}$, respectively, be given. The composition $\sigma \circ \tau$ dominates the product $A B$. Moreover, if $A B=1$, then $n \leq \sigma(\tau(n))$ for all $n \in \mathbb{Z}$.
EXAMPle 4.18. Let $I$ be any subset of $\mathbb{Z}$ and let $X \in \mathcal{Q}(I)$ be a matrix all entries of which belong to the maximal ideal $m$. The matrix $X$ is nilpotent and $1-X \in \mathcal{Q}(I)^{\times}$.
ExAmple 4.19. Let $I$ and $J$ be any subsets of $\mathbb{Z}$. A matrix $A \in$ $\mathcal{Q}(I, J)$ has a left inverse in $\mathcal{Q}(J, I)$ if and only if $A$ has a left inverse modulo $m$. Similarly, $A$ has a right (resp. two-sided) inverse in $\mathcal{Q}(J, I)$ if and only if $A$ has a right (resp. two-sided) inverse modulo $m$.
EXAMPle 4.20 . Let $I$ be any subset of $\mathbb{Z}$ and let $X$ be a strictly upper triangular $I$ by $I$ matrix with scalar entries. One has $(1-X) \in$ $\mathcal{Q}(I)^{\times}$. The sequence
$$
\left\{1+\sum_{n=1}^{N} X^{n}\right\}_{N=1}^{\infty}
$$
is uniformly dominated and converges entrywise to $(1-X)^{-1}$.
ExAmple 4.21. Let $I$ be any subset of $\mathbb{Z}$ and let $A \subseteq \mathcal{Q}(I)$ be an upper triangular matrix. One has $A \in \mathcal{Q}(I)^{\times}$if and only if every diagonal entry of $A$ is an invertible element of $k$. If $A \in \mathcal{Q}(I)^{\times}$, then the inverse $A^{-1} \in \mathcal{Q}(I)$ is again upper triangular. Definition 4.22. Let subsets $I, J \subseteq \mathbb{Z}$ and an $I$ by $J$ matrix $A$ with scalar entries be given. Let $I=\coprod_{i=1}^{p} I_{i}$ and $J=\coprod_{j=1}^{q} J_{j}$ be partitions of $I$ and $J$, respectively. We define the
$$
\left[\begin{array}{c}
I_{1} \\
\vdots \\
I_{p}
\end{array}\right] \times\left[\begin{array}{c}
J_{1} \\
\vdots \\
J_{q}
\end{array}\right]^{T}
$$
block decomposition of $A$ to be the $p$ by $q$ matrix
$$
\left[\begin{array}{ccc}
a_{11} & \ldots & a_{1 q} \\
\vdots & & \vdots \\
a_{p 1} & \ldots & a_{p q}
\end{array}\right]
$$
in which $a_{i j}$ is the $I_{i}$ by $J_{j}$ block of $A$. One has $A \in \mathcal{Q}(I, J)$ if and only one has $a_{i j} \in \mathcal{Q}\left(I_{i}, J_{j}\right)$ for all $i=1, \ldots, p$ and $j=1, \ldots, q$.
ExAmple 4.23 . Let $A \in \mathcal{Q}^{\times}$be given. Let $\mathbb{Z}=I \coprod J$ be any partition of $\mathbb{Z}$ into two sets. Suppose that the
$$
\left[\begin{array}{l}
I \\
J
\end{array}\right] \times\left[\begin{array}{l}
I \\
J
\end{array}\right]^{T}
$$
block decompositions of $A$ and $A^{-1}$ take the form
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right], \quad A^{-1}=\left[\begin{array}{cc}
\bar{a} & \bar{b} \\
\bar{c} & \bar{d}
\end{array}\right] .
$$
The block $a$ is invertible if and only if the block $\bar{d}$ is invertible.
Definition 4.24. The bilateral shift $\mathbf{t} \in \mathcal{Q}$ is defined to be the $\mathbb{Z}$ by $\mathbb{Z}$ matrix with 1 's along the superdiagonal $\{i=j-1\}$ and 0 's elsewhere. Given a Laurent series $f=\sum_{i} a_{i} t^{i} \in k((t))$ and a positive integer $n$, put
$$
f\left(\mathbf{t}^{n}\right)_{i j}:= \begin{cases}a_{\frac{j-i}{n}} & \text { if } i \equiv j \bmod n \\ 0 & \text { otherwise }\end{cases}
$$
for all $i, j \in \mathbb{Z}$, thereby defining a matrix $f\left(\mathbf{t}^{n}\right) \in \mathcal{Q}$.
EXAMPle 4.25. The map
$$
\left(f \mapsto f\left(\mathbf{t}^{n}\right)\right): k((t)) \rightarrow \mathcal{Q}
$$
is an injective $k$-linear ring homomorphism under which the variable $t$ maps to the $n^{\text {th }}$ power $\mathbf{t}^{n}$ of the bilateral shift. The $k$-module $\mathcal{H}$ viewed as a left $k((t))$-module via the homomorphism $f \mapsto f\left(\mathbf{t}^{n}\right)$ is free of rank $n$. A matrix $A \in \mathcal{Q}$ commutes with $\mathbf{t}$ if and only if $A=f(\mathbf{t})$ for some (necessarily unique) $f \in k((t))$. EXAMPle 4.26. Let $\left\{A^{(n)}\right\}_{n=1}^{\infty}$ be a sequence in $\mathcal{Q}^{\times}$. If the sequences $\left\{A^{(n)}\right\}$ and $\left\{\left(A^{(n)}\right)^{-1}\right\}$ are both uniformly dominated and entrywise convergent, the entrywise limit of the sequence $\left\{A^{(n)}\right\}$ belongs to $\mathcal{Q}^{\times}$. The sequence $\left\{\mathbf{t}^{n}\right\}_{n=1}^{\infty}$ is uniformly dominated and converges entrywise to 0 . The sequence $\left\{\mathbf{t}^{-n}\right\}_{n=1}^{\infty}$ fails to be uniformly dominated.
EXAMPle 4.27. Fix a positive integer $n$. Let $\phi \in k((t))$ be a Laurent series of winding number $n$. There exists unique $A \in \mathcal{Q}$ with the following two properties:
- $\phi(\mathbf{t}) A=A \mathbf{t}^{n}$.
- $A_{i j}=\delta_{i j}$ for all $i, j \in \mathbb{Z}$ such that $-n<j \leq 0$.
Modulo $m$ the matrix $A$ is upper triangular and each diagonal entry is nonzero. It follows that $A \in \mathcal{Q}^{\times}$.
ExAmple 4.28. Fix a positive integer $n$. Given $A \in \mathcal{Q}$, let $A^{[n]}$ be the $n$ by $n$ matrix with entries in $\mathcal{Q}$ given by the rule
$$
\left(A_{\mu \nu}^{[n]}\right)_{i j}:=A_{i n-\mu+1, j n-\nu+1}
$$
for $\mu, \nu=1, \ldots, n$ and $i, j \in \mathbb{Z}$. The map
$$
\left(A \mapsto A^{[n]}\right): \mathcal{Q} \rightarrow \operatorname{Mat}_{n}(\mathcal{Q})
$$
is an isomorphism of $k$-algebras. Given $h \in \mathcal{H}$, let $h^{[n]}$ be the column vector of length $n$ with entries in $\mathcal{H}$ given by the rule
$$
\left(h_{\mu}^{[n]}\right)_{i}:=h_{i n-\mu+1}
$$
for $\mu=1, \ldots, n$ and $i \in \mathbb{Z}$. The map
$$
\left(h \mapsto h^{[n]}\right): \mathcal{H} \rightarrow \operatorname{Mat}_{n \times 1}(\mathcal{H})
$$
is bijective. One has
$$
(A h)^{[n]}=A^{[n]} h^{[n]}
$$
for all $A \in \mathcal{Q}$ and $h \in \mathcal{H}$.
ExAmple 4.29. Fix a positive integer $n$. One has
$$
\left(\left(\mathbf{t}^{n}\right)^{[n]}\right)_{\mu \nu}=\mathbf{t} \delta_{\mu \nu}
$$
for $\mu, \nu=1, \ldots, n$ and hence one has a $k$-algebra isomorphism
$$
\left(A \mapsto A^{[n]}\right):\left\{A \in \mathcal{Q} \mid A \mathbf{t}^{n}=\mathbf{t}^{n} A\right\} \rightarrow \operatorname{Mat}_{n}(\{A \in \mathcal{Q} \mid A \mathbf{t}=\mathbf{t} A\}) .
$$
Thus the $k$-algebras
$$
\operatorname{Mat}_{n}(k((t)))
$$
and
$$
\left\{A \in \mathcal{Q} \mid A \mathbf{t}^{n}=\mathbf{t}^{n} A\right\}=\text { commutant of } \mathbf{t}^{n} \text { in } \mathcal{Q}
$$
are canonically isomorphic. ExAmple 4.30. Fix $\phi \in k((t))^{\times}$of positive winding number $n$. There exists a unique $k$-algebra homomorphism
$$
(f \mapsto f \circ \phi): k((t)) \rightarrow k((t))
$$
and a unique multiplicative map
$$
\left(f \mapsto \mathcal{N}_{\phi} f\right): k((t)) \rightarrow k((t))
$$
such that
$$
A f\left(\mathbf{t}^{n}\right) A^{-1}=(f \circ \phi)(\mathbf{t})
$$
and
$$
\operatorname{det}\left(\left(A^{-1} f(\mathbf{t}) A\right)^{[n]}\right)=\left(\mathcal{N}_{\phi} f\right)(\mathbf{t})
$$
for all $f \in k((t))$ and $A \in \mathcal{Q}^{\times}$such that
$$
\phi(\mathbf{t}) A=A \mathbf{t}^{n}
$$
One can verify that
$$
w\left(\mathcal{N}_{\phi} f\right)=w(f), \quad w(f \circ \phi)=n w(f)
$$
for all $f \in k((t))^{\times}$. One can verify that via the map $f \mapsto f \circ \phi$ the ring $k((t))$ becomes a finite $k((t))$-algebra of rank $n$ and that $\mathcal{N}_{\phi}$ is the norm mapping associated in the usual way to that ring extension.
## Degree theory
Definition 5.1. A $k$-submodule $P \subseteq \mathcal{H}$ will be called a parallelotope if $P$ is a free $k$-module and
$$
\mathcal{H}\left(\left\{n \in \mathbb{Z} \mid n \leq n_{0}\right\}\right) \subseteq P \subseteq \mathcal{H}\left(\left\{n \in \mathbb{Z} \mid \leq n_{1}\right\}\right)
$$
for some integers $n_{0} \leq n_{1}$. By Example 4.14, for all parallelotopes $P \subset \mathcal{H}$ and $A \in \mathcal{Q}^{\times}$, again $A P$ is a parallelotope.
Proposition 5.2. There exists a unique function
$$
\left(\left(P_{1}, P_{2}\right) \mapsto\left[P_{1}: P_{2}\right]\right):\{\text { parallelotopes }\} \times\{\text { parallelotopes }\} \rightarrow \mathbb{Z}
$$
with the following properties:
1. $\left[P_{0}: P_{1}\right]$ equals the $k$-rank of $P_{0} / P_{1}$ for all parallelotopes $P_{0} \supseteq P_{1}$.
2. $\left[P_{0}: P_{1}\right]=-\left[P_{1}: P_{0}\right]$ for all parallelotopes $P_{0}$ and $P_{1}$.
3. $\left[P_{0}: P_{2}\right]=\left[P_{0}: P_{1}\right]+\left[P_{1}: P_{2}\right]$ for all parallelotopes $P_{0}, P_{1}, P_{2}$.
(We call $[\cdot: \cdot]$ the index function associated to the family of parallelotopes in $\mathcal{H}$.) Proof. There exists a unique function
$$
i:\{\text { parallelotopes }\} \times\{\text { parallelotopes }\} \rightarrow \mathbb{Z}
$$
such that
$i\left(P_{0}, P_{1}\right)=\left(k\right.$-rank of $\left.P_{0} / \mathcal{H}\left(\left\{n \leq n_{0}\right\}\right)\right)-\left(k\right.$-rank of $\left.P_{1} / \mathcal{H}\left(\left\{n \leq n_{0}\right\}\right)\right)$ for all parallelotopes $P_{0}$ and $P_{1}$ and integers $n_{0} \ll 0$. The function $i$ has all the properties required of an index function. Therefore at least one index function exists. For all parallelotopes $P_{0}, P_{1}$ and $P$ such that $P \subseteq P_{0} \cap P_{1}$, and index functions $[\cdot, \cdot]$, one has
$$
\left[P_{0}: P_{1}\right]=\left[P_{0}: P\right]+\left[P: P_{1}\right]=\left[P_{0}: P\right]-\left[P_{1}: P\right]=i\left(P_{0}, P_{1}\right) .
$$
Therefore at most one index function exists.
Definition 5.3. By Proposition 5.2 one has
$$
\left[A P_{1}: A P_{2}\right]=\left[P_{1}: P_{2}\right]
$$
for all $A \in \mathcal{Q}^{\times}$and parallelotopes $P_{1}, P_{2} \subset \mathcal{H}$. It follows that there exists a unique homomorphism
$$
\operatorname{deg}: \mathcal{Q}^{\times} \rightarrow \mathbb{Z}
$$
such that
$$
\operatorname{deg} A=[A P: P]
$$
for all $A \in \mathcal{Q}^{\times}$and parallelotopes $P$. We call $\operatorname{deg} A$ the degree of $A$.
ExAmPle 5.4. For all $f \in k((t))^{\times}$one has
$$
\operatorname{deg} f(\mathbf{t})=-w(f) .
$$
In particular,
$$
\operatorname{deg} \mathbf{t}=-1 .
$$
For all $A \in \mathcal{Q}^{\times}$such that the $\mathbb{N}$ by $1-\mathbb{N}$ blocks of both $A$ and $A^{-1}$ vanish, one has $\operatorname{deg} A=0$. For all $A \in \mathcal{Q}^{\times}$such that the $1-\mathbb{N}$ by $\mathbb{N}$ blocks of both $A$ and $A^{-1}$ vanish, one has $\operatorname{deg} A=0$.
## Quadratic forms
Definition 6.1. Let $E$ be a free $k$-module. A $k$-quadratic form
$$
q: E \rightarrow k
$$
is a function such that for some $k$-bilinear function $b: E \times E \rightarrow k$ one has
$$
q(e)=b(e, e)
$$
for all $e \in E$. We put
$$
q(e, f):=q(e+f)-q(e)-q(f)=b(e, f)+b(f, e)
$$
for all $e, f \in E$, thus canonically associating a symmetric $k$-bilinear form $q(\cdot, \cdot)$ to the $k$-quadratic form $q(\cdot)$. We say that $k$-linear endomorphisms $\alpha, \beta: E \rightarrow E$ are $q$-adjoint if
$$
q(\alpha e, f)=q(e, \beta f)
$$
for all $e, f \in E$. We say that $q$ is nondegenerate if for every $0 \neq e \in E$ there exists $0 \neq f \in E$ such that $q(e, f) \neq 0$.
ExAmple 6.2. A free $k$-module $E$ equipped with a basis $e, f \in E$ such that $q(x e+y f)=x y$ for all scalars $x$ and $y$ is called a hyperbolic plane. A hyperbolic plane is nondegenerate.
ExAmple 6.3. The quadratic form
$$
\left(h \mapsto \sum_{n \in \mathbb{N}} h_{n} h_{1-n}\right): \mathcal{H} \rightarrow k
$$
is nondegenerate. The quadratic form
$$
\left(\left[\begin{array}{l}
f \\
g
\end{array}\right] \mapsto \sum_{n \in \mathbb{Z}} f_{n} g_{1-n}\right):\left[\begin{array}{l}
\mathcal{H} \\
\mathcal{H}
\end{array}\right] \rightarrow k
$$
is nondegenerate. Composition of the latter quadratic form with the $k$-linear isomorphism $\left(h \mapsto h^{[2]}\right): \mathcal{H} \rightarrow\left[\begin{array}{c}\mathcal{H} \\ \mathcal{H}\end{array}\right]$ yields the former.
ExAMPle 6.4. Let $E$ be a free $k$-module equipped with a nondegenerate $k$-quadratic form $q: E \rightarrow k$. Let $\alpha, \beta: E \rightarrow E$ be idempotent $q$-adjoint $k$-linear endomorphisms of $E$. Let $k$-linearly independent vectors $e_{1}, \ldots, e_{n} \in \alpha E$ be given. Then there exist vectors $f_{1}, \ldots, f_{n} \in \beta E$ such that $q\left(e_{i}, f_{j}\right)=\delta_{i j}$ for $i, j=1, \ldots, n$; necessarily $f_{1}, \ldots, f_{n}$ are $k$ linearly independent.
DeFinition 6.5 . Let $E$ be a free $k$-module equipped with a $k$ quadratic form q. A $k$-linear endomorphism $\pi: E \rightarrow E$ is called a $q$-polarization if
$$
\pi^{2}=\pi, \quad q \circ \pi=0, \quad q \circ(1-\pi)=0 .
$$
For any $q$-polarization $\pi$, one has
$$
\begin{aligned}
0 & =q((1-\pi)(e+f))-q(\pi(e+f)) \\
& =q((1-\pi) e,(1-\pi) f)-q(\pi e, \pi f) \\
& =q(e, f)-q(\pi e, f)-q(e, \pi f) \\
& =q((1-\pi) e, f)-q(e, \pi f)
\end{aligned}
$$
for all $e, f \in E$, i. e., $\pi$ and $(1-\pi)$ are $q$-adjoint. EXAMPLE 6.6. Let $I \subset \mathbb{Z}$ be any subset such that $\mathbb{Z}=I \coprod(1-I)$. Let $\pi: \mathcal{H} \rightarrow \mathcal{H}$ be the $k$-linear endomorphism defined by the rule
$$
(\pi h)_{i}:= \begin{cases}h_{i} & \text { if } i \in I \\ 0 & \text { if } i \notin I\end{cases}
$$
for all $i \in \mathbb{Z}$. Then $\pi$ is a polarization with respect to the quadratic form $h \mapsto \sum_{n \in \mathbb{N}} h_{n} h_{1-n}=\sum_{i \in I} h_{i} h_{1-i}$.
DEFINition 6.7. Let $E$ be a free $k$-module equipped with a $k$ quadratic form $q: E \rightarrow k$. A $k$-linear endomorphism $\mu: E \rightarrow E$ is called is $q$-projector if $\mu^{2}=\mu$ and $\mu$ is $q$-self-adjoint. For each $q$ projector $\mu$ one has $q(e)=q(\mu e)+q((1-\mu) e)$ for all $e \in E$.
ExAmple 6.8 . Let $E$ be a free $k$-module equipped with a $k$-quadratic form $q: E \rightarrow k$ and a $q$-polarization $\pi$. Let $e_{1}, \ldots, e_{n} \in \pi E$ and $f_{1}, \ldots, f_{n} \in(1-\pi) E$ be vectors such that $q\left(e_{i}, f_{j}\right)=\delta_{i j}$ for $i, j=$ $1, \ldots, n$. Then
$$
\mu:=\left(e \mapsto \sum_{i=1}^{n}\left(q\left(e_{i}, e\right) f_{i}+q\left(f_{i}, e\right) e_{i}\right)\right): E \rightarrow E
$$
is a $q$-projector such that $\mu \pi=\pi \mu$.
ExAmple 6.9 . Let $E$ be a free $k$-module equipped with a nondegenerate $k$-quadratic form $q$ and a $q$-polarization $\pi$. Let $e_{1}, \ldots, e_{r} \in \pi E$ and $f_{1}, \ldots, f_{r} \in(1-\pi) E$ be vectors such that $q\left(e_{i}, f_{j}\right)=\delta_{i j}$ for $i, j=1, \ldots, r$. Let $V \subseteq E$ be any finitely generated $k$-submodule. Then there exist vectors $e_{r+1}, \ldots, e_{n} \in \pi E$ and $f_{r+1}, \ldots, f_{n} \in(1-\pi) E$ such that $V$ is contained in the $k$-span of $e_{1}, \ldots, e_{n}, f_{1}, \ldots, f_{n}$ and $q\left(e_{i}, f_{j}\right)=\delta_{i j}$ for $i, j=1, \ldots, n$.
## The big split orthogonal group
Definition 7.1. We say that $A \in \mathcal{Q}$ is split orthogonal if $A \in \mathcal{Q}^{\times}$ and $A$ preserves the quadratic form $\left(h \mapsto \sum_{i \in \mathbb{N}} h_{i} h_{1-i}\right): \mathcal{H} \rightarrow k$. The set consisting of the split orthogonal matrices in $\mathcal{Q}$ forms a group under matrix multiplication called the big split orthogonal group.
Example 7.2. Let a matrix $A \in \mathcal{Q}$ be given. Consider the following conditions:
- $\sum_{n \in \mathbb{Z}} A_{n i} A_{1-n, j}=\delta_{i, 1-j}$ for all distinct $i, j \in \mathbb{Z}$.
- $\sum_{n \in \mathbb{N}} A_{n i} A_{1-n, i}=0$ for all $i \in \mathbb{Z}$.
- $\sum_{n \in \mathbb{Z}}^{n \in \mathbb{N}} A_{i n} A_{j, 1-n}=\delta_{i, 1-j}$ for all distinct $i, j \in \mathbb{Z}$.
- $\sum_{n \in \mathbb{N}} A_{i n} A_{i, 1-n}=0$ for all $i \in \mathbb{Z}$.
These conditions are necessary and sufficient for $A$ to be split orthogonal. Definition 7.3. Given subsets $I, J \subseteq \mathbb{Z}$ and a matrix $A \in \mathcal{Q}(I, J)$, we define $A^{\dagger} \in \mathcal{Q}(1-J, 1-I)$ by the rule
$$
A_{i j}^{\dagger}=A_{1-j, 1-i}
$$
for all $i \in 1-I$ and $j \in 1-J$. In other words, $A^{\dagger}$ is obtained from $A$ by reflection through the anti-diagonal $\{i+j=1\}$. Just like the transpose operation, the dagger operation reverses matrix products. The dagger operation preserves almost upper triangularity, whereas the transpose operation does not.
Definition 7.4. Given a subset $I \subset \mathbb{Z}$ and a matrix $A \in \mathcal{Q}(I, 1-I)$, we say that $A$ is dagger-alternating if $A^{\dagger}+A=0$ and $A_{i, 1-i}=0$ for all $i \in I$. In other words, dagger-alternating matrices are anti-symmetric under reflection through the anti-diagonal $\{i+j=1\}$ and vanish along the anti-diagonal.
Example 7.5. Fix $A \in \mathcal{Q}$. Let $I \subset \mathbb{Z}$ be a subset such that
$$
\mathbb{Z}=I \coprod(1-I),
$$
e. g. $I=\mathbb{N}$ or $I=2 \mathbb{Z}$. The
$$
\left[\begin{array}{c}
1-I \\
I
\end{array}\right] \times\left[\begin{array}{c}
1-I \\
I
\end{array}\right]^{T}
$$
block decompositions of $A$ and $A^{\dagger}$ take the form
$$
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right], \quad A^{\dagger}=\left[\begin{array}{ll}
d^{\dagger} & b^{\dagger} \\
c^{\dagger} & a^{\dagger}
\end{array}\right] .
$$
The matrix $A$ is split orthogonal if and only if
$$
a d^{\dagger}+b c^{\dagger}=1, \quad d^{\dagger} a+b^{\dagger} c=1
$$
and the matrices
$$
d^{\dagger} b, \quad c^{\dagger} a, \quad a b^{\dagger}, \quad c d^{\dagger}
$$
are dagger-alternating. In the special case
$$
A=\left[\begin{array}{ll}
1 & b \\
0 & 1
\end{array}\right]
$$
the matrix $A$ is split orthogonal if and only if the matrix $b$ is daggeralternating. In the special case
$$
A=\left[\begin{array}{ll}
a & 0 \\
0 & d
\end{array}\right]
$$
the matrix $A$ is split orthogonal if and only if $a d^{\dagger}=d^{\dagger} a=1$. ExAmple 7.6. Let $\mathbf{p} \in \mathcal{Q}$ be defined by the rule
$$
\mathbf{p}_{i j}:= \begin{cases}1 & \text { if } i=j \in \mathbb{N} \\ 0 & \text { otherwise }\end{cases}
$$
for all $i, j \in \mathbb{Z}$. One has
$$
\mathbf{p}^{2}=\mathbf{p}, \quad \mathbf{p}+\mathbf{p}^{\dagger}=1 .
$$
We call $\mathbf{p}$ the standard polarization. The $k$-linear endomorphism
$$
(h \mapsto \mathbf{p} h): \mathcal{H} \rightarrow \mathcal{H}
$$
is a $\left(h \mapsto \sum_{n \in \mathbb{N}} h_{n} h_{1-n}\right)$-polarization.
Example 7.7. A matrix $A \in \mathcal{Q}$ is split orthogonal if and only if
$$
A A^{\dagger}=A^{\dagger} A=1
$$
and
$$
X^{\dagger} A^{\dagger} \mathbf{p} A X=X^{\dagger} \mathbf{p} X
$$
for all $X \in \mathcal{Q}$ supported in a single column.
ExAmple 7.8. Let $X \in \mathcal{Q}$ be strictly upper triangular and supported in a single column. The matrices
$$
1-\mathbf{p} X+X^{\dagger} \mathbf{p}^{\dagger}, \quad 1-\mathbf{p}^{\dagger} X+X^{\dagger} \mathbf{p}
$$
and
$$
1-X+X^{\dagger}-X^{\dagger} \mathbf{p}^{\dagger} X=\left(1-\mathbf{p} X+X^{\dagger} \mathbf{p}^{\dagger}\right)\left(1-\mathbf{p}^{\dagger} X+X^{\dagger} \mathbf{p}\right)
$$
are split orthogonal.
EXAMPle 7.9. The entrywise limit of a uniformly dominated entrywise convergent sequence of split orthogonal elements of $\mathcal{Q}$ is split orthogonal.
Example 7.10. Let $A \in \mathcal{Q}$ be given and write
$$
A^{[2]}=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \in \operatorname{Mat}_{2}(\mathcal{Q}) .
$$
One has
$$
\left(A^{\dagger}\right)^{[2]}=\left[\begin{array}{cc}
d^{\dagger} & b^{\dagger} \\
c^{\dagger} & a^{\dagger}
\end{array}\right] .
$$
Definition 7.11. The following conditions on a matrix
$$
\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \in \operatorname{Mat}_{2}(\mathcal{Q})
$$
are equivalent: - One has $A^{[2]}=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ for some split orthogonal $A \in \mathcal{Q}$.
- The matrix $\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ belongs to $\mathrm{GL}_{2}(\mathcal{Q})$ and preserves the quadratic form $\left(\left[\begin{array}{l}f \\ g\end{array}\right] \mapsto \sum_{n \in \mathbb{Z}} f_{n} g_{1-n}\right):\left[\begin{array}{l}\mathcal{H} \\ \mathcal{H}\end{array}\right] \rightarrow k$.
- One has
$$
a d^{\dagger}+b c^{\dagger}=1, \quad d^{\dagger} a+b^{\dagger} c=1
$$
and the matrices
$$
d^{\dagger} b, \quad c^{\dagger} a, \quad a b^{\dagger}, \quad c d^{\dagger}
$$
are dagger-alternating.
Under the equivalent conditions above we say that $\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ is a split orthogonal element of $\operatorname{Mat}_{2}(\mathcal{Q})$.
## Tame maps
Definition 8.1. Let subsets $I, J \subseteq \mathbb{Z}$ be given. Let $A$ be an $I$ by $J$ matrix with scalar entries. We say that $A$ is almost diagonal if both $A$ and its transpose $A^{T}$ are almost upper triangular. Let
$$
\omega: J \rightarrow I
$$
be a map and let $W$ be the $I$ by $J$ matrix defined by the rule
$$
W_{i j}:=\delta_{i, \omega(j)}
$$
for all $i \in I$ and $j \in J$. In this situation we say that $W$ represents $\omega$. The following conditions are equivalent:
- The matrix $W$ is almost diagonal.
- $\forall n \in \mathbb{Z}|\{j \in J \mid \omega(j) \geq n \geq j\} \cup\{j \in J \mid j \geq n \geq \omega(j)\}|<\infty$.
- There exist gauges $\sigma$ and $\tau$ such that
$$
\sigma(j) \leq \omega(j) \leq \tau(j)
$$
for all $j \in J$.
Under these equivalent conditions we say that the map $\omega$ is tame. EXAmple 8.2. Let $I$ and $J$ be subsets of $\mathbb{Z}$. Let $\omega: J \rightarrow I$ be a map. Assuming that $J$ is neither bounded above nor below, the map $\omega$ is tame if and only if
$$
\lim _{j \rightarrow-\infty} \omega(j)=-\infty \text { and } \lim _{j \rightarrow+\infty} \omega(j)=+\infty .
$$
Assuming that $J$ is either bounded above or bounded below, the map $\omega$ is tame if and only if the set $\{j \in J \mid \omega(j)=n\}$ is finite for all $n \in \mathbb{Z}$.
ExAmple 8.3. Let subsets $I, J \subseteq \mathbb{Z}$ be given. Let $I=\coprod_{i=1}^{n} I_{i}$ be a disjoint decomposition of $I$. A map $\omega: I \rightarrow J$ is tame if and only if the restriction of $\omega$ to $I_{i}$ is tame for $i=1, \ldots, n$.
ExAmple 8.4. Let subsets $I, J, K \subseteq \mathbb{Z}$ and tame maps $\eta: K \rightarrow J$ and $\omega: J \rightarrow I$ be given. Let $W$ be the $I$ by $J$ matrix representing $\omega$. Let $Y$ be the $J$ by $K$ matrix representing $\eta$. The composite map $\omega \circ \eta: K \rightarrow I$ is tame and the product matrix $W Y$ represents the composition $\omega \circ \eta$. If $\omega$ is bijective, the inverse function $\omega^{-1}: I \rightarrow J$ is tame and the transposed matrix $W^{T}$ represents $\omega^{-1}$.
ExAMPle 8.5. Let $\omega: \mathbb{Z} \rightarrow \mathbb{Z}$ be a tame one-to-one map and let $W$ be the almost diagonal $\mathbb{Z}$ by $\mathbb{Z}$ matrix representing $\omega$. One has
$$
\left(W^{T} W\right)_{i j}=\delta_{i j}, \quad\left(W W^{T}\right)_{i j}= \begin{cases}1 & \text { if } i=j \in \omega(\mathbb{Z}) \\ 0 & \text { otherwise }\end{cases}
$$
for all $i, j \in \mathbb{Z}$. For each $A \in \mathcal{Q}^{\times}$, put
$$
\omega_{*} A:=1-W W^{T}+W A W^{T} .
$$
One has
$$
\left(\omega_{*} A\right)_{i j}=\left\{\begin{array}{cc}
A_{\omega^{-1}(i), \omega^{-1}(j)} & \text { if } i, j \in \omega(\mathbb{Z}) \\
\delta_{i j} & \text { otherwise }
\end{array}\right.
$$
for all $i, j \in \mathbb{Z}$. One has
$$
\omega_{*} 1=1, \quad \omega_{*}(A B)=\left(\omega_{*} A\right)\left(\omega_{*} B\right), \quad(\omega \circ \eta)_{*} A=\omega_{*}\left(\eta_{*} A\right)
$$
and
$$
\omega(\mathbb{Z}) \cap \eta(\mathbb{Z})=\emptyset \Rightarrow\left(\omega_{*} A\right)\left(\eta_{*} B\right)=\left(\eta_{*} B\right)\left(\omega_{*} A\right)
$$
for all $A, B \in \mathcal{Q}^{\times}$and tame one-to-one maps $\omega, \eta: \mathbb{Z} \rightarrow \mathbb{Z}$. EXAMPle 8.6. Fix a positive integer $n$. One has
$$
\left((\ell \mapsto n \ell)_{*} A\right)^{[n]}=\left[\begin{array}{cccc}
A & & & \\
& 1 & & \\
& & \ddots & \\
& & & 1
\end{array}\right]
$$
for all $A \in \mathcal{Q}^{\times}$. One has
$$
g\left(\mathbf{t}^{n}\right)^{[n]}=\left[\begin{array}{ccc}
g(\mathbf{t}) & & \\
& \ddots & \\
& & g(\mathbf{t})
\end{array}\right], \quad g\left(\mathbf{t}^{n}\right)=\prod_{i=1}^{n}(\ell \mapsto \ell n+1-i)_{*} g(\mathbf{t})
$$
for all $g \in k((t))^{\times}$.
ExAmPle 8.7. For all tame one-to-one maps $\omega: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$$
\omega(1-n)=1-\omega(n)
$$
for all $n \in \mathbb{Z}$ and split orthogonal matrices $A \in \mathcal{Q}$, again the matrix $\omega_{*} A$ is split orthogonal.
ExAmple 8.8. Let $f, \phi \in k((t))^{\times}$be given. Assume that the winding number $n$ of $\phi$ is positive. Let $A \in \mathcal{Q}^{\times}$be given such that
$$
\phi(\mathbf{t}) A=A \mathbf{t}^{n}
$$
By Examples 1.6, 4.29 and 8.6, one has a factorization
$$
A^{-1} f(\mathbf{t}) A=\left((\ell \mapsto n \ell)_{*}\left(\mathcal{N}_{\phi} f\right)(\mathbf{t})\right) C
$$
where $C$ belongs to the commutator subgroup of the commutant of $\mathbf{t}^{n}$ in $\mathcal{Q}^{\times}$.
## Partitions, wedge indices and diamond indices
Definition 9.1. A partition $\lambda$ is an infinite nonincreasing sequence
$$
\lambda_{1} \geq \lambda_{2} \geq \lambda_{3} \geq \ldots
$$
of nonnegative integers almost all terms of which vanish. Put
$$
|\lambda|:=\sum_{i} \lambda_{i}, \quad \ell(\lambda):=\left|\left\{i \in \mathbb{N} \mid \lambda_{i}>0\right\}\right|,
$$
thereby defining the weight and length of $\lambda$, respectively. See [Macdonald SFHP] for background concerning partitions. Definition 9.2. The diagram of partition $\lambda$ is defined to be the set $\left\{(i, j) \in \mathbb{N} \times \mathbb{N} \mid j \leq \lambda_{i}\right\}$ and (by abuse of notation) is again denoted by $\lambda$. We always visualize the diagram of a partition according to the row-column convention of matrix theory. The weight of a partition is the number of nodes in its diagram. The partition with diagram the transpose of the diagram of $\lambda$ is denoted $\lambda^{\prime}$ and said to be conjugate to $\lambda$. One has has $|\lambda|=\left|\lambda^{\prime}\right|$ and $\ell(\lambda)=\lambda_{1}^{\prime}$.
Definition 9.3. Given a partition $\lambda$, one writes
$$
\lambda=\left(\alpha_{1}>\cdots>\alpha_{r} \mid \beta_{1}>\cdots>\beta_{r}\right)
$$
under the following three conditions:
- The main diagonal of $\lambda$ consists of $r$ nodes.
- For $i=1, \ldots, r$, there are $\alpha_{i}$ nodes of $\lambda$ to the right of $(i, i)$.
- For $i=1, \ldots, r$, there are $\beta_{i}$ nodes of $\lambda$ below $(i, i)$.
This notation for partitions is due to Frobenius.
EXAMPLE 9.4.
$$
\begin{aligned}
& =(4>2>1 \mid 3>1>0), \\
& (5 \geq 4 \geq 4 \geq 1 \geq 0 \geq \ldots)^{\prime}= \\
& =(3>1>0 \mid 4>2>1) \\
& =(4 \geq 3 \geq 3 \geq 3 \geq 1 \geq 0 \geq \ldots) \text {. }
\end{aligned}
$$
ExAmple 9.5. Let a $\mathbb{Z}$ by $\mathbb{N}$ matrix $A$ with scalar entries be given such that $A_{i j}=\delta_{i j}$ for all $i, j \in \mathbb{N}$. Let a partition
$$
\lambda=\left(\lambda_{1} \geq \lambda_{2} \geq \ldots\right)=\left(\alpha_{1}>\cdots>\alpha_{r} \mid \beta_{1}>\cdots>\beta_{r}\right)
$$
be given. One has
$$
\operatorname{det}_{i, j=1}^{N} A_{i-\lambda_{i}, j}=(-1)^{\sum_{j} \beta_{j}} \operatorname{det}_{i, j=1}^{r} A_{-\alpha_{i}, \beta_{j}+1}
$$
for all $N \geq \ell(\lambda)$.
EXAMPle 9.6. Let $x_{1}, \ldots, x_{N}$ be independent variables and put
$$
\left.\begin{array}{rl}
\sum_{n} h_{n} t^{n} & =\prod_{i=1}^{N}\left(1-x_{i} t\right)^{-1} \\
\sum_{n}(-1)^{n} e_{n} t^{n} & =\prod_{i=1}^{N}\left(1-x_{i} t\right)
\end{array}\right\} \in \mathbb{Z}\left[x_{1}, \ldots, x_{N}\right][[t]] .
$$
The $h$ 's and the $e$ 's are the so called complete and elementary symmetric functions of the $x$ 's, respectively. One has
$$
\operatorname{det}_{i, j=1}^{\ell(\lambda)} h_{\lambda_{i}-i+j}=\frac{\operatorname{det}_{i, j=1}^{N} x_{i}^{\lambda_{j}+N-j}}{\operatorname{det}_{i, j=1}^{N} x_{i}^{N-j}}=\operatorname{det}_{i, j=1}^{\ell\left(\lambda^{\prime}\right)} e_{\lambda_{i}^{\prime}-i+j}
$$
for all partitions $\lambda$ such that $N \geq \ell(\lambda)$. The determinant on the left is by definition the $S$-function of the $x$ 's indexed by $\lambda$. See [Macdonald SFHP] for background and proof.
Example 9.7. Put
$$
h_{n}:= \begin{cases}1 / n ! & \text { if } n \geq 0 \\ 0 & \text { if } n<0\end{cases}
$$
One has
$$
\stackrel{\ell(\lambda)}{\operatorname{det}} h_{\lambda_{i}-i+j}=1 / \prod_{(i, j) \in \lambda}\left(\lambda_{i}+\lambda_{j}^{\prime}-i-j+1\right)
$$
for all partitions $\lambda$. For example, for the partition
one has
$$
{ }_{i, j=1}^{\ell(\lambda)} h_{\lambda_{j}+i-j}=\frac{1}{8 \cdot 6 \cdot 5 \cdot 4 \cdot 1 \cdot 6 \cdot 4 \cdot 3 \cdot 2 \cdot 5 \cdot 3 \cdot 2 \cdot 1 \cdot 1} .
$$
See Macdonald [Macdonald SFHP] for background and proof. Definition 9.8. A subset $I \subset \mathbb{Z}$ such that
$$
\sup I<\infty, \quad \inf (\mathbb{Z} \backslash I)>-\infty
$$
will be called a wedge index. For each wedge index $I$, put
$$
\operatorname{deg} I:=|I \cap \mathbb{N}|-|(1-\mathbb{N}) \backslash I|,
$$
thereby defining the degree of $I$.
Definition 9.9. A subset $I \subset \mathbb{Z}$ such that
$$
\mathbb{Z}=I \coprod(1-I), \quad \text { sup } I<\infty
$$
will be called a diamond index. The diamond indices are in bijective correspondence with the finite subsets of $\mathbb{N}$ under the map $I \mapsto I \cap \mathbb{N}$. Put
$$
\text { parity } I \equiv|I \cap \mathbb{N}| \bmod 2
$$
thereby defining the parity of $I$.
ExAmple 9.10. For each partition $\lambda$, the set $\left\{\lambda_{i}-1+i \mid i \in \mathbb{N}\right\}$ is a wedge index and one has
$$
\mathbb{Z}=\left\{\lambda_{i}-1+i \mid i \in \mathbb{N}\right\} \coprod\left\{i-\lambda_{i}^{\prime} \mid i \in \mathbb{N}\right\}
$$
The construction $\lambda \mapsto\left\{\lambda_{i}-1+i \mid i \in \mathbb{N}\right\}$ puts the partitions in bijective correspondence with the wedge indices of degree 0 . More generally, the construction $(n, \lambda) \mapsto\left\{n+\lambda_{i}-1+i \mid i \in \mathbb{N}\right\}$ puts the cartesian product of $\mathbb{Z}$ and the set of partitions in bijective correspondence with the wedge indices.
EXAmple 9.11. Given any subset $I \subset \mathbb{Z}$, put
$$
I^{\diamond}:=\{2 i \mid i \in I\} \cup\{1-2 i \mid i \in \mathbb{Z} \backslash I\}
$$
One has
$$
(\mathbb{Z} \backslash I)^{\diamond}=1-I^{\diamond}
$$
for any subset $I \subset \mathbb{Z}$. The construction $I \mapsto I^{\diamond}$ puts the wedge indices in bijective correspondence with the diamond indices. if
Definition 9.12. We say that a diamond index $I$ is parity-balanced
$$
|\{i \in I \cap \mathbb{N} \mid i \equiv 0 \bmod 2\}|=|\{i \in I \cap \mathbb{N} \mid i \equiv 1 \bmod 2\}| .
$$
A diamond index $I$ is parity-balanced if and only if $I=J^{\diamond}$ for some wedge index $J$ of degree 0 . EXAMPle 9.13. Let $I \subset \mathbb{Z}$ be a parity-balanced diamond index and put
$$
\begin{aligned}
I \cap \mathbb{N} & =\left\{2 \alpha_{1}+2>\cdots>2 \alpha_{r}+2\right\} \cup\left\{2 \beta_{1}+1>\cdots>2 \beta_{r}+1\right\}, \\
I \cap 2 \mathbb{Z} & =\left\{2 \lambda_{1}>2\left(\lambda_{2}-1\right)>2\left(\lambda_{3}-2\right)>\ldots\right\} .
\end{aligned}
$$
One obtains in this way a partition
$$
\lambda=\left(\lambda_{1} \geq \lambda_{2} \geq \ldots\right)=\left(\alpha_{1}>\cdots>\alpha_{r} \mid \beta_{1}>\cdots>\beta_{r}\right) .
$$
The construction $I \mapsto \lambda$ puts the parity-balanced diamond indices in bijective correspondence with the partitions.
## The big Weyl group and its split orthogonal analogue
DeFinition 10.1. The big Weyl group is by definition the group of $\mathbb{Z}$ by $\mathbb{Z}$ almost diagonal permutation matrices. The big split orthogonal Weyl group is by definition the group of $\mathbb{Z}$ by $\mathbb{Z}$ almost diagonal split orthogonal permutation matrices.
EXAMPle 10.2 . Let $\omega: \mathbb{Z} \rightarrow \mathbb{Z}$ be a bijective map and let $W$ be the $\mathbb{Z}$ by $\mathbb{Z}$ permutation matrix representing $\omega$. The following properties are equivalent:
- The map $\omega$ is tame.
- The matrix $W$ belongs to the big Weyl group.
- The matrices $\mathbf{p} W \mathbf{p}^{\dagger}$ and $\mathbf{p}^{\dagger} W \mathbf{p}$ are finitely supported, where $\mathbf{p} \in \mathcal{Q}$ is the standard polarization matrix.
- For some wedge index $I \subset \mathbb{Z}$, again $\omega(I)$ is a wedge index.
- For every wedge index $I \subset \mathbb{Z}$, again $\omega(I)$ is a wedge index.
- $\lim _{j \rightarrow+\infty} \omega(j)=+\infty$ and $\lim _{j \rightarrow-\infty} \omega(j)=-\infty$.
Under the equivalent conditions above we say that $\omega$ is a tame permutation of $\mathbb{Z}$. The tame permutations of $\mathbb{Z}$ form a group under composition isomorphic to the big Weyl group. The group of tame permutations of $\mathbb{Z}$ acts transitively on the family of wedge indices.
EXAmple 10.3 . Let $\omega: \mathbb{Z} \rightarrow \mathbb{Z}$ be a tame permutation of $\mathbb{Z}$. One has
$$
\operatorname{deg} I-\operatorname{deg} J=|I \backslash J|-|J \backslash I|
$$
for all wedge indices $I$ and $J$, and hence exists a unique integer $\operatorname{deg} \omega$ such that
$$
\operatorname{deg} \omega(I)=\operatorname{deg} \omega+\operatorname{deg} I
$$
for all wedge indices $I$. The group of tame permutations of $\mathbb{Z}$ is generated by the shift $n \mapsto n-1$ and the subgroup consisting of permutations stabilizing $\mathbb{N}$ and $1-\mathbb{N}$. EXAMPLE 10.4. The big Weyl group is generated by the bilateral shift $\mathbf{t}$ and elements $W$ such that $\mathbf{p} W \mathbf{p}^{\dagger}=0=\mathbf{p}^{\dagger} W \mathbf{p}$ where $\mathbf{p}$ is the standard polarization. One has
$$
\operatorname{deg} W=\left|\operatorname{supp} \mathbf{p} W \mathbf{p}^{\dagger}\right|-\left|\operatorname{supp} \mathbf{p}^{\dagger} W \mathbf{p}\right|
$$
for all matrices $W$ belonging to the big Weyl group.
EXAMPle 10.5 . Let $\omega: \mathbb{Z} \rightarrow \mathbb{Z}$ be a tame permutation and let $W$ be the almost diagonal permutation matrix representing $\omega$. The following statements are equivalent:
- $W$ is split orthogonal.
- $W^{T}=W^{\dagger}$.
- $\omega(1-n)=1-\omega(n)$ for all $n \in \mathbb{Z}$.
Under these equivalent conditions, we say that the tame permutation $\omega$ is split orthogonal. The split orthogonal tame permutations of $\mathbb{Z}$ form a group under composition isomorphic to the big split orthogonal Weyl group. The group of tame split orthogonal permutations acts transitively on the family of diamond indices. The group of tame split orthogonal permutations is generated by the transposition of 0 and 1 and the subgroup consisting of maps fixing $\mathbb{N}$ and $1-\mathbb{N}$.
EXAMPLE 10.6. Let $\omega: \mathbb{Z} \rightarrow \mathbb{Z}$ be a tame split orthogonal permutation of $\mathbb{Z}$. One has
$$
\text { parity } I-\text { parity } J \equiv|I \backslash J| \bmod 2
$$
for all diamond indices $I$ and $J$, and hence there exists a unique congruence class parity $\omega$ modulo 2 such that
$$
\operatorname{parity} \omega(I) \equiv \operatorname{parity} \omega+\text { parity } I \bmod 2
$$
for all diamond indices $I$.
EXAMPLE 10.7. The big split orthogonal Weyl group is generated by the fundamental reflection $\mathbf{r}$ and the subgroup consisting of matrices $W$ such that $\mathbf{p}^{\dagger} W \mathbf{p}=0$ (and hence also $\mathbf{p} W \mathbf{p}^{\dagger}=0$ ), where $\mathbf{p}$ is the standard polarization. The map
$$
W \mapsto\left|\operatorname{supp} \mathbf{p} W \mathbf{p}^{\dagger}\right| \bmod 2
$$
is a surjective homomorphism from the big split orthogonal Weyl group to $\mathbb{Z} / 2 \mathbb{Z}$. EXAmPle 10.8. Temporarily let $\mathcal{W}$ denote the big Weyl group and let $\mathcal{W}_{0}$ denote subgroup consisting of matrices $W$ such that $\mathbf{p}^{\dagger} W \mathbf{p}=0$ and $\mathbf{p} W \mathbf{p}^{\dagger}=0$. The map
$$
W \mapsto\left(\left|\operatorname{supp} \mathbf{p} W \mathbf{p}^{\dagger}\right|,\left|\operatorname{supp} \quad \mathbf{p}^{\dagger} W \mathbf{p}\right|\right)
$$
puts the double coset space $\mathcal{W}_{0} \backslash \mathcal{W} / \mathcal{W}_{0}$ in bijective correspondence with the set of pairs of nonnegative integers.
ExAmple 10.9. Temporarily let $\mathcal{W}$ denote the big split orthogonal Weyl group and let $\mathcal{W}_{0}$ denote the subgroup consisting of matrices $W$ such that $\mathbf{p}^{\dagger} W \mathbf{p}=0$ (and hence also $\mathbf{p} W \mathbf{p}^{\dagger}=0$ ). The map
$$
W \mapsto\left|\operatorname{supp} \mathbf{p} W \mathbf{p}^{\dagger}\right|
$$
puts the double coset space $\mathcal{W}_{0} \backslash \mathcal{W} / \mathcal{W}_{0}$ in bijective correspondence with the nonnegative integers.
EXAMPle 10.10. Let $\omega: \mathbb{Z} \rightarrow \mathbb{Z}$ be a tame one-to-one map. There exists a tame permutation $\sigma: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $\omega \circ \sigma$ is strictly increasing. If $\omega(1-n)=1-\omega(n)$ for all $n \in \mathbb{Z}$, there exists a unique tame split orthogonal permutation $\sigma: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $\omega \circ \sigma$ is strictly increasing.
## Pivots
Throughout the discussion of pivots, unless explicitly noted otherwise, we assume that $k$ is a field.
Definition 11.1. Let subsets $I, J \subseteq \mathbb{Z}$ and a matrix $A \in \mathcal{Q}(I, J)$ be given. Let $\left(i_{0}, j_{0}\right) \in I \times J$ be given. If the
$$
\left[\begin{array}{c}
I \cap\left\{i<i_{0}\right\} \\
\left\{i_{0}\right\} \\
I \cap\left\{i>i_{0}\right\}
\end{array}\right] \times\left[\begin{array}{c}
J \cap\left\{j<j_{0}\right\} \\
\left\{j_{0}\right\} \\
J \cap\left\{j>j_{0}\right\}
\end{array}\right]^{T}
$$
block decomposition of $A$ takes the form
$$
A=\left[\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
0 & a_{22} & a_{23} \\
a_{31} & 0 & a_{33}
\end{array}\right], \quad a_{22} \neq 0,
$$
we say that $\left(i_{0}, j_{0}\right)$ is a pivot of $A$. If this block decomposition takes the form
$$
A=\left[\begin{array}{ccc}
a_{11} & 0 & a_{13} \\
0 & a_{22} & 0 \\
a_{31} & 0 & a_{33}
\end{array}\right], \quad a_{22} \neq 0,
$$
we say that the pivot $\left(i_{0}, j_{0}\right)$ of $A$ is cleared. ExAmple 11.2. Fix $\left(i_{0}, j_{0}\right) \in \mathbb{Z}^{2}$ and let $\left\{A^{(n)}\right\}_{n=1}^{\infty}$ be a uniformly dominated sequence in $\mathcal{Q}$ converging entrywise to $A \in \mathcal{Q}$.
- There exists $n_{0} \in \mathbb{N}$ such that $\left(i_{0}, j_{0}\right)$ is a pivot of $A^{(n)}$ for all $n \geq n_{0}$ if and only if $\left(i_{0}, j_{0}\right)$ is a pivot of $A$.
- If $\left(i_{0}, j_{0}\right)$ is an uncleared pivot of $A$, then there exists $n_{0} \in \mathbb{N}$ such that $\left(i_{0}, j_{0}\right)$ is an uncleared pivot of $A^{(n)}$ for all $n \geq n_{0}$.
- If there exists $n_{0} \in \mathbb{N}$ such that $\left(i_{0}, j_{0}\right)$ is a cleared pivot of $A^{(n)}$ for all $n \geq n_{0}$, then $\left(i_{0}, j_{0}\right)$ is a cleared pivot of $A$.
The converses of the latter two statements do not hold in general.
Proposition 11.3. Let $A \in \mathcal{Q}$ be given. Let $\left(i_{0}, j_{0}\right)$ be a pivot of A. Let I (resp. J) be the set of integers indexing rows (resp. columns) of $A$ wherein cleared pivots of $A$ distinct from $\left(i_{0}, j_{0}\right)$ appear. There exist unique matrices $X, Y \in \mathcal{Q}$ with the following properties:
1. $X$ is strictly upper triangular and $\operatorname{supp} X \subseteq(\mathbb{Z} \backslash I) \times\left\{i_{0}\right\}$.
2. $Y$ is strictly upper triangular and $\operatorname{supp} Y \subseteq\left\{j_{0}\right\} \times(\mathbb{Z} \backslash J)$.
3. $\left(i_{0}, j_{0}\right)$ is a cleared pivot of $A^{\prime}:=(1-X) A(1-Y)$.
4. Every cleared pivot of $A$ remains a cleared pivot of $A^{\prime}$.
(We call $X$ the left pivot-clearing operator and $Y$ the right pivotclearing operator associated to $A$ and its pivot $\left(i_{0}, j_{0}\right)$.)
Proof. By hypothesis, the
$$
\left[\begin{array}{c}
\left\{i<i_{0}\right\} \backslash I \\
\left\{i_{0}\right\} \\
\left\{i_{0}<i\right\} \backslash I \\
I
\end{array}\right] \times\left[\begin{array}{c}
J \\
\left\{j<j_{0}\right\} \backslash J \\
\left\{j_{0}\right\} \\
\left\{j_{0}<j\right\} \backslash J
\end{array}\right]^{T}
$$
block decomposition of $A$ takes the form
$$
A=\left[\begin{array}{cccc}
0 & a_{12} & a_{13} & a_{14} \\
0 & 0 & a_{23} & a_{24} \\
0 & a_{32} & 0 & a_{34} \\
a_{41} & 0 & 0 & 0
\end{array}\right], \quad a_{23} \neq 0
$$
To satisfy condition 1 the matrix $X$ we seek must have a
$$
\left[\begin{array}{c}
\left\{i<i_{0}\right\} \backslash I \\
\left\{i_{0}\right\} \\
\left\{i_{0}<i\right\} \backslash I \\
I
\end{array}\right] \times\left[\begin{array}{c}
\left\{i<i_{0}\right\} \backslash I \\
\left\{i_{0}\right\} \\
\left\{i_{0}<i\right\} \backslash I \\
I
\end{array}\right]^{T}
$$
block decomposition of the form
$$
X=\left[\begin{array}{llll}
0 & x & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
To satisfy condition 2 the matrix $Y$ we seek must have a
$$
\left[\begin{array}{c}
J \\
\left\{j<j_{0}\right\} \backslash J \\
\left\{j_{0}\right\} \\
\left\{j_{0}<j\right\} \backslash J
\end{array}\right] \times\left[\begin{array}{c}
J \\
\left\{j<j_{0}\right\} \backslash J \\
\left\{j_{0}\right\} \\
\left\{j_{0}<j\right\} \backslash J
\end{array}\right]^{T}
$$
block decomposition of the form
$$
Y=\left[\begin{array}{llll}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & y \\
0 & 0 & 0 & 0
\end{array}\right]
$$
One has
$$
(1-X) A(1-Y)=\left[\begin{array}{cccc}
0 & a_{12} & a_{13}-x a_{23} & \left(a_{13}-x a_{23}\right) y+a_{14}-x a_{24} \\
0 & 0 & a_{23} & a_{24}-a_{23} y \\
0 & a_{32} & 0 & a_{34} \\
a_{41} & 0 & 0 & 0
\end{array}\right]
$$
Conditions 1 and 2 granted, condition 3 holds if and only if $x=a_{13} a_{23}^{-1}$ and $y=a_{23}^{-1} a_{24}$. Thus conditions $1-3$ uniquely determine $X$ and $Y$. Clearly conditions 1-3 imply condition 4.
Lemma 11.4. Let subsets $I, J \subseteq \mathbb{Z}$ be given. Let $A \in \mathcal{Q}(I, J)$ and a nonpivot $\left(i_{0}, j_{0}\right) \in \operatorname{supp} A$ be given. Let $A^{\prime}$ be the
$$
\left\{(i, j) \in I \times J \mid \begin{array}{l}
i \geq i_{0} \\
j \leq j_{0}
\end{array}\right\}
$$
block of $A$. There exists an uncleared pivot $\left(i^{*}, j^{*}\right)$ of $A^{\prime}$.
Proof. For each $n \in \mathbb{N}$, put
$$
j_{n}:=\min \left\{j \in J \mid A_{i_{n-1} j} \neq 0\right\}, \quad i_{n}:=\max \left\{i \in I \mid A_{i j_{n}} \neq 0\right\},
$$
thereby defining a sequence of points in supp $A^{\prime}$ fitting into zigzag pattern thus:
$$
\begin{array}{ccc}
& \left(i_{0}, j_{1}\right) & \left(i_{0}, j_{0}\right) \\
\left(i_{1}, j_{2}\right) & \left(i_{1}, j_{1}\right) & \\
\left(i_{2}, j_{3}\right) \quad\left(i_{2}, j_{2}\right) &
\end{array}
$$
Since the matrix $A^{\prime}$ is finitely supported there exists a unique pivot $\left(i^{*}, j^{*}\right)$ of $A^{\prime}$ such that $i^{*}=i_{n}$ and $j^{*}=j_{n}$ for all $n \gg 0$. By hypothesis $\left(i_{0}, j_{0}\right)$ is not a pivot of $A$, hence $\left(i_{0}, j_{0}\right)$ is not a pivot of $A^{\prime}$, hence the zigzag sequence is nonconstant, and hence $\left(i^{*}, j^{*}\right)$ is an uncleared pivot of $A^{\prime}$.
EXAMPle 11.5. Let $A \in \mathcal{Q}^{\times}$be given such that every pivot of $A$ is cleared. Then there exists a unique factorization of the form $A=D W$ where $D \in \mathcal{Q}^{\times}$is diagonal and $W$ is an element of the big Weyl group. Moreover, if $A$ is split orthogonal, then the matrices $D$ and $W$ are likewise split orthogonal.
Example 11.6. Put
$$
\lambda:=((i, j) \mapsto \max (-2 i, 2 j-1)): \mathbb{Z}^{2} \rightarrow \mathbb{Z} .
$$
The $\mathbb{Z}$ by $\mathbb{Z}$ matrix with entry $\lambda(i, j)$ in position $(i, j)$ looks like this:
The level sets of $\lambda$ thus fit together in a sort of herring bone pattern. For every $A \in \mathcal{Q}$, the restriction of $\lambda$ to the set of pivots of $A$ is oneto-one and bounded below. In other words, the function $\lambda$ well orders the pivots of $A$. ExAmPle 11.7. Let $W$ and $Y$ be elements of the big Weyl group. Let $B, C \in \mathcal{Q}^{\times}$be upper triangular with nonzero diagonal entries. If $B W C=Y$, then $W=Y$.
Theorem 11.8. Let $A \in \mathcal{Q}^{\times}$be given. There exist $B, C, D, W \in$ $\mathcal{Q}^{\times}$with the following properties:
- $B-1$ and $C-1$ are strictly upper triangular.
- $D$ is diagonal.
- W belongs to the big Weyl group.
- $B A C=D W$.
Moreover, the matrix $W$ thus arising is uniquely determined by $A$.
Proof. The uniqueness of $W$ follows from Example 11.7. We turn now to the proof of the existence of $B, C, D$ and $W$. We define sequences
$$
\left\{A^{(n)}\right\}_{n=0}^{\infty}, \quad\left\{B^{(n)}\right\}_{n=0}^{\infty}, \quad\left\{C^{(n)}\right\}_{n=0}^{\infty}, \quad\left\{X^{(n)}\right\}_{n=1}^{\infty}, \quad\left\{Y^{(n)}\right\}_{n=1}^{\infty}
$$
in $\mathcal{Q}$ by the following inductive procedure. Put
$$
A^{(0)}:=A, \quad B^{(0)}:=1, \quad C^{(0)}:=1 .
$$
For each $n \in \mathbb{N}$ such that $A^{(n)}$ has at least one uncleared pivot, let $X^{(n)}$ (resp. $Y^{(n)}$ ) be the left (resp. right) pivot-clearing operator associated to the matrix $A^{(n-1)}$ and the unique $\lambda$-minimal uncleared pivot of $A^{(n-1)}$ by Proposition 11.3 , where $\lambda$ is the function considered in Example 11.6. For each $n \in \mathbb{N}$ such that $A^{(n)}$ has no uncleared pivot, simply set $X^{(n)}:=0$ and $Y^{(n)}:=0$. For every $n \in \mathbb{N}$, set
$$
\begin{aligned}
A^{(n)} & :=\left(1-X^{(n)}\right) A^{(n-1)}\left(1-Y^{(n)}\right), \\
B^{(n)} & :=\left(1-X^{(n)}\right) B^{(n-1)}, \\
C^{(n)} & :=C^{(n-1)}\left(1-Y^{(n)}\right) .
\end{aligned}
$$
By Lemma 4.12 and Example 4.15, the sequences
$$
\left\{A^{(n)}\right\}, \quad\left\{\left(A^{(n)}\right)^{-1}\right\}, \quad\left\{B^{(n)}\right\}, \quad\left\{\left(B^{(n)}\right)^{-1}\right\}, \quad\left\{C^{(n)}\right\}, \quad\left\{\left(C^{(n)}\right)^{-1}\right\}
$$
are uniformly dominated and entrywise convergent. Let $M$ be the entrywise limit of the sequence $\left\{A^{(n)}\right\}$. Let $B$ (resp. $C$ ) be the entrywise limit of the sequence $\left\{B^{(n)}\right\}$ (resp. $\left\{C^{(n)}\right\}$ ). One has $M, B, C \in \mathcal{Q}^{\times}$ by Example 4.26. The differences $B-1$ and $C-1$ are strictly upper triangular by Example 4.15. One has $B A C=M$ by Lemma 4.12. We claim that $M$ has no uncleared pivots. Suppose to the contrary that $M$ has an uncleared pivot $\left(i^{*}, j^{*}\right)$. By Example 11.2 there exists $n_{0} \in \mathbb{N}$ such that $\left(i^{*}, j^{*}\right)$ is an uncleared pivot of $A^{(n)}$ for all $n \geq n_{0}$. For each $n \geq n_{0}$, let $\left(i_{n}, j_{n}\right)$ be the unique $\lambda$-minimal uncleared pivot of $A^{(n)}$. By the definitions, for all $n^{\prime}>n \geq n_{0}$, the pair $\left(i_{n}, j_{n}\right)$ is a cleared pivot of $A^{\left(n^{\prime}\right)}$ and in particular is distinct from the pair $\left(i_{n^{\prime}}, j_{n^{\prime}}\right)$. By Example 11.2 it follows that each of the infinitely many points in the sequence $\left\{\left(i_{n}, j_{n}\right)\right\}_{n=n_{0}}^{\infty}$ is a cleared pivot of $M$. By construction one has $\lambda\left(i_{n}, j_{n}\right) \leq \lambda\left(i^{*}, j^{*}\right)$ for all $n \geq n_{0}$. But the set $\left\{(i, j) \in \operatorname{supp} M \mid \lambda(i, j) \leq \lambda\left(i^{*}, j^{*}\right)\right\}$ is finite because $M$ is almost upper triangular. This contradiction proves the claim. By Example 11.5 one has a factorization $B A C=M=D W$ where $D$ is diagonal and $W$ is an element of the big Weyl group.
ExAmple 11.9. Let split orthogonal $A \in \mathcal{Q}$ and a pivot $\left(i_{0}, j_{0}\right)$ of $A$ be given. If $A_{i_{0} j}=0$ for all $j \notin\left\{j_{0}, 1-j_{0}\right\}$ and $A_{i j_{0}}=0$ for all $i \notin\left\{i_{0}, 1-i_{0}\right\}$, then $\left(i_{0}, j_{0}\right)$ is a cleared pivot of $A$. If $\left(i_{0}, j_{0}\right)$ is a cleared pivot of $A$, then $\left(1-i_{0}, 1-j_{0}\right)$ is a cleared pivot of $A$, and $A_{i_{0} j_{0}} A_{1-i_{0}, 1-j_{0}}=1$. If $I \subseteq \mathbb{Z}$ is the set of integers indexing rows (resp. columns) wherein cleared pivots of $A$ appear, then $I=1-I$.
EXAmple 11.10. Let split orthogonal $A \in \mathcal{Q}$ be given. Let $\left(i_{0}, j_{0}\right)$ be a pivot of $A$. Let $I$ (resp. $J$ ) be the set of integers indexing rows (resp. columns) wherein cleared pivots of $A$ distinct from $\left(i_{0}, j_{0}\right)$ and $\left(1-i_{0}, 1-j_{0}\right)$ appear. Let $X$ and $Y$ be the left and right pivot-clearing operators associated to $A$ and its pivot $\left(i_{0}, j_{0}\right)$, respectively. Put
$$
X^{\star}:=X-X^{\dagger}+X^{\dagger} \mathbf{p}^{\dagger} X, \quad Y^{\star}:=Y-Y^{\dagger}+Y \mathbf{p}^{\dagger} Y^{\dagger}
$$
where $\mathbf{p}$ is the standard polarization. The matrices $X^{\star}$ and $Y^{\star}$ have the following properties:
- The matrices $X^{\star}$ and $Y^{\star}$ are strictly upper triangular.
- $\operatorname{supp} X^{\star} \subseteq\left((\mathbb{Z} \backslash I) \times\left\{i_{0}\right\}\right) \cup\left(\left\{1-i_{0}\right\} \times(\mathbb{Z} \backslash I)\right)$.
- $\operatorname{supp} Y^{\star} \subseteq\left(\left\{j_{0}\right\} \times(\mathbb{Z} \backslash J)\right) \cup\left((\mathbb{Z} \backslash J) \times\left\{1-j_{0}\right\}\right)$.
- The matrices $1-X^{\star}, 1-Y^{\star}$ and $A^{\prime}:=\left(1-X^{\star}\right) A\left(1-Y^{\star}\right)$ are split orthogonal.
- The pairs $\left(i_{0}, j_{0}\right)$ and $\left(1-i_{0}, 1-j_{0}\right)$ are cleared pivots of $A^{\prime}$.
- Every cleared pivot of $A$ remains a cleared pivot of $A^{\prime}$.
This is the split orthogonal version of Proposition 11.3. ExAmple 11.11. Let $A \in \mathcal{Q}$ be split orthogonal. There exist split orthogonal matrices $B, C, D, W \in \mathcal{Q}$ such that $B-1$ and $C-1$ are strictly upper triangular, $D$ is diagonal and split orthogonal, $W$ belongs to the big split orthogonal Weyl group, and $B A C=D W$. This is the split orthogonal version of Theorem 11.8.
ExAMPle 11.12. In this example $k$ is any artinian local ring. Let $\mathcal{B} \subseteq \mathcal{Q}^{\times}$be the subgroup consisting of upper triangular matrices. Let $\mathcal{U} \subseteq \mathcal{Q}^{\times}$be the subgroup consisting of matrices differing from the identity matrix by a strictly lower triangular matrix all entries of which belong to $m$. Let $\mathcal{W}$ be the big Weyl group. One has a disjoint union decomposition $\mathcal{Q}^{\times}=\coprod_{W \in \mathcal{W}} \mathcal{U} \mathcal{B} W \mathcal{B}$. Let $\mathcal{P} \subseteq \mathcal{Q}^{\times}$be the subgroup consisting of matrices $A$ such that the $\mathbb{N}$ by $1-\mathbb{N}$ (lower left) blocks of both $A$ and $A^{-1}$ vanish. The group $\mathcal{Q}^{\times}$is generated by $\mathcal{P}$ and $\mathbf{t}$.
Example 11.13. Again $k$ is any artinian local ring. Temporarily let $\mathcal{G}$ denote the group of split orthogonal elements of $\mathcal{Q}^{\times}$. Let $\mathcal{B}$ be the group of upper triangular elements of $\mathcal{G}$. Let $\mathcal{U}$ be the group of elements of $\mathcal{G}$ differing from the identity matrix by a strictly lower triangular matrix all entries of which belong to $m$. Let $\mathcal{W}$ be the big split orthogonal Weyl group. One has a disjoint union decomposition $\mathcal{G}=\coprod_{W \in \mathcal{W}} \mathcal{U} \mathcal{B} W \mathcal{B}$. Let $\mathcal{P} \subseteq \mathcal{G}$ be the subgroup consisting matrices with vanishing $\mathbb{N}$ by $1-\mathbb{N}$ (lower left) block. The group $\mathcal{G}$ is generated by $\mathcal{P}$ and $\mathbf{r}$.
## Clifford algebras
Definition 12.1 . Let $E$ be a free $k$-module equipped with a $k$ quadratic form $q: E \rightarrow k$. The Clifford algebra $\mathcal{C}(E, q)$ is by definition the quotient of the tensor algebra $\mathcal{T}(E)$ by the two-sided ideal $\mathcal{I}$ generated by the set $\{e \otimes e-q(e) \mid e \in E\}$. The Clifford algebra $\mathcal{C}(E, 0)$ is canonically isomorphic to the exterior algebra $\bigwedge(E)$. We denote the product in $\mathcal{C}(E, q)$ simply by juxtaposition, and we put
$$
e^{\sharp}:=e+\mathcal{I} \in \mathcal{C}(E, q)
$$
for each $e \in E$. One has
$$
\left(e^{\sharp}\right)^{2}=q(e)
$$
for all $e \in E$. The Clifford algebra $\mathcal{C}(E, q)$ has the following universal property:
- For all $k$-algebras $\mathcal{A}$ and $k$-linear maps $\phi: E \rightarrow \mathcal{A}$ such that
$$
\phi(e)^{2}=q(e)
$$
for all $e \in E$, there exists a unique $k$-algebra homomorphism
$$
\phi^{\natural}: \mathcal{C}(E, q) \rightarrow \mathcal{A}
$$
such that
$$
\phi^{\natural}\left(e^{\sharp}\right)=\phi(e)
$$
for all $e \in E$.
As will be explained presently, the natural map $\left(e \mapsto e^{\sharp}\right): E \rightarrow \mathcal{C}(E, q)$ is always injective, but we do not identify $E$ with a $k$-submodule of $\mathcal{C}(E, q)$ with respect to the map $e \mapsto e^{\sharp}$.
ExAmple 12.2 . Let $\mathcal{A}$ be a $k$-algebra. Let $E$ be a free $k$-module equipped with a $k$-quadratic form $q: E \rightarrow k$ and a $k$-basis $\left\{e_{i}\right\}_{i \in S}$ indexed by a linearly ordered set $S$. Let $\phi: E \rightarrow \mathcal{A}$ be a $k$-linear map such that $\phi(e)^{2}=q(e)$ for all $e \in E$. For each finite subset $I=\left\{i_{1}<\cdots<i_{r}\right\} \subseteq S$, put
$$
a_{I}:=\phi\left(e_{i_{1}}\right) \cdots \phi\left(e_{i_{r}}\right) \in \mathcal{A} .
$$
One has
$$
a_{\{i\}} a_{I}=\sum_{j \in I}(-1)^{|\{\ell \in I \mid \ell<j\}|} \begin{cases}q\left(e_{i}, e_{j}\right) a_{I \backslash\{j\}} & \text { if } j<i \\ q\left(e_{i}\right) a_{I \backslash\{i\}} & \text { if } j=i \in I \\ a_{I \cup\{i\}} & \text { if } j=i \notin I \\ 0 & \text { if } j>i\end{cases}
$$
for all $i \in S$ and finite subsets $I \subseteq S$. If the family $\left\{e_{i}\right\}$ indexed by $i \in S$ spans the $k$-module $E$, then the family $\left\{a_{I}\right\}$ indexed by finite subsets $I \subseteq S$ spans the $\mathcal{A}$ as $k$-module.
ExAmple 12.3. Let $E$ be a free $k$-module equipped with a $k$ quadratic form $q: E \rightarrow k$ and a $k$-basis $\left\{e_{i}\right\}_{i \in S}$ indexed by a linearly ordered set $S$. Let $V$ be a free $k$-module equipped with a basis $\left\{v_{I}\right\}$ indexed by finite subsets $I \subseteq S$. For each $i \in S$, let $\partial_{i}, \delta_{i}: V \rightarrow V$ be the $k$-linear endomorphisms defined by the rules
$$
\begin{aligned}
& \partial_{i} v_{I}:= \begin{cases}(-1)^{|\{\ell \in I \mid \ell<i\}|} v_{I \backslash\{i\}} & \text { if } i \in I \\
0 & \text { if } i \notin I\end{cases} \\
& \delta_{i} v_{I}:= \begin{cases}0 & \text { if } i \in I \\
(-1)^{|\{\ell \in I \mid \ell<i\}|} v_{I \cup\{i\}} & \text { if } i \notin I\end{cases}
\end{aligned}
$$
for all finite subsets $I \subseteq S$. One has
$$
\partial_{i}^{2}=0, \quad \delta_{i}^{2}=0, \quad \partial_{i} \delta_{i}+\delta_{i} \partial_{i}=1
$$
for all $i \in S$, and
$$
\partial_{i} \partial_{j}+\partial_{j} \partial_{i}=0, \quad \delta_{i} \delta_{j}+\delta_{j} \delta_{i}=0, \quad \partial_{i} \delta_{j}+\delta_{j} \partial_{i}=0
$$
for all distinct $i, j \in S$. Put
$$
q_{i j}:= \begin{cases}0 & \text { if } i<j \\ q\left(e_{i}\right) & \text { if } i=j \\ q\left(e_{i}, e_{j}\right) & \text { if } i>j\end{cases}
$$
for all $i, j \in S$. Let $\phi$ be the unique $k$-linear map from $E$ to the $k$-algebra of $k$-linear endomorphisms of $V$ such that
$$
\phi\left(e_{i}\right) v_{I}:=\delta_{i} v_{I}+\sum_{j \in I} q_{i j} \partial_{j} v_{I}
$$
for all $i \in S$ and finite subsets $I \subseteq S$. One verifies by a straightforward calculation that $\phi(e)^{2}=q(e)$ for all $e \in E$; let $\phi^{\natural}$ be the induced $k$-algebra homomorphism from the Clifford algebra $\mathcal{C}(E, q)$ to the $k$ algebra of $k$-linear endomorphisms of $V$. For each finite subset
$$
I=\left\{i_{1}<\cdots<i_{n}\right\} \subseteq S
$$
put
$$
e_{I}:=e_{i_{1}}^{\sharp} \cdots e_{i_{n}}^{\sharp} \in \mathcal{C}(E, q),
$$
and observe that
$$
\phi^{\natural}\left(e_{I}\right) v_{\emptyset}=\phi\left(e_{i_{1}}\right) \cdots \phi\left(e_{i_{n}}\right) v_{\emptyset}=\delta_{i_{1}} \cdots \delta_{i_{n}} v_{\emptyset}=v_{I} .
$$
It follows that the family $\left\{e_{I}\right\}$ is $k$-linearly independent. In particular, the canonical map $\left(e \mapsto e^{\sharp}\right): E \rightarrow \mathcal{C}(E, q)$ is injective.
THEOREM 12.4. Let $E$ be a free $k$-module equipped with a $k$-quadratic form $q: E \rightarrow k$ and let $\mathcal{C}=\mathcal{C}(E, q)$ be the associated Clifford algebra. Let $E_{0}$ be a free $k$-submodule of $E$. Let $\mathcal{C}_{0} \subseteq \mathcal{C}$ be the $k$-subalgebra generated by $E_{0}^{\sharp}$. Let $\left\{e_{i}\right\}_{i \in S}$ be a $k$-basis for $E_{0}$ indexed by a linearly ordered set $S$. For each finite subset
$$
I=\left\{i_{1}<\cdots<i_{n}\right\} \subseteq S
$$
put
$$
e_{I}:=e_{i_{1}}^{\sharp} \cdots e_{i_{n}}^{\sharp} \in \mathcal{C}_{0} .
$$
The family $\left\{e_{I}\right\}$ indexed by finite subsets $I \subseteq S$ is a $k$-basis for $\mathcal{C}_{0}$ and the natural map $\mathcal{C}\left(E_{0},\left.q\right|_{E_{0}}\right) \rightarrow \mathcal{C}_{0}$ is bijective.
Proof. Example 12.2 shows that the family $\left\{e_{I}\right\}$ spans $\mathcal{C}_{0}$ over $k$. To prove the $k$-linear independence of the family $\left\{e_{I}\right\}$ there is no loss of generality in assuming that $\left\{e_{i}\right\}_{i \in S}$ is a $k$-basis of $E$, in which case Example 12.3 does the job. Since the $k$-basis $\left\{e_{I}\right\}$ of $\mathcal{C}_{0}$ is the image of a $k$-basis for $\mathcal{C}\left(E_{0},\left.q\right|_{E_{0}}\right)$, the natural map $\mathcal{C}\left(E_{0},\left.q\right|_{E_{0}}\right) \rightarrow \mathcal{C}_{0}$ is bijective as claimed. Example 12.5. Consider the Pauli spin matrices
$$
\sigma_{x}:=\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right], \quad \sigma_{y}:=\left[\begin{array}{rr}
0 & -i \\
i & 0
\end{array}\right], \quad \sigma_{z}:=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]
$$
of quantum mechanics. One has
$$
\left(a \sigma_{x}+b \sigma_{y}+c \sigma_{z}\right)^{2}=a^{2}+b^{2}+c^{2}
$$
for any real numbers $a, b$ and $c$. The natural $\mathbb{R}$-algebra homomorphism
$$
\mathcal{C}\left(\mathbb{R}^{3},(a, b, c) \mapsto a^{2}+b^{2}+c^{2}\right) \rightarrow \operatorname{Mat}_{2}(\mathbb{C})
$$
is bijective.
Definition 12.6. Let $A$ be an $n$ by $n$ alternating matrix with scalar entries. (A matrix is said to be alternating if anti-symmetric and zero along the diagonal.) Put
$$
\text { pfaff } A:= \begin{cases}1 & \text { if } n=0 \\ 0 & \text { if } n \equiv 1 \bmod 2 \\ \sum_{j=2}^{n}(-1)^{j} A_{1 j} \text { pfaff } A^{\{1, j\}} & \text { if } n \equiv 0 \bmod 2 \text { and } n>0\end{cases}
$$
where for each $I \subseteq\{1, \ldots, n\}$ we denote by $A^{I}$ the alternating matrix obtained by striking row $i$ and column $i$ from $A$ for all $i \in I$. One calls pfaff $A$ the pfaffian of $A$. For example, one has
$$
\text { pfaff } A=A_{12}, \quad \text { pfaff } A=A_{12} A_{24}-A_{13} A_{24}+A_{14} A_{23}
$$
in the cases $n=2$ and $n=4$, respectively. It is well known that
$$
(\operatorname{pfaff} A)^{2}=\operatorname{det} A,
$$
and
$$
\operatorname{pfaff}\left(B^{T} A B\right)=(\operatorname{det} B)(\operatorname{pfaff} A)
$$
for any $n$ by $n$ matrix $B$ with scalar entries.
EXAmple 12.7. Let $E$ be a free $k$-module equipped with a $k$ quadratic form $q: E \rightarrow k$ and let $\mathcal{C}=\mathcal{C}(E, q)$ be the associated Clifford algebra. Let $\pi$ be a $q$-polarization. Let $\mathcal{A}$ (resp. $\mathcal{B}$ ) be the $k$-subalgebra of $\mathcal{C}$ generated by $(\pi E)^{\sharp}$ (resp. $\left.((1-\pi) E)^{\sharp}\right)$. One has canonical $k$-algebra isomorphisms
$$
\bigwedge(\pi E)=\mathcal{A}, \quad \bigwedge((1-\pi) E)=\mathcal{B}
$$
Let $\left\{e_{i}\right\}_{i \in S}$ be a $k$-basis for $\pi E$ indexed by a linearly ordered set $S$. Let $\left\{f_{j}\right\}_{j \in T}$ be a $k$-basis for $(1-\pi) E$ indexed by a linearly ordered set T. Put
$$
e_{I}:=e_{i_{1}}^{\sharp} \cdots e_{i_{r}}^{\sharp} \in \mathcal{A}, \quad f_{J}:=f_{j_{1}}^{\sharp} \cdots f_{j_{s}}^{\sharp} \in \mathcal{B}
$$
for all finite subsets
$$
I=\left\{i_{1}<\cdots<i_{r}\right\} \subset S, \quad J=\left\{j_{1}<\cdots<j_{s}\right\} \subset T .
$$
Let $\mathcal{I}^{\star}$ be the right ideal of $\mathcal{C}$ generated by $(\pi E)^{\sharp}$. Let $\mathcal{J}$ be the left ideal of $\mathcal{C}$ generated by $((1-\pi) E)^{\sharp}$. The families
$$
\left\{e_{I}\right\}, \quad\left\{f_{J}\right\}, \quad\left\{e_{I} f_{J}\right\}, \quad\left\{e_{I} f_{J}\right\}_{I \neq \emptyset}, \quad\left\{e_{I} f_{J}\right\}_{J \neq \emptyset}
$$
are $k$-bases for $\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{I}^{\star}$ and $\mathcal{J}$, respectively. One has direct sum decompositions
$$
\mathcal{C}=\mathcal{A} \oplus \mathcal{J}=\mathcal{I}^{\star} \oplus \mathcal{B}=k \oplus\left(\mathcal{I}^{\star}+\mathcal{J}\right)
$$
One has
$$
g_{1}^{\sharp} \cdots g_{n}^{\sharp} \equiv \underset{i, j=1}{n} \operatorname{pfaff}^{n} q\left((1-\pi) g_{i}, g_{j}\right) \bmod \mathcal{I}^{\star}+\mathcal{J}
$$
for any vectors $g_{1}, \ldots, g_{n} \in E$ spanning a $k$-submodule on which the quadratic form $q$ vanishes identically. Thus the theory of Clifford algebras is linked to the theory of pfaffians.
ExAmple 12.8. Let $E$ be a free $k$-module equipped with a $k$ quadratic form $q: E \rightarrow k$ and let $\mathcal{C}=\mathcal{C}(E, q)$ be the associated Clifford algebra. Let $\pi$ be a $q$-polarization. Let $e_{1} \in \pi E$ and $f_{1} \in(1-\pi) E$ be given such that $q\left(e_{1}, f_{1}\right)=1$. The $k$-linear endomorphism
$$
\mu:=\left(e \mapsto q\left(e_{1}, e\right) f_{1}+q\left(f_{1}, e\right) e_{1}\right): E \rightarrow E
$$
is a $q$-projector commuting with $\pi$. Let $E_{1}$ be the $k$-span of $e_{1}$ and $f_{1}$. One has $\mu E=E_{1}$. Put $E_{0}:=(1-\mu) E$. Then $E=E_{0} \oplus E_{1}$. For $i=0,1$, let $\mathcal{C}_{i}$ be the $k$-subalgebra of $\mathcal{C}$ generated by $E_{i}^{\sharp}$. Let $\mathcal{I}^{\star} \subset \mathcal{C}$ be the right ideal generated by $(\pi E)^{\sharp}$. Let $\mathcal{I} \subset \mathcal{C}$ be the left ideal generated by $((1-\pi) E)^{\sharp}$. Let $\mathcal{I}_{0}^{\star} \subset \mathcal{C}_{0}$ be the right ideal generated by $\left(\pi E_{0}\right)^{\sharp}$. Let $\mathcal{I}_{0} \subset \mathcal{C}_{0}$ be the left ideal generated by $\left((1-\pi) E_{0}\right)^{\sharp}$. Let $\phi: E \rightarrow \operatorname{Mat}_{2}\left(\mathcal{C}_{0}\right)$ be the unique $k$-linear map such that
$$
\phi\left(e_{0}+x e_{1}+y f_{1}\right)=\left[\begin{array}{cc}
e_{0}^{\sharp} & y \\
x & -e_{0}^{\sharp}
\end{array}\right]
$$
for all $e_{0} \in E_{0}$ and scalars $x$ and $y$. Then
$$
\phi(e)^{2}=\left[\begin{array}{cc}
q(e) & 0 \\
0 & q(e)
\end{array}\right]
$$
for all $e \in E$, the induced $k$-algebra homomorphism
$$
\phi^{\natural}: \mathcal{C} \rightarrow \operatorname{Mat}_{2}\left(\mathcal{C}_{0}\right)
$$
is bijective, and one has
$$
\phi^{\natural}\left(\mathcal{I}^{\star}\right)=\left[\begin{array}{cc}
\mathcal{I}_{0}^{\star} & \mathcal{I}_{0}^{\star} \\
\mathcal{C}_{0} & \mathcal{C}_{0}
\end{array}\right], \quad \phi^{\natural}(\mathcal{I})=\left[\begin{array}{cc}
\mathcal{I}_{0} & \mathcal{C}_{0} \\
\mathcal{I}_{0} & \mathcal{C}_{0}
\end{array}\right] .
$$
EXAMPle 12.9 . Let $E$ be a free $k$-module equipped with a $k$-basis $\left\{e_{i}, f_{i}\right\}_{i=1}^{n}$ and a $k$-quadratic form $q: E \rightarrow k$ such that
$$
q\left(\sum_{i=1}^{n} x_{i} e_{i}+y_{i} f_{i}\right)=\sum_{i=1}^{n} x_{i} y_{i}
$$
for all scalars $x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n} \in k$. Let $\mathcal{C}=\mathcal{C}(E, q)$ be the associated Clifford algebra and put $\mathcal{A}:=\operatorname{Mat}_{2^{n}}(k)$. Let $\mathcal{J}^{\star} \subset \mathcal{A}$ be the right ideal consisting of matrices with vanishing first row. Let $\mathcal{J} \subset \mathcal{A}$ be the left ideal consisting of matrices with vanishing first column. Let $\mathcal{I}^{\star} \subset \mathcal{C}$ be the right ideal generated by $e_{1}^{\sharp}, \ldots, e_{n}^{\sharp}$. Let $\mathcal{I} \subset \mathcal{C}$ be the left ideal generated by $f_{1}^{\sharp}, \ldots, f_{n}^{\sharp}$. There exists a $k$-algebra isomorphism $\phi: \mathcal{C} \stackrel{\sim}{\rightarrow} \mathcal{A}$ such that $\phi\left(\mathcal{I}^{\star}\right)=\mathcal{J}^{\star}$ and $\phi(\mathcal{I})=\mathcal{J}$.
EXAMPle 12.10 . Let $E$ be a free $k$-module equipped with a $k$ quadratic form $q: E \rightarrow k$. We say that an element $x$ of the Clifford algebra $\mathcal{C}=\mathcal{C}(E, q)$ is even (resp. odd) if $x$ belongs to the $k$-span of all products of even (resp. odd) numbers of elements of $E^{\sharp}$. We say that $x \in \mathcal{C}$ is parity-homogeneous if $x$ is either even or odd. If $x \in \mathcal{C}$ is parity-homogeneous, we write $\operatorname{deg} x=0,1$ according as $x$ is even or odd. Every $x \in \mathcal{C}$ has a unique decomposition of the form $x=y+z$ where $y \in \mathcal{C}$ is even and $z \in \mathcal{C}$ is odd.
ExAmple 12.11. Let $E$ be a free $k$-module equipped with a $k$ quadratic form $q: E \rightarrow k$ and let $\mathcal{C}=\mathcal{C}(E, q)$ be the associated Clifford algebra. Let $\mu$ be a $q$-projector. Let $\mathcal{C}_{0} \subseteq \mathcal{C}$ be the $k$-subalgebra generated by $(\mu E)^{\sharp}$. Let $\mathcal{C}_{1} \subseteq \mathcal{C}$ be the $k$-subalgebra generated by $((1-\mu) E)^{\sharp}$. The natural map
$$
\left(x_{0} \otimes x_{1} \mapsto x_{0} x_{1}\right): \mathcal{C}_{0} \otimes_{k} \mathcal{C}_{1} \rightarrow \mathcal{C}
$$
is a $k$-linear isomorphism but not quite a $k$-algebra isomorphism: one has
$$
x_{1} x_{0}=(-1)^{\left(\operatorname{deg} x_{0}\right)\left(\operatorname{deg} x_{1}\right)} x_{0} x_{1}
$$
for all parity-homogeneous $x_{0} \in \mathcal{C}_{0}$ and $x_{1} \in \mathcal{C}_{1}$.
## The infinite wedge model of fermionic Fock space
Definition 13.1. Let $\mathcal{F}$ be the free $k$-module on the basis
$|I\rangle \quad(I$ : a wedge index $)$
For each wedge index $I$ we define $\langle I| \in \mathcal{F}^{*}$ by the rule
$$
\langle I \mid J\rangle= \begin{cases}1 & \text { if } I=J \\ 0 & \text { if } I \neq J\end{cases}
$$
for all wedge indices $J$. The abbreviations
$$
|\bullet\rangle=|\mathbb{Z} \backslash \mathbb{N}\rangle, \quad\langle\bullet|=\langle\mathbb{Z} \backslash \mathbb{N}|
$$
will sometimes be employed. Given $f, g \in \mathcal{H}$, let
$$
f^{\sharp}, g^{b}: \mathcal{F} \rightarrow \mathcal{F}
$$
be the unique $k$-linear endomorphisms such that
$$
f^{\sharp}|I\rangle=\sum_{n \in \mathbb{Z} \backslash I}(-1)^{|\{i \in I \mid i>n\}|} f_{n}|I \cup\{n\}\rangle
$$
and
$$
g^{b}|I\rangle=\sum_{n \in I}(-1)^{|\{i \in I \mid i>n\}|} g_{1-n}|I \backslash\{n\}\rangle
$$
for all wedge indices $I$. One readily verifies that
$$
\left(f^{\sharp}\right)^{2}=0, \quad\left(g^{b}\right)^{2}=0, \quad f^{\sharp} g^{b}+g^{b} f^{\sharp}=\sum_{n \in \mathbb{Z}} f_{n} g_{1-n} .
$$
We call $\mathcal{F}$ the infinite wedge model of fermionic Fock space.
EXAMPLE 13.2. We continue working with the infinite wedge model of fermionic Fock space. For each $n \in \mathbb{Z}$ let $e_{n}$ be the $n^{\text {th }}$ column of the $\mathbb{Z}$ by $\mathbb{Z}$ identity matrix. One has
$$
\begin{aligned}
e_{n}^{\sharp}|I\rangle & = \begin{cases}(-1)^{|\{i \in I \mid i>n\}|}|I \cup\{n\}\rangle & \text { if } n \notin I \\
0 & \text { otherwise }\end{cases} \\
e_{1-n}^{b}|I\rangle & = \begin{cases}(-1)^{|\{i \in I \mid i>n\}|}|I \backslash\{n\}\rangle & \text { if } n \in I \\
0 & \text { otherwise }\end{cases} \\
\langle I| e_{n}^{\sharp} & = \begin{cases}(-1)^{|\{i \in I \mid i>n\}|}\langle I \backslash\{n\}| & \text { if } n \in I \\
0 & \text { otherwise }\end{cases} \\
\langle I| e_{1-n}^{b} & = \begin{cases}(-1)^{|\{i \in I \mid i>n\}|}\langle I \cup\{n\}| & \text { if } n \notin I \\
0 & \text { otherwise }\end{cases}
\end{aligned}
$$
for all wedge indices $I$ and $n \in \mathbb{Z}$. REMARK 13.3. Strictly speaking the equation
$$
\left|\left\{i_{1}>i_{2}>i_{3}>\ldots\right\}\right\rangle=e_{i_{1}} \wedge e_{i_{2}} \wedge e_{i_{3}} \wedge \ldots
$$
is nonsense, but it nonetheless provides a valuable guide to intuition. This bit of nonsense is the rationale behind the terminology.
ExAmPle 13.4. For each $n \in \mathbb{Z}$ again let $e_{n}$ denote the $n^{\text {th }}$ column of the $\mathbb{Z}$ by $\mathbb{Z}$ identity matrix. One has
$$
|I\rangle=(-1)^{j_{1}+\cdots+j_{s}} e_{i_{1}}^{\sharp} \cdots e_{i_{r}}^{\sharp} e_{1-j_{1}}^{b} \cdots e_{1-j_{s}}^{b}|\bullet\rangle
$$
for all wedge indices $I$ where
$$
I \cap \mathbb{N}=\left\{i_{1}>\cdots>i_{r}\right\}, \quad(\mathbb{Z} \backslash \mathbb{N}) \backslash I=\left\{j_{1}>\cdots>j_{s}\right\} .
$$
One has
$$
\begin{aligned}
|I\rangle & = \pm e_{i_{1}}^{\sharp} \cdots e_{i_{r}}^{\sharp} e_{1-j_{1}}^{b} \cdots e_{1-j_{s}}^{b}|J\rangle \\
\langle I| & = \pm\langle J| e_{j_{s}}^{\sharp} \cdots e_{j_{1}}^{\sharp} e_{1-i_{r}}^{b} \cdots e_{1-i_{1}}^{b} \\
1=\langle I \mid J\rangle & =\left\langle J\left|e_{j_{s}}^{\sharp} \cdots e_{j_{1}}^{\sharp} e_{1-i_{r}}^{b} \cdots e_{1-i_{1}}^{b} e_{i_{1}}^{\sharp} \cdots e_{i_{r}}^{\sharp} e_{1-j_{1}}^{b} \cdots e_{1-j_{s}}^{b}\right| J\right\rangle
\end{aligned}
$$
for all wedge indices $I$ and $J$ where $I \backslash J=\left\{i_{1}>\cdots>i_{r}\right\}$ and $J \backslash I=\left\{j_{1}>\cdots>j_{s}\right\}$.
Proposition 13.5. Fix a wedge index I. Let $L \subseteq \mathcal{F}$ be the $k$ submodule consisting of all vectors $\psi$ such that $f^{\sharp} \psi=0$ and $g^{b} \psi=0$ for all $f \in \mathcal{H}(I)$ and $g \in \mathcal{H}(\mathbb{Z} \backslash(1-I))$. Let $V \subseteq \mathcal{F}$ be the intersection of all $k$-submodules of $\mathcal{F}$ containing $L$, stable under $f^{\sharp}$ for all $f \in \mathcal{H}$, and stable under $g^{b}$ for all $g \in \mathcal{H}$. Let $H \subseteq \mathcal{F}$ be the $k$-submodule spanned by all vectors of the form $f^{\sharp} \psi$ or $g^{b} \psi$ where $\psi \in \mathcal{F}, f \in \mathcal{H}(\mathbb{Z} \backslash I)$ and $g \in \mathcal{H}(1-I)$. The $k$-module $L$ is spanned over $k$ by the vector $|I\rangle$. One has $V=\mathcal{F}$. A vector $\psi \in \mathcal{F}$ is annihilated by $\langle I|$ if and only if $\psi \in H$.
Proof. This follows directly from Examples 13.2 and 13.4.
Proposition 13.6. The only $k$-linear endomorphisms of $\mathcal{F}$ commuting with all operators of the form $f^{\sharp}$ or $g^{b}$ with $f, g \in \mathcal{H}$ are the scalar multiplication operators.
Proof. This follows formally from the preceding proposition. EXAmple 13.7. Let $\mathcal{C}$ be the Clifford algebra associated to the free $k$-module $\left[\begin{array}{l}\mathcal{H} \\ \mathcal{H}\end{array}\right]$ and the $k$-quadratic form $\left[\begin{array}{l}g \\ h\end{array}\right] \mapsto \sum_{n \in \mathbb{Z}} g_{n} h_{1-n}$. By writing
$$
\left[\begin{array}{l}
g \\
0
\end{array}\right]^{\sharp}:=g^{\sharp}, \quad\left[\begin{array}{l}
0 \\
h
\end{array}\right]^{\sharp}:=h^{b}
$$
for all $g, h \in \mathcal{H}$, we make the $k$-module $\mathcal{F}$ naturally into a left $\mathcal{C}$-module. Let $\mathcal{I}^{\star} \subset \mathcal{C}$ be the right ideal generated by
$$
\mathcal{H}(\mathbb{N})^{\sharp}+\mathcal{H}(\mathbb{N})^{b} .
$$
Let $\mathcal{J} \subset \mathcal{C}$ be the left ideal generated by
$$
\mathcal{H}(\mathbb{Z} \backslash \mathbb{N})^{\sharp}+\mathcal{H}(\mathbb{Z} \backslash \mathbb{N})^{b} .
$$
The sequences
$$
0 \rightarrow \mathcal{J} \subset \mathcal{C} \stackrel{x \mapsto x|\bullet\rangle}{\rightarrow} \mathcal{F} \rightarrow 0
$$
and
$$
0 \rightarrow \mathcal{I}^{\star}+\mathcal{J} \subset \mathcal{C} \stackrel{x \mapsto\langle\bullet|x| \bullet\rangle}{\rightarrow} k \rightarrow 0
$$
are exact.
## The diamond model of fermionic Fock space
Definition 14.1. Let $\mathcal{F}_{\diamond}$ be the free $k$-module on the basis
$$
|I\rangle \quad \text { ( } I \text { : a diamond index). }
$$
For each diamond index $I$ let $\langle I|$ denote the unique $k$-linear functional on $\mathcal{F}_{\diamond}$ such that
$$
\langle I \mid J\rangle= \begin{cases}1 & \text { if } I=J \\ 0 & \text { if } I \neq J\end{cases}
$$
for all diamond indices $J$. The abbreviations
$$
|\bullet\rangle=|1-\mathbb{N}\rangle, \quad\langle\bullet|=\langle 1-\mathbb{N}|
$$
will sometimes be employed. For each $h \in \mathcal{H}$, let
$$
h^{\sharp}: \mathcal{F}_{\diamond} \rightarrow \mathcal{F}_{\diamond}
$$
be the unique $k$-linear endomorphism such that
$$
h^{\sharp}|I\rangle:=\sum_{n \in \mathbb{Z} \backslash I}(-1)^{|\{i \in I \mid \max (n, 1-n)<i\}|} h_{n}|I \cup\{n\} \backslash\{1-n\}\rangle
$$
for all diamond indices $I$. One readily verifies that
$$
\left(h^{\sharp}\right)^{2}=\sum_{n \in \mathbb{N}} h_{n} h_{1-n} .
$$
We call $\mathcal{F}_{\diamond}$ the diamond model of fermionic Fock space. ExAmple 14.2. For each $n \in \mathbb{Z}$ let $e_{n} \in \mathcal{H}$ be the $n^{\text {th }}$ column of the $\mathbb{Z}$ by $\mathbb{Z}$ identity matrix. One has
$$
\begin{aligned}
& e_{n}^{\sharp}|I\rangle= \begin{cases}(-1)^{|\{i \in I \mid \max (n, 1-n)<i\}|}|I \cup\{n\} \backslash\{1-n\}\rangle & \text { if } n \notin I \\
0 & \text { otherwise }\end{cases} \\
& \langle I| e_{n}^{\sharp}= \begin{cases}(-1)^{|\{i \in I \mid \max (n, 1-n)<i\}|}\langle I \backslash\{n\} \cup\{1-n\}| & \text { if } n \in I \\
0 & \text { otherwise }\end{cases}
\end{aligned}
$$
for all $n \in \mathbb{Z}$ and diamond indices $I$. One has
$$
\left\langle I\left|=\left\langle\bullet\left|e_{1-i_{r}}^{\sharp} \cdots e_{1-i_{1}}^{\sharp}, \quad\right| I\right\rangle=e_{i_{1}}^{\sharp} \cdots e_{i_{r}}^{\sharp}\right| \bullet\right\rangle
$$
for all diamond indices $I$, where
$$
\left\{i_{1}>\cdots>i_{r}\right\}=I \cap \mathbb{N} .
$$
One has
$$
\begin{aligned}
\pm|I\rangle & =e_{i_{1}}^{\sharp} \cdots e_{i_{r}}^{\sharp}|J\rangle, \\
\pm\langle I| & =\langle J| e_{1-i_{r}}^{\sharp} \cdots e_{1-i_{1}}^{\sharp}, \\
1 & =\left\langle J\left|e_{1-i_{r}}^{\sharp} \cdots e_{1-i_{1}}^{\sharp} e_{i_{1}}^{\sharp} \cdots e_{i_{r}}^{\sharp}\right| J\right\rangle
\end{aligned}
$$
for all diamond indices $I$ and $J$, where
$$
\left\{i_{1}>\cdots>i_{r}\right\}=I \cap(1-J) .
$$
ExAmple 14.3. One naturally regards $\mathcal{F}_{\diamond}$ as a left module over the Clifford algebra $\mathcal{C}_{\diamond}$ associated to the free $k$-module $\mathcal{H}$ and the $k$ quadratic form $h \mapsto \sum_{n \in \mathbb{N}} h_{n} h_{1-n}$. Let $\mathcal{I}_{\diamond}^{\star} \subset \mathcal{C}_{\diamond}$ be the right ideal generated by
$$
\mathcal{H}(\mathbb{N})^{\sharp}
$$
Let $\mathcal{J}_{\diamond} \subset \mathcal{C}_{\diamond}$ be the left ideal generated by
$$
\mathcal{H}(1-\mathbb{N})^{\sharp} .
$$
The sequences
$$
0 \rightarrow \mathcal{J}_{\diamond} \rightarrow \mathcal{C}_{\diamond} \stackrel{x \mapsto x|\bullet\rangle}{\longrightarrow} \mathcal{F}_{\diamond} \rightarrow 0
$$
and
$$
0 \rightarrow \mathcal{I}_{\diamond}^{\star}+\mathcal{J}_{\diamond} \subset \mathcal{C}_{\diamond} \stackrel{x \mapsto\langle\bullet|x| \bullet\rangle}{\rightarrow} k \rightarrow 0
$$
are exact. Proposition 14.4. Fix a diamond index I. Let $L \subseteq \mathcal{F}_{\diamond}$ be the $k$-submodule consisting of all vectors $\psi$ such that $h^{\sharp} \psi=0$ for all $h \in \mathcal{H}(I)$. Let $V \subseteq \mathcal{F}_{\diamond}$ be the intersection of all $k$-submdules of $\mathcal{F}$ containing $L$ and stable under $h^{\sharp}$ for all $h \in \mathcal{H}$. Let $H \subseteq \mathcal{F}_{\diamond}$ be the $k$-submodule spanned by all vectors of the form $h^{\sharp} \psi$ where $\psi \in \mathcal{F}_{\diamond}$ and $h \in \mathcal{H}(1-I)$. The $k$-module $L$ is spanned over $k$ by the vector $|I\rangle$. One has $V=\mathcal{F}_{\diamond}$. A vector $\psi \in \mathcal{F}_{\diamond}$ is annihilated by the $k$-linear functional $\langle I|$ if and only if $\psi \in H$.
Proof. This follows directly from Example 14.2.
Proposition 14.5. The only k-linear endomorphisms of $\mathcal{F}_{\diamond}$ commuting with all operators of the form $h^{\sharp}$ with $h \in \mathcal{H}$ are the scalar multiplication operators.
Proof. This is a formal consequence of Proposition 14.4.
Example 14.6. Temporarily put
$$
\Phi:=\left(h \mapsto h^{[2]}\right): \mathcal{H} \stackrel{\sim}{\rightarrow}\left[\begin{array}{l}
\mathcal{H} \\
\mathcal{H}
\end{array}\right] .
$$
We keep the notation introduced in Examples 13.7 and 14.3. The map $\phi$ induces a $k$-algebra isomorphism
$$
\Phi^{\natural}: \mathcal{C}_{\diamond} \stackrel{\sim}{\rightarrow} \mathcal{C}
$$
such that
$$
\Phi^{\natural}\left(\mathcal{I}_{\diamond}^{\star}\right)=\mathcal{I}^{\star}, \quad \Phi^{\natural}\left(\mathcal{J}_{\diamond}\right)=\mathcal{J} .
$$
It follows that there exists a unique $k$-linear isomorphism
$$
\tilde{\Phi}: \mathcal{F}_{\diamond} \stackrel{\sim}{\rightarrow} \mathcal{F}
$$
such that
and
$$
\tilde{\Phi}|\bullet\rangle=|\bullet\rangle, \quad\langle\bullet| \tilde{\Phi}=\langle\bullet|,
$$
$$
\tilde{\Phi}\left(h^{\sharp} \psi\right)=\left(f^{\sharp}+g^{b}\right) \tilde{\Phi}(\psi)
$$
for all $f, g, h \in \mathcal{H}$ such that
$$
\Phi(h)=\left[\begin{array}{l}
f \\
g
\end{array}\right]
$$
and $\psi \in \mathcal{F}_{\diamond}$. One can verify that
$$
\tilde{\Phi}\left|I^{\diamond}\right\rangle= \pm|I\rangle
$$
for all wedge indices $I$. (We omit the somewhat unpleasant description of the sign rule because fortunately we will not need to use it.) Thus the infinite wedge and diamond models of fermionic Fock space are equivalent.
## The fundamental theorem
Definition 15.1. Let $A \in \mathcal{Q}$ be a split orthogonal matrix. A $k$ linear automorphism $\tilde{A}$ of $\mathcal{F}_{\diamond}$ is called a diamond representation of $A$ if
$$
\tilde{A}\left(h^{\sharp} \psi\right)=(A h)^{\sharp} \tilde{A} \psi
$$
for all $h \in \mathcal{H}$ and $\psi \in \mathcal{F}_{\diamond}$. By Proposition 14.5 diamond representations are unique up to an invertible scalar multiple.
Definition 15.2. Let $A \in \mathcal{Q}^{\times}$be given. A $k$-linear automorphism $\tilde{A}$ of $\mathcal{F}$ is called an infinite wedge representation of $A$ if
$$
\tilde{A}\left(g^{\sharp} \psi\right)=(A g)^{\sharp} \tilde{A} \psi, \quad \tilde{A}\left(h^{b} \psi\right)=\left(A^{-\dagger} h\right)^{b} \tilde{A} \psi
$$
for all $g, h \in \mathcal{H}$ and $\psi \in \mathcal{F}$, where $A^{-\dagger}:=\left(A^{-1}\right)^{\dagger}$. By Proposition 13.6, infinite wedge representations are unique up to an invertible scalar multiple.
EXAMPLE 15.3. In fact the notion of diamond representation encompasses the notion of infinite wedge representation. Let $\tilde{\Phi}: \mathcal{F}_{\diamond} \sim \mathcal{\sim} \mathcal{F}$ be the $k$-linear isomorphism constructed in Example 14.6. Let $A \in \mathcal{Q}^{\times}$ be given and let $B \in \mathcal{Q}$ be the unique split orthogonal matrix such that
$$
B^{[2]}=\left[\begin{array}{cc}
A & 0 \\
0 & A^{-\dagger}
\end{array}\right]
$$
(Look back at Definition 7.11.) Given a diamond representation $\tilde{B}$ of $B$, there exists a unique infinite wedge representation $\tilde{A}$ of $A$ such that the diagram
commutes, and conversely, given an infinite wedge representation $\tilde{A}$ of $A$, there exists a unique diamond representation $\tilde{B}$ of $B$ rendering the diagram above commutative. ExAmple 15.4. As in Example 14.3, let $\mathcal{C}_{\diamond}$ be the Clifford algebra associated to the free $k$-module $\mathcal{H}$ and the $k$-quadratic form $h \mapsto \sum_{n \in \mathbb{N}} h_{n} h_{1-n}$ and let $\mathcal{J}_{\diamond}$ be the left ideal generated by $\mathcal{H}(1-\mathbb{N})^{\sharp}$. Let $A \in \mathcal{Q}$ be a split orthogonal matrix. Let
$$
A^{\natural}: \mathcal{C}_{\diamond} \stackrel{\sim}{\rightarrow} \mathcal{C}_{\diamond}
$$
be the induced $k$-algebra automorphism induced by the $k$-linear automorphism
$$
(h \mapsto A h): \mathcal{H} \stackrel{\sim}{\rightarrow} \mathcal{H} .
$$
There exist $x, y \in \mathcal{C}_{\diamond}$ such that
$$
\left(A^{\natural} \mathcal{J}_{\diamond}\right) x \subseteq \mathcal{J}_{\diamond}, \quad \mathcal{J}_{\diamond} y \subseteq A^{\natural} \mathcal{J}_{\diamond}, \quad x y \equiv 1 \bmod A^{\natural} \mathcal{J}_{\diamond}, \quad y x \equiv 1 \bmod \mathcal{J}_{\diamond}
$$
if and only if there exists a diamond representation $\tilde{A}$ of $A$ such that
$$
\tilde{A}|\bullet\rangle=x|\bullet\rangle .
$$
Example 15.5. As in Example 13.7, let $\mathcal{C}$ be the Clifford algebra associated to the free $k$-module $\left[\begin{array}{l}\mathcal{H} \\ \mathcal{H}\end{array}\right]$ and the quadratic form
$$
\left[\begin{array}{l}
g \\
h
\end{array}\right] \mapsto \sum_{n \in \mathbb{Z}} g_{n} h_{1-n}
$$
write
$$
\left[\begin{array}{l}
g \\
0
\end{array}\right]^{\sharp}=g^{\sharp}, \quad\left[\begin{array}{l}
0 \\
h
\end{array}\right]^{\sharp}=h^{b},
$$
and let $\mathcal{J} \subset \mathcal{C}$ be the left ideal generated by
$$
\mathcal{H}(\mathbb{Z} \backslash \mathbb{N})^{\sharp}+\mathcal{H}(\mathbb{Z} \backslash \mathbb{N})^{b} .
$$
Let $A \in \mathcal{Q}^{\times}$be given. Let
$$
A^{\natural}: \mathcal{C} \stackrel{\sim}{\rightarrow} \mathcal{C}
$$
be the $k$-algebra automorphism induced by the $k$-linear automorphism
$$
\left(\left[\begin{array}{l}
g \\
h
\end{array}\right] \mapsto\left[\begin{array}{c}
A g \\
A^{-\dagger} h
\end{array}\right]\right):\left[\begin{array}{c}
\mathcal{H} \\
\mathcal{H}
\end{array}\right] \stackrel{\sim}{\rightarrow}\left[\begin{array}{c}
\mathcal{H} \\
\mathcal{H}
\end{array}\right] .
$$
There exist $x, y \in \mathcal{C}$ such that
$$
\left(A^{\natural} \mathcal{J}\right) x \subseteq \mathcal{J}, \quad \mathcal{J} y \subseteq A^{\natural} \mathcal{J}, \quad x y \equiv 1 \bmod A^{\natural} \mathcal{J}, \quad y x \equiv 1 \bmod \mathcal{J}
$$
if and only if there exists an infinite wedge representation $\tilde{A}$ of $A$ such that
$$
\tilde{A}|\bullet\rangle=x|\bullet\rangle
$$
EXAmPle 15.6. The triple
$$
A=\mathbf{r}, \quad x=e_{1}^{\sharp}, \quad y=e_{0}^{\sharp}
$$
satisfies the conditions enunciated in Example 15.4, and hence there exists a unique diamond representation $\tilde{\mathbf{r}}$ of $\mathbf{r}$ such that
$$
\tilde{\mathbf{r}}|\bullet\rangle=e_{1}^{\sharp}|\bullet\rangle .
$$
EXAmple 15.7. If $A \in \mathcal{Q}$ is a split orthogonal matrix with vanishing $\mathbb{N}$ by $1-\mathbb{N}$ block, the triple
$$
A, \quad x=1, \quad y=1
$$
satisfies the conditions enunciated in Example 15.4, and hence there exists a unique diamond representation $\tilde{A}$ of $A$ such that
$$
\tilde{A}|\bullet\rangle=|\bullet\rangle .
$$
EXAMPle 15.8. The triple
$$
A=\mathbf{t}, \quad x=e_{1}^{b}, \quad y=e_{0}^{\sharp}
$$
satisfies the conditions enunciated in Example 15.5 and hence there exists a unique infinite wedge representation $\tilde{\mathbf{t}}$ of $\mathbf{t}$ such that
$$
\tilde{\mathbf{t}}|\bullet\rangle=e_{1}^{b}|\bullet\rangle .
$$
ExAmple 15.9. If $A \in \mathcal{Q}^{\times}$is such that the $\mathbb{N}$ by $1-\mathbb{N}$ blocks of both $A$ and $A^{-1}$ vanish, the triple
$$
A, \quad x=1, \quad y=1
$$
satisfies the conditions enunciated in Example 15.5 and hence there exists a unique infinite wedge representation $\tilde{A}$ of $A$ such that
$$
\tilde{A}|\bullet\rangle=|\bullet\rangle .
$$
ThEOREM 15.10.
1. Let $A$ be an almost upper triangular split orthogonal $\mathbb{Z}$ by $\mathbb{Z} m a$ trix with scalar entries. There exists a diamond representation of A unique up to invertible scalar multiple.
2. Let $A \in \mathcal{Q}^{\times}$be given. There exists an infinite wedge representation of $A$ unique up to invertible scalar multiple.
Proof. In both diamond and infinite wedge cases, uniqueness has already been noted. Existence in the diamond case follows from Examples 11.13, 15.6 and 15.7. Existence in the infinite wedge case follows from Examples 11.12, 15.8, and 15.9.
## Matrix coefficient calculations in the diamond model
ExAmple 16.1. Let $\omega: \mathbb{Z} \rightarrow \mathbb{Z}$ be a tame split orthogonal permutation. There exists a diamond representation $\tilde{W}$ of the matrix $W$ representing $\omega$ such that
$$
\tilde{W}|I\rangle= \pm|\omega(I)\rangle
$$
for all diamond indices $I$.
ExAmple 16.2. Fix diamond indices $I$ and $J$. Let a split orthogonal matrix $A \in \mathcal{Q}$ be given with a
$$
\left[\begin{array}{c}
I \\
1-I
\end{array}\right] \times\left[\begin{array}{c}
J \\
1-J
\end{array}\right]^{T}
$$
block decomposition of the form
$$
A=\left[\begin{array}{ll}
a & b \\
0 & d
\end{array}\right] \text { or } A=\left[\begin{array}{ll}
a & 0 \\
c & d
\end{array}\right]
$$
Let $\tilde{A}$ be a diamond representation of $A$ and put
$$
x:=\langle I|\tilde{A}| J\rangle
$$
Under the first assumption one has
$$
A \mathcal{H}(J)=\mathcal{H}(I)
$$
and hence
$$
\tilde{A}|J\rangle=x|I\rangle .
$$
Under the second assumption one has
$$
A \mathcal{H}(1-J)=\mathcal{H}(1-I)
$$
and hence
$$
\langle J| \tilde{A}=x\langle I| .
$$
Both conclusions are justified by Fundamental Theorem combined with Proposition 14.4. 16. MATRIX COEFFICIENT CALCULATIONS IN THE DIAMOND MODEL 55
ExAmple 16.3. Let $A \in \mathcal{Q}^{\times}$be split orthogonal and assume that the
$$
\left[\begin{array}{c}
1-\mathbb{N} \\
\mathbb{N}
\end{array}\right] \times\left[\begin{array}{c}
1-\mathbb{N} \\
\mathbb{N}
\end{array}\right]^{T}
$$
block decomposition of $A$ takes the form
$$
A=\left[\begin{array}{ll}
a & b \\
0 & d
\end{array}\right] \quad \text { or } \quad A=\left[\begin{array}{ll}
a & 0 \\
c & d
\end{array}\right] .
$$
Let $I$ be any diamond index and write
$$
I \cap \mathbb{N}=\left\{i_{1}>\cdots>i_{r}\right\} .
$$
For each $n \in \mathbb{Z}$ let $e_{n}$ be the $n^{\text {th }}$ column of the $\mathbb{Z}$ by $\mathbb{Z}$ identity matrix, let $a_{n}$ be the $n^{\text {th }}$ column of $A$, and let $\bar{a}_{n}$ be the $n^{\text {th }}$ column of $A^{\dagger}$. Let $\tilde{A}$ be any diamond representation of $A$. One has
$$
\begin{aligned}
\langle\bullet|\tilde{A}| I\rangle & =\left\langle\bullet\left|\tilde{A} e_{i_{1}}^{\sharp} \cdots e_{i_{r}}^{\sharp}\right| \bullet\right\rangle \\
& =\left\langle\bullet\left|a_{i_{1}}^{\sharp} \cdots a_{i_{r}}^{\sharp}\right| \bullet\right\rangle\langle\bullet|\tilde{A}| \bullet\rangle \\
& =\left(\begin{array}{c}
r \\
\operatorname{pfaff} \\
\mu, \nu=1
\end{array} \sum_{n \in \mathbb{N}} A_{1-n, i_{\mu}} A_{n, i_{\nu}}\right)\langle\bullet|\tilde{A}| \bullet\rangle .
\end{aligned}
$$
under the first assumption. One has
$$
\begin{aligned}
\langle I|\tilde{A}| \bullet\rangle & =\left\langle\bullet\left|e_{1-i_{r}}^{\sharp} \cdots e_{1-i_{1}}^{\sharp} \tilde{A}\right| \bullet\right\rangle \\
& =\left\langle\bullet\left|\bar{a}_{1-i_{r}}^{\sharp} \cdots \bar{a}_{1-i_{1}}^{\sharp}\right| \bullet\right\rangle\langle\cdot \tilde{A} \mid \bullet\rangle \\
& =\left(\begin{array}{c}
r \\
\operatorname{pfaff} \\
\mu, \nu=1
\end{array} \sum_{n \in \mathbb{N}} A_{n, 1-i_{\mu}}^{\dagger} A_{1-n, 1-i_{\nu}}^{\dagger}\right)\langle\bullet|\tilde{A}| \bullet\rangle \\
& =\left(\begin{array}{c}
r \\
\operatorname{pfaff} \\
\mu, \nu=1
\end{array} \sum_{n \in \mathbb{N}} A_{i_{\mu}, 1-n} A_{i_{\nu}, n}\right)\langle\bullet|\tilde{A}| \bullet\rangle
\end{aligned}
$$
under the second assumption. EXAmPle 16.4. Let split orthogonal $A \in \mathcal{Q}$ be given. Let $\tilde{A}$ be a diamond representation of $A$. Fix a diamond index $J$ and put
$$
x:=\langle J|\tilde{A}| J\rangle .
$$
Make one of the following assumptions:
- $\tilde{A}|J\rangle=x|J\rangle$.
- $\langle J| \tilde{A}=x\langle J|$.
Let $I$ be another diamond index and write
$$
I \backslash J=\left\{i_{1}>\cdots>i_{r}\right\}, \quad J \backslash I=\left\{1-i_{1}>\cdots>1-i_{r}\right\} .
$$
For each $n \in \mathbb{Z}$, let $e_{n}$ denote the $n^{\text {th }}$ column of the $\mathbb{Z}$ by $\mathbb{Z}$ identity matrix, let $a_{n}$ denote the $n^{\text {th }}$ column of $A$, and let $\bar{a}_{n}$ denote the $n^{\text {th }}$ column of $A^{\dagger}$. One has
$$
\begin{aligned}
\langle I|\tilde{A}| I\rangle & =\left\langle J\left|e_{1-i_{r}}^{\sharp} \cdots e_{1-i_{1}}^{\sharp} \tilde{A} e_{i_{1}}^{\sharp} \cdots e_{i_{r}}^{\sharp}\right| J\right\rangle \\
& =\left\langle J\left|e_{1-i_{r}}^{\sharp} \cdots e_{1-i_{1}}^{\sharp} a_{i_{1}}^{\sharp} \cdots a_{i_{r}}^{\sharp}\right| J\right\rangle\langle J|\tilde{A}| J\rangle \\
& =\left(\operatorname{det}_{i, j \in I \backslash J} A_{i j}\right)\langle J|\tilde{A}| J\rangle
\end{aligned}
$$
under the first assumption. One has
$$
\begin{aligned}
\langle I|\tilde{A}| I\rangle & =\left\langle J\left|e_{1-i_{r}}^{\sharp} \cdots e_{1-i_{1}}^{\sharp} \tilde{A} e_{i_{1}}^{\sharp} \cdots e_{i_{r}}^{\sharp}\right| J\right\rangle \\
& =\left\langle J\left|\bar{a}_{1-i_{r}}^{\sharp} \cdots \bar{a}_{1-i_{1}}^{\sharp} e_{i_{1}}^{\sharp} \cdots e_{i_{r}}\right| J\right\rangle\langle J|\tilde{A}| J\rangle \\
& =\left(\operatorname{det}_{i, j \in I \backslash J} A_{1-j, 1-i}^{\dagger}\right)\langle J|\tilde{A}| J\rangle \\
& =\left(\operatorname{det}_{i, j \in I \backslash J} A_{i j}\right)\langle J|\tilde{A}| J\rangle
\end{aligned}
$$
under the second assumption.
## The transversality theorem
Theorem 17.1. Let split orthogonal $A \in \mathcal{Q}$ be given. Let $\tilde{A}$ be any diamond representation of $A$. The following assertions are equivalent:
- $\langle\bullet|\tilde{A}| \bullet\rangle \not \equiv 0 \bmod m$.
- The $1-\mathbb{N}$ by $1-\mathbb{N}$ block of $A$ is invertible.
- The $\mathbb{N}$ by $\mathbb{N}$ block of $A^{-1}$ is invertible.
- $\mathcal{H}=\mathcal{H}(\mathbb{N}) \oplus A \mathcal{H}(1-\mathbb{N})$.
- $\mathcal{H}=A^{-1} \mathcal{H}(\mathbb{N}) \oplus \mathcal{H}(1-\mathbb{N})$.
Proof. The equivalence of the last four conditions is trivial. Without loss of generality we may assume that $m=0$ and hence that $k$ is a field. After making the evident reductions based on Examples 10.9, 11.13 and 16.3 , we may assume that for some positive integer $n$ one has a factorization
$$
A=B W
$$
where $B, W \in \mathcal{Q}$ have the following properties:
- $B$ and $W$ are split orthogonal.
- $B-1$ vanishes outside the $1-\mathbb{N}$ by $\mathbb{N}$ block.
- The tame split orthogonal permutation
$$
\omega:=\left(i \mapsto\left\{\begin{array}{ll}
1-i & \text { if }-n<i \leq n \\
i & \text { otherwise }
\end{array}\right): \mathbb{Z} \stackrel{\sim}{\rightarrow} \mathbb{Z}\right.
$$
is represented by $W$.
Let $\tilde{B}$ and $\tilde{W}$ be Fock representations of $B$ and $W$. Without loss of generality we may assume that $\tilde{A}=\tilde{B} \tilde{W}$. Put
$$
I:=\omega(1-\mathbb{N})=\{1, \ldots, n\} \cup\{-n,-n-1,-n-2, \ldots,\} .
$$
Given $x, y \in k$ we write $x \sim y$ if $x$ and $y$ generate the same ideal of $k$. We now calculate with
$$
\left[\begin{array}{c}
\{i \leq-n\} \\
\{-n<i \leq 0\} \\
\{0<i \leq n\} \\
\{n<i\}
\end{array}\right] \times\left[\begin{array}{c}
\{j \leq-n\} \\
\{-n<j \leq 0\} \\
\{0<j \leq n\} \\
\{n<j\}
\end{array}\right]^{T}
$$
block decompositions. Write
$$
B=\left[\begin{array}{cccc}
1 & 0 & b_{13} & b_{14} \\
0 & 1 & b_{23} & b_{24} \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right], \quad W=\left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 0 & w_{23} & 0 \\
0 & w_{32} & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] .
$$
Then
$$
A=\left[\begin{array}{cccc}
1 & b_{13} w_{32} & 0 & b_{14} \\
0 & b_{23} w_{23} & w_{23} & b_{24} \\
0 & w_{32} & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}\right], \quad w_{23}=\underbrace{\left[\begin{array}{lll}
& & \\
& . & \\
& . & \\
1 & &
\end{array}\right]}_{n}
$$
The matrix $b_{23}$ is dagger-alternating, hence the matrix $b_{23} w_{23}$ alternating. One has
$$
\text { pfaff } b_{23} w_{23} \sim\langle\bullet|\tilde{B}| I\rangle \sim\langle\bullet|\tilde{B} \tilde{W}| \bullet\rangle=\langle\bullet|\tilde{A}| \bullet\rangle
$$
by Examples 16.1 and 16.3. Finally, $\operatorname{det} b_{23} w_{23}=\left(\text { pfaff } b_{23} w_{23}\right)^{2}$ is a unit of $k$ if and only if the $1-\mathbb{N}$ by $1-\mathbb{N}$ block of $A$ is invertible. We are done.
Corollary 17.2. Let $A \in \mathcal{Q}^{\times}$be given. Let
$$
\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right], \quad\left[\begin{array}{ll}
\bar{a} & \bar{b} \\
\bar{c} & \bar{d}
\end{array}\right]
$$
be the
$$
\left[\begin{array}{c}
\mathbb{Z} \backslash \mathbb{N} \\
\mathbb{N}
\end{array}\right] \times\left[\begin{array}{c}
\mathbb{Z} \backslash \mathbb{N} \\
\mathbb{N}
\end{array}\right]^{T}
$$
block decompositions of $A$ and $A^{-1}$, respectively. Let $\tilde{A}$ be any infinite wedge representation of $A^{\diamond}$. The following assertions are equivalent:
- $\langle\bullet|\tilde{A}| \bullet\rangle \not \equiv 0 \bmod m$.
- The block a is invertible.
- The block $\bar{d}$ is invertible.
- $\mathcal{H}=\mathcal{H}(\mathbb{N}) \oplus A \mathcal{H}(1-\mathbb{N})$
- $\mathcal{H}=A^{-1} \mathcal{H}(\mathbb{N}) \oplus \mathcal{H}(1-\mathbb{N})$.
Proof. The equivalence of the last four conditions is trivial. Let $B \in \mathcal{Q}$ be the unique split orthogonal matrix such that
$$
B^{[2]}=\left[\begin{array}{cc}
A & 0 \\
0 & A^{-\dagger}
\end{array}\right] .
$$
By Example 15.3 and the theorem, invertibility of the $1-\mathbb{N}$ by $1-\mathbb{N}$ block of $B$ is equivalent to condition 1. By direct calculation one verifies that the $1-\mathbb{N}$ by $1-\mathbb{N}$ block of $B$ is invertible if and only if both $a$ and $\bar{d}$ are invertible.
## Matrix coefficient calculations in the infinite wedge model
EXAMPle 18.1. Let $\omega: \mathbb{Z} \stackrel{\sim}{\rightarrow} \mathbb{Z}$ be a tame permutation. Let $W \in$ $\mathcal{Q}^{\times}$be the permutation matrix representing $\omega$. There exists an infinite wedge representation $\tilde{W}$ of $W$ such that
$$
\tilde{W}|I\rangle= \pm|\omega(I)\rangle
$$
for all wedge indices $I$.
ExAmple 18.2. Fix wedge indices $I$ and $J$. Let $A \in \mathcal{Q}^{\times}$be given such that the
$$
\left[\begin{array}{c}
I \\
\mathbb{Z} \backslash I
\end{array}\right] \times\left[\begin{array}{c}
J \\
\mathbb{Z} \backslash J
\end{array}\right]^{T}
$$
block decompositions of $A$ and $A^{-1}$ are of one of the following two forms:
$$
\begin{aligned}
& \text { - } A=\left[\begin{array}{ll}
a & b \\
0 & d
\end{array}\right], \quad A^{-1}=\left[\begin{array}{cc}
\bar{a} & \bar{b} \\
0 & \bar{d}
\end{array}\right] \\
& \text { - } A=\left[\begin{array}{ll}
a & 0 \\
c & d
\end{array}\right], \quad A^{-1}=\left[\begin{array}{cc}
\bar{a} & 0 \\
\bar{c} & \bar{d}
\end{array}\right]
\end{aligned}
$$
Let $\tilde{A}$ be an infinite wedge representation of $A$ and put
$$
x:=\langle I|\tilde{A}| J\rangle
$$
Under the first assumption one has
$$
A \mathcal{H}(J)=\mathcal{H}(I)
$$
and hence
$$
\tilde{A}|J\rangle=x|I\rangle .
$$
Under the second assumption one has
$$
A \mathcal{H}(\mathbb{Z} \backslash J)=\mathcal{H}(\mathbb{Z} \backslash I)
$$
and hence
$$
\langle J| \tilde{A}=x\langle I| .
$$
Both conclusions are justified by Fundamental Theorem combined with Proposition 13.5 . ExAmple 18.3. Let $A \in \mathcal{Q}^{\times}$and a wedge index $J$ be given. Let $\tilde{A}$ be an infinite wedge representation of $A$ and put
$$
x:=\langle J|\tilde{A}| J\rangle .
$$
Assume that
$$
\tilde{A}|J\rangle=x|J\rangle .
$$
Let $I$ be another wedge index and write
$$
I \backslash J=\left\{i_{1}>\cdots>i_{r}\right\}, \quad J \backslash I=\left\{j_{1}>\cdots>j_{s}\right\} .
$$
For each $n \in \mathbb{Z}$, let $e_{n}$ denote the $n^{\text {th }}$ column of the $\mathbb{Z}$ by $\mathbb{Z}$ identity matrix, let $a_{n}$ denote the $n^{\text {th }}$ column of $A$, and let $\bar{a}_{n}$ denote the $n^{\text {th }}$ column of $A^{-\dagger}$. One has
$$
\begin{aligned}
\langle I|\tilde{A}| I\rangle & =\left\langle J\left|e_{j_{s}}^{\sharp} \cdots e_{j_{1}}^{\sharp} e_{1-i_{r}}^{b} \cdots e_{1-i_{1}}^{b} \tilde{A} e_{i_{1}}^{\sharp} \cdots e_{i_{r}}^{\sharp} e_{1-j_{1}}^{b} \cdots e_{1-j_{s}}^{b}\right| J\right\rangle \\
& =\left\langle J\left|e_{j_{s}}^{\sharp} \cdots e_{j_{1}}^{\sharp} e_{1-i_{r}}^{b} \cdots e_{1-i_{1}}^{b} a_{i_{1}}^{\sharp} \cdots a_{i_{r}}^{\sharp} \bar{a}_{1-j_{1}}^{b} \cdots \bar{a}_{1-j_{s}}^{b}\right| J\right\rangle\langle J|\tilde{A}| J\rangle \\
& =\left(\operatorname{det}_{i, j \in I \backslash J} A_{i j}\right)\left(\operatorname{det}_{i, j \in J \backslash I} A_{1-j, 1-i}^{-\dagger}\right)\langle J|\tilde{A}| J\rangle \\
& =\left(\operatorname{det}_{i, j \in I \backslash J} A_{i j}\right)\left(\operatorname{det}_{i, j \in J \backslash I} A_{i j}^{-1}\right)\langle J|\tilde{A}| J\rangle .
\end{aligned}
$$
This formula links fermionic Fock space to the Kyoto school notion of $\tau$-function.
DEFINITION 18.4. We introduce the abbreviated notation
$$
\begin{aligned}
& \langle N|:=\langle\{n \in \mathbb{Z} \mid n \leq N\}|, \\
& |N\rangle:=|\{n \in \mathbb{Z} \mid n \leq N\}\rangle
\end{aligned}
$$
where $N$ is any integer. In particular, one has $\langle\bullet|=\langle 0|$ and $|\bullet\rangle=|0\rangle$.
Definition 18.5. We say that $A \in \mathcal{Q}$ belongs to the deformation class if $A-1$ is strictly lower triangular with entries in $m$. The set of deformation class matrices forms a group under matrix multiplication. For each $A \in \mathcal{Q}^{\times}$of the deformation class, there exists a unique infinite wedge representation $\tilde{A}$ of $A$ such that
$$
\langle N| \tilde{A}=\langle N|
$$
for all $N \in \mathbb{Z}$; in this situation we say that $\tilde{A}$ is left-normalized. Definition 18.6. Let matrix $A \in \mathcal{Q}$ be given. We say that $A$ belongs to the test class if for some positive integer $N$ the
$$
\left[\begin{array}{c}
\{i \leq-N\} \\
\{-N<i \leq N\} \\
\{N<i\}
\end{array}\right] \times\left[\begin{array}{c}
\{j \leq-N\} \\
\{-N<j \leq N\} \\
\{N<j\}
\end{array}\right]^{T}
$$
block decomposition of $A$ takes the form
$$
A=\left[\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
0 & a_{22} & a_{23} \\
0 & 0 & a_{33}
\end{array}\right]
$$
where the blocks have the following properties:
- $a_{11}-1$ and $a_{33}-1$ are strictly upper triangular.
- The matrix $a_{22}$ is invertible.
The matrices of the test class form a group under matrix multiplication. For each $A \in \mathcal{Q}^{\times}$of the test class there exists a unique infinite wedge representation $\tilde{A}$ of $A$ such that
$$
\tilde{A}|N\rangle=|N\rangle
$$
for all integers $N \ll 0$; in this situation we say that $\tilde{A}$ is rightnormalized.
ExAmple 18.7. If a matrix $A \in \mathcal{Q}^{\times}$belongs both to the test class and to the deformation class, an infinite wedge representation of $A$ is right-normalized if and only if left-normalized.
EXAmple 18.8. Let $A \in \mathcal{Q}^{\times}$of the deformation class and $B \in \mathcal{Q}^{\times}$ of the test class be given. Let
$$
A=\left[\begin{array}{cc}
a_{11} & 0 \\
a_{21} & a_{22}
\end{array}\right]
$$
be the
$$
\left[\begin{array}{c}
\mathbb{Z} \backslash \mathbb{N} \\
\mathbb{N}
\end{array}\right] \times\left[\begin{array}{c}
\mathbb{Z} \backslash \mathbb{N} \\
\mathbb{N}
\end{array}\right]
$$
block decomposition of $A$ and put
$$
U:=\left[\begin{array}{cc}
a_{11} & 0 \\
0 & a_{22}
\end{array}\right], \quad C:=A U^{-1}=\left[\begin{array}{cc}
1 & 0 \\
a_{21} a_{11}^{-1} & 1
\end{array}\right] .
$$
Let $\tilde{B}$ be the right-normalized infinite wedge representation of $B$. Let $\tilde{A}, \tilde{U}$ and $\tilde{C}$ be the left-normalized infinite wedge representations of $A, U$ and $C$, respectively. Necessarily one has $\tilde{A}=\tilde{C} \tilde{U}$. Moreover the matrix $C$ belongs to the test class and hence $\tilde{C}$ is also right-normalized. The upshot is that
$$
\langle\bullet|\tilde{B} \tilde{A}| \bullet\rangle=\langle\bullet|\tilde{B} \tilde{C}| \bullet\rangle=\underset{i, j=1}{N}(B C)_{1-i, 1-j}=\underset{i, j=1}{N}(B A)_{1-i, 1-j}
$$
for all $N \gg 0$ by Example 18.3.
Example 18.9. Let $A, B, C, \tilde{A}$ and $\tilde{B}$ be as in Example 18.8. Suppose also that for some positive integer $N$ the
$$
\left[\begin{array}{c}
\{i \leq-N\} \\
\{-N<i \leq N\} \\
\{N<i\}
\end{array}\right] \times\left[\begin{array}{c}
\{j \leq-N\} \\
\{-N<j \leq N\} \\
\{N<j\}
\end{array}\right]^{T}
$$
block decompositions of $B$ and $C$ take the form
$$
B=\left[\begin{array}{ccc}
b_{11} & b_{12} & b_{13} \\
0 & 1 & b_{23} \\
0 & 0 & b_{33}
\end{array}\right], \quad C=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & c_{22} & 0 \\
0 & 0 & 1
\end{array}\right],
$$
respectively. Then $\langle\bullet|\tilde{B} \tilde{A}| \bullet\rangle=1$.
Definition 18.10. Let $A \in \mathcal{Q}$ be given. Let $c$ be a scalar. Let $I$ and $J$ be wedge indices. Write
$$
I=\left\{i_{1}>i_{2}>\ldots\right\}, \quad J=\left\{j_{1}>j_{2}>\ldots\right\} .
$$
We write
$$
c=A_{I J}
$$
if one of the following conditions hold:
- $|I \backslash J| \neq|J \backslash I|$ and $c=0$.
- $|I \backslash J|=|J \backslash I|$ and
$$
c=\underset{\mu, \nu=1}{N} A_{i_{\mu}, j_{\nu}}
$$
for all but finitely many positive integers $N$.
Otherwise the expression $A_{I J}$ is not defined.
ExAmple 18.11. If a matrix $A \in \mathcal{Q}^{\times}$belongs either to the test class or to the deformation class, the minor $A_{I J}$ is defined for all wedge indices $I$ and $J$. Example 18.12. Let $A \in \mathcal{Q}$ of the deformation class be given, and let $\tilde{A}$ be the left-normalized infinite wedge representation of $A$. Let $I$ and $J$ be wedge indices. We claim that
$$
\langle I|\tilde{A}| J\rangle=A_{I J}
$$
To prove the claim, write
$$
I=\left\{i_{1}>\cdots>i_{r}\right\} \coprod\{n \leq N\}, \quad J=\left\{j_{1}>\cdots>j_{s}\right\} \coprod\{n \leq N\}
$$
where $N$ is any sufficiently negative integer. For each $n \in \mathbb{Z}$, let $e_{n}$ denote the $n^{\text {th }}$ column of the $\mathbb{Z}$ by $\mathbb{Z}$ identity matrix, and let $\bar{a}_{n}$ denote the $n^{t h}$ column of $A^{\dagger}$. One has
$$
\begin{aligned}
\langle I|\tilde{A}| J\rangle & =\left\langle N\left|e_{1-i_{r}}^{b} \cdots e_{1-i_{1}}^{b} \tilde{A} e_{j_{1}}^{\sharp} \cdots e_{j_{s}}^{\sharp}\right| N\right\rangle \\
& =\left\langle N\left|\tilde{A}\left(\left(A^{-1}\right)^{-\dagger} e_{1-i_{r}}\right)^{b} \cdots\left(\left(A^{-1}\right)^{-\dagger} e_{1-i_{1}}\right)^{b} e_{j_{1}}^{\sharp} \cdots e_{j_{s}}^{\sharp}\right| N\right\rangle \\
& =\left\langle N\left|\bar{a}_{1-i_{r}}^{b} \cdots \bar{a}_{1-i_{1}}^{b} e_{j_{1}}^{\sharp} \cdots e_{j_{s}}^{\sharp}\right| N\right\rangle \\
& =A_{I J} .
\end{aligned}
$$
The claim is proved.
ExAmple 18.13. Let $A \in \mathcal{Q}$ of the test class be given, and let $\tilde{A}$ be the right-normalized infinite wedge representation of $A$. Let $I$ and $J$ be wedge indices. We claim that
$$
\langle I|\tilde{A}| J\rangle=A_{I J} .
$$
As above, to prove the claim, write
$$
I=\left\{i_{1}>\cdots>i_{r}\right\} \coprod\{n \leq N\}, \quad J=\left\{j_{1}>\cdots>j_{s}\right\} \coprod\{n \leq N\}
$$
where $N$ is any sufficiently negative integer. For each $n \in \mathbb{Z}$ let $e_{n}$ be the $n^{t h}$ column of the $\mathbb{Z}$ by $\mathbb{Z}$ identity matrix and let $a_{n}$ be the $n^{t h}$ column of $A$. One has
$$
\begin{aligned}
\langle I|\tilde{A}| J\rangle & =\left\langle N\left|e_{1-i_{r}}^{b} \cdots e_{1-i_{1}}^{b} \tilde{A} e_{j_{1}}^{\sharp} \cdots e_{j_{s}}^{\sharp}\right| N\right\rangle \\
& =\left\langle N\left|e_{1-i_{r}}^{b} \cdots e_{1-i_{1}}^{b} a_{j_{1}}^{\sharp} \cdots a_{j_{s}}^{\sharp}\right| N\right\rangle \\
& =A_{I J} .
\end{aligned}
$$
The claim is proved.
Example 18.14. If $A \in \mathcal{Q}^{\times}$belongs to the deformation class or to the test class, then for each wedge index $J$ there exist only finitely many wedge indices $I$ such that $A_{I J} \neq 0$. EXAMPLE 18.15. Let
$$
h=\sum_{i} h_{i} t^{i} \in 1+t k[[t]]
$$
be given and put
$$
H:=h(\mathbf{t})=\left[\begin{array}{rrrrrr}
\ddots & \ddots & \ddots & \ddots & & \\
\ddots & 1 & h_{1} & h_{2} & h_{3} & \\
\ddots & 0 & 1 & h_{1} & h_{2} & \ddots \\
\ddots & 0 & 0 & 1 & h_{1} & \ddots \\
& 0 & 0 & 0 & 1 & \ddots \\
& & \ddots & \ddots & \ddots & \ddots
\end{array}\right]
$$
thereby defining a matrix of the test class, and let $\tilde{H}$ be the rightnormalized infinite wedge representation of $H$. Let an integer $N$ and partitions $\lambda$ and $\mu$ be given. Put
$$
I:=\left\{N+1-i+\lambda_{i} \mid i \in \mathbb{N}\right\}, \quad J:=\left\{N+1-j+\mu_{j} \mid j \in \mathbb{N}\right\}
$$
thereby defining wedge indices such that $|I \backslash J|=|J \backslash I|$. One has
$$
\underset{i, j=1}{\max (\ell(\lambda), \ell(\mu))} h_{\mu_{j}-\lambda_{i}+i-j}=H_{I J}=\langle I|\tilde{H}| J\rangle
$$
The determinant on the left is the so called skew $S$-function associated to $\lambda$ and $\mu$, taking the $h$ 's as the complete symmetric function. One has
$$
\operatorname{det}_{i, j=1}^{\ell(\mu)} h_{\mu_{i}-i+j}=\langle\bullet|\tilde{H}| J\rangle
$$
in the special case $N=0$ and $\lambda=0$; thus Schur functions are recovered as matrix coefficients. 18. MATRIX COEFFICIENT CALCULATIONS IN THE INFINITE WEDGE MODEb
EXAMPLE 18.16. Let
$$
h=\sum_{i} h_{i} t^{-i} \in 1+t^{-1} m\left[t^{-1}\right]
$$
be given and put
$$
H:=h(\mathbf{t})=\left[\begin{array}{cccccc}
\ddots & \ddots & \ddots & \ddots & & \\
\ddots & 1 & 0 & 0 & 0 & \\
\ddots & h_{1} & 1 & 0 & 0 & \ddots \\
\ddots & h_{2} & h_{1} & 1 & 0 & \ddots \\
& h_{3} & h_{2} & h_{1} & 1 & \ddots \\
& & \ddots & \ddots & \ddots & \ddots
\end{array}\right]
$$
thereby defining a matrix of the deformation class, and let $\tilde{H}$ be the left-normalized infinite wedge representation of $H$. Let an integer $N$ and partitions $\lambda$ and $\mu$ be given. Put
$$
I:=\left\{N+1-i+\lambda_{i} \mid i \in \mathbb{N}\right\}, \quad J:=\left\{N+1-j+\mu_{j} \mid j \in \mathbb{N}\right\}
$$
thereby defining wedge indices such that $|I \backslash J|=|J \backslash I|$. One has
$$
\underset{i, j=1}{\max (\ell(\lambda), \ell(\mu))} h_{\lambda_{i}-i-\mu_{j}+j}=H_{I J}=\langle I|\tilde{H}| J\rangle
$$
Again skew $S$-functions appear as matrix coefficients. One has
$$
\underset{i, j=1}{\ell(\lambda)} h_{\lambda_{i}-i+j}=\langle I|\tilde{H}| \bullet\rangle
$$
in the special case $\mu=0$ and $N=0$; thus again Schur functions appear as matrix coefficients. EXAMPLE 18.17. Let $k_{0}$ be an artinian local ring and assume that
$$
k=k_{0}[\epsilon] /\left(\epsilon^{N}\right),
$$
where $\epsilon$ is a variable and $N$ is a large positive integer which eventually we let tend to infinity. Let $\mathcal{Q}_{0} \subseteq \mathcal{Q}$ be the $k_{0}$-subalgebra consisting of matrices with entries in $k_{0}$. Fix $A \in \mathcal{Q}_{0}$ of the test class and put
$$
H=\sum_{i=0}^{\infty} \epsilon^{i} \mathbf{t}^{-i}=\left[\begin{array}{cccccc}
\ddots & \ddots & \ddots & \ddots & & \\
\ddots & 1 & 0 & 0 & 0 & \\
\ddots & \epsilon & 1 & 0 & 0 & \ddots \\
\ddots & \epsilon^{2} & \epsilon & 1 & 0 & \ddots \\
& \epsilon^{3} & \epsilon^{2} & \epsilon & 1 & \ddots \\
& & \ddots & \ddots & \ddots & \ddots
\end{array}\right]
$$
which is a matrix of the deformation class. Let $\tilde{A}$ be the right-normalized infinite wedge representation of $A$, and let $\tilde{H}$ be the left-normalized infinite wedge representation of $H$. Put
$$
b_{n}:=(-1)^{n} A_{I_{n}, I_{0}}, \quad c_{n}:=A_{J_{0}, J_{n}}
$$
where
$$
I_{n}:=\mathbb{Z} \backslash \mathbb{N} \backslash\{0\} \backslash\{n\}, \quad J_{n}:=\mathbb{Z} \backslash \mathbb{N} \backslash\{0\} \cup\{n\}
$$
for all nonnegative integers $n$. Note that $b_{n}=0$ for all $n \gg 0$. One has
$$
\langle\bullet \tilde{A} \tilde{H} \mid \bullet\rangle=\sum_{n=0}^{N-1} c_{n} \epsilon^{n} \quad\left(c_{n} \in k_{0}\right) .
$$
Abusing notation in what is we hope an understandable fashion, one has
$$
\begin{aligned}
& {\left[\begin{array}{llllll}
\ldots & b_{2} & b_{1} & b_{0} & 0 & \ldots
\end{array}\right] A=\left[\begin{array}{llllll}
\ldots & 0 & c_{0} & c_{1} & c_{2} & \ldots
\end{array}\right]} \\
& \text { position } 0 \quad \text { position } 0
\end{aligned}
$$
This is the essence of the Kyoto school method for recovering the Baker function from the $\tau$-function. See [Segal-Wilson 1985] for background. 18. MATRIX COEFFICIENT CALCULATIONS IN THE INFINITE WEDGE MODGE
ExAmple 18.18. Let $A \in \mathcal{Q}$ of the deformation class and $B \in \mathcal{Q}$ of the test class be given. Choose any positive integer $N$ such that the
$$
\left[\begin{array}{c}
\{i \leq-N\} \\
\{-N<i \leq N\} \\
\{N<i\}
\end{array}\right] \times\left[\begin{array}{c}
\{j \leq-N\} \\
\{-N<j \leq N\} \\
\{N<j\}
\end{array}\right]^{T}
$$
block decomposition of $B \in \mathcal{Q}$ takes the form
$$
B=\left[\begin{array}{ccc}
b_{11} & b_{12} & b_{13} \\
0 & b_{22} & b_{23} \\
0 & 0 & b_{33}
\end{array}\right]
$$
where $b_{11}-1$ and $b_{33}-1$ are strictly upper triangular and put
$$
V:=\left[\begin{array}{ccc}
b_{11} & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & b_{33}
\end{array}\right]
$$
Suppose that the
$$
\left[\begin{array}{c}
\mathbb{Z} \backslash \mathbb{N} \\
\mathbb{N}
\end{array}\right] \times\left[\begin{array}{c}
\mathbb{Z} \backslash \mathbb{N} \\
\mathbb{N}
\end{array}\right]^{T}
$$
block decomposition of $A$ take the form
$$
A=\left[\begin{array}{cc}
a_{11} & 0 \\
a_{21} & a_{22}
\end{array}\right]
$$
and put
$$
U:=\left[\begin{array}{cc}
a_{11} & 0 \\
0 & a_{22}
\end{array}\right] .
$$
Then the matrix $U^{-1} V^{-1} B A$ belongs to the test class. 1. ESSENTIAL TOOLS
## CHAPTER 2
## Commuting differential operators and $\tau$-functions
In chapter (except in the very last section) the scalar ring $k$ is assumed to be a field of characteristic 0 .
## Commuting differential operators
Put
$$
K:=k((x)), \quad D:=\frac{d}{d x} .
$$
Let $K[D]$ denote the ring of differential operators. The typical element of $K[D]$ can be written in the form
$$
\sum_{i=0}^{n} a_{i} D^{i} \quad\left(a_{i} \in K, \quad a_{n} \neq 0\right)
$$
and multiplication of such expressions is effected by repeatedly applying the Leibniz rule
$$
D^{n} a=\sum_{i}\left(\begin{array}{c}
n \\
i
\end{array}\right) a^{(i)} D^{n-i}
$$
from freshman calculus.
The problem that interests us is that of constructing commuting pairs
$$
L=D^{p}+\sum_{i=0}^{p-1} a_{i} D^{i}, \quad M=D^{q}+\sum_{j=0}^{q-1} b_{j} D^{j}
$$
of differential operators where $(p, q)=1$. This problem was originally investigated in the 1920's by Burchnall, Chaundy and Baker, later and independently in the 1970's by Krichever. See [Mumford 1978] for background. Example 1.1. Consider the case $p=2$ and $q=3$. Write
$$
L=D^{2}-2 u, \quad M=D^{3}-3 a D-3 b / 2 \quad(u, a, b \in K)
$$
One has
$$
\begin{aligned}
L M= & D^{5}-3 a^{\prime \prime} D-6 a^{\prime} D^{2}-3 a D^{3}-3 b^{\prime \prime} / 2-3 b^{\prime} D \\
& -3 b D^{2} / 2-2 u D^{3}+6 a u D+3 b u \\
= & D^{5}+(-3 a-2 u) D^{3}+\left(-6 a^{\prime}-3 b / 2\right) D^{2} \\
& +\left(-3 a^{\prime \prime}-3 b^{\prime}+6 a u\right) D+\left(-3 b^{\prime \prime} / 2+3 b u\right), \\
M L= & D^{5}-2 u^{\prime \prime \prime}-6 u^{\prime \prime} D-6 u^{\prime} D^{2}-2 u D^{3}-3 a D^{3}+ \\
& 6 a u^{\prime}+6 a u D-3 b D^{2} / 2+3 b u \\
= & D^{5}+(-2 u-3 a) D^{3}+\left(-6 u^{\prime}-3 b / 2\right) D^{2} \\
& +\left(-6 u^{\prime \prime}+6 a u\right) D+\left(-2 u^{\prime \prime \prime}+6 a u^{\prime}+3 b u\right)
\end{aligned}
$$
and hence
$$
L M=M L \Leftrightarrow\left\{\begin{array}{l}
u^{\prime}=a^{\prime} \\
b^{\prime}=a^{\prime \prime} \\
a^{\prime \prime \prime}=12 a a^{\prime} \leftarrow \text { stationary Korteweg de Vries. }
\end{array}\right.
$$
The stationary Korteweg de Vries equation has the Weierstrass $\wp$ function as a solution, as well as the rational function $1 / x^{2}$.
## Dressing operators
The Leibniz rule makes sense also for negative values of $n$ and thus one can equip the set of expressions
$$
\sum_{i=-\infty}^{n} a_{i} D^{i} \quad\left(a_{i} \in K, \quad a_{n} \neq 0\right)
$$
with the structure of associative $k$-algebra; this larger ring is denoted $K\left(\left(D^{-1}\right)\right)$ and its elements are called pseudo-differential operators. Let $k((D))^{-1}$ be the subring consisting of pseudo-differential operators with constant coefficients. Let $K\left[\left[D^{-1}\right]\right]$ be the subring of pseudo differential operators of order $\leq 0$ and let $k\left[\left[D^{-1}\right]\right]$ be the subring consisting of such with constant coefficients. EXAmple 2.1. Each pseudo-differential operator
$$
\Psi=1+\sum_{i=1}^{\infty} c_{i} D^{-i} \quad\left(c_{i} \in K\right)
$$
of degree zero with leading coefficient 1 is invertible. One has
$$
\begin{array}{rrr}
\Psi D \Psi^{-1}= & D & +O\left(D^{-1}\right) \\
\Psi D^{2} \Psi^{-1}= & D^{2}-2 c_{1}^{\prime} & +O\left(D^{-1}\right) \\
\Psi D^{3} \Psi^{-1}= & D^{3}-3 c_{1}^{\prime} D+3 c_{1}^{\prime} c_{1}-3 c_{1}^{\prime \prime}-3 c_{2}^{\prime} & +O\left(D^{-1}\right)
\end{array}
$$
Elements of $K[D]$ of the form
$$
\text { "differential operator part" of } \Psi D^{n} \Psi^{-1}
$$
have an extremely important role to play in the theory of the KadomtsevPetviashvili (KP) hierarchy. See [Segal-Wilson 1985] for background.
A superior reformulation of the original "commuting differential operators" problem is as follows. Let
$$
A \subseteq k((D))^{-1}
$$
be a $k$-subalgebra such that
$$
A \cap k\left[\left[D^{-1}\right]\right]=k, \quad \operatorname{dim}_{k} \frac{k\left(\left(D^{-1}\right)\right)}{A+k\left[\left[D^{-1}\right]\right]}<\infty .
$$
Such a $k$-algebra $A$ is the coordinate ring of an affine curve over $k$ with a unique nonsingular $k$-rational point at infinity. We ask for a pseudodifferential operator
$$
\Psi=1+\sum_{i=1}^{\infty} c_{i} D^{-i} \quad\left(c_{i} \in K\right)
$$
of order zero such that
$$
\Psi A \Psi^{-1} \subseteq K[D] .
$$
The operator $\Psi$ is called a dressing operator for the ring $A$. From the dressing operator $\Psi$ and constant coefficient pseudodifferential operators
$$
f=D^{p}+O\left(D^{p-1}\right), \quad g=D^{q}+O\left(D^{q-1}\right)
$$
of relatively prime order one obtains commuting differential operators
$$
\Psi f \Psi^{-1}=D^{p}+O\left(D^{p-1}\right), \quad \Psi g \Psi^{-1}=D^{q}+O\left(D^{q-1}\right)
$$
of the sort we originally set out to study. Hereafter we focus our attention on the construction of dressing operators. ExAmPle 2.2. Let $A \subset k\left(\left(D^{-1}\right)\right)$ be a $k$-subalgebra generated by constant coefficient pseudo-differential operators
$$
f=D^{2}+O\left(D^{-1}\right), \quad g=D^{3}+O\left(D^{-1}\right)
$$
for which there exists a dressing operator
$$
\Psi=1+\sum_{i=1}^{\infty} c_{i} D^{-i} \quad\left(c_{i} \in K\right) .
$$
One has
$$
\Psi f \Psi^{-1}=D^{2}-2 c_{1}^{\prime}, \quad \Psi g \Psi^{-1}=D^{3}-3 c_{1}^{\prime} D+3 c_{1}^{\prime} c_{1}-3 c_{1}^{\prime \prime}-3 c_{2}^{\prime}
$$
by Example 2.1 and hence
$$
c_{1}^{\prime \prime \prime \prime}=12 c_{1}^{\prime \prime} c_{1}^{\prime}
$$
by Example 1.1, i. e., $c_{1}^{\prime}$ is in this situation a solution of the stationary Korteweg de Vries equation.
## Baker functions
Let $K\{\{z\}\}$ be the ring of series
$$
\sum_{i \in \mathbb{Z}} c_{i} z^{i} \quad\left(c_{i} \in K\right)
$$
where $c_{i}$ is $x$-adically bounded and $c_{i} \rightarrow 0$ as $i \rightarrow-\infty$. In a more or less obvious way this is an ultrametric Banach algebra over $K$. The function
$$
e^{x z^{-1}}=\sum_{n=0}^{\infty} \frac{x^{n} z^{-n}}{n !}
$$
lives in $K\{\{z\}\}$. The Laurent series field $k((z))$ lives in $K\{\{z\}\}$.
Now fix a $k$-subspace
$$
W \subset k((z))
$$
such that
$$
\operatorname{dim}_{k} W \cap k\left[\left[z^{-1}\right]\right]=\operatorname{dim}_{k} \frac{k((z))}{W+k\left[\left[z^{-1}\right]\right]}<\infty ;
$$
in this situation we say that $W$ is of index zero. The collection of such subspaces is the famous Sato Grassmannian. Put
$$
\tilde{W}:=\left(\text { closure of } e^{-x z^{-1}}(K \otimes W) \text { in } K\{\{z\}\}\right) \cap K((z))
$$
where $K \otimes W$ denotes the $K$-span in $K\{\{z\}\}$ of $W$. Facts: - The natural map
$$
\tilde{W} \rightarrow \frac{K((z))}{K\left[\left[z^{-1}\right]\right]}
$$
is bijective.
- One has
and the rule
$$
\frac{\partial}{\partial x}\left(e^{x z^{-1}} \tilde{W}\right) \subseteq e^{x z^{-1}} \tilde{W}
$$
$$
\left(\sum_{i} a_{i} D^{i}\right) w:=\sum_{i} a_{i} \frac{\partial^{i} w}{\partial x^{i}}
$$
makes $e^{x z^{-1}} \tilde{W}$ into a free left $K[D]$-module of rank 1 .
In particular, there exists unique
$$
\psi_{W}:=1+\sum_{i>0} c_{i} z^{i} \in z \tilde{W}
$$
The function $\psi_{W}$ is called the Baker function associated to the index zero subspace $W \subset k((z))$. Put
$$
\Psi_{W}=1+\sum_{i>0} c_{i} D^{-i} \in K\left[\left[D^{-1}\right]\right]
$$
It follows that
$$
\Psi_{W}\left(\sum_{i} a_{i} D^{-i}\right) \Psi_{W}^{-1} \in K[D] \Leftrightarrow\left(\sum_{i} a_{i} z^{i}\right) W \subseteq W
$$
for all $\sum_{i} a_{i} z^{i} \in k((z))$. In particular, if there exist functions
$$
f=\sum_{i} a_{i} z^{i}=z^{p}+O\left(z^{p-1}\right), \quad g=\sum_{j} b_{j} z^{j}=z^{q}+O\left(z^{q-1}\right)
$$
in $k((z))$ of relatively prime order such that
$$
f W, g W \subseteq W
$$
the corresponding differential operators
$$
\Psi\left(\sum_{i} a_{i} D^{-i}\right) \Psi^{-1}=D^{p}+O\left(D^{p-1}\right), \quad \Psi\left(\sum_{j} b_{j} D^{-j}\right) \Psi^{-1}=D^{q}+O\left(D^{q-1}\right)
$$
commute and are of the form we originally interested ourselves in. Thus the highly nonlinear problem of constructing of commuting differential operators reduces to the essentially linear problem of constructing Baker functions.
## 4. "Bare-handed" construction of Baker functions
We continue in the setting of the previous section. Put
$$
\mathcal{H}:=\left(\begin{array}{l}
\text { the space of column vectors } \\
h \text { with entries } h_{i} \text { in } k \text { indexed } \\
\text { by } \mathbb{Z} \text { vanishing for } i \gg 0
\end{array}\right)
$$
Identify $k((z))$ with $\mathcal{H}$ by the rule
$$
\sum_{i} a_{i} z^{i} \leftrightarrow\left[\begin{array}{c}
\vdots \\
a_{2} \\
a_{1} \\
a_{0} \\
a_{-1} \\
a_{-2} \\
\vdots
\end{array}\right] \leftarrow \text { position } 0
$$
There exists a $k$-basis
$$
\left\{w_{n}\right\}_{n=1}^{\infty}
$$
of $W$, an integer $N$ and a partition
$$
\lambda: \lambda_{1} \geq \lambda_{2} \geq \cdots \geq \lambda_{\ell}>0=\lambda_{\ell+1} \geq \cdots
$$
such that
$$
w_{n}=z^{-n+\lambda_{n}}+O\left(z^{-n+\lambda_{n}+1}\right)=z^{-n}+O\left(z^{N}\right)
$$
for all $n$; this follows from the index zero condition. We arrange the sequence $w_{1}, w_{2}, \ldots$ into a $\mathbb{Z}$ by $\mathbb{N}$ matrix with entries in $k$ which we denote again by $W$. The
$$
\left[\begin{array}{c}
\mathbb{Z} \backslash \mathbb{N} \\
\mathbb{N}
\end{array}\right] \times \mathbb{N}
$$
block decomposition of $W$ takes the form
$$
W=\left[\begin{array}{c}
P \\
1+Q
\end{array}\right]
$$
where $Q$ is an $\mathbb{N}$ by $\mathbb{N}$ matrix with only finitely many nonvanishing rows.
tions
EXAmple 4.1 . Let $A \subset k((z))$ be the $k$-algebra generated by func-
$$
f=z^{-2}, \quad g=z^{-3}\left(1-\left(g_{2} / 4\right) z^{4}-\left(g_{3} / 4\right) z^{6}\right)^{1 / 2} .
$$
If
$$
g_{2}^{3}-27 g_{3}^{2} \neq 0
$$
the $k$-algebra $A$ is a copy of the affine coordinate ring of the elliptic curve
$$
E: \quad Y^{2}=X^{3}-g_{2} X / 4-g_{3} / 4
$$
defined over $k$. There exists a unique $k$-basis $\left\{a_{n}\right\}_{n=1}^{\infty}$ of $A$ such that
$$
a_{n}= \begin{cases}1 & \text { if } n=1 \\ z^{-n}+O(z) & \text { if } n>1\end{cases}
$$
for all $n \in \mathbb{N}$. Let $A$ denote also the $\mathbb{Z}$ by $\mathbb{N}$ matrix obtained by arranging the vectors $\left\{a_{n}\right\}_{n=1}^{\infty}$ in the manner described above. The matrix
$$
\left[\begin{array}{ccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -\frac{1}{128} g_{2}^{2} & 0 & -\frac{1}{64} g_{2} g_{3} & 0 & -\frac{1}{512} g_{2}{ }^{3}-\frac{1}{128} g_{3}{ }^{2} \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -\frac{1}{8} g_{3} & 0 & -\frac{1}{128} g_{2}{ }^{2} & 0 & -\frac{1}{32} g_{2} g_{3} \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -\frac{1}{8} g_{2} & 0 & -\frac{1}{8} g_{3} & 0 & -\frac{3}{128} g_{2}{ }^{2} \\
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -\frac{1}{8} g_{2} & 0 & -\frac{1}{8} g_{3} \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array}\right]
$$
is the $\{-6, \ldots, 7\} \times\{1, \ldots, 7\}$ block of $A$. Let $\mathbf{t}$ be the $\mathbb{Z}$ by $\mathbb{Z}$ matrix with 1 's along the superdiagonal $j-i=1$ and 0's elsewhere. One has
$$
\mathbf{t}=\left[\begin{array}{cccccc}
\ddots & \ddots & \ddots & \ddots & & \\
\ddots & 0 & 1 & 0 & 0 & \\
\ddots & 0 & 0 & 1 & 0 & \ddots \\
\ddots & 0 & 0 & 0 & 1 & \ddots \\
& 0 & 0 & 0 & 0 & \ddots \\
& & \ddots & \ddots & \ddots & \ddots
\end{array}\right]
$$
main diagonal
and
$$
e^{-x \mathbf{t}^{-1}}=\left[\begin{array}{rrrrrr}
\ddots & \ddots & \ddots & \ddots & & \\
\ddots & 1 & 0 & 0 & 0 & \\
\ddots & -x & 1 & 0 & 0 & \ddots \\
\ddots & x^{2} / 2 & -x & 1 & 0 & \ddots \\
& -x^{3} / 6 & x^{2} / 2 & -x & 1 & \ddots \\
& & \ddots & \ddots & \ddots & \ddots
\end{array}\right]
$$
main diagonal.
The problem of constructing the Baker function
$$
\psi_{W}=1+\sum_{i=1}^{\infty} c_{i} z^{i}
$$
comes down to constructing sequences
$$
\left\{b_{i}\right\}_{i=1}^{\infty}, \quad\left\{c_{i}\right\}_{i=1}^{\infty}
$$
in $K$ tending $x$-adically to 0 such that
$$
e^{-x \mathbf{t}^{-1} W}\left[\begin{array}{c}
b_{1} \\
b_{2} \\
b_{3} \\
\vdots
\end{array}\right]=\left[\begin{array}{c}
\vdots \\
c_{3} \\
c_{2} \\
c_{1} \\
1 \\
0 \\
0 \\
\vdots
\end{array}\right] \leftarrow \text { position } 1
$$
Now we are going to cut to the chase. Put
$$
\tau_{W}(x, z)=\lim _{N \rightarrow \infty} \operatorname{det}_{i, j=1}^{N}\left(e^{x \mathbf{t}^{-1}}\left(1-z \mathbf{t}^{-1}\right)^{-1} W\right)_{i j}
$$
where
$$
\left(1-z \mathbf{t}^{-1}\right)=\left[\begin{array}{cccccc}
\ddots & \ddots & \ddots & \ddots & & \\
\ddots & 1 & 0 & 0 & 0 & \\
\ddots & z & 1 & 0 & 0 & \ddots \\
\ddots & z^{2} & z & 1 & 0 & \ddots \\
& z^{3} & z^{2} & z & 1 & \ddots \\
& & \ddots & \ddots & \ddots & \ddots
\end{array}\right]
$$
main diagonal.
The limit exists $(x, z)$-adically in $k[[x, z]]$. If one replaces $W$ by another matrix representing the same index zero subspace of $k((z)), \tau_{W}(x, z)$ is multiplied by a nonzero constant factor. Put
$$
\tau_{W}(x):=\tau_{W}(x, 0)
$$
The power series $\tau_{W}(x)$ is the determinant of the system of linear equations we are trying to solve. The great discovery of the Kyoto school is that
$$
\tau_{W}(x) \propto x^{\sum_{i} \lambda_{i}}(1+O(x)), \quad \psi_{W}=\frac{\tau_{W}(x, z)}{\tau_{W}(x)}, \quad c_{1}=-\frac{d}{d x} \log \tau_{W}(x) .
$$
The order of vanishing of $\tau_{W}(x)$ was (in a different language) also calculated by Fay; for background see [Segal-Wilson 1985]. ExAmple 4.2. As in Example 4.1, assume that $A \subset k((z))$ is generated as a $k$-algebra by functions
$$
f=z^{-2}, \quad g=z^{-3}\left(1-\left(g_{2} / 4\right) z^{4}-\left(g_{3} / 4\right) z^{6}\right)^{1 / 2} .
$$
Let $A$ be the matrix constructed in Example 4.1. One has
$$
\tau_{A}(x)=-\left(x-\frac{1}{240} g_{2} x^{5}-\frac{1}{840} g_{3} x^{7}\right)+O\left(x^{8}\right)
$$
and
$$
c_{1}^{\prime}=-\frac{d^{2}}{d x^{2}} \log \tau_{A}(x)=\frac{1}{x^{2}}+\frac{g_{2} x^{2}}{20}+\frac{g_{3} x^{4}}{28}+O\left(x^{5}\right)
$$
Since $c_{1}^{\prime}$ satisfies the stationary Korteweg-deVries equation by Example 2.2 , it follows that
$$
\left(c_{1}^{\prime \prime}\right)^{2}=4\left(c_{1}^{\prime}\right)^{3}-g_{2} c_{1}^{\prime}-g_{3} .
$$
Finally, if
$$
g_{2}^{3}-27 g_{3}^{2} \neq 0
$$
then $A$ is the affine coordinate ring of the elliptic curve
$$
E: y^{2}=x^{3}-g_{2} x / 4-g_{3} / 4
$$
and
$$
c_{1}^{\prime}(x)=\wp(x),
$$
where $\wp$ is the Weierstrass $\wp$-function attached to $E$.
## Complements in characteristic $p>0$
Versions of the preceding game can be played in characteristic $p$ and also p-adically. I attempted to play it in the papers [Anderson 1994a] and [Anderson 1994b]; of the latter paper I will not speak here. Assume now that $k$ is a finite field of $q$ elements, but otherwise leave everything the same: fix a subspace $W \subset k((z))$, basis $\left\{w_{n}\right\}_{n=1}^{\infty}$ and partition $\lambda$ as above. Consider the matrix
$$
E(x, y, z):=(1-z \mathbf{t})^{-1} \prod_{i=0}^{\infty}\left(1-y^{q^{i}} \mathbf{t}^{-1}\right)\left(1-x^{q^{i}} \mathbf{t}^{-1}\right)^{-1}
$$
The limit
$$
\tau_{W}(x, y, z):=\lim _{N \rightarrow \infty} \operatorname{det}_{i, j=1}^{N}(E(x, y, z) W)_{1-i, 1-j}
$$
exists $(x, y, z)$-adically in $k[[x, y, z]]$. One has
$$
\tau_{W}(x, y, 0) \sim \prod_{(i, j) \in \lambda}\left(x^{q^{i}}-y^{q^{j}}\right)
$$
where $\sim$ means that the left side is equal to the right side times a power series in $x$ and $y$ with constant term 1 , and $(i, j) \in \lambda$ means that $i$ and $j$ are positive integers such that $j \leq \lambda_{i}$; in particular,
$$
\tau_{W}(x, 0,0) \neq 0 .
$$
Now suppose that $A \subseteq k((z))$ is a $k$-subalgebra such that
$$
A \cap k[[z]]=k, \quad \operatorname{dim}_{k} \frac{k((z))}{A+k[[z]]}<\infty .
$$
Suppose that $W$ is a rank one projective $A$-module. Then the quotient
$$
\psi_{W}(x, z)=\frac{\tau_{W}(x, 0, z)}{\tau_{W}(x, 0,0)}
$$
admits interpretation as the Baker function associated to a rank one elliptic $A$-module. See [Anderson 1994a] for background and details.
## CHAPTER 3
## Reciprocity
We work exclusively with the infinite wedge model of fermionic Fock space in this chapter. As usual, $k$ denotes an artinian local ring and $m$ the maximal ideal thereof. The main result of this section is Theorem 3.12.
## Commutator calculations
Definition 1.1. Given $A, B \in \mathcal{Q}^{\times}$such that $A B=B A$, there exists a unique invertible scalar $\{A, B\}$ such that
$$
\tilde{B} \tilde{A}=\{A, B\} \tilde{A} \tilde{B}
$$
for all infinite wedge representations $\tilde{A}$ and $\tilde{B}$ of $A$ and $B$, respectively.
Example 1.2. For all $A, B, C \in \mathcal{Q}^{\times}$the following assertions hold:
- $\{A, A\}=1$.
- If $A B=B A$, then $\{A, B\}=\{B, A\}^{-1}$.
- If $A C=C A$ and $B C=C B$, then $\{A, C\}\{B, C\}=\{A B, C\}$.
- If $A B=B A$, then $\{A, B\}=\left\{C A C^{-1}, C B C^{-1}\right\}$.
ExAmple 1.3. Let $A, B \in \mathcal{Q}^{\times}$be commuting matrices. If the
$$
\left[\begin{array}{c}
\mathbb{Z} \backslash \mathbb{N} \\
\mathbb{N}
\end{array}\right] \times\left[\begin{array}{c}
\mathbb{Z} \backslash \mathbb{N} \\
\mathbb{N}
\end{array}\right]^{T}
$$
block decompositions of $A, A^{-1}, B$ and $B^{-1}$ are
$$
\text { all of the form }\left[\begin{array}{ll}
* & * \\
0 & *
\end{array}\right], \quad \text { or all of the form }\left[\begin{array}{ll}
* & 0 \\
* & *
\end{array}\right],
$$
then $\{A, B\}=1$. ExAmple 1.4. Let $\omega: \mathbb{Z} \rightarrow \mathbb{Z}$ be a tame one-to-one map. One has
$$
\{A, B\}=\left\{\omega_{*} A, \omega_{*} B\right\}
$$
for all commuting matrices $A, B \in \mathcal{Q}^{\times}$. Example 15.5 of Chapter 1 can be exploited to give a proof of this fact.
ExAmple 1.5 . Let $\omega, \eta: \mathbb{Z} \rightarrow \mathbb{Z}$ be tame one-to-one maps with disjoint images. One has
$$
\left\{\omega_{*} A, \eta_{*} B\right\}=(-1)^{\operatorname{deg} A \cdot \operatorname{deg} B}
$$
for all $A, B \in \mathcal{Q}^{\times}$. Again Example 15.5 of Chapter 1 can be exploited to give a proof of this fact.
EXAMPLE 1.6. Let
$$
f=\sum_{i} a_{i} t^{i} \in 1+t^{-1} m\left[t^{-1}\right], \quad g=\sum_{j} b_{j} t^{j} \in 1+t k[[t]]
$$
be given. Put
$$
A:=f(\mathbf{t})=\left[\begin{array}{cccccc}
\ddots & \ddots & \ddots & \ddots & & \\
\ddots & 1 & 0 & 0 & 0 & \\
\ddots & a_{-1} & 1 & 0 & 0 & \ddots \\
\ddots & a_{-2} & a_{-1} & 1 & 0 & \ddots \\
& a_{-3} & a_{-2} & a_{-1} & 1 & \ddots \\
& & \ddots & \ddots & \ddots & \ddots
\end{array}\right] \in \mathcal{Q}^{\times}
$$
and
One has
$$
B:=g(\mathbf{t})=\left[\begin{array}{cccccc}
\ddots & \ddots & \ddots & \ddots & & \\
\ddots & 1 & b_{1} & b_{2} & b_{3} & \\
\ddots & 0 & 1 & b_{1} & b_{2} & \ddots \\
\ddots & 0 & 0 & 1 & b_{1} & \ddots \\
& 0 & 0 & 0 & 1 & \ddots \\
& & \ddots & \ddots & \ddots & \ddots
\end{array}\right] \in \mathcal{Q}^{\times} .
$$
$$
\{A, B\}=\frac{\langle\bullet|\tilde{B} \tilde{A}| \bullet\rangle}{\langle\bullet|\tilde{A} \tilde{B}| \bullet\rangle}=\operatorname{det}_{i, j=1}^{n}(B A)_{1-i, 1-j}=\operatorname{det}_{i, j=1}^{n}\left(A^{-1} B^{-1}\right)_{i j}
$$
for all integers $n \gg 0$ by Example 18.8. Also by Examples 18.16 and ? one has
where the sum is
Example 1.7. Fix scalars $x$ and $y$. Assume that $x$ is nilpotent. Put
$$
\begin{aligned}
A:=1-x \mathbf{t}^{-1}= & {\left[\begin{array}{ccccc}
\ddots & & & & \\
\ddots & 1 & & & \\
& -x & 1 & & \\
& & -x & 1 & \\
& & & \ddots & \ddots
\end{array}\right] \in \mathcal{Q}^{\times}, } \\
B:=1-y \mathbf{t}= & {\left[\begin{array}{ccccc}
\ddots & \ddots & & & \\
& 1 & -y & & \\
& & 1 & -y & \\
& & & 1 & \ddots \\
& & & & \ddots
\end{array}\right] \in \mathcal{Q}^{\times} . }
\end{aligned}
$$
One has
$$
\underbrace{\left|\begin{array}{ccccc}
1+x y & -y & & & \\
-x & 1+x y & -y & & \\
& -x & 1+x y & \ddots & \\
& & \ddots & \ddots & -y \\
& & -x & 1+x y
\end{array}\right|}_{n}=1+x y+\cdots+(x y)^{n} .
$$
We conclude that
$$
\{A, B\}=\left\{1-x \mathbf{t}^{-1}, 1-y \mathbf{t}\right\}=(1-x y)^{-1}
$$
via Example 1.6.
ExAmple 1.8. Fix scalars $x, y \in k$. Assume that $x$ is nilpotent. By a calculation similar to that presented in Example 1.7, one can verify that
$$
\left\{1-x \mathbf{t}^{-p}, 1-y \mathbf{t}^{q}\right\}=\left(1-x^{q /(p, q)} y^{p /(p, q)}\right)^{-(p, q)}
$$
for all positive integers $p$ and $q$, where $(p, q)$ denotes the greatest common divisor of $p$ and $q$. It is also possible to deduce this formula by factoring $1-x t^{-p}$ and $1-y t^{q}$ in $k^{\prime}((t))$ where $k^{\prime} / k$ is a suitably constructed finite flat $k$-algebra. EXample 1.9. Let $f \in k[[t]]^{\times}$be given. Put
$$
A:=f(\mathbf{t}), \quad B:=\mathbf{t} .
$$
We claim that
$$
\{A, B\}=c:=\text { constant term of } f .
$$
By Example 18.2 of Chapter 1, there exist unique infinite wedge representations $\tilde{A}$ and $\tilde{B}$ of $A$ and $B$, respectively, such that
$$
\tilde{A}|0\rangle=|0\rangle, \quad \tilde{B}|0\rangle=|-1\rangle=e_{1}^{b}|0\rangle,
$$
where $e_{n}$ denotes the $n^{\text {th }}$ column of the $\mathbb{Z}$ by $\mathbb{Z}$ identity matrix. One has
$$
\tilde{B} \tilde{A}|0\rangle=\tilde{B}|0\rangle=e_{1}^{b}|0\rangle=c \tilde{A} e_{1}^{b}|0\rangle=c \tilde{A} \tilde{B}|0\rangle .
$$
The claim is proved.
Example 1.10. Let $f \in k[t]^{\times}$be given. Put
$$
A:=f(\mathbf{t}), \quad B:=\mathbf{t} .
$$
We claim that
$$
\{A, B\}=c:=\text { constant term of } f .
$$
By Example 18.2 of Chapter 1, there exist unique infinite wedge representations $\tilde{A}$ and $\tilde{B}$ of $A$ and $B$, respectively, such that
$$
\langle 0| \tilde{A}=\langle 0|, \quad\langle 0| \tilde{B}=\langle 1|=\langle 0| e_{0}^{b}
$$
where $e_{n}$ denotes the $n^{\text {th }}$ column of the $\mathbb{Z}$ by $\mathbb{Z}$ identity matrix. One has
$$
\langle 0| \tilde{B} \tilde{A}=\langle 0| e_{0}^{b} \tilde{A}=c\langle 0| \tilde{A} e_{0}^{b}=c\langle 0| \tilde{A} \tilde{B} .
$$
The claim is proved.
EXAMPLE 1.11. Let
$$
f \in 1+t^{-1} m\left[t^{-1}\right]
$$
be given. By Example 18.8 there exists a positive integer $N$ such that one has
$$
\{f(\mathbf{t}), g(\mathbf{t})\}=1
$$
for all $g \in 1+t^{N} k[[t]]$.
## The Contou-Carrère symbol
Definition 2.1. Given $f, g \in k((t))^{\times}$, put
$$
\{f, g\}=(-1)^{w(f) w(g)} \frac{a_{0}^{w(g)}}{b_{0}^{w(f)}} \frac{\prod_{i=1}^{\infty} \prod_{j=1}^{\infty}\left(1-a_{i}^{j /(i, j)} b_{-j}^{i /(i, j)}\right)^{(i, j)}}{\prod_{i=1}^{\infty} \prod_{j=1}^{\infty}\left(1-a_{-i}^{j /(i, j)} b_{j}^{i /(i, j)}\right)^{(i, j)}},
$$
where $\left\{a_{i}\right\}$ and $\left\{b_{j}\right\}$ are the systems of Witt parameters associated to $f$ and $g$, respectively. All but finitely many of the terms appearing in the products differ from 1 and thus $\{f, g\}$ is a well defined invertible scalar. The function
$$
\{\cdot, \cdot\}: k((t))^{\times} \times k((t))^{\times} \rightarrow k^{\times}
$$
is an elementary version of the Contou-Carrère symbol [Contou-Carrère 1994].
EXAMPLE 2.2. By combining all the calculations carried out above, one finds that
$$
\left\{\omega_{*} f(\mathbf{t}), \omega_{*} g(\mathbf{t})\right\}=(-1)^{w(f) w(g)}\{f, g\}
$$
for all $f, g \in k((t))^{\times}$and any tame one-to-one $\operatorname{map} \omega: \mathbb{Z} \rightarrow \mathbb{Z}$. One also has
$$
\left\{\omega_{*} f(\mathbf{t}), \eta_{*} g(\mathbf{t})\right\}=(-1)^{w(f) w(g)}
$$
for all $f, g \in k((t))^{\times}$and any tame one-to-one maps $\omega, \eta: \mathbb{Z} \rightarrow \mathbb{Z}$ with disjoint images.
ExAmple 2.3. One has
$$
\begin{aligned}
\{f, f\} & =(-1)^{w(f)} \\
\{f, g\} & =\{g, f\}^{-1} \\
\{f g, h\} & =\{f, h\}\{g, h\}
\end{aligned}
$$
for all $f, g, h \in k((t))^{\times}$.
EXAMPle 2.4. If $k$ is a field, the Contou-Carrère symbol coincides with the tame symbol.
ExAmple 2.5. If $k=k_{0}[\epsilon]$ where $k_{0}$ is a field and $\epsilon^{3}=0$, one has
$$
\{1-\epsilon f, 1-\epsilon g\}=1-\epsilon^{2} \operatorname{Res}\left(f^{\prime} g d t\right)
$$
for all $f, g \in k_{0}((t))$. Thus one recovers the residue from the ContouCarrère symbol. EXAMPle 2.6. We claim that the Contou-Carrère symbol satisfies the adjunction formula
$$
\left\{\mathcal{N}_{\phi} f, g\right\}=\{f, g \circ \phi\}
$$
for all $f, g, \phi \in k((t))^{\times}$such that $\phi$ has a positive winding number. In particular, the Contou-Carrère symbol is reparameterization invariant. Let $n$ be the winding number of $\phi$. By Example 4.27, there exists $A \in \mathcal{Q}^{\times}$such that
$$
\phi(\mathbf{t}) A=A \mathbf{t}^{n}
$$
By definition (see Example 4.30) one has
$$
\left(\mathcal{N}_{\phi} f\right)(\mathbf{t})=\operatorname{det}\left(\left(A^{-1} f(\mathbf{t}) A\right)^{[n]}\right), \quad(g \circ \phi)(\mathbf{t})=A g\left(\mathbf{t}^{n}\right) A^{-1}
$$
By Example 8.8 one has a factorization
$$
A^{-1} f(\mathbf{t}) A=\left((\ell \mapsto n \ell)_{*}\left(\mathcal{N}_{\phi} f\right)(\mathbf{t})\right) C
$$
where $C$ is an element of the commutator subgroup of the subgroup of $\mathcal{Q}^{\times}$consisting of matrices commuting with $\mathbf{t}^{n}$. Exploiting all the previous calculations, we have
$$
\begin{aligned}
& (-1)^{w(f) w(g \circ \phi)}\{f, g \circ \phi\} \\
& =\{f(\mathbf{t}),(g \circ \phi)(\mathbf{t})\} \\
& =\left\{f(\mathbf{t}), A\left(g\left(\mathbf{t}^{n}\right)\right) A^{-1}\right\} \\
& =\left\{A^{-1} f(\mathbf{t}) A, g\left(\mathbf{t}^{n}\right)\right\} \\
& =\left\{(\ell \mapsto n \ell)_{*}\left(\mathcal{N}_{\phi} f\right)(\mathbf{t}), \prod_{i=1}^{n}(\ell \mapsto n \ell+1-i)_{*} g(\mathbf{t})\right\} \\
& =(-1)^{w\left(\mathcal{N}_{\phi} f\right) n w(g)}\left\{\mathcal{N}_{\phi} f, g\right\} \text {. }
\end{aligned}
$$
Finally, the signs cancel. The claim is proved.
## An adelic formalism and a reciprocity law
Under this heading we fix a map
$$
\tau: \mathbb{Z} \rightarrow \mathbb{Z}
$$
with the following properties:
- $\tau$ is bijective.
- $\tau(n)>n$ for all $n \in \mathbb{Z}$.
- For each integer $n_{0}$ maximal among the nonpositive elements of its $\tau$-orbit one has
$$
\tau^{1-i}\left(n_{0}\right)=1-\tau^{i}\left(n_{0}\right)
$$
for all $i \in \mathbb{Z}$.
We call such a map $\tau$ a multi-Toeplitz structure. All the constructions under this heading depend on the choice of $\tau$, but for brevity's sake we usually suppress reference to $\tau$ in the notation. We usually refer to $\tau$-orbits as standard places.
ExAmPle 3.1. For each positive integer $N$, the map
$$
(n \mapsto n+N): \mathbb{Z} \rightarrow \mathbb{Z}
$$
is a multi-Toeplitz structure with $N$ orbits.
ExAmple 3.2. For each $n \in \mathbb{Z}$, let $e(n)$ be the largest power of 2 dividing $\max (n, 1-n)$. Put
$$
\alpha(n):= \begin{cases}n+2 e(n) & \text { if } n \neq 1-e(n) \\ e(n) & \text { if } n=1-e(n)\end{cases}
$$
for all $n \in \mathbb{Z}$, thereby defining a map $\alpha: \mathbb{Z} \rightarrow \mathbb{Z}$ that turns out to be a multi-Toeplitz structure. Here's the orbit structure of $\alpha$ :
$$
\begin{aligned}
& \ldots \rightarrow-4 \rightarrow-2 \rightarrow 0 \quad \rightarrow \quad 1 \rightarrow 3 \rightarrow 5 \rightarrow \ldots \\
& \ldots \rightarrow-9 \rightarrow-5 \rightarrow-1 \rightarrow 2 \rightarrow 6 \rightarrow 10 \rightarrow \ldots \\
& \ldots \rightarrow-19 \rightarrow-11 \rightarrow-3 \rightarrow 4 \rightarrow 12 \rightarrow 20 \rightarrow \ldots
\end{aligned}
$$
The sequence $\left\{1-2^{n}\right\}_{n=0}^{\infty}$ meets each $\alpha$-orbit in $\mathbb{Z}$ exactly once. Definition 3.3. We say that a matrix $A \in \mathcal{Q}$ is multi-Toeplitz under the following conditions:
- $A_{\tau(i), \tau(j)}=A_{i j}$ for all $i, j \in \mathbb{Z}$.
- $A_{i j}=0$ for all $i, j \in \mathbb{Z}$ belonging to distinct $\tau$-orbits.
We define the universal adele ring $\mathbb{A} \subset \mathcal{Q}$ to be the $k$-subalgebra consisting of multi-Toeplitz matrices. We define $\mathcal{O} \subset \mathbb{A}$ to be the $k$ subalgebra of consisting of upper triangular matrices. Every element of $\mathbb{A}$ is fixed by the dagger involution and consequently $\mathbb{A}$ is a commutative $k$-algebra.
Definition 3.4. Given $A \in \mathbb{A}$ and a standard place $P$ there exists a unique Laurent series
$$
A_{P}=\sum_{n} a_{n} t^{n} \in k((t))
$$
such that
$$
A_{i, \tau^{n}(i)}=a_{n}
$$
for all $i \in P$ and $n \in \mathbb{Z}$. The map
$$
\left(A \mapsto A_{P}\right): \mathbb{A} \rightarrow k((t))
$$
is a surjective $k$-algebra homomorphism. For each $A \in \mathbb{A}$ and standard place $P$, one has $A=0$ (resp. $A \in \mathcal{O}$ ) if and only $A_{P}=0$ (resp. $A_{P} \in$ $k[[t]])$ for all standard places $P$. For each collection $\left\{f_{P}\right\}$ of elements of $k((t))$ indexed by standard places $P$, there exists $A \in \mathbb{A}$ such that $A_{P}=f_{P}$ for all standard places $P$ if and only if $f_{P} \in k[[t]]$ for all but finitely many standard places $P$.
EXAMPle 3.5. One has
$$
\operatorname{deg} f=\sum_{P} w\left(f_{P}\right)
$$
for all $f \in \mathbb{A}^{\times}$where the sum is extended over standard places $P$. Definition 3.6. Let $\mathbf{e} \in \mathcal{H}$ be defined by the rule
$$
\mathbf{e}_{n}:= \begin{cases}1 & \text { if } n=\max \left(\left\{\tau^{i}(n) \mid i \in \mathbb{Z}\right\} \backslash \mathbb{N}\right) \\ 0 & \text { otherwise }\end{cases}
$$
for all $n \in \mathbb{Z}$. The maps
$$
(f \mapsto f \mathbf{e}): \mathbb{A} \rightarrow \mathcal{H}, \quad(f \mapsto f \mathbf{e}): \mathcal{O} \rightarrow \mathcal{H}(\mathbb{Z} \backslash \mathbb{N})
$$
are bijective. Put
$$
\operatorname{Res}:=\left(\sum_{i} a_{i} t^{i} \mapsto a_{-1}\right): k((t)) \rightarrow k .
$$
One has
$$
\sum_{n \in \mathbb{Z}}(f \mathbf{e})_{n}(g \mathbf{e})_{1-n}=\sum_{P} \operatorname{Res}\left(f_{P} g_{P}\right)
$$
for all $f, g \in \mathbb{A}$. We introduce the abbreviated notation
$$
f^{\sharp}:=(f \mathbf{e})^{\sharp}, \quad g^{b}:=(g \mathbf{e})^{b}
$$
for all $f, g \in \mathbb{A}$. One then has
$$
\left(f^{\sharp}\right)^{2}=0, \quad\left(g^{b}\right)^{2}=0, \quad f^{\sharp} g^{b}+g^{b} f^{\sharp}=\sum_{P} \operatorname{Res}\left(f_{P} g_{P}\right)
$$
for all $f, g \in \mathbb{A}$. For each $a \in \mathbb{A}^{\times}$, the corresponding infinite wedge representations $\tilde{a}$ of $a$ are characterized by the identities
$$
\tilde{a}\left(f^{\sharp} \psi\right)=(a f)^{\sharp}(\tilde{a} \psi), \quad \tilde{a}\left(g^{b} \psi\right)=\left(a^{-1} g\right)^{b}(\tilde{a} \psi)
$$
for all $f, g \in \mathbb{A}$ and $\psi \in \mathcal{F}$.
Definition 3.7. For any infinite subset $P \subseteq \mathbb{Z}$ such that $P=1-P$, e. g., any standard place $P$, there exists a unique strictly increasing bijective map $[P]: \mathbb{Z} \rightarrow P$ such that
$$
[P](1-n)=1-[P](n)
$$
for all $n \in \mathbb{Z}$.
EXAmple 3.8. Fix $f \in \mathbb{A}^{\times}$and let $P_{1}, \ldots, P_{n}$ be any finite collection distinct standard places such that $f_{P} \in k[[t]]^{\times}$for all $P \notin\left\{P_{1}, \ldots, P_{n}\right\}$. Then one has a unique factorization of the form
$$
f=[Q]_{*} A \cdot\left[P_{1}\right]_{*} f_{1}(\mathbf{t}) \cdots\left[P_{n}\right]_{*} f_{n}(\mathbf{t})
$$
where $f_{1}, \ldots, f_{n} \in k((t))^{\times}, Q:=\mathbb{Z} \backslash\left(P_{1} \cup \cdots \cup P_{n}\right)$, and $A \in \mathcal{Q}^{\times}$is a matrix such that both $A$ and $A^{-1}$ are upper triangular. ExAmple 3.9. Let $f, g \in \mathbb{A}^{\times}$be given. We claim that
$$
\{f, g\}=(-1)^{\operatorname{deg} f \cdot \operatorname{deg} g} \prod_{P}\left\{f_{P}, g_{P}\right\}
$$
where the product is extended over all standard places. The claim is verified by factoring $f$ and $g$ in the fashion described in Example 3.8 and calculating according to the rules worked out above.
ExAMPle 3.10. Let $f, g \in \mathbb{A}^{\times}$and a matrix $\Omega \in \mathcal{Q}^{\times}$be given such that the $\mathbb{Z} \backslash \mathbb{N}$ by $\mathbb{N}$ blocks of the matrices
$$
\Omega^{-1} f \Omega, \quad \Omega^{-1} f^{-1} \Omega, \quad \Omega^{-1} g \Omega, \quad \Omega^{-1} g^{-1} \Omega
$$
vanish. Then
$$
1=\prod_{P}\left\{f_{P}, g_{P}\right\} \in k^{\times}
$$
where the product is extended over all standard places $P$. Again this is verified by a straightforward calculation based upon the rules worked out above.
ExAmPle 3.11. Let $K \subset \mathbb{A}$ be a flat $k$-subalgebra such that the $k$ modules $K \cap \mathcal{O}$ and $\mathbb{A} /(K+\mathcal{O})$ are finitely generated. By Example 4.4 of Chapter 1 and the hypothesis, one has
$$
\mathcal{H}=K \mathbf{e} \oplus \mathcal{H}(I)
$$
for some wedge index $I \subset \mathbb{Z}$. It follows that for each $n \in \mathbb{Z} \backslash I$ there exists unique $f^{(n)} \in K$ such that
$$
f^{(n)} \mathbf{e}-e_{n} \in \mathcal{H}(I)
$$
where $e_{n}$ denotes the $n^{\text {th }}$ column of the $\mathbb{Z}$ by $\mathbb{Z}$ identity matrix. Let $A \in \mathcal{Q}$ be defined by the rule
$$
A e_{n}= \begin{cases}f^{(n)} \mathbf{e} & \text { if } n \in \mathbb{Z} \backslash I \\ e_{n} & \text { if } n \in I\end{cases}
$$
for all $n \in \mathbb{Z}$. By construction the
$$
\left[\begin{array}{c}
I \\
\mathbb{Z} \backslash I
\end{array}\right] \times\left[\begin{array}{c}
I \\
\mathbb{Z} \backslash I
\end{array}\right]^{T}
$$
block decomposition of the matrix $A$ is of the form
$$
A=\left[\begin{array}{ll}
1 & * \\
0 & 1
\end{array}\right],
$$
and hence $A \in \mathcal{Q}^{\times}$. Choose now a tame permutation $\omega: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $\omega(\mathbb{N})=\mathbb{Z} \backslash I$ and let $W$ be the element of the big Weyl group representing $\omega$. Put
$$
\Omega:=A W .
$$
The matrix $\Omega$ the following properties:
- $\Omega \mathcal{H}(\mathbb{N})=K \mathbf{e}$
- For each $f \in K$ the $\mathbb{Z} \backslash \mathbb{N}$ by $\mathbb{N}$ block of $\Omega^{-1} f \Omega$ vanishes.
- For each $b \in \mathbb{A}^{\times}$, the $\mathbb{N}$ by $\mathbb{N}$ block of $b^{-1} \Omega$ is invertible if and only if $\mathbb{A}=b \mathcal{O} \oplus K$.
We call $\Omega$ a Riemann matrix for $K$.
THEOrem 3.12. Let $K \subset \mathbb{A}$ be a flat $k$-subalgebra such that the $k$-modules $K \cap \mathcal{O}$ and $\mathbb{A} /(K+\mathcal{O})$ are finitely generated. Then for all $f, g \in K^{\times}$one has $\prod_{P}\left\{f_{P}, g_{P}\right\}=1$.
Proof. There is nothing left to prove.
## Recovery of some classical reciprocity laws
We assume under this heading that $k$ is a countable algebraically closed field and we suppose that a multi-Toeplitz structure with infinitely many orbits has been fixed.
Definition 4.1. A finitely generated field extension $K / k$ of transcendence degree 1 will be called a function field.
Definition 4.2. An adelization of a function field $K / k$ is a $k$ algebra embedding
$$
f \mapsto f^{\iota}: K \rightarrow \mathbb{A}
$$
such that
$$
\mathbb{A}=a \mathcal{O} \oplus K^{\iota}
$$
for some $a \in \mathbb{A}^{\times}$.
Definition 4.3. Let $K / k$ be a function field. A function
$$
v: K \rightarrow \mathbb{Z} \coprod\{+\infty\}
$$
is called a normalized additive valuation if it has the following properties:
$$
\begin{aligned}
v^{-1}(+\infty) & =\{0\} \\
v\left(k^{\times}\right) & =\{0\} \\
v(f g) & =v(f)+v(g) \\
v(f+g) & \geq \min (v(f), v(g))
\end{aligned}
$$
Under our hypothesis that $k$ is countable, the set of normalized additive valuations of $k$ is a countably infinite set. EXAMPle 4.4. Let $K / k$ be a function field. Let $\iota: K \rightarrow \mathbb{A}$ be a $k$ algebra embedding. Then $\iota$ is an adelization if and only if the following two conditions hold:
- For each standard place $P$, the map
$$
\left(f \mapsto w\left(f_{P}^{\iota}\right)\right): K \rightarrow \mathbb{Z} \coprod\{+\infty\}
$$
is a normalized additive valuation of $K / k$.
- The map $P \mapsto\left(f \mapsto w\left(f_{P}^{\iota}\right)\right)$ puts the standard places in bijective correspondence with the normalized additive valuations of $K / k$.
Since we are assuming that $k$ is a countable algebraically closed field, every function field $K / k$ can be adelized.
ExAmple 4.5 . Let $K / k$ be a function field. Let $\iota_{1}, \iota_{2}: K \rightarrow \mathbb{A}$ be adelizations. Then there exists a matrix $A \in \mathcal{Q}^{\times}$such that the
$$
\left[\begin{array}{c}
\mathbb{Z} \backslash \mathbb{N} \\
\mathbb{N}
\end{array}\right] \times\left[\begin{array}{c}
\mathbb{Z} \backslash \mathbb{N} \\
\mathbb{N}
\end{array}\right]^{T}
$$
block decompositions of $A$ and $A^{-1}$ take the form
$$
\left[\begin{array}{ll}
* & * \\
0 & *
\end{array}\right]
$$
and
$$
A f^{\iota_{1}} A^{-1}=f^{\iota_{2}}
$$
for all $f \in K$. Thus all adelizations of $K / k$ are conjugate.
EXAMPLE 4.6. Let $K / k$ be a function field equipped with an adelization $\iota$. Let $k[\epsilon]$ be a (commutative) $k$-algebra generated by a finite collection $\left\{\epsilon_{i}\right\}$ of nilpotent elements; then $k[\epsilon]$ is an artinian local ring. Put $K[\epsilon]:=K \otimes_{k} k[\epsilon]$. For each standard place $P$, we extend the embedding $\left(f \mapsto f_{P}^{\iota}\right)$ by $k[\epsilon]$-linearity to an embedding $K[\epsilon] \rightarrow k[\epsilon]((t))$. Let $\{\cdot, \cdot\}$ denote the version of the Contou-Carrère symbol defined over $k[\epsilon]$ rather than over $k$. By Theorem 3.12, for any $f, g \in K[\epsilon]^{\times}$, one has
$$
\prod_{v}\left\{f_{P}^{\iota}, g_{P}^{\iota}\right\}=1 \in k[\epsilon]^{\times}
$$
where the product is extended over all standard places $P$; the product is well defined because all but finitely many terms are equal to 1 .
ExAmple 4.7. When all the $\epsilon$ 's vanish, Example 4.6 reduces to the Weil reciprocity law obeyed by the tame symbol; thus the general reciprocity law for the Contou-Carrère symbol can be viewed as a "deformation" of the Weil reciprocity law. EXAMPle 4.8. Assume that the collection $\left\{\epsilon_{i}\right\}$ consists of just one element, which we denote now by $\epsilon$. Assume that $\epsilon^{3}=0$ and that $1, \epsilon, \epsilon^{2}$ are $k$-linearly independent. Assume that $f=1-\epsilon f_{0}$ and $g=1-\epsilon g_{0}$ where $f_{0}, g_{0} \in K$. In this special case Example 4.6 reduces to the assertion that the residues of the differential $g_{0} d f_{0}$ sum to 0 .
ExAmple 4.9. The relationship of the Contou-Carrère symbol to the ring of Witt vectors in characteristic $p$ deserves further discussion; we would like to include such a discussion in a later draft of the notes. 3. RECIPROCITY
## CHAPTER 4
## Calculation of genus 1 Jacobians
Under this heading $k$ is assumed to be a field of characteristic 0 .
## The logarithmic derivative yoga
Let $V$ be a vector space over $k$ and let $\mathbf{1}$ denote the identity map $V \rightarrow V$. Let $\mathfrak{g} \subseteq \operatorname{End}_{k}(V)$ be a Lie $k$-subalgebra and let $\mathfrak{a}$ and $\mathfrak{b}$ be Lie $k$-subalgebras. Assume that
$$
\mathfrak{g}, \mathfrak{a}, \mathfrak{b} \supseteq k \cdot \mathbf{1}, \quad[\mathfrak{g}, \mathfrak{g}] \subseteq k \cdot \mathbf{1}
$$
Let $L \subset V$ be a one-dimensional $k$-subspace (a line). Let $H \subset V$ be a $k$-subspace of codimension 1 (a hyperplane). Assume that
$$
\mathfrak{a} H \subseteq H, \quad \mathfrak{b} L \subseteq L .
$$
Fix $0 \neq \ell \in L$ and $0 \neq h^{*} \in H^{\perp} \subseteq V^{*}$. For any endomorphism $X: V \rightarrow V$, put
$$
e^{t X}=\sum_{n=0}^{\infty} \frac{X^{n}}{n !} x^{n} \in \operatorname{End}_{k}(V)[[x]] .
$$
For all $X, Y \in \mathfrak{g}$ one has
$$
e^{x(X+Y)}=e^{-\lambda x^{2} / 2} e^{x X} e^{x Y}
$$
where
$$
[X, Y]=\lambda \cdot \mathbf{1}
$$
The identity is proved by verifying that both sides are equal if $x=0$, and satisfy the same first order linear differential equation. (I thank Dennis Stanton for explaining this to me.) For each $X \in \mathfrak{g}$ put
$$
\tau_{X}:=h^{*} e^{x X} \ell \in k[[x]] .
$$
One has
$$
\tau_{A+X}=e^{-\lambda x^{2} / 2+\alpha x} \tau_{X}
$$
for all $A \in \mathfrak{a}$ and $X \in \mathfrak{g}$ where
$$
[A, X]=\lambda \cdot \mathbf{1}, \quad A \ell=\alpha \ell \quad(\lambda, \alpha \in k) .
$$
One has
$$
\tau_{X+B}=e^{-\mu x^{2} / 2+\beta x} \tau_{X}
$$
for all $X \in \mathfrak{g}$ and $B \in \mathfrak{b}$ where
$$
[X, B]=\mu \cdot \mathbf{1}, \quad h^{*} B=\beta h^{*} \quad(\mu, \beta \in k) .
$$
Assume now that the coset $X+\mathfrak{a}+\mathfrak{b} \in \mathfrak{g} /(\mathfrak{a}+\mathfrak{b})$ contains some $Y$ such that $\tau_{Y} \neq 0$. The Laurent series
$$
-\frac{d^{3}}{d x^{3}} \log \tau_{X} \in k((x))
$$
is then well-defined, depends only on the coset $X+\mathfrak{a}+\mathfrak{b}$, and is also independent of the choice of $\ell$ and $h^{*}$.
## The basic construction
Let $K / k$ be a genus 1 function field and let $\omega$ be a nonzero $k$-rational everywhere regular differential of $K / k$. Let $\mathbb{A}=\mathbb{A}_{K / k}$ denote the adèle ring of $K / k$ and let $\mathcal{O}=\mathcal{O}_{K / k}$ be the subring consisting of integeral ad'eles. Let $\mathcal{C}=\mathcal{C}(K / k, \omega)$ be the Clifford algebra associated to the quadratic space
$$
\left(\left[\begin{array}{l}
\mathbb{A} \\
\mathbb{A}
\end{array}\right],\left[\begin{array}{l}
f \\
g
\end{array}\right] \mapsto \operatorname{Res}(f g \omega)\right)
$$
and write
$$
\left[\begin{array}{l}
f \\
0
\end{array}\right]^{\sharp}=f^{\sharp}, \quad\left[\begin{array}{l}
0 \\
g
\end{array}\right]^{\sharp}=g^{b}
$$
for all $f, g \in \mathbb{A}$. Let $\mathcal{J}=\mathcal{J}(K / k, \omega) \subset \mathcal{C}$ be the left ideal generated by $\mathcal{O}_{K / k}^{\sharp}+\mathcal{O}_{K / k}^{b}$. Let $\mathcal{I}=\mathcal{I}(K / k, \omega) \subset \mathcal{C}$ be the right ideal generated by $K^{\sharp}+K^{b}$. It can be shown that the quotient $\mathcal{C} /(\mathcal{I}+\mathcal{J})$ is onedimensional over $k$ and is generated by $f^{\sharp}$ where $f \in \mathbb{A}$ is any adele such that $\operatorname{Res}(f \omega)=1$. Finally, put
$$
\mathcal{F}=\mathcal{F}(K / k, \omega)=\mathcal{C} / \mathcal{J}
$$
thereby defining a left $\mathcal{C}$-module. Put $|\bullet\rangle=1+\mathcal{J} \in \mathcal{F}$, and choose any nonzero $k$-linear function $\langle\bullet|$ on $\mathcal{F}$ killing $\mathcal{I} \mathcal{F}$. For each $X \in \mathbb{A}$ there exists a unique derivation $X^{\natural}$ of $\mathcal{C}$ such that
$$
X^{\natural} f^{\sharp}=(X f)^{\sharp}, \quad X^{\natural} g^{b}=-(X g)^{b}
$$
for all $f, g \in \mathbb{A}$. It can be shown that for each $X \in \mathbb{A}$ there exists a $k$-linear endomorphism $\tilde{X}$ of $\mathcal{F}$ well defined up to a scalar addend such that
$$
\tilde{X}(x \psi)=\left(X^{\natural} x\right) \psi+x \tilde{X} \psi
$$
for all adeles $x \in \mathcal{C}$ and $\psi \in \mathcal{F}$. Now choose any $X \in \mathbb{A}$ such that $\operatorname{Res}(X \omega)=1$, lifting $\tilde{X}$ as above, and put
$$
\tau(x)=\sum_{i=0}^{\infty}\left\langle\bullet\left|\tilde{X}^{n}\right| \bullet\right\rangle \frac{x^{n}}{n !} \in k[[x]] .
$$
Presently it will be explained that $\tau \neq 0$; this granted, it follows that the Laurent series
$$
\wp_{K / k, \omega}^{\prime}:=-\frac{d^{3}}{d x^{3}} \log \tau(x) \in k((x))
$$
is well defined, i. e., independent of all the choices made to define it.
The question now arises as to just what interpretation the power series $\wp_{K / k, \omega}^{\prime}$ has. Let $\tilde{K} / \bar{k}$ be the base-change of the extension $K / k$ to an algebraic closure $\bar{k}$ of $k$ and let $\bar{\omega}$ be the differential of $\tilde{K} / \bar{k}$ obtained from $\omega$ by base-change. Let $L / k$ be another genus one function field equipped with a nonzero $k$-rational everywhere regular differential $\eta$. Suppose that we can identify the extensions $\bar{L} / \bar{k}$ and $\bar{K} / \bar{k}$ in such a way that $\bar{\omega}=\bar{\eta}$. Then it is clear that $\wp_{K / k, \omega}^{\prime}=\wp_{L / k, \omega}^{\prime}$. If $L / k$ happens to be an elliptic function field, one can verify by the methods developed above that $\wp_{L / k, \eta}^{\prime}$ is the derivative the Weierstrass $\wp$-function associated to $L / k$ and $\eta$. In short, calculation of $\wp_{K / k, \omega}^{\prime}$ is tantamount to computing the Jacobian of the genus one curve whose function field is $K / k$.
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[Artin GA] Artin, E.: Geometric Algebra. Interscience Publishers, New York 1957
[Artin ANAF] Artin, E.: Algebraic Numbers and Algebraic Functions. Gordon and Breach, New York 1967
[Brylinski 1997] Brylinski, Jean-Luc: Central extensions and reciprocity laws. Cahiers Topologie Gèom. Diffèrentielle Catèg. 38(1997), no. 3, 193-215
[Chevalley ATS] Chevalley, C.: The Algebraic Theory of Spinors. Columbia University Press, New York 1954
[Contou-Carrère 1994] Contou-Carrère, C.: Jacobienne locale, groupe de bivecteurs de Witt universel, et symbol modéré. C. R. Acad. Sci. Paris Ser. I Math. 318(1994) no. 8, 743-746
[Macdonald SFHP] Symmetric Functions and Hall Polynomials.
[Matsumura CRT] Matsumura, H.: Commutative ring theory. Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, New York 1986, ISBN 0-521-36764-6.
[Mumford 1978] Mumford, D.: An algebra-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-deVries equation, and related nonlinear equations, in: Proceedings of the International Symposium on Algebraic Geometry, ed. by M. Nagata, Kinokuniya, Tokyo 1978, 115-153.
[Segal-Wilson 1985] Segal, G., Wilson, G.: Loop groups and equations of KdV type. Inst. Hautes Etudes Sci. Publ. Math. 61(1985)5-65
[Tate 1968] Tate, J.: Residues of differentials on curves. Ann. Sci. Ecole Norm. Sup. (4) 1 (1968) 149-159
[Weil 1954] Weil, A.: Remarques sur un mémoire d'Hermite. Arch. d. Math. 5(1954)197-202 (=[1954a], Collected Papers, Vol. II, Springer Verlag, New York 1980, ISBN 0-387-90330-5.)
| Textbooks |
Algorithms for Molecular Biology
Biologically feasible gene trees, reconciliation maps and informative triples
Marc Hellmuth1,2
Algorithms for Molecular Biology volume 12, Article number: 23 (2017) Cite this article
The history of gene families—which are equivalent to event-labeled gene trees—can be reconstructed from empirically estimated evolutionary event-relations containing pairs of orthologous, paralogous or xenologous genes. The question then arises as whether inferred event-labeled gene trees are biologically feasible, that is, if there is a possible true history that would explain a given gene tree. In practice, this problem is boiled down to finding a reconciliation map—also known as DTL-scenario—between the event-labeled gene trees and a (possibly unknown) species tree.
In this contribution, we first characterize whether there is a valid reconciliation map for binary event-labeled gene trees T that contain speciation, duplication and horizontal gene transfer events and some unknown species tree S in terms of "informative" triples that are displayed in T and provide information of the topology of S. These informative triples are used to infer the unknown species tree S for T. We obtain a similar result for non-binary gene trees. To this end, however, the reconciliation map needs to be further restricted. We provide a polynomial-time algorithm to decide whether there is a species tree for a given event-labeled gene tree, and in the positive case, to construct the species tree and the respective (restricted) reconciliation map. However, informative triples as well as DTL-scenarios have their limitations when they are used to explain the biological feasibility of gene trees. While reconciliation maps imply biological feasibility, we show that the converse is not true in general. Moreover, we show that informative triples neither provide enough information to characterize "relaxed" DTL-scenarios nor non-restricted reconciliation maps for non-binary biologically feasible gene trees.
The evolutionary history of genes is intimately linked with the history of the species in which they reside. Genes are passed from generation to generation to the offspring. Some of those genes are frequently duplicated, mutate, or get lost—a mechanism that also ensures that new species can evolve. In particular, genes that share a common origin (homologs) can be classified into the type of their "evolutionary event relationship", namely orthologs, paralogs and xenologs [1, 2]. Two homologous genes are orthologous if at their most recent point of origin the ancestral gene is transmitted to two daughter lineages; a speciation event happened. They are paralogous if the ancestor gene at their most recent point of origin was duplicated within a single ancestral genome; a duplication event happened. Horizontal gene transfer (HGT) refers to the transfer of genes between organisms in a manner other than traditional reproduction and across different species and yield so-called xenologs. In contrast to orthology and paralogy, the definition of xenology is less well established and by no means consistent in the biological literature. One definition stipulates that two genes are xenologs if their history since their common ancestor involves horizontal transfer of at least one of them [2, 3]. The mathematical framework for evolutionary event-relations relations in terms of symbolic ultrametrics, cographs and two-structures [4,5,6,7], on the other hand, naturally accommodates more than two types of events associated with the internal nodes of the gene tree. We follow the notion in [1, 6] and call two genes xenologous, whenever their least common ancestor was a HGT event.
The knowledge of evolutionary event relations such as orthology, paralogy or xenology is of fundamental importance in many fields of mathematical and computational biology, including the reconstruction of evolutionary relationships across species [8,9,10,11,12], as well as functional genomics and gene organization in species [13,14,15]. The type of event relationship is determined by the true history of the genes and species. However, events of the past cannot be observed directly and hence, must be inferred from the genomic data available today. Tree-reconciliation methods are widely studied in the literature [9, 16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] and provide one way to address this problem. Here, a gene tree is mapped into a species tree such that certain optimization criteria are fulfilled. This mapping, eventually, identifies inner vertices of the gene tree as a duplication, speciation or HGT. These methods usually require a gene and species tree as input. In most practical applications, however, neither the gene tree nor the species tree can be determined unambiguously. Intriguingly, there are methods to infer orthologs [14, 32,33,34,35,36,37,38,39,40] or to detect HGT [41,42,43,44,45] without the need to construct gene or species trees. Given empirical estimated event-relations one can infer the history of gene families which are equivalent to event-labeled gene trees [5, 6, 11, 46,47,48].
The crucial point is the following important result: For (tree-free estimated) event-relations there is an event-labeled gene tree that represents this estimate if and only if the respective event-relations are directed cographs [5, 6]. Usually, estimated event-relations violate this condition and must, therefore, be corrected [33, 36, 46,47,48,49,50,51]. Such corrected event-relations can, in most cases, be represented by an event-labeled gene tree. However, these trees can still be error-prone in the sense that there is no species tree on which they can evolve. The latter strongly depends on the applied correction method, the presence or absence of HGT events and, in particular, the theoretical model that is used to define that "a gene tree evolves along a species tree" (reconciliation map). The method ParaPhylo [11] already uses many of the latter mentioned ideas for the reconstruction of species trees and event-labeled gene trees without HGT-events. ParaPhylo is based on the knowledge of estimated orthology relations which are cleaned up to the closest cograph and, afterwards, corrected to obtain biologically feasible gene trees.
For an event-labeled gene tree to be biologically feasible there must be a putative "true" history that can explain the inferred gene tree. However, in practice it is not possible to observe the entire evolutionary history as, e.g. gene losses eradicate the entire information on parts of the history. Therefore, the problem of determining whether an event-labeled gene tree is biologically feasible is reduced to the problem of finding a valid reconciliation map, also known as DTL-scenario [29, 31], between the event-labeled gene trees and an arbitrary (possibly unknown) species tree. DTL-scenarios and its variants have been extensively studied [22, 29, 52,53,54] and have also applications in the context of the host-parasite cophylogeny problem [55,56,57,58,59,60,61,62].
In this contribution, we assume that we have a given event-labeled gene tree T and wish to answer the question: Is T biologically feasible and how much information about the unknown species tree S and the reconciliation between T and S is already contained in the gene tree T?
To this end, we first provide a mathematical definition of the term "biologically feasible" and two types of reconciliation maps: DTL-scenarios (as used in, e.g. [29, 31, 63]) and a restricted version (as used in, e.g. [12, 48]). Given the event-labeled gene-trees, it is possible to derive "informative" triples that are displayed in the gene tree T and provide information on the topology of the species tree S. In particular, we prove that consistency of informative triple sets characterize whether there are DTL-scenarios and restricted maps for binary and non-binary gene trees, respectively. The latter generalizes results established for binary gene trees that do not contain HGT-events by Hernandez et al. [10]. Furthermore, we provide a polynomial-time algorithm to decide whether there is a species tree for a given event-labeled gene tree and, in the positive case, to construct the species tree and a respective (restricted) reconciliation map.
In addition to the established results, we discuss limitations of reconciliation maps to explain biological feasibility of gene trees. While any (restricted) reconciliation map gives an idea of a putative true history that can explain the given gene tree, the converse is in general not true. We provide simple examples that show that not all biologically feasible gene trees can be explained by (restricted) DTL-scenarios. This immediately raises the question whether generalization of reconciliation maps might be used to explain biological feasibility. We shortly discuss a mild generalization, so-called "relaxed" reconciliation maps. However, as it turns out such general maps cannot be characterized by informative triples. We close this contribution with a couple of open problems.
A rooted tree \(T=(V,E)\) (on L) is an acyclic connected simple graph with leaf set \(L\subseteq V\), set of edges E, and set of interior vertices \(V^0=V\setminus L\) such that there is one distinguished vertex \(\rho _T \in V\), called the root of T.
A vertex \(v\in V\) is called a descendant of \(u\in V\), \(v \preceq _T u\), and u is an ancestor of v, \(u \succeq _T v\), if u lies on the path from \(\rho _T\) to v. As usual, we write \(v \prec _T u\) and \(u \succ _T v\) to mean \(v \preceq _T u\) and \(u\ne v\). If \(u \preceq _T v\) or \(v \preceq _T u\) then u and v are comparable and otherwise, incomparable. For \(x\in V\), we write \(L_T(x):=\{ y\in L \mid y\preceq x\}\) for the set of leaves in the subtree T(x) of T rooted in x.
It will be convenient to use a notation for edges e that implies which of the vertex in e is closer to the root. Thus, the notation for edges (u, v) of a tree is always chosen such that \(u\succ _T v\).
For our discussion below we need to extend the ancestor relation \(\preceq _T\) on V to the union of the edge and vertex sets of T. More precisely, for the edge \(e=(u,v)\in E\) we put \(x \prec _T e\) if and only if \(x\preceq _T v\) and \(e \prec _T x\) if and only if \(u\preceq _T x\). For edges \(e=(u,v)\) and \(f=(a,b)\) in T we put \(e\preceq _T f\) if and only if \(v \preceq _T b\). In the latter case, the edges e and f are called comparable.
For a non-empty subset of leaves \(A\subseteq L\), we define \({\text {lca}}_T(A)\), or the least common ancestor of A, to be the unique \(\preceq _T\)-minimal vertex of T that is an ancestor of every vertex in A. In case \(A=\{x,y \}\), we put \({\text {lca}}_T(x,y):={\text {lca}}_T(\{x,y\})\) and if \(A=\{x,y,z \}\), we put \({\text {lca}}_T(x,y,z):={\text {lca}}_T(\{x,y,z\})\). We will make frequent use that for two non-empty vertex sets A, B of a tree, it always holds that \({\text {lca}}(A\cup B) = {\text {lca}}({\text {lca}}(A),{\text {lca}}(B))\).
A phylogenetic tree T (on L) is a rooted tree \(T=(V,E)\) (on L) such that no interior vertex \(v\in V^0\) has degree two, except possibly the root \(\rho _T\). If L corresponds to a set of genes \(\mathbb {G}\) or species \(\mathbb {S}\), we call a phylogenetic tree on L gene tree and species tree, respectively. The restriction \(T_{|L'}\) of a phylogenetic tree T to \(L'\subseteq L\) is the rooted tree with leaf set \(L'\) obtained from T by first forming the minimal subtree of T with leaf set \(L'\) and then by suppressing all vertices of degree two with the exception of the root \(\rho _{T_{|L'}}\). By construction, \(V(T_{|L'})\subseteq V(T)\). If \(T=(V,E)\) is equipped with a map \(\ell :V\cup E \rightarrow M\), then the restriction of \(\ell\) to \(T_{|L'}=(V',E')\) is the map \(\ell _{|L'}:V'\cup E' \rightarrow M\) that satisfies
$$\begin{aligned} \ell _{|L'}(\alpha ) =\left\{ \begin{array}{ll} \ell (v) &{}\quad \text{ if } \alpha = v\in V',\ \\ \ell (e) &{}\quad \text{ if } \alpha =(u,b)\in E', e=(u,v)\in E \\ &{}\quad \text {and either } v=b \text { or } v \text { is suppressed} \\ &{} \quad \text {in } T_{|L'} \text { and lies on the path from } \\ &{} \quad u \text { to } { b} \text { in } T.\ \end{array}\right. \end{aligned}$$
In other words, \(\ell _{|L'}\) keeps the vertex-labels of all non-suppressed vertices and assigns the edge-label of the edge (u, v) in T to the edge (u, v) in \(T_{|L'}\), if \(v=b\) and otherwise, to the edge (u, b) in \(T_{|L'}\), where b is the first non-suppressed vertex that lies on the unique path from v to b in T.
Rooted triples are phylogenetic trees on three leaves with precisely two interior vertices. They constitute an important concept in the context of supertree reconstruction [64,65,66] and will also play a major role here. A rooted tree T on L displays a triple \(\mathsf {(xy|z)}\) if, \(x,y,z\in L\) and the path from x to y does not intersect the path from z to the root \(\rho _T\) and thus, having \({\text {lca}}_T(x,y)\prec _T {\text {lca}}_T(x,y,z)\). We denote by \(\mathcal {R}(T)\) the set of all triples that are displayed by the rooted tree T.
A set R of triples is consistent if there is a rooted tree T on \(L_R= \cup _{r\in R} L_r(\rho _r)\) such that \(R\subseteq \mathcal {R}(T)\) and thus, T displays each triple in R. Not all sets of triples are consistent of course. Nevertheless, given a triple set R there is a polynomial-time algorithm, referred to in [64, 67] as BUILD, that either constructs a phylogenetic tree T that displays R or that recognizes that R is not consistent [68]. The runtime of BUILD is \(\mathcal {O}(|L_R||R|)\) [64]. Further practical implementations and improvements have been discussed in [69,70,71,72].
We will consider rooted trees \(T=(V,E)\) from which particular edges are removed. Let \(\mathcal {E}\subseteq E\) and consider the forest \(T_{\mathcal {\overline{E}}}:=(V,E\setminus \mathcal {E})\). We can preserve the order \(\preceq _T\) for all vertices within one connected component of \(T_{\mathcal {\overline{E}}}\) and define \(\preceq _{T_{\mathcal {\overline{E}}}}\) as follows: \(x\preceq _{T_{\mathcal {\overline{E}}}}y\) iff \(x\preceq _{T}y\) and x, y are in same connected component of \(T_{\mathcal {\overline{E}}}\). Since each connected component \(T'\) of \(T_{\mathcal {\overline{E}}}\) is a tree, the ordering \(\preceq _{T_{\mathcal {\overline{E}}}}\) also implies a root \(\rho _{T'}\) for each \(T'\), that is, \(x\preceq _{T_{\mathcal {\overline{E}}}} \rho _{T'}\) for all \(x\in V(T')\). If \(L(T_{\mathcal {\overline{E}}})\) is the leaf set of \(T_{\mathcal {\overline{E}}}\), we define \(L_{T_{\mathcal {\overline{E}}}}(x) = \{y\in L(T_{\mathcal {\overline{E}}}) \mid y\prec _{T_{\mathcal {\overline{E}}}} x\}\) as the set of leaves in \(T_{\mathcal {\overline{E}}}\) that are reachable from x. Hence, all \(y\in L_{T_{\mathcal {\overline{E}}}}(x)\) must be contained in the same connected component of \(T_{\mathcal {\overline{E}}}\). We say that the forest \(T_{\mathcal {\overline{E}}}\) displays a triple r, if r is displayed by one of its connected components. Moreover, \(\mathcal {R}(T_{\mathcal {\overline{E}}})\) denotes the set of all triples that are displayed by the forest \(T_{\mathcal {\overline{E}}}\).
Biologically feasible and observable gene trees
A gene tree arises through a series of events (speciation, duplication, HGT, and gene loss) along a species tree. In a "true history" the gene tree \({\widehat{T}} = (V,E)\) on a set of genes \({\widehat{\mathbb {G}}}\) is equipped with an event-labeling map \({\widehat{t}}:V\cup E\rightarrow {\widehat{I}}\cup \{0,1\}\) with \({\widehat{I}}=\{\mathfrak {s},\mathfrak {d},\mathfrak {t},\odot ,\varvec{\mathsf {x}}\}\) that assigns to each vertex v of \({\widehat{T}}\) a value \({\widehat{t}}(v)\in {\widehat{I}}\) indicating whether v is a speciation event (\(\mathfrak {s}\)), duplication event (\(\mathfrak {d}\)), HGT event (\(\mathfrak {t}\)), extant leaf (\(\odot\)) or a loss event (\(\varvec{\mathsf {x}}\)). Note, in the figures we omitted the symbol \(\odot\) and used \(\bullet , \square\) and \(\triangle\) for \(\mathfrak {s}, \mathfrak {d}\) and \(\mathfrak {t}\), respectively.
Left an example of a "true" history of a gene tree that evolves along the (tube-like) species tree. The set of extant genes \(\mathbb {G}\) comprises a,a′,b,b′,c,c′,c″ and e and \(\sigma\) maps each gene in \(\mathbb {G}\) to the species (capitals below the genes) \(A,B,C,E\in \sigma (\mathbb {G})\). For simplicity all speciation events followed by a loss along the path from v to \(a'\) in T are omitted. Left the observable gene tree \((T;t,\sigma )\) is shown. Since there is a true scenario which explains \((T;t,\sigma )\), the gene tree is biologically feasible. In particular, \((T;t,\sigma )\) satisfies (O1), (O2) and (O3)
Horizontal gene transfer is intrinsically a directional event, i.e., there is a clear distinction between the horizontally transferred "copy" and the "original" that continues to be vertically transferred. To this end, the edges in the gene tree are annotated by associating a label to the edge that points from the horizontal transfer event to the next event in the history of the copy. To be more precise, to each edge e a value \({\widehat{t}}(e)\in \{0,1\}\) is assigned that indicates whether e is a transfer edge (1) or not (0). Hence, \(e=(x,y)\) and \({\widehat{t}}(e) =1\) iff \({\widehat{t}}(x)=\mathfrak {t}\) and the genetic material is transferred from the species containing x to a species containing y. We remark that the restriction of t to the vertex set V was introduced as "symbolic dating map" in [4] and that there is a close relationship to so-called cographs [5, 73, 74]. Let \(\mathbb {G}\subseteq {\widehat{\mathbb {G}}}\) be the set of all extant genes in \({\widehat{T}}\), i.e., \(\mathbb {G}\) contains all genes v of \({\widehat{\mathbb {G}}}\) with \({\widehat{t}}(v)\ne \varvec{\mathsf {x}}\). Hence, there is a map \(\sigma :\mathbb {G}\rightarrow \mathbb {S}\) that assigns to each extant gene the extant species in which it resides.
We assume that the gene tree and its event labels are inferred from (sequence) data, i.e., T is restricted to those labeled trees that can be constructed at least in principle from observable data. Gene losses eradicate the entire information on parts of the history and thus, cannot directly be observed from extant sequences. Hence, in our setting the (observable) gene tree T is the restriction \({\widehat{T}}_{|\mathbb {G}}\) to the set of extant genes equipped with the event-label \(t={\widehat{t}}_{|\mathbb {G}}\), see Fig. 1. Since all leaves of T are extant genes in \(\mathbb {G}\) we don't need to specially label the leaves in \(\mathbb {G}\), and thus simplify the event-labeling map \(t:V^0\cup E\rightarrow I\cup \{0,1\}\) by assigning only to the interior vertex an event in \(I=\{\mathfrak {s},\mathfrak {d},\mathfrak {t}\}\). We assume here that all non-transfer edges transmit the genetic material vertically, that is, from an ancestral species to its descendants.
Definition 1
We write \((T;t,\sigma )\) for the tree \(T=(V,E)\) with event-labeling t and corresponding map \(\sigma\). The set \(\mathcal {E}= \{e\in E\mid t(e)=1\}\) will always denote the set of transfer edges in \((T;t,\sigma )\).
Additionally, we consider gene trees \((T=(V,E);t,\sigma )\) from which the transfer edges have been removed, resulting in the forest \(T_{\mathcal {\overline{E}}}= (V, E\setminus \mathcal {E})\) in which we preserve the event-labeling t of all vertices.
We call a gene tree \((T;t,\sigma )\) on \(\mathbb {G}\) biologically feasible, if there is a true scenario such that \(T = {\widehat{T}}_{|\mathbb {G}}\) and \(t={\widehat{t}}_{|\mathbb {G}}\), that is, there is a true history that can explain \((T;t,\sigma )\). By way of example, the gene tree in Fig. 1 (right) is biologically feasibly. However, so-far it is unknown whether there are gene trees \((T;t,\sigma )\) that are not biologically feasible. Answering the latter might be a hard task, as many HGT or duplication vertices followed by losses can be inserted into T that may result in a putative true history that explains the event-labeled gene tree.
Following Nøjgaard et al. [63], we additionally restrict the set of observable gene trees \((T;t,\sigma )\) to those gene trees that satisfy the following observability axioms:
(O1)
Every internal vertex v has degree at least three, except possibly the root which has degree at least two.
Every HGT node has at least one transfer edge, \(t(e)=1\), and at least one non-transfer edge, \(t(e)=0\).
(a) If \(x\in V\) is a speciation vertex, then there are distinct children v, w of x in T with \(\sigma _{T_{\mathcal {\overline{E}}}}(v)\cap \sigma _{T_{\mathcal {\overline{E}}}}(w) = \emptyset\). (b) If \((x,y) \in \mathcal {E}\), then \(\sigma _{T_{\mathcal {\overline{E}}}}(x)\cap \sigma _{T_{\mathcal {\overline{E}}}}(y) = \emptyset\).
Condition (O1) is justified by the restriction \(T={\widehat{T}}_{|\mathbb {G}}\) of the true binary gene tree \({\widehat{T}}\) to the set of extant genes \(\mathbb {G}\), since \(T={\widehat{T}}_{|\mathbb {G}}\) is always a phylogenetic tree. In particular, (O1) ensures that every event leaves a historical trace in the sense that there are at least two children that have survived in at least two of its subtrees. Condition (O2) ensures that for an HGT event a historical trace remains of both the transferred and the non-transferred copy.
Condition (O3.a) is a consequence of (O1), (O2) and a stronger Condition (O3.a') claimed in [63]: If x is a speciation vertex, then there are at least two distinct children v, w of x such that the species V and W that contain v and w, resp., are incomparable in S. Note, a speciation vertex x cannot be observed from data if it does not "separate" lineages, that is, there are two leaf descendants of distinct children of x that are in distinct species. Condition (O3.a') is even weaker and ensures that any "observable" speciation vertex x separates at least locally two lineages. As a result of (O3.a') one can obtain (O3.a) [63]. Intuitively, (O3.a) is satisfied since within a connected component of \(T_{\mathcal {\overline{E}}}\) no genetic material is exchanged between non-comparable nodes. Thus, a gene separated in a speciation event necessarily ends up in distinct species in the absence of the transfer edges.
Condition (O3.b) is a consequence of (O1), (O2) and a stronger Condition (O3.b') claimed in [63]: If (v, w) is a transfer edge in T, then \(t(v)=\mathfrak {t}\) and the species V and W that contain v and w, resp., are incomparable in S. Note, if \((v,w)\in \mathcal {E}\) then v signifies the transfer event itself but w refers to the next (visible) event in the gene tree T. In a "true history" v is contained in a species V that transmits its genetic material (maybe along a path of transfers) to a contemporary species Z that is an ancestor of the species W containing w. In order to have evidence that this transfer happened, Condition (O3.b') is used and as a result one obtains (O3.b). The intuition behind (O3.b) is as follows: observe that \({T_{\mathcal {\overline{E}}}}(x)\) and \({T_{\mathcal {\overline{E}}}}(y)\) are subtrees of distinct connected components of \(T_{\mathcal {\overline{E}}}\) whenever \((x,y) \in \mathcal {E}\). Since HGT amounts to the transfer of genetic material across distinct species, the genes x and y are in distinct species, cf. (O3.b). However, since \(T_{\mathcal {\overline{E}}}\) does not contain transfer edges and thus, there is no genetic material transferred across distinct species between distinct connected components in \(T_{\mathcal {\overline{E}}}\). We refer to [63] for further details.
In what follows, we only consider gene trees \((T;t,\sigma )\) that satisfy (O1), (O2) and (O3).
We simplify the notation a bit and write \(\sigma _{T_{\mathcal {\overline{E}}}}(u):=\sigma (L_{T_{\mathcal {\overline{E}}}}(u))\).
Based on Axiom (O2) the following results was established in [63].
Lemma 3.1
Let \((T;t,\sigma )\) be an event-labeled gene tree. Let \(\mathcal {T}_1, \dots , \mathcal {T}_k\) be the connected components of \(T_{\mathcal {\overline{E}}}\) with roots \(\rho _1, \dots , \rho _k\) , respectively. Then, \(\{L_{T_{\mathcal {\overline{E}}}}(\rho _1), \dots , L_{T_{\mathcal {\overline{E}}}}(\rho _k)\}\) forms a partition of \(\mathbb {G}\).
Lemma 3.1 particularly implies that \(\sigma _{T_{\mathcal {\overline{E}}}}(x) \ne \emptyset\) for all \(x\in V(T)\). Note, \(T_{\mathcal {\overline{E}}}\) might contain interior vertices (distinct from the root) that have degree two. Nevertheless, for each \(x\preceq _{T_{\mathcal {\overline{E}}}} y\) in \(T_{\mathcal {\overline{E}}}\) we have \(x\preceq _T y\) in T. Hence, partial information (that in particular is "undisturbed" by transfer edges) on the partial ordering of the vertices in T can be inferred from \(T_{\mathcal {\overline{E}}}\).
Reconciliation map
Before we define a reconciliation map that "embeds" a given gene tree into a given species tree we need a slight modification of the species tree. In order to account for duplication events that occurred before the first speciation event, we need to add an extra vertex and an extra edge "above" the last common ancestor of all species: hence, we add an additional vertex to W (that is now the new root \(\rho _S\) of S) and the additional edge \((\rho _S,{\text {lca}}_S(\mathbb {S}))\in F\). Note that strictly speaking S is not a phylogenetic tree anymore. In case there is no danger of confusion, we will from now on refer to a phylogenetic tree on \(\mathbb {S}\) with this extra edge and vertex added as a species tree on \(\mathbb {S}\).
(DTL-scenario) Suppose that \(\mathbb {S}\) is a set of species, \(S=(W,F)\) is a phylogenetic tree on \(\mathbb {S}\), \(T=(V,E)\) is a gene tree with leaf set \(\mathbb {G}\) and that \(\sigma :\mathbb {G}\rightarrow \mathbb {S}\) and \(t:V^0\rightarrow \{\mathfrak {s},\mathfrak {d},\mathfrak {t}\} \cup \{0,1\}\) are the maps described above. Then we say that S is a species tree for \((T;t,\sigma )\) if there is a map \(\mu :V\rightarrow W\cup F\) such that, for all \(x\in V\):
Leaf constraint. If \(x\in \mathbb {G}\) then \(\mu (x)=\sigma (x)\).
Event constraint.
If \(t(x)=\mathfrak {s}\), then \(\mu (x) = {\text {lca}}_S(\sigma _{T_{\mathcal {\overline{E}}}}(x))\).
If \(t(x) \in \{\mathfrak {d}, \mathfrak {t}\}\), then \(\mu (x)\in F\).
If \(t(x)=\mathfrak {t}\) and \((x,y)\in \mathcal {E}\), then \(\mu (x)\) and \(\mu (y)\) are incomparable in S.
Ancestor constraint. Let \(x,y\in V\) with \(x\prec _{T_{\mathcal {\overline{E}}}} y\). Note, the latter implies that the path connecting x and y in T does not contain transfer edges. We distinguish two cases:
If \(t(x),t(y)\in \{\mathfrak {d}, \mathfrak {t}\}\), then \(\mu (x)\preceq _S \mu (y)\),
otherwise, i.e., at least one of t(x) and t(y) is a speciation \(\mathfrak {s}\), \(\mu (x)\prec _S\mu (y)\).
We call \(\mu\) the reconciliation map from \((T;t,\sigma )\) to S.
Definition 2 is a natural generalization of the map defined in [10], that is, in the absence of horizontal gene transfer, Condition (M2.iii) vanishes and thus, the proposed reconciliation map precisely coincides with the one given in [10]. In case that the event-labeling of T is unknown, but a species tree S is given, the authors in [31, 54] gave an axiom set, called DTL-scenario, to reconcile T with S. This reconciliation is then used to infer the event-labeling t of T. The "usual" DTL axioms explicitly refer to binary, fully resolved gene and species trees. We therefore use a different axiom set that is, nevertheless, equivalent to DTL-scenarios in case the considered gene trees are binary [63].
Condition (M1) ensures that each leaf of T, i.e., an extant gene in \(\mathbb {G}\), is mapped to the species in which it resides. Condition (M2.i) and (M2.ii) ensure that each vertex of T is either mapped to a vertex or an edge in S such that a vertex of T is mapped to an interior vertex of S if and only if it is a speciation vertex. We will discuss (M2.i) in further detail below. Condition (M2.iii) maps the vertices of a transfer edge in a way that they are incomparable in the species tree and is used to satisfy axiom (O3). Condition (M3) refers only to the connected components of \(T_{\mathcal {\overline{E}}}\) and is used to preserve the ancestor order \(\preceq _T\) of T along the paths that do not contain transfer edges is preserved.
It needs to be discussed, why one should map a speciation vertex x to \({\text {lca}}_S(\sigma _{T_{\mathcal {\overline{E}}}}(x))\) as required in (M2.i). The next lemma shows, that one can put \(\mu (x) = {\text {lca}}_S(\sigma _{T_{\mathcal {\overline{E}}}}(x))\).
Nøjgaard et al. [63] Let \(\mu\) be a reconciliation map from \((T;t,\sigma )\) to S that satisfies (M1) and (M3), then \(\mu (u)\succeq _S {\text {lca}}_S(\sigma _{T_{\mathcal {\overline{E}}}}(u))\) for any \(u\in V(T)\).
Condition (M2.i) implies in particular the weaker property "(M2.i') if \(t(x)=\mathfrak {s}\) then \(\mu (x)\in W\)". In the light of Lemma 4.1, \(\mu (x)={\text {lca}}_S(\sigma _{T_{\mathcal {\overline{E}}}}(x))\) is the lowest possible choice for the image of a speciation vertex. Note that there are possibly exponentially many reconciliation maps, whenever \(\mu (x)\succ _S{\text {lca}}_S(\sigma _{T_{\mathcal {\overline{E}}}}(x))\) is allowed for speciation vertices x. First, we we restrict our attention to those maps that satisfy (M2.i) only. In particular, as we shall see in "Binary gene trees" section, there is a neat characterization of maps that satisfy (M2.i) that does, however, not work for maps with "relaxed" (M2.i), as discussed in "Limitations of informative triples and reconciliation maps" section.
Moreover, we have the following result, which is a mild generalization of [63].
Let \(\mu\) be a reconciliation map from a gene tree \((T;t,\sigma )\) to S.
If \(v,w\in V(T)\) are in the same connected component of \(T_{\mathcal {\overline{E}}}\), then \(\mu ({\text {lca}}_{T_{\mathcal {\overline{E}}}}(v,w)) \succeq _S {\text {lca}}_S(\mu (v),\mu (w))\).
If \((T;t,\sigma )\) is a binary gene tree and x a speciation vertex with children v, w in T, then then \(\mu (v)\) and \(\mu (w)\) are incomparable in S.
Let \(v,w\in V(T)\) be in the same connected component of \(T_{\mathcal {\overline{E}}}\). Assume that v and w are comparable in \(T_{\mathcal {\overline{E}}}\) and that w.l.o.g. \(v\succ _{T_{\mathcal {\overline{E}}}} w\). Condition (M3) implies that \(\mu (v)\succeq _S\mu (w)\). Hence, \(v = {\text {lca}}_{T_{\mathcal {\overline{E}}}}(v,w)\) and \(\mu (v) = {\text {lca}}_S(\mu (v),\mu (w))\) and we are done.
Now assume that v and w are incomparable in \(T_{\mathcal {\overline{E}}}\). Consider the unique path P connecting w with v in \(T_{\mathcal {\overline{E}}}\). This path P is uniquely subdivided into a path \(P'\) and a path \(P''\) from \({\text {lca}}_{T_{\mathcal {\overline{E}}}}(v,w)\) to v and w, respectively. Condition (M3) implies that the images of the vertices of \(P'\) and \(P''\) under \(\mu\), resp., are ordered in S with regards to \(\preceq _S\) and hence, are contained in the intervals \(Q'\) and \(Q''\) that connect \(\mu ({\text {lca}}_{T_{\mathcal {\overline{E}}}}(v,w))\) with \(\mu (v)\) and \(\mu (w)\), respectively. In particular, \(\mu ({\text {lca}}_{T_{\mathcal {\overline{E}}}}(v,w))\) is the largest element (w.r.t. \(\preceq _S\)) in the union of \(Q'\cup Q''\) which contains the unique path from \(\mu (v)\) to \(\mu (w)\) and hence also \({\text {lca}}_S(\mu (v),\mu (w))\).
Item 2 was already proven in [63]. \(\square\)
Assume now that there is a reconciliation map \(\mu\) from \((T;t,\sigma )\) to S. From a biological point of view, however, it is necessary to reconcile a gene tree with a species tree such that genes do not "travel through time", a see Fig. 4 for an example.
(Time Map) The map \(\tau _T: V(T) \rightarrow \mathbb {R}\) is a time map for the rooted tree T if \(x\prec _T y\) implies \(\tau _T(x)>\tau _T(y)\) for all \(x,y\in V(T)\).
A reconciliation map \(\mu\) from \((T;t,\sigma )\) to S is time-consistent if there are time maps \(\tau _T\) for T and \(\tau _S\) for S for all \(u\in V(T)\) satisfying the following conditions:
(T1)
If \(t(u) \in \{\mathfrak {s}, \odot \}\), then \(\tau _T(u) = \tau _S(\mu (u))\).
If \(t(u)\in \{\mathfrak {d},\mathfrak {t}\}\) and, thus \(\mu (u)=(x,y)\in E(S)\), then \(\tau _S(y)>\tau _T(u)>\tau _S(x)\).
Condition (T1) is used to identify the time-points of speciation vertices and leaves u in the gene tree with the time-points of their respective images \(\mu (u)\) in the species trees. Moreover, duplication or HGT vertices u are mapped to edges \(\mu (u)=(x,y)\) in S and the time point of u must thus lie between the time points of x and y which is ensured by Condition (T2). Nøjgaard et al. [63] designed an \(\mathcal {O}(|V(T)|\log (|V(S)|))\)-time algorithm to check whether a given reconciliation map \(\mu\) is time-consistent, and an algorithm with the same time complexity for the construction of a time-consistent reconciliation map, provided one exists. Clearly, a necessary condition for the existence of time-consistent reconciliation maps from \((T;t,\sigma )\) to S is the existence of some reconciliation map from \((T;t,\sigma )\) to S. In the next section, we first characterize the existence of reconciliation maps and discuss open time-consistency problems.
From gene trees to species trees
Left an example of a "true" history of a gene tree that evolves along the (tube-like) species tree (taken from [11]). The set of extant genes \(\mathbb {G}\) comprises \(a,b,c_1,c_2\) and d and \(\sigma\) maps each gene in \(\mathbb {G}\) to the species (capitals below the genes) \(A,B,C,D\in \mathbb {S}\). Upper right the observable gene tree \((T;t,\sigma )\) is shown. To derive \(\mathcal {S}(T;t,\sigma )\) we cannot use the triples \(\mathcal {R}_0(T)\), that is, we need to remove the transfer edges. To be more precise, if we would consider \(\mathcal {R}_0(T)\) we obtain the triples \(\mathsf {(ac_1|d)}\) and \(\mathsf {(c_2d|a)}\) which leads to the two contradicting species triples \(\mathsf {(AC|D)}\) and \(\mathsf {(CD|A)}\). Thus, we restrict \(\mathcal {R}_0\) to \(T_{\mathcal {\overline{E}}}\) and obtain \(\mathcal {R}_0(T_{\mathcal {\overline{E}}}) = \{\mathsf {(ac_1|d)}\}\). However, this triple alone would not provide enough information to obtain a species tree such that a valid reconciliation map \(\mu\) can be constructed. Hence, we take \(\mathcal {R}_1(T_{\mathcal {\overline{E}}})=\{\mathsf {(bc_2|d)}\}\) into account and obtain \(\mathcal {S}(T;t,\sigma ) = \{\mathsf {(AC|D)},\mathsf {(BC|D)}\}\). Lower right a least resolved species tree S (obtained with BUILD) that displays all triples in \(\mathcal {S}(T;t,\sigma )\) together with the reconciled gene tree \((T;t,\sigma )\) is shown. Although S does not display the triple \(\mathsf {(AB|C)}\) as in the true history, this tree S does not pretend a higher resolution than actually supported by \((T;t,\sigma )\). Clearly, as more gene trees (gene families) are available as more information about the resolution of the species tree can be provided
Since a gene tree T is uniquely determined by its induced triple set \(\mathcal {R}(T)\), it is reasonable to expect that a lot of information on the species tree(s) for \((T;t, \sigma )\) is contained in the images of the triples in \(\mathcal {R}(T)\), or more precisely their leaves under \(\sigma\). However, not all triples in \(\mathcal {R}(T)\) are informative, see Fig. 2 for an illustrative example. In the absence of HGT, it has already been shown by Hernandez-Rosales et al. [10] that the informative triples \(r\in \mathcal {R}(T)\) are precisely those that are rooted at a speciation event and where the genes in r reside in three distinct species. However, in the presence of HGT we need to further subdivide the informative triples as follows.
Let \((T;t,\sigma )\) be a given event-labeled gene tree with respective set of transfer-edges \(\mathcal {E}= \{e_1,\dots ,e_h\}\) and \(T_{\mathcal {\overline{E}}}\) as defined above. We define
$$\begin{aligned} \mathcal {R}_{\sigma }(T_{\mathcal {\overline{E}}}) =& \{\mathsf {(ab|c)} \in \mathcal {R}(T_{\mathcal {\overline{E}}}) :\sigma (a),\sigma (b),\sigma (c)&\quad \\&\text {are pairwise distinct} \} \end{aligned}$$
as the subset of all triples displayed in \(T_{\mathcal {\overline{E}}}\) such that the leaves are from pairwise distinct species.
$$\begin{aligned} \mathcal {R}_0(T_{\mathcal {\overline{E}}}) :=\{\mathsf {(ab|c)} \in \mathcal {R}_{\sigma }(T_{\mathcal {\overline{E}}}) :t({\text {lca}}_{T_{\mathcal {\overline{E}}}}(a,b,c)) = \mathfrak {s}\} \end{aligned}$$
be the set of triples in \(\mathcal {R}_{\sigma }(T_{\mathcal {\overline{E}}})\) that are rooted at a speciation event.
For each \(e_i=(x,y) \in \mathcal {E}\) define
$$\begin{aligned} \begin{array}{ll} \mathcal {R}_i(T_{\mathcal {\overline{E}}}) :=\{ \mathsf {(ab|c)} :&{} \sigma (a),\sigma (b),\sigma (c) \text { are pairwise distinct} \\ &{}\text {and either } a,b\in L_{T_{\mathcal {\overline{E}}}}(x), c\in L_{T_{\mathcal {\overline{E}}}}(y) \\ &{}\text {or } c\in L_{T_{\mathcal {\overline{E}}}}(x), a,b\in L_{T_{\mathcal {\overline{E}}}}(y) \}. \end{array} \end{aligned}$$
Hence, \(\mathcal {R}_i(T_{\mathcal {\overline{E}}})\) contains a triple \(\mathsf {(ab|c)}\) for every \(a,b\in L_{T_{\mathcal {\overline{E}}}}(x), c\in L_{T_{\mathcal {\overline{E}}}}(y)\) that reside in pairwise distinct species. Analogously, for any \(a,b\in L_{T_{\mathcal {\overline{E}}}}(y), c\in L_{T_{\mathcal {\overline{E}}}}(x)\) there is a triple \(\mathsf {(ab|c)}\in \mathcal {R}_i(T_{\mathcal {\overline{E}}})\), if \(\sigma (a),\sigma (b),\sigma (c)\) are pairwise distinct.
The informative triples of T are comprised in the set \(\mathcal {R}(T;t,\sigma ) = \cup _{i=0}^h \mathcal {R}_i(T_{\mathcal {\overline{E}}})\).
Finally, we define the informative species triple set
$$\begin{aligned} \mathcal {S}(T;t,\sigma ):=\{\mathsf {(\sigma (a)\sigma (b)|\sigma (c))} :\mathsf {(ab|c)} \in \mathcal {R}(T;t,\sigma ) \} \end{aligned}$$
that can be inferred from the informative triples of \((T;t,\sigma )\).
Binary gene trees
In this section, we will be concerned only with binary, i.e., "fully resolved" gene trees, if not stated differently. This is justified by the fact that a speciation or duplication event instantaneously generates exactly two offspring. However, we will allow also non-binary species tree to model incomplete knowledge of the exact species phylogeny. Non-binary gene trees are discussed in "Non-binary gene trees" section.
Hernandez et al. [10] established the following characterization for the HGT-free case.
Theorem 5.1
For a given gene tree \((T;t, \sigma )\) on \(\mathbb {G}\) that does not contain HGT and \(\mathfrak {S}:=\{\mathsf {(\sigma (a)\sigma (b)|\sigma (c))} :\mathsf {(ab|c)} \in \mathcal {R}_0(T)\}\), the following statement is satisfied:
There is a species tree on \(\mathbb {S}= \sigma (\mathbb {G})\) for \((T;t, \sigma )\) if and only if the triple set \(\mathfrak {S}\) is consistent.
We emphasize that the results established in [10] are only valid for binary gene trees, although this was not explicitly stated. For an example that shows that Theorem 5.1 is not always satisfied for non-binary gene trees see Fig. 3. Lafond and El-Mabrouk [12, 48] established a similar result as in Theorem 5.1 by using only species triples that can be obtained directly from a given orthology/paralogy-relation. However, they require a stronger version of axiom (O3.a), that is, the images of all children of a speciation vertex must be pairwisely incomparable in the species tree. We, too, will use this restriction in "Non-binary gene trees" section.
In what follows, we generalize the latter result and show that consistency of \(\mathcal {S}(T;t,\sigma )\) characterizes whether there is a species tree S for \((T;t,\sigma )\) even if \((T;t,\sigma )\) contains HGT.
If \(\mu\) is a reconciliation map from a gene tree \((T;t,\sigma )\) to a species tree S and \(\mathsf {(ab|c)} \in \mathcal {R}(T;t,\sigma )\) , then \(\mathsf {(\sigma (a)\sigma (b)|\sigma (c))}\) is displayed in S.
Recall that \(\mathbb {G}\) is the leaf set of \(T=(V,E)\) and, by Lemma 3.1, of \(T_{\mathcal {\overline{E}}}\). Let \(\{a,b,c\} \in \left( {\begin{array}{c}\mathbb {G}\\ 3\end{array}}\right)\) and assume w.l.o.g. \(\mathsf {(ab|c)} \in \mathcal {R}(T;t,\sigma )\).
First assume that \(\mathsf {(ab|c)} \in \mathcal {R}_0\), that is \(\mathsf {(ab|c)}\) is displayed in \(T_{\mathcal {\overline{E}}}\) and \(t({\text {lca}}_{T_{\mathcal {\overline{E}}}}(a,b,c)) = \mathfrak {s}\). For simplicity set \(u={\text {lca}}_{T_{\mathcal {\overline{E}}}}(a,b,c)\) and let x, y be its children in \(T_{\mathcal {\overline{E}}}\). Since \(\mathsf {(ab|c)} \in \mathcal {R}_0\), we can assume that w.l.o.g. \(a,b\in L_{T_{\mathcal {\overline{E}}}}(x)\) and \(c\in L_{T_{\mathcal {\overline{E}}}}(y)\). Hence, \(x\succeq _{T_{\mathcal {\overline{E}}}} {\text {lca}}_{T_{\mathcal {\overline{E}}}}(a,b)\) and \(y\succeq _{T_{\mathcal {\overline{E}}}} c\). Condition (M3) implies that \(\mu (y)\succeq _S \mu (c) = \sigma (c)\). Moreover, Condition (M3) and Lemma 4.2(1) imply that \(\mu (x)\succeq _S \mu ({\text {lca}}_{T_{\mathcal {\overline{E}}}}(a,b)) \succeq _S {\text {lca}}_S(\mu (a),\mu (b)) = {\text {lca}}_S(\sigma (a),\sigma (b))\). Since \(t(u)=\mathfrak {s}\), we can apply Lemma 4.2(2) and conclude that \(\mu (x)\) and \(\mu (y)\) are incomparable in S. Hence, \(\sigma (c)\) and \({\text {lca}}_S(\sigma (a),\sigma (b))\) are incomparable. Thus, the triple \(\mathsf {(\sigma (a)\sigma (b)|\sigma (c))}\) must be displayed in S.
Now assume that \(\mathsf {(ab|c)} \in \mathcal {R}_i\) for some transfer edge \(e_i = (x,y)\in \mathcal {E}\). For \(e_i = (x,y)\) we either have \(a,b\in L_{T_{\mathcal {\overline{E}}}}(x)\) and \(c\in L_{T_{\mathcal {\overline{E}}}}(y)\) or \(c\in L_{T_{\mathcal {\overline{E}}}}(x)\) and \(a,b\in L_{T_{\mathcal {\overline{E}}}}(y)\). W.l.o.g. let \(a,b\in L_{T_{\mathcal {\overline{E}}}}(x)\) and \(c\in L_{T_{\mathcal {\overline{E}}}}(y)\). Thus, \(x\succeq _{T_{\mathcal {\overline{E}}}} {\text {lca}}_{T_{\mathcal {\overline{E}}}}(a,b)\) and \(y\succeq _{T_{\mathcal {\overline{E}}}} c\). Condition (M3) implies that \(\mu (y)\succeq _S \mu (c) = \sigma (c)\). Moreover, Condition (M3) and Lemma 4.2(1) imply that \(\mu (x)\succeq _S \mu ({\text {lca}}_{T_{\mathcal {\overline{E}}}}(a,b)) \succeq _S {\text {lca}}_S(\mu (a),\mu (b)) = {\text {lca}}_S(\sigma (a),\sigma (b))\). Since \(t(x)=\mathfrak {t}\), we can apply (M2.iii) and conclude that \(\mu (x)\) and \(\mu (y)\) are incomparable in S. Hence, \(\sigma (c)\) and \({\text {lca}}_S(\sigma (a),\sigma (b))\) are incomparable. Thus, the triple \(\mathsf {(\sigma (a)\sigma (b)|\sigma (c))}\) must be displayed in S. \(\square\)
Let \(S=(W,F)\) be a species tree on \(\mathbb {S}\) . Then there is a reconciliation map \(\mu\) from a gene tree \((T;t,\sigma )\) to S whenever S displays all triples in \(\mathcal {S}(T;t,\sigma )\).
Recall that \(\mathbb {G}\) is the leaf set of \(T=(V,E)\) and, by Lemma 3.1, of \(T_{\mathcal {\overline{E}}}\). In what follows, we write \(\mathcal {L}(u)\) instead of the more complicated writing \(L_{T_{\mathcal {\overline{E}}}}(u)\) and, for consistency and simplicity, we also often write \(\sigma (\mathcal {L}(u))\) instead of \(\sigma _{T_{\mathcal {\overline{E}}}}(u)\). Put \(S=(W,F)\) and \(\mathcal {S} = \mathcal {S}(T;t,\sigma )\). We first consider the subset \(U=\{x\in V \mid x\in \mathbb {G}\text { or } t(x) = \mathfrak {s}\}\}\) of V comprising the leaves and speciation vertices of T.
In what follows we will explicitly construct \(\mu : V \rightarrow W\cup F\) and verify that \(\mu\) satisfies Conditions (M1), (M2) and (M3). To this end, we first set for all \(x\in U\):
\(\mu (x) = \sigma (x)\), if \(x\in \mathbb {G}\),
\(\mu (x)= {\text {lca}}_S(\sigma (\mathcal {L}(x)))\), if \(t(x)=\mathfrak {s}\).
Conditions (S1) and (M1), as well as (S2) and (M2.i) are equivalent.
For later reference, we show that \({\text {lca}}_S(\sigma (\mathcal {L}(x))) \in W^0 = W\setminus \mathbb {S}\) and that there are two leaves \(a,b\in \mathcal {L}(x)\) such that \(\sigma (a) \ne \sigma (b)\), whenever \(t(x)=\mathfrak {s}\). By Condition (O3.a), for the two children v and w of x in T we have \(\sigma (\mathcal {L}(v)) \cap \sigma (\mathcal {L}(w)) = \emptyset\). Moreover, Lemma 3.1 implies that both \(\mathcal {L}(v)\) and \(\mathcal {L}(w)\) are non-empty subsets of \(\mathbb {G}\) and hence, neither \(\sigma (\mathcal {L}(v))=\emptyset\) nor \(\sigma (\mathcal {L}(w))=\emptyset\). Thus, there are two leaves \(a, b\in \mathcal {L}(x)\) such that \(\sigma (a) \ne \sigma (b)\). Hence, \({\text {lca}}_S(\sigma (\mathcal {L}(x))) \in W^0 = W\setminus \mathbb {S}\).
Claim 1: For all \(x,y\in U\) with \(x\prec _{T_{\mathcal {\overline{E}}}} y\) we have \(\mu (x)\prec _S \mu (y)\).
Note, y must be an interior vertex, since \(x\prec _{T_{\mathcal {\overline{E}}}} y\).
Hence \(t(y)=\mathfrak {s}\).
If x is a leaf, then \(\mu (x)=\sigma (x)\in \mathbb {S}\). As argued above, \(\mu (y) \in W\setminus \mathbb {S}\). Since \(x\in \mathcal {L}(y)\) and \(\sigma (\mathcal {L}(y))\ne \emptyset\), we have \(\sigma (x) \in \sigma (\mathcal {L}(y))\subseteq \mathbb {S}\) and thus, \(\mu (x)\prec _S \mu (y)\).
Now assume that x is an interior vertex and hence, \(t(x)=\mathfrak {s}\). Again, there are leaves \(a,b \in \mathcal {L}(x)\) with \(A = \sigma (a)\ne \sigma (b)=B\). Since \(t(y)=\mathfrak {s}\), vertex y has two children in \(T_{\mathcal {\overline{E}}}\). Let \(y'\) denote the child of y with \(x\preceq _{T_{\mathcal {\overline{E}}}} y'\). Since \(\mathcal {L}(x)\subseteq \mathcal {L}(y')\subsetneq \mathcal {L}(y)\), we have \(\mathcal {L}(y)\setminus \mathcal {L}(y')\ne \emptyset\) and, by Condition (O3.a), there is a gene \(c\in \mathcal {L}(y)\setminus \mathcal {L}(y') \subseteq \mathcal {L}(y)\setminus \mathcal {L}(x)\) with \(\sigma (c)=C\ne A,B\). By construction, \(\mathsf {(ab|c)}\in \mathcal {R}_0\) and hence, \(\mathsf {(AB|C)}\in \mathcal {S}(T;t,\sigma )\). Hence, \({\text {lca}}_S(A,B)\prec _S {\text {lca}}_S(A,B,C)\). Since this holds for all triples \(\mathsf {(x'x''|z)}\) with \(x',x''\in \mathcal {L}(x)\) and \(z\in \mathcal {L}(y)\setminus \mathcal {L}(y')\), we can conclude that
$$\begin{aligned} \mu (x)&= {\text {lca}}_S(\sigma (\mathcal {L}(x))) \\&\prec _S {\text {lca}}_S(\sigma (\mathcal {L}(x))\cup \sigma (\mathcal {L}(y) \setminus \mathcal {L}(y'))). \end{aligned}$$
Since \(\sigma (\mathcal {L}(x))\cup \sigma (\mathcal {L}(y) \setminus \mathcal {L}(y')) \subseteq \sigma (\mathcal {L}(y))\) we obtain
$$\begin{aligned} & {\text{lca}}_S(\sigma (\mathcal {L}(x))\cup \sigma (\mathcal {L}(y) \setminus \mathcal {L}(y^{\prime}))) \\ & \quad \preceq _S {\text{lca}}_S(\sigma (\mathcal {L}(y))) = \mu (y). \end{aligned}$$
Hence, \(\mu (x)\prec _S\mu (y)\). Thus, the claim is proven. \(\square\)
We continue to extend \(\mu\) to the entire set V. To this end, observe first that if \(t(x) \in \{\mathfrak {t}, \mathfrak {d}\}\) then we wish to map x on an edge \(\mu (x) = (u,v) \in F\) such that Lemma 4.1 is satisfied: \(v\succeq _S {\text {lca}}_S(\sigma (\mathcal {L}(x)))\). Such an edge exists for \(v = {\text {lca}}_S(\sigma (\mathcal {L}(x)))\) in S by construction. Every speciation vertex y with \(y\succ _{T_{\mathcal {\overline{E}}}} x\) therefore necessarily maps on the vertex u or above, i.e., \(\mu (y) \succeq _S u\) must hold. Thus, we set:
\(\mu (x) = (u,{\text {lca}}_S(\sigma (\mathcal {L}(x))))\), if \(t(x)\in \{\mathfrak {t}, \mathfrak {d}\}\),
which now makes \(\mu\) a map from V to \(W\cup F\).
By construction of \(\mu\), Conditions (M1), (M2.i), (M2.ii) are satisfied by \(\mu\).
We proceed to show that (M3) is satisfied.
Claim 2: For all \(x,y\in V\) with \(x\prec _{T_{\mathcal {\overline{E}}}} y\), Condition (M3) is satisfied.
If both x and y are speciation vertices, then we can apply the Claim 1 to conclude that \(\mu (x)\prec _S \mu (y)\). If x is a leaf, then we argue similarly as in the proof of Claim 1 to conclude that \(\mu (x)\preceq _S \mu (y)\).
Now assume that both x and y are interior vertices of T and at least one vertex of x, y is not a speciation vertex. Since, \(x\prec _{T_{\mathcal {\overline{E}}}} y\) we have \(\mathcal {L}(x) \subseteq \mathcal {L}(y)\) and thus, \(\sigma (\mathcal {L}(x)) \subseteq \sigma (\mathcal {L}(y))\).
We start with the case \(t(y)=\mathfrak {s}\) and \(t(x)\in \{\mathfrak {d}, \mathfrak {t}\}\). Since \(t(y)=\mathfrak {s}\), vertex y has two children in \(T_{\mathcal {\overline{E}}}\). Let \(y'\) be the child of y with \(x\preceq _{T_{\mathcal {\overline{E}}}} y'\). If \(\sigma (\mathcal {L}(x))\) contains only one species A, then \(\mu (x) = (u,A)\prec _S u\preceq _S {\text {lca}}_S(\sigma (\mathcal {L}(y))) = \mu (y)\). If \(\sigma (\mathcal {L}(x))\) contains at least two species, then there are \(a,b\in \mathcal {L}(x)\) with \(\sigma (a)=A\ne \sigma (b)=B\) Moreover, since \(\mathcal {L}(x)\subseteq \mathcal {L}(y')\subsetneq \mathcal {L}(y)\), we have \(\mathcal {L}(y)\setminus \mathcal {L}(y')\ne \emptyset\) and, by Condition (O3.a), there is a gene \(c\in \mathcal {L}(y)\setminus \mathcal {L}(y') \subseteq \mathcal {L}(y)\setminus \mathcal {L}(x)\) with \(\sigma (c)=C\ne A,B\). By construction, \(\mathsf {(ab|c)}\in \mathcal {R}_0\) and hence \(\mathsf {(AB|C)}\in \mathcal {S}(T;t,\sigma )\). Now we can argue similar as in the proof of the Claim 1, to see that
$$\begin{aligned} \mu (x)&= (u,{\text {lca}}_S(\sigma (\mathcal {L}(x)))) \prec _S u \\&\preceq _S {\text {lca}}_S(\sigma (\mathcal {L}(y))) = \mu (y). \end{aligned}$$
If \(t(x)=\mathfrak {s}\) and \(t(y)\in \{\mathfrak {d}, \mathfrak {t}\}\), then \(\sigma (\mathcal {L}(x)) \subseteq \sigma (\mathcal {L}(y))\) implies that
$$\begin{aligned} \mu (x)&= {\text {lca}}_S(\sigma (\mathcal {L}(x)))\preceq _S {\text {lca}}_S(\sigma (\mathcal {L}(y))) \\&\prec _S(u,{\text {lca}}_S(\sigma (\mathcal {L}(y)))) = \mu (y). \end{aligned}$$
Finally assume that \(t(x),t(y)\in \{\mathfrak {d}, \mathfrak {t}\}\). If \(\sigma (\mathcal {L}(x)) = \sigma (\mathcal {L}(y))\), then \(\mu (x) = \mu (y)\). Now let \(\sigma (\mathcal {L}(x)) \subsetneq \sigma (\mathcal {L}(y))\) which implies that \({\text {lca}}_S(\sigma (\mathcal {L}(x)))\preceq _S {\text {lca}}_S(\sigma (\mathcal {L}(y)))\). If \({\text {lca}}_S(\sigma (\mathcal {L}(x))) = {\text {lca}}_S(\sigma (\mathcal {L}(y)))\), then \(\mu (x) = \mu (y)\). If \({\text {lca}}_S(\sigma (\mathcal {L}(x)))\prec _S {\text {lca}}_S(\sigma (\mathcal {L}(y)))\), then
$$\begin{aligned} \mu (x)&=(u,{\text {lca}}_S(\sigma (\mathcal {L}(x)))) \prec _S u \\&\preceq _S {\text {lca}}_S(\sigma (\mathcal {L}(y))) \prec (u',{\text {lca}}_S(\sigma (\mathcal {L}(y)))) \\&=\mu (y). \end{aligned}$$
It remains to show (M2.iii), that is, if \(e_i=(x,y)\) is a transfer-edge, then \(\mu (x)\) and \(\mu (y)\) are incomparable in S. Since (x, y) is a transfer edge and by Condition (O3.b), \(\sigma (\mathcal {L}(x)) \cap \sigma (\mathcal {L}(y)) = \emptyset\). If \(\sigma (\mathcal {L}(x))=\{A\}\) and \(\sigma (\mathcal {L}(y))=\{C\}\), then \(\mu (x) = (u,A)\) and \(\mu (y) = (u',C)\). Since A and C are distinct leaves in S, \(\mu (x)\) and \(\mu (y)\) are incomparable. Assume that \(|\sigma (\mathcal {L}(x))|>1\). Hence, there are leaves \(a,b \in \mathcal {L}(x)\) with \(A = \sigma (a)\ne \sigma (b)=B\) and \(c\in \mathcal {L}(y)\) with \(\sigma (c)=C\ne A,B\). By construction, \(\mathsf {(ab|c)}\in \mathcal {R}_i\) and hence, \(\mathsf {(AB|C)}\in \mathcal {S}(T;t,\sigma )\). The latter is fulfilled for all triples \(\mathsf {(x'x''|c)}\in \mathcal {R}_i\) with \(x',x''\in \mathcal {L}(x)\), and, therefore, \({\text {lca}}_S(\sigma (\mathcal {L}(x))\cup \{C\}) \succ _S {\text {lca}}_S(\sigma (\mathcal {L}(x)))\). Set \(v={\text {lca}}_S(\sigma (\mathcal {L}(x))\cup \{C\})\). Thus, there is an edge \((v,v')\) in S with \(v'\succeq _S {\text {lca}}_S(\sigma (\mathcal {L}(x)))\) and an edge \((v,v'')\) such that \(v''\succeq _S C\). Hence, either \(\mu (x) = (v,v')\) or \(\mu (x) = (u,{\text {lca}}_S(\sigma (\mathcal {L}(x)))\) and \(v'\succeq _S u\). Assume that \(\sigma (\mathcal {L}(y))\) contains only the species C and thus, \(\mu (y) = (u',C)\). Since \(v''\succeq _S C\), we have either \(v'' = C\) which implies that \(\mu (y) = (v,v'')\) or \(v'' \succ _S C\) which implies that \(\mu (y) = (u',C)\) and \(v''\succeq _S u'\). Since both vertices \(v'\) and \(v''\) are incomparable in S, so \(\mu (x)\) and \(\mu (y)\) are. If \(|\sigma (\mathcal {L}(y))|>1\), then we set \(v={\text {lca}}_S(\sigma (\mathcal {L}(x))\cup \sigma (\mathcal {L}(y)))\) and we can argue analogously as above and conclude that there are edges \((v,v')\) and \((v,v'')\) in S such that \(v'\succeq _S {\text {lca}}_S(\sigma (\mathcal {L}(x)))\) and \(v''\succeq _S {\text {lca}}_S(\sigma (\mathcal {L}(y)))\). Again, since \(v'\) and \(v''\) are incomparable in S and by construction of \(\mu\), \(\mu (x)\) and \(\mu (y)\) are incomparable. Thus, the claim is proven. \(\square\)
Consider the "true" history (left) that is also shown in Fig. 1. The center-left gene tree \((T;t,\sigma )\) is biologically feasible and obtained as the observable part of the true history. There is no reconciliation map for \((T;t,\sigma )\) to any species tree according to Def. 2 because \(\mathcal {S}(T;t,\sigma )\) is inconsistent (cf. Thm. 5.4). The graph in the lower-center depicts the orthology-relation that comprises all pairs (x, y) of vertices for which \(t({\text {lca}}(x,y)) =\mathfrak {s}\). The center-right gene tree \((T';t,\sigma )\) is non-binary and can directly be computed from the orthology-relation. Although \(\mathcal {S}(T';t,\sigma )\) is inconsistent, there is a valid reconciliation map \(\mu\) to a species tree for \((T';t,\sigma )\) according to Def. 2 (right). Note, both trees \((T;t,\sigma )\) and \((T';t,\sigma )\) satisfy axioms (O1)–(O3) and even (O3.A). However, the reconciliation map \(\mu\) does not satisfy the extra Condition (M2.iv), since \(\mu (z)\) and \(\mu (a')=A\) are comparable, although z and \(a'\) are children of a common speciation vertex. Therefore, Axioms (O1)–(O3) and (O3.A) do not imply (M2.iv). Moreover, Thm. 5.7 implies that there is no restricted reconciliation map for \((T;t,\sigma )\) as well as \((T';t,\sigma )\) and any species tree, since \(\mathcal {S}(T;t,\sigma )\) and \(\mathcal {S}(T';t,\sigma )\) are inconsistent. See text for further details
Lemma 5.2 implies that consistency of the triple set \(\mathcal {S}(T; t,\sigma )\) is necessary for the existence of a reconciliation map from \((T; t,\sigma )\) to a species tree on \(\mathbb {S}\). Lemma 5.3, on the other hand, establishes that this is also sufficient. Thus, we have
There is a species tree on \(\mathbb {S}= \sigma (\mathbb {G})\) for a binary gene tree \((T;t, \sigma )\) on \(\mathbb {G}\) if and only if the triple set \(\mathcal {S}(T; t,\sigma )\) is consistent.
Non-binary gene trees
Now, we consider arbitrary, possibly non-binary gene trees that might be used to model incomplete knowledge of the exact genes phylogeny. Consider the "true" history of a gene tree that evolves along the (tube-like) species tree in Fig. 3 (left). The observable gene tree \((T;t,\sigma )\) is shown in Fig. 3 (center-left). Since \(\mathsf {(ab|c)},\mathsf {(b'c'|a')} \in \mathcal {R}_0\), we obtain a set of species triples \(\mathcal {S}(T;t,\sigma )\) that contain the pair of inconsistent species triple \(\mathsf {(AB|C)},\mathsf {(BC|A)}\). Thus, there is no reconciliation map for \((T;t,\sigma )\) and any species tree, although \((T;t,\sigma )\) is biologically feasible. Consider now the "orthology" graph G (shown below the gene trees) that has as vertex set \(\mathbb {G}\) and two genes x, y are connected by an edge if \({\text {lca}}(x,y)\) is a speciation vertex. Such graphs can be obtained from orthology inference methods [14, 36,37,38] and the corresponding non-binary gene tree \((T';t,\sigma )\) (center-right) is constructed from such estimates (see [5,6,7] for further details). Still, we can see that \(\mathcal {S}(T';t,\sigma )\) contains the two inconsistent species triples \(\mathsf {(AB|C)},\mathsf {(BC|A)}\). However, there is a reconciliation map \(\mu\) according to Definition 2 and a species tree S, as shown in Fig. 3 (right). Thus, consistency of \(\mathcal {S}(T';t,\sigma )\) does not characterize whether there is a valid reconciliation map for non-binary gene trees.
In order to obtain a similar result as in Theorem 5.4 for non-binary gene trees we have to strengthen observability axiom (O3.a) to
(O3.A)
If x is a speciation vertex with children \(v_1,\dots ,v_k\), then \(\sigma _{T_{\mathcal {\overline{E}}}}(v_i) \cap \sigma _{T_{\mathcal {\overline{E}}}}(v_j) =\emptyset\), \(1\le i<j\le k\);
and to add an extra event constraint to Definition 2:
(M2.iv)
Let \(v_1,\dots ,v_k\) be the children of the speciation vertex x. Then, \(\mu (v_i)\) and \(\mu (v_j)\) are incomparable in S, \(1\le i<j\le k\).
We call a reconciliation map that additionally satisfies (M2.iv) a restricted reconciliation map. Such restricted reconciliation maps satisfy the condition as required in [12, 48] for the HGT-free case. It can be shown that restricted reconciliation maps imply Condition (O3.A), however, the converse is not true in general, see Fig. 3. Hence, we cannot use the axioms (O1)-(O3) and (O3.A) to derive Condition (M2.iv)—similar to Lemma 4.2(2)—and thus, need to claim it.
In particular, Condition (M2.iv) forbids ancestral relationships of the images \(\mu (v_i)\) and \(\mu (v_i)\) in S for any two distinct children \(v_i\) and \(v_j\) of a speciation vertex x. In Fig. 3 (right) a map \(\mu\) is shown that violates Condition (M2.iv). Here, the images \(\mu (z)\) and \(\mu (a')\) are comparable. The latter might happen, if there are unrecognized HGT events followed by a loss. Condition (M2.iv) is a quite strong restriction, however, it is indispensable for the characterization of reconciliation maps for non-binary gene trees in terms of informative triples, as we shall see soon.
It is now straightforward to obtain the next result.
If \(\mu\) is a restricted reconciliation map from \((T;t,\sigma )\) to S and \(\mathsf {(ab|c)} \in \mathcal {R}(T;t,\sigma )\) , then \(\mathsf {(\sigma (a)\sigma (b)|\sigma (c))}\) is displayed in S.
Let \(\{a,b,c\} \in \left( {\begin{array}{c}\mathbb {G}\\ 3\end{array}}\right)\) and assume w.l.o.g. \(\mathsf {(ab|c)} \in \mathcal {R}(T;t,\sigma )\).
First assume that \(\mathsf {(ab|c)} \in \mathcal {R}_0\), that is \(\mathsf {(ab|c)}\) is displayed in \(T_{\mathcal {\overline{E}}}\) and \(t({\text {lca}}_{T_{\mathcal {\overline{E}}}}(a,b,c)) = \mathfrak {s}\). For simplicity set \(u={\text {lca}}_{T_{\mathcal {\overline{E}}}}(a,b,c)\). Hence, there are two children x, y of u in \(T_{\mathcal {\overline{E}}}\) such that w.l.o.g. \(a,b\in L_{T_{\mathcal {\overline{E}}}}(x)\) and \(c\in L_{T_{\mathcal {\overline{E}}}}(y)\). Now we can argue analogously as in the proof of Lemma 5.2 after replacing "we can apply Lemma 4.2(2)" by "we can apply Condition (M2.iv)". The proof for \(\mathsf {(ab|c)} \in \mathcal {R}_i\) remains the same as in Lemma 5.2. \(\square\)
Let S be a species tree on \(\mathbb {S}\) . Then, there is a restricted reconciliation map \(\mu\) from a gene tree \((T;t,\sigma )\) that satisfies also (O3.A) to S whenever S displays all triples in \(\mathcal {S}(T;t,\sigma )\).
The proof is similar to the proof of Lemma 5.6. However, note that a speciation vertex might have more than two children. In these cases, one simply has to apply Axiom (O3.A) instead of Lemma (O3.a) to conclude that (M1), (M2.i)–(M2.iii), (M3) are satisfied.
It remains to show that (M2.iv) is satisfied. To this end, let x be a speciation vertex in T and the set of its children \(\mathsf {Ch}(x) = \{v_1,\dots ,v_k\}\). By axiom (O3.A) we have \(\sigma _{T_{\mathcal {\overline{E}}}}(v_i) \cap \sigma _{T_{\mathcal {\overline{E}}}}(v_j) =\emptyset\) for all \(i\ne j\). Consider the following partition of \(\mathsf {Ch}(x)\) into \(\mathsf {Ch}_1\) and \(\mathsf {Ch}_2\) that contain all vertices \(v_i\) with \(|\sigma _{T_{\mathcal {\overline{E}}}}(v_i)|=1\) and \(|\sigma _{T_{\mathcal {\overline{E}}}}(v_i)|>1\), respectively. By construction of \(\mu\), for all vertices in \(v_i,v_j\in \mathsf {Ch}_1\), \(i\ne j\) we have that \(\mu (v_i)\in \{\sigma (v_i), (u,\sigma (v_i)) \}\) and \(\mu (v_j)\in \{\sigma (v_j), (u',\sigma (v_j)) \}\) are incomparable. Now let \(v_i\in \mathsf {Ch}_1\) and \(v_j\in \mathsf {Ch}_2\). Thus, there are \(A,B\in \sigma _{T_{\mathcal {\overline{E}}}}(v_j)\) and \(\sigma (v_i)=C\). Hence, \(\mathsf {(AB|C)} \in \mathcal {S}(T;t,\sigma )\) Therefore, \({\text {lca}}_S(A,B)\) must be incomparable to C in S. Since the latter is satisfied for all species in \(\sigma _{T_{\mathcal {\overline{E}}}}(v_j)\), \({\text {lca}}_S( \sigma _{T_{\mathcal {\overline{E}}}}(v_j))\) and C must be incomparable in S. Again, by construction of \(\mu\), we see that \(\mu (v_i)\in \{C, (u,C) \}\) and \(\mu (v_j)\in \{{\text {lca}}_S( \sigma _{T_{\mathcal {\overline{E}}}}(v_j)), (u',{\text {lca}}_S( \sigma _{T_{\mathcal {\overline{E}}}}(v_j))) \}\) are incomparable in S. Analogously, if \(v_i,v_j\in \mathsf {Ch}_2\), \(i\ne j\), then all triples \(\mathsf {(AB|C)}\) and \(\mathsf {(CD|A)}\) for all \(A,B\in \sigma _{T_{\mathcal {\overline{E}}}}(v_j)\) and \(C,D\in \sigma _{T_{\mathcal {\overline{E}}}}(v_j)\) are contained in \(\mathcal {S}(T;t,\sigma )\) and thus, displayed by S. Hence, \({\text {lca}}_S( \sigma _{T_{\mathcal {\overline{E}}}}(v_i))\) and \({\text {lca}}_S( \sigma _{T_{\mathcal {\overline{E}}}}(v_j))\) must be incomparable in S. Again, by construction of \(\mu\), we obtain that \(\mu (v_i)\in \{{\text {lca}}_S( \sigma _{T_{\mathcal {\overline{E}}}}(v_i)), (u,{\text {lca}}_S( \sigma _{T_{\mathcal {\overline{E}}}}(v_i))) \}\) and \(\mu (v_j)\in \{{\text {lca}}_S( \sigma _{T_{\mathcal {\overline{E}}}}(v_j)), (u',{\text {lca}}_S( \sigma _{T_{\mathcal {\overline{E}}}}(v_j))) \}\) are incomparable in S. Therefore, (M2.iv) is satisfied. \(\square\)
As in the binary case, we obtain
There is a restricted reconciliation map for a gene tree \((T;t, \sigma )\) on \(\mathbb {G}\) that satisfies also (O3.A) and some species tree on \(\mathbb {S}= \sigma (\mathbb {G})\) if and only if the triple set \(\mathcal {S}(T; t,\sigma )\) is consistent.
From the binary gene tree \((T;t,\sigma )\) (right) we obtain the species triples \(\mathcal {S}(T;t,\sigma ) = \{\mathsf {(AB|D)},\mathsf {(AC|D)}\}\). Shown are two (tube-like) species trees (left and middle) that display \(\mathcal {S}(T;t,\sigma )\). The respective reconciliation maps for T and S are given implicitly by drawing T within the species tree S. The left tree S is least resolved for \(\mathcal {S}(T;t,\sigma )\). Although there is even a unique reconciliation map from T to S, this map is not time-consistent. Thus, no time-consistent reconciliation between T and S exists. On the other hand, for T and the middle species tree \(S'\) (that is a refinement of S) there is a time-consistent reconciliation map. Fig. 2 provides an example that shows that also least-resolved species trees can have a time-consistent reconciliation map with gene trees
The proof of Lemmas 5.3 and 5.6 is constructive and we summarize the latter findings in Algorithm 1, see Fig. 2 for an illustrative example.
Algorithm 1 returns a species tree S for a binary gene tree \((T;t,\sigma )\) and a reconciliation map \(\mu\) in polynomial time, if one exists and otherwise, returns that there is no species tree for \((T;t,\sigma )\).
If \((T;t,\sigma )\) is non-binary but satisfies Condition (O3.A), then Algorithm 1 returns a species tree S for \((T;t,\sigma )\) and a restricted reconciliation map \(\mu\) in polynomial time, if one exists and otherwise, returns that there is no species tree for \((T;t,\sigma )\).
Theorem 5.4 and the construction of \(\mu\) in the proof of Lemmas 5.3 and 5.6 implies the correctness of the algorithm.
For the runtime observe that all tasks, computing \(\mathcal {S}(T;t,\sigma )\), using the BUILD algorithm [64, 68] and the construction of the map \(\mu\) [10, Cor.7] can be done in polynomial time. \(\square\)
In our examples, the species trees that display \(\mathcal {S}(T; t,\sigma )\) is computed using the \(\mathcal {O}(|L_R||R|)\) time algorithm BUILD, that either constructs a tree S that displays all triples in a given triple set R or recognizes that R is not consistent. However, any other supertree method might be conceivable, see [65] for an overview. The tree T returned by \(\texttt {BUILD}\) is least resolved, i.e., if \(T'\) is obtained from T by contracting an edge, then \(T'\) does not display R anymore. However, the trees generated by \(\texttt {BUILD}\) do not necessarily have the minimum number of internal vertices, i.e., the trees may resolve multifurcations in an arbitrary way that is not implied by any of the triples in R. Thus, depending on R, not all trees consistent with R can be obtained from \(\texttt {BUILD}\). Nevertheless, in [11, Prop. 2(SI)] the following result was established.
Let R be a consistent triple set. If the tree T obtained with \(\texttt {BUILD}\) applied on R is binary, then T is a unique tree on \(L_R\) that displays R, i.e., for any tree \(T'\) on \(L_R\) that displays R we have \(T'\simeq T\).
Shown is a binary and biologically feasible gene tree \((T;t,\sigma )\) (center) that is obtained as the observable part of the true scenario (left). However, there is no reconciliation map for \((T;t,\sigma )\) to any species tree according to Def. 2 because \(\mathcal {S}(T;t,\sigma )\) is inconsistent. Nevertheless, a relaxed reconciliation map \(\mu\) between \((T;t,\sigma )\) and the species tree exists (right). However, this map does not satisfy Lemma 4.2(2) since \(\mu (a')=A\) and \(\mu ({\text {lca}}_{T_{\mathcal {\overline{E}}}}(b',c'))\) are comparable. See text for further details
So-far, we have shown that event-labeled gene trees \((T;t,\sigma )\) for which a species tree exists can be characterized by a set of species triples \(\mathcal {S}(T;t,\sigma )\) that is easily constructed from a subset of triples displayed in T. From a biological point of view, however, it is necessary to reconcile a gene tree with a species tree such that genes do not "travel through time". In [63], the authors gave algorithms to check whether a given reconciliation map \(\mu\) is time-consistent and for the construction of a time-consistent reconciliation maps, provided one exists. These algorithms require as input an event-labeled gene tree and species tree. Hence, a necessary condition for the existence of time-consistent reconciliation maps is given by consistency of the species triple \(\mathcal {S}(T;t,\sigma )\) derived from \((T;t,\sigma )\). However, there are possibly exponentially many species trees that are consistent with \(\mathcal {S}(T;t,\sigma )\) for which some of them have a time-consistent reconciliation map with T and some not, see Fig. 4. The question therefore arises as whether there is at least one species tree S with time-consistent map, and if so, construct S.
Limitations of informative triples and reconciliation maps
Shown is a (tube-like) species trees S with reconciled gene tree \((T;t,\sigma )\) (taken from [63]). The informative triple set \(\mathcal {S}(T;t,\sigma )\) is consistent and application of Lemma 5.9 shows that S is unique. Moreover, the reconciliation map \(\mu\) is unique, however, not time-consistent. Thus, although \(\mathcal {S}(T;t,\sigma )\) is consistent, there is no time-consistent reconciliation map for \((T;t,\sigma )\) and S. Nevertheless, it can be shown that \((T;t,\sigma )\) is biologically feasible
In "Non-binary gene trees" section we have already discussed that consistency of \(\mathcal {S}(T;t,\sigma )\) cannot be used to characterize whether there is a reconciliation map that doesn't need to satisfy (M2.iv) for some non-binary gene tree, see Fig. 3. In particular, Fig. 3 shows a biologically feasible binary gene trees (center-left) for which, however, neither a reconciliation map nor a restricted reconciliation map exists.
A further simple example is given in Fig. 5. Consider the "true" history of the gene tree that evolves along the (tube-like) species tree in Fig. 5 (left). The set of extant genes \(\mathbb {G}\) comprises \(a,a',b,b',c\) and \(c'\) and \(\sigma\) maps each gene in \(\mathbb {G}\) to the species (capitals below the genes) \(A,B,C\in \mathbb {S}\). For the observable gene tree \((T;t,\sigma )\) in Fig. 5 (center) we observe that \(\mathcal {R}_0 = \{\mathsf {(ab|c)},\mathsf {(b'c'|a')}\}\) and thus, one obtains the inconsistent species triples \(\mathcal {S}(T;t,\sigma ) = \{\mathsf {(AB|C)},\mathsf {(BC|A)}\}\). Hence, Theorem 5.4 implies that there is no species tree for \((T;t,\sigma )\). Note, \((T;t,\sigma )\) satisfies also Condition (O3.A). Hence, Theorem 5.7 implies that no restricted reconciliation map to any species tree exists for \((T;t,\sigma )\). Nevertheless, \((T;t,\sigma )\) is biologically feasible as there is a true scenario that explains the gene tree.
Now consider the gene tree \((T;t,\sigma )\) in Fig. 6 (right). The set \(\mathcal {S}(T;t,\sigma )\) is consistent. Both, the species trees S that displays all informative triples and the reconciliation map \(\mu\) from \((T;t,\sigma )\) to S, are unique. However, \(\mu\) is not time-consistent. Uniqueness of S and \(\mu\) implies that there is no time-consistent reconciliation map for \((T;t,\sigma )\) to any species tree. Thus, consistency of \(\mathcal {S}(T;t,\sigma )\) does not imply the existence of time-consistent reconciliation maps. It can be shown that \((T;t,\sigma )\) is biologically feasible.
Finally, we shortly discuss a relaxation of Condition (M2.i). Lemma 4.1 implies that \(\mu (x)={\text {lca}}_S(\sigma _{T_{\mathcal {\overline{E}}}}(x))\) is the lowest possible choice for the image of a speciation vertex. Nevertheless, it is possible to relax this condition, i.e., we could allow for speciation vertices x that \(\mu (x) \succeq _S {\text {lca}}_S(\sigma _{T_{\mathcal {\overline{E}}}}(x))\). Indeed, there are relaxed reconciliation maps for \((T;t,\sigma )\) and S, although no (restricted) reconciliation map for \((T;t,\sigma )\) to any species tree exists, see Fig. 5 (right). This example shows that the existence of relaxed reconciliation maps cannot be characterized by means of consistency of the informative triples in \(\mathcal {S}(T;t,\sigma )\). However, it might be of interest for future research to investigate this generalization in more detail and to understand to what extent relaxed reconciliation maps imply biologically feasibility.
We summarize the latter observations:
Consistency of \(\mathcal {S}(T;t,\sigma )\) is equivalent to the existence of a (restricted) reconciliation map. Thus, if \(\mathcal {S}(T;t,\sigma )\) is consistent, then there is also a relaxed reconciliation map. The converse is not true in general.
Existence of time-consistent (restricted) reconciliation maps implies the existence of (restricted) reconciliation maps and thus, consistency of \(\mathcal {S}(T;t,\sigma )\). The converse is not true in general.
If \((T;t,\sigma )\) does not contain HGT-events and \(\mathcal {S}(T;t,\sigma )\) is consistent, then \((T;t,\sigma )\) is biologically feasible. The converse is not true in general.
If \((T;t,\sigma )\) contains HGT-events and there is a time-consistent reconciliation map for \((T;t,\sigma )\) to some species tree, then \((T;t,\sigma )\) is biologically feasible. The converse is not true in general.
Conclusion and open problems
Event-labeled gene trees can be obtained by combining the reconstruction of gene phylogenies with methods for orthology and HGT detection. We showed that event-labeled gene trees \((T;t,\sigma )\) for which a species tree exists can be characterized by a set of species triples \(\mathcal {S}(T;t,\sigma )\) that is easily constructed from a subset of triples displayed in T.
We have shown that biological feasibility of gene trees cannot be explained in general by reconciliation maps, that is, there are biologically feasible gene trees for which no reconciliation map to any species tree exists.
We close this contribution by stating some open problems that need to be solved in future work.
Are all event-labeled gene trees \((T;t,\sigma )\) biologically feasible? If not, how are biologically feasible gene trees characterized and what is the computational complexity to recognize them?
The results established here are based on informative triples provided by the gene trees. If it is desired to find "non-restricted" reconciliation maps (those for which Condition (M2.iv) is not required) for non-binary gene trees the following question needs to be answered: How much information of a non-restricted reconciliation map and a species tree is already contained in non-binary event-labeled gene trees \((T;t,\sigma )\)? The latter might also be generalized by considering relaxed reconciliation maps (those for which \(\mu (x)\succ _S {\text {lca}}_S(\sigma _{T_{\mathcal {\overline{E}}}}(x))\) for speciation vertices x or any other relaxation is allowed).
Our results depend on three axioms (O1)–(O3) on the event-labeled gene trees that are motivated by the fact that event-labels can be assigned to internal vertices of gene trees only if there is observable information on the event. The question which event-labeled gene trees are actually observable given an arbitrary, true evolutionary scenario deserves further investigation in future work, since a formal theory of observability is still missing.
The definition of reconciliation maps is by no means consistent in the literature. For the results established here we considered three types of reconciliation maps, that is, the "usual" map as in Def. 2 (as used in, e.g. [10, 31, 54, 63]), a restricted version (as used in, e.g. [12, 48]) and a relaxed version. However, a unified framework for reconciliation maps is desirable and might be linked with a formal theory of observability.
"Satisfiable" event-relations \(R_1,\dots ,R_k\) are those for which there is a representing gene tree \((T;t,\sigma )\) such that \((x,y)\in R_i\) if and only if \(t({\text {lca}}(x,y))=i\). They are equivalent to so-called unp two-structures [6]. In particular, if event-relations consist of orthologs, paralogs and xenologs only, then satisfiable event-relations are equivalent to directed cographs [6]. Satisfiable event-relations \(R_1,\dots ,R_k\) are "S-consistent" if there is a species tree S for the representing gene tree \((T;t,\sigma )\) [12, 48]. However, given the unavoidable noise in the input data and possible uncertainty about the true relationship between two genes, one might ask to what extent the work of Lafond et al. [12, 48] can be generalized to determine whether given "partial" event-relations are S-consistent or not. It is assumable that subsets of the informative species triples \(\mathcal {S}(T;t,\sigma )\) that might be directly computed from such event-relations can offer an avenue to the latter problem. Characterization and complexity results for "partial" event-relations to be satisfiable have been addressed in [74].
In order to determine whether there is a time-consistent reconciliation map for some given event-labeled gene tree and species trees fast algorithms have been developed [63]. However, these algorithms require as input a gene tree \((T;t,\sigma )\) and a species tree S. A necessary condition to a have time-consistent (restricted) reconciliation map to some species tree is given by the consistency of the species triples \(\mathcal {S}(T;t,\sigma )\). However, in general there might be exponentially many species trees that display \(\mathcal {S}(T;t,\sigma )\) for which some of them may have a time-consistent reconciliation map with \((T;t,\sigma )\) and some might have not (see Fig. 4 or [63]). Therefore, additional constraints to determine whether there is at least one species tree S with time-consistent map, and if so, construct S, must be established.
A further key problem is the reliable identification of horizontal transfer events. In principle, likely genes that have been introduced into a genome by HGT can be identified directly from sequence data [75]. Sequence composition often identifies a gene as a recent addition to a genome. In the absence of horizontal transfer, the similarities of pairs of true orthologs in the species pairs (A,B) and (A,C) are expected to be linearly correlated. Outliers are likely candidates for HGT events and thus can be "relabeled". However, a more detailed analysis of the relational properties of horizontally transferred genes is needed.
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I would like to thank Nikolai Nøjgaard and Nicolas Wieseke for all the outstanding and fruitful discussions!
The author declares no competing interests.
Institute of Mathematics and Computer Science, University of Greifswald, Walther-Rathenau-Strasse 47, 17487, Greifswald, Germany
Marc Hellmuth
Center for Bioinformatics, Saarland University, Building E 2.1, P.O. Box 151150, 66041, Saarbrücken, Germany
Correspondence to Marc Hellmuth.
Hellmuth, M. Biologically feasible gene trees, reconciliation maps and informative triples. Algorithms Mol Biol 12, 23 (2017). https://doi.org/10.1186/s13015-017-0114-z
DTL-scenario
Horizontal gene transfer
Phylogenetic tree
Event-label
Submission enquiries: [email protected] | CommonCrawl |
Nonparametric estimation of a primary care production function in urban Brazil
Bruno Wichmann ORCID: orcid.org/0000-0003-1104-00351 &
Roberta Wichmann2,3
The Brazilian public health system is one of the largest health systems in the world, with a mandate to deliver medical care to more than 200 million Brazilians. The objective of this study is to estimate a production function for primary care in urban Brazil. Our goal is to use flexible estimates to identify heterogeneous returns and complementarities between medical capital and labor.
We use a large dataset from 2012 to 2016 (with more than 400 million consultations, 270 thousand physicians, and 11 thousand clinics) to nonparametrically estimate a primary care production function and calculate the elasticity of doctors' visits (output) to two inputs: capital stock (number of clinics) and labor (number of physicians). We benchmark our nonparametric estimates against estimates of a Cobb-Douglas (CD) production function. The CD model was chosen as a baseline because it is arguably the most popular parametric production function model. By comparing our nonparametric results with those from the CD model, our paper shed some light on the limitations of the parametric approach, and on the novelty of nonparametric insights.
The nonparametric results show significantly heterogeneity of returns to both capital and labor, depending on the scale of operation. We find that diseconomies of scale, diminishing returns to scale, and increasing returns to scale are possible, depending on the input range.
The nonparametric model identifies complementarities between capital and labor, which is essential in designing efficient policy interventions. For example, we find that the response of primary care consultations to labor is steeper when capital level is high. This means that, if the goal is to allocate labor to maximize increases in consultations, adding physicians in cities with a high number of clinics is preferred to allocating physicians to low medical infrastructure municipalities. The results highlight how the CD model hides useful policy information by not accounting for the heterogeneity in the data.
Primary health care represents a broad approach to the promotion of individual and societal health and well-being, and as such it includes health services employed to deliver prevention, treatment, rehabilitation, and palliative care. According to the World Health Organization (WHO), the delivery of quality primary care can have significant short-term impacts in reducing risk factors and poor health conditions [1]. The WHO recognizes that health systems based on primary health care are of paramount importance in achieving sustainable health goals. This architecture is especially important in developing countries, where primary care systems often need to be further developed. WHO works with many countries to implement primary health care policies that integrate health-promoting and preventive interventions thereby reducing health care delivery costs and improving efficiency through lower hospital admissions.
Many developing and middle-income countries have been struggling to maintain adequate levels of public services in light of increasingly tighter budget constraints. For example, since 2010, Latin American countries have experienced declining GDP per capita growth rates, which dropped from 4.67% to − 0.44% in 2018. In fact, the region has been averaging a negative growth rate since 2014 [2]. This trend highlights the struggle of financing improvements in standards of living when wealth grows at a rate slower than that of population. A study by Varela and co-authors shows that only 6% of the municipalities in Sao Paulo, the largest Brazilian state, efficiently allocate health care expenditures to the delivery of primary care [3]. Lobo et al. examine a sample of 104 Brazilian teaching hospitals finding similar results; only 5% of the hospitals efficiently allocate resources [4].
As population grows, so does the demand for health care. In this context, it is essential for managers and policy makers to carefully understand the determinants of medical services uptake. One approach is to estimate health care production as a function of capital infrastructure in the healthcare system and the availability of health professionals. Such estimates allow for an evaluation of the returns to investments in capital and labor. Moreover, it is possible, and even probable, that the outcomes of health care investments are nonlinear, and the impact of health policies may vary depending on the scale of capital and labor. For example, it is possible for investments on medical personnel to have differentiated impacts on output as we move along the distribution of both labor and capital. Are returns to labor higher when labor is low? Are returns to labor investments a function of the stock of capital? Having an approach to answer these types of questions is imperative for the efficient design of health policy.
The goal of this paper is to estimate a primary care production function for urban Brazil. We employ nonparametric methods to estimate the elasticities of the uptake of primary health care services to capital infrastructure (number of clinics) and labor (number of physicians). We contrast our approach with a popular parametric baseline model to highlight the advantages of nonparametric approaches, and to show how the more flexible nonparametric estimates can provide additional support to evidence-informed public policies.
Literature and contributions
Examination of health production using parametric and nonparametric approaches has been an ongoing and fruitful area of research in the health economics and policy fields. The literature is large and evolved in many directions. For example, the efficiency of health care delivery systems has been studied using parametric models such as the Stochastic Frontier Approach [5,6,7] and nonparametric models such as Data Envelopment Analysis [8,9,10]. Some papers explore the benefits of parametric estimation trying to limit the disadvantages of the parametric assumption by using flexible functional forms to estimate primary care or health production functions [11, 12]. Other papers examine how the choice of functional form affect parametric estimation of health care technical efficiency [13]. Health economics research has also compared parametric and nonparametric methods [14, 15].
We add to this literature by using both parametric and nonparametric approaches to estimate elasticities of primary health care delivery in a large urban setting: the Brazilian public health system. Our contribution is twofold. First, by studying primary care in Brazil we are examining one of the largest public health systems in the world, with a mandate to deliver universal health care free of charges to more than 200 million individuals. We examine the five-year period from 2012 to 2016 where Brazil faced unfavorable economic conditions. As the country (and many others in the region) continues to struggle to revert the economic environment, data-driven and evidence-based policy recommendations are in high demand.
Second, we contribute by constrasting simple parametric estimates with a set of rich estimates from nonparametric local-linear models. Our application takes advantage of large datasets (with millions of observations) to mitigate issues related to the curse of the dimenssionality (although at the expense of computing time). Our paper highlights the insights that nonparametric estimates are able to deliver regarding heterogeneity, nonlinearities, and cross-effects. In doing so, the paper showcases the potential of local-linear models in informing public health policy.
The Brazilian health system
The Brazilian Unified Health System (Sistema Único de Saúde – SUS) is a public health system created in 1989 that offers access to health care, free of charge, to all Brazilians. SUS serves a population of more than 200 million people, ranking among the largest government funded and managed health care systems of the world. SUS network is responsible for all levels of health care, including all levels of medical care, laboratory and diagnostic care, physical and occupational therapy, nutritional support, pharmaceutical care, etc.
While all Brazilians are covered by SUS, private medical services and health insurances are also available for purchase in Brazil. Table 1 shows total Brazilian population and the number of individuals with supplemental private medical insurance. The data show that Brazilians increasingly depend on publicly provided health care. In the 2012–2016 period, population growth paired with a decrease in private insurance enrolment led to the coverage of the Brazilian population with supplemental private health insurance to decrease from 15.1 to 12.8%.
Table 1 Brazilian Population with Private Medical Insurance
Both parametric and nonparametric approaches can be used to examine health care delivery productivity and technical efficiency. In productivity studies the interest lies mainly on returns to input usage, or elasticities. Such estimates are valuable from a policy perspective as they can guide the allocation of health care resources. In efficiency studies, researchers are interested in measuring how producers deviate from an estimate of the production function, viewed as the state-of-the-art technology frontier. Therefore, efficiency studies are useful in informing policies with a focus on minimizing waste [16].
Regardless of the focus, i.e. production or efficiency, these studies require the estimation of a production function, which begs the question of which method to use. An important methodological decision is choosing between parametric and nonparametric approaches. There is no consensus in the literature and pros and cons have been reported about both methods. Approaches based on parametric functions are simple and can be easily implemented. Under the proper assumptions, parametric approaches have desirable statistical properties (e.g. fast convergence rates, which is important in small samples). Nonparametric approaches are more flexible as no functional form is pre-specified and the shape of the relationship between health output and inputs is determined by the data. On the downside, nonparametric estimators have low convergence rates and require larger amounts of data to deliver estimation errors equivalent to those from correctly specified parametric counterparts [16].
We conceptualize the delivery of medical services in terms of a medical care production function [17]. We consider the following random production function for healthcare delivery:
$$ Y=f\left(K,L\right)+\varepsilon $$
where Y is the delivery of medical services (output) and is determined by two components. The first component is deterministic and depends on health care inputs, while the second is a random component. The deterministic portion of output is given by a production function f that depends on capital K and labor L. The random term ε captures unobserved determinants of output and is assumed to be zero-mean.
Our goal is to use data on Y, K, and L to estimate the elasticities of primary care delivery with respect to capital and labor, which are determined by, respectively,
\( \frac{\partial Y}{\partial K}\frac{K}{Y}\ \mathrm{and}\ \frac{\partial Y}{\partial L}\frac{L}{Y} \).
We are also interested in examining estimates of the conditional mean of output, E(Y| K, L). These estimates allow us to visualize partial prediction plots, which can be of great value to policy makers. These types of plots are a simple way to illustrate nonlinearities and, as a result, they help inform policy by highlighting complementarities between capital and labor (see Fig. 2).
Our approach follows closely the production function estimation described by Henderson and Parmeter [18].Footnote 1 We consider two approaches to estimate a cross-city primary care production function in Brazil. The first is a parametric approach based on a Cobb-Douglas production function. The second is a nonparametric approach where no functional form is specified for primary care production.
Parametric model
This section presents a typically used parametric production model to establish a baseline for comparison with the nonparametric estimates. The Cobb-Douglas production function is arguably the most common parameterization of production in the literature and has been used to model production processes for more than one hundred years [19]. For the case of two inputs, capital and labor, the Cobb-Douglas model with an additively separable error term assumes the primary care function (1) takes the form
$$ Y=A{K}^{\alpha }{L}^{\beta }+\varepsilon $$
where A is a technology parameter, α is the elasticity of primary care output with respect to medical capital, and β is the elasticity of output with respect to the number of physicians. The typical approach to estimate our parameters of interest α and β is to log-linearize the model and use Ordinary Least Squares (OLS) to estimate the parameters in a log-linear regression. Nevertheless, it has been shown in the literature that such an approach can introduce bias [18, 20]. To avoid such a bias, we estimate α and β using Nonlinear Least Squares (NLS).
The NLS estimation procedure is as follows. To simplify notation, let f(X, θ) = AKαLβ, where X = (K, L) and θ = (A, α, β). The nonlinear least squares estimator \( \hat{\theta} \) is the value of θ that minimizes the sum of the squared residuals [21]:
$$ \underset{\theta }{\min\ }\left(Y-f\left(X,\theta \right)\right)^{\prime}\left(Y-f\left(X,\theta \right)\right). $$
The problem can be solved numerically using the Gauss–Newton algorithm, and standard errors are computed using a wild bootstrap procedure [18].
Nonparametric model
Consistent parametric estimation of the elasticities of capital and labor relies on the assumption that the parametric functional form chosen, in many cases the Cobb-Douglas, is the correct or true functional form. However, there is no consensus in the health economics literature regarding the correct functional form for the delivery of primary health care. Nonparametric estimation of function (1) allows us to avoid the bias of incorrectly imposing a certain parametric shape to the relationship between inputs K and L and output Y. In fact, one of the main advantages of the nonparametric approach is that it recovers the relationship between inputs and output directly from the data.
We use a Local-Linear Least Squares model (LLLS) to approximate the function f(K, L) in eq. (1).Footnote 2 The LLLS estimator is perhaps the most popular nonparametric regression estimator. To simplify notation, let X denote the input matrix (K, L). The LLLS estimator fits a line on the neighborhood of a point X0, where the concept of "neighborhood" is determined by a bandwidth vector h. The estimator re-writes the original model by considering a Taylor approximation around X0:
$$ Y=f\left({X}_0\right)+\left(X-{X}_0\right)\beta \left({X}_0\right)+\varepsilon, $$
where β is the gradient and is treated like a parameter to be estimated, and so is f(X0). Denoting a = f(X0) and b = β, the LLLS chooses θ = (a, b) to minimize the weighted sum of squared residuals:
$$ \underset{\theta }{\min}\left(Y-\overset{\sim }{X}\theta \right)^{\prime }K\left(X,{X}_0,h\right)\left(Y-\overset{\sim }{X}\theta \right), $$
where \( \overset{\sim }{X}=\left(1,X-{X}_0\right) \), 1 is a column vector of ones and \( K\left(X,{X}_0,h\right)=\frac{1}{\sqrt{2\pi }}{e^{-\left(\frac{1}{2}\right)\left(\frac{X-{X}_0}{h}\right)}}^2 \) is the Gaussian Kernel. When the Kernel is the identity matrix, the estimator reduces to the OLS estimator. By using Kernel weights we mitigate the effect of poor approximations of points far from X0. Standard errors and confidence intervals can be computed using a wild bootstrap. To avoid issues with numerical optimization and facilitate estimation, we standardize the data by dividing each variable by their mean.
The final step required to implement the above LLLS nonparametric estimator is to choose bandwidths in h. Typically, the literature relies on data-driven methods to determine the appropriate set h [18]. We use least squares cross validation (LSCV) to determine the bandwidths in h. The idea is simple and relies on choosing h that minimizes the sum of the squared errors of the model prediction, which in turn depends on h. Formally, the bandwidths in h minimize \( \sum {\left(Y-\hat{Y}(h)\right)}^2 \).
In summary, the concept of the LLLS is to fit a linear model around the neighborhood of inputs K and L, where this neighborhood is determined by a bandwidth chosen using cross validation (LSCV). The nonparametric model moves along the distributions of K and L estimating local linear regressions, connecting the predicted outputs (or conditional mean) from these various regressions to generate the relationship between inputs and output.
The nonparametric estimates of the elasticities of capital and labor are computed as.
$$ \hat{\beta}(K)K/\hat{Y}\ \mathrm{and}\ \hat{\beta}(L)\ L/\hat{Y} $$
respectively, where \( \hat{\beta}(K) \) is the gradient of the conditional mean with respect to capital, \( \hat{\beta}(L) \) is the gradient of the conditional mean with respect to labor, and \( \hat{Y} \) is the fitted value. Standard errors for elasticities, returns to scale, the predictions are computed via wild bootstrap as it is consistent under both homoskedasticity and heteroskedasticity [18]. Estimation was done using the R software [22] using codes provided by Henderson and Parmeter [18].
Our analysis is based on a sample of SUS users from the 100 largest cities in Brazil. The data contains information about SUS medical care delivery, physical medical infrastructure, and supply of healthcare professionals, at the city-level, monthly, from 2012 to 2016. Preliminary examination of the data revealed that the two largest cities (São Paulo and Rio de Janeiro) represent extreme values when compared against the remainder 98. We therefore exclude these two cities from our analysis. Our working sample offers a fair representation of urban Brazil. Collectively, the 98 cities considered in this study account for approximately one third of Brazil's total population.
The data comes from DATASUS – SUS IT department. DATASUS has several SUS databases that are publicly available for download.Footnote 3 Our output data were collected from the SIA database -- System of Outpatient Information (Sistema de Informação Ambulatorial – SIA). We measure primary care output as the number of doctor's visits in the SUS system, for each city-month observation. The number of patient visits (per unit of time) is a typical measure of primary care output [11]. Our input data are from the CNES database -- National Registry of Healthcare Facilities (Cadastro Nacional de Estabelecimentos de Saúde – CNES). Our proxy for health capital infrastructure is the city's number of clinics and similar health care delivery units. The city's stock of labor is measured as the number of physicians working in the city's SUS system.Footnote 4
In the 98 cities considered in this study, from 2012 to 2016, the sample contains 274,175 distinct physicians, 11,203 distinct clinics, and 407,259,570 primary care consultations. Figure 1 shows the (city-month) average number of consultations, clinics, and doctors, along with the interquartile range (IQR), by year. The data show that the representative city-month observation hovers around 60 thousand consultations (with a slight decline from 2012 to 2016), just under 100 clinics (steady trend in the period), and approximately 1500 doctors (with a slight upward trend).
Mean and IQR of output, capital, and labor, by year
We start by reporting the elasticities of capital and labor estimated by the parametric model. The simplicity of the Cobb-Douglas model is a feature that makes such a parametrization attractive and contributes to the popularity of this model. On the downside, the Cobb-Douglas model does not allow the elasticities to vary along different levels of capital and labor. The model's functional form leads to a single elasticity measure for each input, and therefore does not account for nonlinearities and cross-effects.Footnote 5
Table 2 shows the Cobb-Douglas elasticities estimates along with their standard errors. We find elasticities of output with respect to capital and labor of similar magnitude. The elasticities are precisely estimated, as indicated by the narrow standard errors. Both inputs have elasticity around 0.38, which suggests diminishing returns to scale in the order of 0.76. As a result, we find that primary care delivery increases by less than the proportional increase in both medical capital infrastructure and labor requirements. The Cobb-Douglas results suggest that if policy makers double capital and labor, primary care output will less than double and increase by a factor of 1.5.
Table 2 Estimates of the Cobb-Douglas model
As discussed above, consistency of parametric results relies on strong assumptions. To be unbiased, the Cobb-Douglas functional form must be the appropriate functional form for primary health care production. That would imply constant elasticities, and there exists no evidence that this is the case in primary care delivery.
Next, we use nonparametric models to obtain estimates that are not plagued by parametric misspecifications. The elasticities estimates are driven by the data and are allowed to vary along the distribution of capital and labor. As a result, the nonparametric approach captures nonlinear effects allowing policy makers to have a deeper understanding of the heterogeneity in returns to scale. It is possible, for example, to capture diminishing returns to scale in some input range and increasing returns to scale in another.
We estimate a nonparametric LLLS model using LSCV to select bandwidths. Table 3 reports the bandwidth estimates. We find bandwidths of 0.037 and 0.453, for capital and labor, respectively. For both values, the estimates are significantly lower than the upper bound suggested in the literature of two times the standard deviation of the corresponding input [23]. This indicates that both capital and labor have a nonlinear effect on output.
Table 3 LSCV Bandwidths
The nonparametric model produces observation-specific elasticities. Table 4 summarizes results by showing the LLLS elasticities estimates at their mean, 25th quantile, median, and 75th quantile. The table reveals several important lessons. First, focusing at the mean, while the nonparametric estimate of the elasticity of capital is similar to that of the parametric model, the LLLS estimate of the elasticity of labor is significantly smaller than the Cobb-Douglas counterpart. Table 4 shows that mean returns to capital (0.391) are more than seven times larger than mean returns to labor (0.050). That has a significant effect on the nonparametric estimates of returns to scale. The nonparametric model estimates diminishing returns to scale of 0.441, i.e. returns diminishing stronger than those of the parametric model. These estimates suggest that doubling input usage leads to an increase in output of a factor of 0.88 (as opposed to 1.5 in the Cobb-Douglas model).
Table 4 Results of the nonparametric model
Next, we find significant heterogeneity on both sets of elasticities. In the 25th quantile (i.e. the bottom of the distribution), the elasticities of both capital and labor are negative and represent diseconomies of scale. This is an indication of the existence of dysfunctional local health systems where investments in medical inputs actually drive patients away. Diseconomies of scale in the healthcare sector have been reported both in developed countries, e.g. the United States [24], and in developing countries, e.g. Turkey [13]. At the median, returns to capital are 28% smaller than the mean, and returns to labor are almost doubled.
Finally, elasticity heterogeneity becomes even more evident when we examine the top of the distribution. In the 75th quantile, the elasticity of output with respect to capital is 1.174, which suggests increasing returns to capital. Labor continues to exhibit diminishing returns. Comparing parametric and nonparametric estimates, only at the 75th quantile the LLLS estimate of the elasticity of labor is similar to that of the Cobb-Douglas model. In general, the 75th quantile results indicate that urban Brazil has high functioning local health care systems with strong increasing returns to scale.
When examining production functions, another interesting way to present nonparametric findings is to plot the estimate of the conditional mean of output (and its confidence interval) against one input. When production depends on two inputs, as in our case, this plot needs to be constructed by holding the other input constant at some (arbitrary) level [18]. The exercise we pursue in this paper is to plot output estimates obtained using one variable input (capital or labor), holding the other input fixed. For each case (variable capital or variable labor), we estimate three models that differ by the level of the fixed input (which is fixed at 25th, 50th, and 75th quantiles). This examination allows us to uncover cross-effects and assess whether there exist complementarities between the medical capital infrastructure and the supply of physicians.
Results are shown in Fig. 2. The graphs in left column show the counterfactual exercise of displaying output prediction as a function of capital, holding labor fixed at the 25th quantile (PANEL a), 50th quantile (PANEL b), and 75th quantile (PANEL c). The right column displays the same exercise for varying labor. We normalize output and inputs by their mean to avoid issues with numerical optimizations. As a result, an output (input) value of 1 represents mean output (input). Similarly, a value of 2 represents a level twice the mean. In addition to facilitate optimization, this normalization simplifies the scale of the graphs, making them easier to read while retaining economic meaning by allowing for cross comparisons.
Conditional mean output estimates versus a single input. PANEL a – Prediction holds the other input fixed at the 0.25 quantile. PANEL b -- Prediction holds the other input fixed at the median. PANEL c – Prediction holds the other input fixed at the 0.75 quantile
The counterfactual plots of output against capital (holding labor constant) are far more volatile than the ones of output and labor (holding capital constant). This finding suggests that there is more uncertainty in health care capital investments than in investments to expand the number of physicians. In general, as input usage increases, output increases. But there are downward-sloping regions, especially on the left-hand side graphs for high levels of capital. The right-hand side column shows a similar effect for labor, however with significant less variance and a shallower negative output response at the upper tail of the labor distribution.
The counterfactual analysis complements the previously discussed elasticity analysis. The graphs allow us to gauge the slope of the production function at different points, which shows information about the general regions with diseconomies of scale, diminishing returns, and increasing returns. While results in Table 4 show that there are negative elasticities in the bottom of the elasticity distribution (25th quantile), Fig. 2 suggests that many of these negative results occur at the top of both the capital and labor distributions.
In Fig. 2, a comparison of rows within a column reveals complementarities between capital and labor, but these are not uniform. For instance, on the right-hand side column, in PANEL a, the entire prediction of output lies below the mean output (or below 1). The behavior of the conditional mean as labor varies does not change much when capital increases from the 25th quantile to the median (i.e. comparing right-hand sides of PANELS a and b). One exception is that negative returns to labor at the top of the labor distribution disappear when capital increases from 25th to 50th quantile. However, moving down to PANEL c (i.e. output estimates hold capital fixed at the 75th quantile), we see that output is at a higher level and many points are above the mean. Moreover, output response to labor is steeper when capital level is high. In general, we conclude that medical infrastructure complements physicians mainly when capital levels are high, but high capital levels may also induce diseconomies of scale for expansion of physicians at the top of the labor distribution.
We estimate a primary care production function using data from the public health care systems of the largest cities in Brazil. In doing so, we compare the insights from the popular Cobb-Douglas parametric model against those from a nonparametric model. Our results highlight the rich set of policy-related information that can be generated by flexible nonparametric estimates. With the fast-pace development of information systems making large datasets available to managers, the application of nonparametric methods offers flexibility in an environment where the standard "curse of dimensionality" can be circumvented.
Nevertheless, the study has limitations. While the nonparametric model is able to deliver a variety of interesting insights by exploring the heterogeneity in the data, in our application for primary care in urban Brazil, we find estimates that are less efficient than those from the parametric model. For instance, while Table 3 reports a parametric elasticity of output to capital of 0.378 with standard error of 0.016, Table 4 reports a similar nonparametric mean elasticity of capital of 0.391, however, with much larger standard error of 0.246.
The paper's output is the city's number of primary care medical consultations. Although it is not always necessarily the case that efforts to increase the number of consultations are needed or desired, it seems reasonable to expect that increases on primary care consultations in urban Brazil are welfare enhancing. In our sample, the average number of primary care consultations per person per year is 0.4, which is significantly below the SUS recommendation of 2–3 visits/person/year [25].
While the approaches presented in the paper are applied to local health care systems (i.e. cities), the measurement of health care production and its associated efficiency can also be applied at the health care delivery unit level. This method would require output and input data at the hospital or clinic level. Such an approach would offer micro-level information that could inform budgetary decisions. Here, a word of caution is warranted. The direct application of production and efficiency estimates to reimbursement frameworks can be problematic as production models are limited by their ability to capture nuances of the health care delivery process. For example, if output measures are not adjusted for quality of care, the models may underestimate the return to investments and high-quality health care facilities could appear to be unproductive [26].
Adopting a more aggregated perspective that examines local healthcare systems, our models estimate elasticities and identify the existence of significant heterogeneities in urban Brazil. Our approach is able to identify that some local healthcare systems function well, while others do not. However, our approach does not inform the drivers of these results, nor the reasons and specific mechanisms underlying undesirable outcomes. In that, this study represents only the first step of a more comprehensive analysis of primary care. Our methods are able to identify cities with high-return health systems that may represent investment opportunities, and cities with diseconomies of scale whose local healthcare sector should be examined more closely. Future research is needed to investigate the role of characteristics of the city, such as sociodemographic composition (e.g. age, education, income), on determining the returns to scale in health care.
Many countries struggle with fiscal challenges and must overcome great difficulties to finance health care systems. This highlights the growing importance of efficient investments in primary health care, especially in fiscally challenged countries with large health care systems like Brazil. It is, therefore, fundamental to identify investments opportunities to leverage budgets and achieve maximum outcomes.
Flexible nonparametric production methods can help design health care policy. For the most part, the nonparametric literature applied to health care production has focused on envelopment techniques such as Data Envelopment Analysis (DEA) and Free Disposal Hull (FDH) to estimate a health care production frontier [10, 27,28,29,30,31,32]. However, envelopment estimators are sensitive to outliers and extreme values and, as a result, may deliver biased estimates of returns to scales, which compromises the ability to inform health care investments [33, 34].
This study shows how nonparametric methods can be used to inform public health policy. We estimate a LLLS production model that is, by construction, less sensitive to extreme values and outliers than envelopment estimators. The model is applied to a large dataset of Brazilian cities with more than 400 million consultations, 270 thousand physicians, and 11 thousand clinics, from 2012 to 2016.
We find that, while the results of the typically used CD parametric model suggest that average returns to medical capital and labor in urban Brazil are similar, the more flexible nonparametric estimates indicate that average returns to capital are almost 8 times larger than returns to labor. That is, capital investments promote, on average, higher uptake of primary care services. In order words, nonparametric results suggest that, on average, expanding the number of clinics, holding the number of physicians constant, is more effective than expanding the number of physicians, holding the number of clinics constant.
The nonparametric model allows us to go beyond average estimates and explore the heterogeneity in returns to capital and labor. We find that when the goal is to increase the uptake of primary care services in Brazil, investments in health care capital are more uncertain than expanding the number of physicians. Results reveal significant heterogeneity on returns to scale along the distribution of medical capital and labor, however, returns to capital infrastructure are generally higher than returns to physicians. Medical infrastructure complements physicians mainly when capital levels are high, but high capital levels may also induce diseconomies of scale when expanding the number of physicians at the top of the labor distribution.
A recent World Health Organization report shows that global health spending is on an upward trajectory [35]. This is especially important for Brazil. During the period 2012–2016, Brazil's annual health spending averages $1325 (international dollars) per capita, which ranks Brazil as the 58th country in health spending, globally. This level of investment represents only 28% of that from OECD countries, or 52% of the investments of Europe and Central Asia [2]. These statistics are striking considering that Brazil was the 7th largest economy in the world in 2012 (behind only U.S., China, Japan, Germany, U.K., and France). In fact, despite having economies of similar size, Brazil's health expenditure (per capita) is only 29% that of France.
Currently, the Brazilian federal government faces a tremendous amount of political and popular pressure to increase health investments and expand the SUS network. In a scenario of health expansion, understanding nonlinearities and complementarities between medical inputs is important as it allows policy makers to predict the returns of increasing the availability of an input, depending on the profile of each local market (e.g. scale and level of input complementarity). This information can be harnessed to determine, for instance, priority regions where investments have higher productivity.
The datasets generated and/or analysed during the current study are available in the DATASUS website: datasus.saude.gov.br/informacoes-de-saude-tabnet .
Henderson and Parmeter [18], in Chapter 5, offer further details and a comprehensive discussion of alternative parametric (Constant Elasticity of Substitution, Generalized Quadratic, and Generalized Leontief) and nonparametric (Local-Constant and Local-Polynomial) models.
Refer to Henderson & Parmeter for a detailed discussion of the LLLS estimator [18].
SUS data are available online at datasus.saude.gov.br/informacoes-de-saude-tabnet.
Refer to Electronic Supplemental Material for further details on measurements of capital, labor, and output.
It can be easily verified that \( \left(\frac{\partial A{K}^{\alpha }{L}^{\beta }}{\partial K}\right)\left(\frac{K}{A{K}^{\alpha }{L}^{\beta }}\right)=\alpha \) and \( \left(\frac{\partial A{K}^{\alpha }{L}^{\beta }}{\partial L}\right)\left(\frac{L}{A{K}^{\alpha }{L}^{\beta }}\right)=\beta \).
CNES:
National Registry of Healthcare Facilities
DATASUS:
SUS IT department
DEA:
Data Envelopment Analysis
FDH:
Free Disposal Hull
GDP:
LLLS:
Local-Linear Least Squares
LSCV:
Least Squares Cross Validation
NLS:
Nonlinear Least Squares
OLS:
SIA:
System of Outpatient Information
SUS:
Brazilian Unified Health System
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This work has benefited from comments and suggestions from Felipe Garcia, Marina Martins, Vitor Pereira, and Marcus Tolentino. All errors and omissions are our own.
No funding was received.
Department of Resource Economics and Environmental Sociology, University of Alberta, 515 General Services Building, Edmonton, AB, T6G 2H1, Canada
Bruno Wichmann
World Bank, SCN Quadra 2, Lote A, Ed. Corporate Financial Center, 7o Andar, Brasília, DF, CEP 70712-900, Brazil
Roberta Wichmann
Department of Public Health University of Brasília, Brasília, DF, Brazil
Both authors equally contributed to the manuscript.
Correspondence to Bruno Wichmann.
Disclaimer Statement: The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. The World Bank does not guarantee the accuracy of the data included in this work.
Wichmann, B., Wichmann, R. Nonparametric estimation of a primary care production function in urban Brazil. Health Econ Rev 10, 37 (2020). https://doi.org/10.1186/s13561-020-00294-9
Public healthcare investment
Returns to capital and labor
Heterogeneity
Nonlinearities
Complementarities | CommonCrawl |
\begin{document}
\title{Remarks on the $\Gamma $--regularization \\ of Non--convex and Non--semi--continuous \\ Functions on Topological Vector Spaces} \author{J.-B. Bru and W. de Siqueira Pedra} \date{\today } \maketitle
\begin{abstract} We show that the minimization problem of any non--convex and non--lower semi--continuous function on a compact convex subset of a locally convex real topological vector space can be studied via an associated convex and lower semi--continuous function $\Gamma \left( h\right) $. This observation uses the notion of $\Gamma $--regularization as a key ingredient. As an application we obtain, on any locally convex real space, a generalization of the Lanford III--Robinson theorem which has
only been proven for separable real Banach spaces. The latter is a characterization of subdifferentials of convex continuous functions. \\[0.3ex] {\small \textit{Keywords:} variational problems, non--linear analysis, non--convexity, \\ $\Gamma $--regularization, Lanford III -- Robinson theorem.}\\[0.3ex] {\small \textit{Mathematics subject classifications:} 58E30, 46N10, 52A07.} \end{abstract}
\section{Introduction and Main Results}
\setcounter{equation}{0}
Minimization problems $\inf \,h\left( K\right) $ on compact convex subsets $ K $ of a locally convex real (topological vector) space\footnote{ We assume throughout this paper that topological vector spaces are Hausdorff spaces, i.e., points in those spaces define closed sets.}
$\mathcal{X}$ are extensively studied for convex and lower semi--continuous real--valued functions $h$. See, for instance, \cite{Zeidler3}.
Such variational problems are, however, not systematically studied for \emph{ non}--convex and \emph{non}--lower semi--continuous real--valued functions $h$, except for a few specific functions. See for instance \cite{Mueller}. The aim of this paper is to show that -- independently of convexity or lower semi--continuity of functions $h$ -- the minimization problem $\inf \,h\left( K\right) $ on compact convex subsets $K$ of a locally convex real space $\mathcal{X}$ can be analyzed via another minimization problem $\inf \,\Gamma \left( h\right) \left( K\right) $ associated with a convex and lower semi--continuous function $\Gamma \left( h\right) $, for which various methods of analysis are available.
We are particularly interested in characterizing the following set of generalized minimizers of any real--valued function $h$ on a compact convex set $K $:
\begin{definition}[Set of generalized minimizers] \label{gamm regularisation copy(4)}\mbox{ }\newline Let $K$ be a (non--empty) compact convex subset of a locally convex real space $\mathcal{X}$ and $h:K\rightarrow \left( -\infty ,\infty \right] $ be any extended real--valued function. Then the set $\overline{\mathit{\Omega }\left( h,K\right) }\subset K$ of generalized minimizers of $h$ is the closure of the set \begin{equation*} \mathit{\Omega }\left( h,K\right) :=
\Big\{
x\in K:\exists \{x_{i}\}_{i\in I}\subset K\mathrm{\ \ with\ } x_{i}\rightarrow x\;\mathrm{and\;}\lim_{I}h(x_{i})=\inf \,h(K)
\Big\}
\end{equation*} of all limit points of approximating minimizers of $h$. \end{definition}
\noindent Here, $\{x_{i}\}_{i\in I}\subset K$ is per definition a net of \emph{approximating minimizers} when \begin{equation*} \underset{I}{\lim }\ h(x_{i})=\inf \,h(K). \end{equation*} Note that, for any compact set $K$, $\mathit{\Omega }\left( h,K\right) $ is non--empty because any net $\{x_{i}\}_{i\in I}\subset K$ converges along a subnet.
In order to motivate the issue here, observe that $\inf \,h\left( K\right) $ can always be studied via a minimization problem associated with a (possibly not convex, but) lower semi--continuous function $h_{0}$, known as the \emph{lower semi--continuous hull} of $h$:
\begin{lemma}[Minimization of real--valued functions -- I] \label{theorem trivial sympa 1 copy(2)}\mbox{ }\newline Let $K$ be any (non--empty) compact, convex, and metrizable subset of a locally convex real space $\mathcal{X}$ and $h:K\rightarrow \lbrack \mathrm{k },\infty ]$ be any extended real--valued
function with $\mathrm{k}\in \mathbb{R}$. Then there is a lower semi--continuous extended function
$h_{0}:K\rightarrow \lbrack \mathrm{k},\infty ]$ such that \begin{equation*} \inf \,h\left( K\right) =\inf \,h_{0}\left( K\right) \quad \RIfM@\expandafter\text@\else\expandafter\mbox\fi{and}\quad \mathit{\Omega }\left( h_{0},K\right) =\mathit{\Omega }\left( h,K\right) . \end{equation*} \end{lemma}
\noindent By lower semi--continuity, note that $\mathit{\Omega }\left( h_{0},K\right) $ corresponds to the set of usual minimizers of $h_{0}$. Note further that Lemma \ref{theorem trivial sympa 1 copy(2)} implies -- \ in the case $K$ is metrizable -- that $\mathit{\Omega }\left( h,K\right) $ is closed, again by lower semi--continuity of $h_{0}$. The proof of this lemma is straightforward and is given in Section \ref{Section Proofs-I} for completeness.
This result has two drawbacks: The compact convex set $K$ must be \emph{ metrizable} in the elementary proof we give here and, more important, the lower semi--continuous hull $h_{0}$ of $h$ is \emph{generally not convex}. We give below a more elaborate result and show that both problems mentioned above can be overcome by using the so--called $\Gamma $--regularization of extended real--valued functions.
The last is defined from the space $\mathrm{A}\left( \mathcal{X}\right) $ of all affine continuous real--valued functions on a locally convex real space\ $\mathcal{X}$ as follows (cf. \cite[Eq. (1.3) in Chapter I]{Alfsen}):
\begin{definition}[$\Gamma $--regularization of real--valued functions] \label{gamm regularisation}\mbox{ }\newline For any extended real--valued function $h:K\rightarrow \lbrack \mathrm{k},\infty ]$ defined on a (non--empty) compact convex subset $K\subset \mathcal{X}$, its $\Gamma $ --regularization $\Gamma \left( h\right) $ on $K$ is the function defined as the supremum over all affine and continuous minorants $m:\mathcal{X} \rightarrow \mathbb{R}$ of $h$, i.e., for all $x\in K$, \begin{equation*} \Gamma \left( h\right) \left( x\right) :=\sup \left\{ m(x):m\in \mathrm{A}
\left( \mathcal{X}\right) \;\RIfM@\expandafter\text@\else\expandafter\mbox\fi{and }m|_{K}\leq h\right\} . \end{equation*} \end{definition}
\noindent Since the $\Gamma $--regularization $\Gamma \left( h\right) $ of a extended real--valued function $h$ is a supremum over continuous functions, $\Gamma \left( h\right) $ is a \emph{convex} and \emph{lower semi--continuous} function on $K$. For convenience, note that we identify extended
real--valued functions $g $
only defined on a\ convex compact subset $K\subset \mathcal{X}$ of the locally convex real space\ $\mathcal{X}$ with its (trivial) extension
$g_{\mathrm{ext}}$ to the whole space\ $\mathcal{X}$ defined by \begin{equation*} g_{\mathrm{ext}}(x):=\left\{ \begin{array}{c} g(x) \\ \infty \end{array} \begin{array}{l} \mathrm{for}\ x\in K, \\ \mathrm{otherwise.} \end{array} \right. \end{equation*} Clearly, with this prescription $g$ is lower semi--continuous (resp. convex) on $K$ iff $g$ is lower semi--continuous (resp. convex) on $\mathcal{X}$.
We prove in Section \ref{Section Proofs-II} the main result of this paper:
\begin{theorem}[Minimization of real--valued functions -- II] \label{theorem trivial sympa 1}\mbox{ }\newline Let $K$ be any (non--empty) compact convex subset of a locally convex real space $\mathcal{X}$ and $h:K\rightarrow \lbrack \mathrm{k},\infty ]$ be any extended real--valued function with $\mathrm{k}\in \mathbb{R}$. Then we have that:\newline \emph{(i)} \begin{equation*} \inf \,h\left( K\right) =\inf \,\Gamma \left( h\right) \left( K\right) . \end{equation*} \emph{(ii) }The set $\mathit{M}$ of minimizers of $\Gamma \left( h\right) $ over $K$ equals the closed convex hull of the set $\mathit{\Omega }\left( h,K\right) $ of generalized minimizers of $h$ over $K$, i.e., \begin{equation*} \mathit{M}=\overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) . \end{equation*} \end{theorem}
This general fact related to the minimization of non--convex and non--lower semi--continuous real--valued
functions on compact convex sets has not been observed\footnote{Assertion (i) is, however, trivial.} before, at least to our knowledge. Note that related results were obtained in \cite{Benoist}\footnote{We thank the referee for pointing out this reference.} for $\mathcal{X}=\mathbb{R}^n$. It turns out to be extremely useful. It is, for instance, an essential argument in the proof given in \cite{BruPedra2} of the validity of the so--called Bogoliubov approximation on the level of states for a class of models for fermions on the lattice. This problem, well--known in mathematical physics, was first addressed by Ginibre \cite[p. 28]{Ginibre} in 1968 and is still open for many physically important models.
Then, by using the theory of compact convex subsets of locally convex real spaces $\mathcal{X}$ (see, e.g., \cite{Alfsen}), Theorem \ref{theorem trivial sympa 1} yields a characterization of the set $\overline{\mathit{ \Omega }\left( h,K\right) }$ of all generalized minimizers of $h$ over $K$. Indeed, one important observation concerning locally convex real spaces $ \mathcal{X}$ is that any compact convex subset $K\subset \mathcal{X}$ is the closure of the convex hull of the (non--empty) set $\mathcal{E}(K)$ of its extreme points, i.e., of the points which cannot be expressed as (non--trivial) convex combinations of other elements in $K$. This is the Krein--Milman theorem, see, e.g., \cite[Theorems 3.4 (b) and 3.23]{Rudin}. In fact, among all subsets $Z\subset K$ generating $K$, $\mathcal{E}(K)$ is -- in a sense -- the smallest one. This is the Milman theorem, see, e.g., \cite[Theorem 3.25]{Rudin}. It follows from Theorem \ref{theorem trivial sympa 1} together with \cite[Theorems 3.4 (b), 3.23, 3.25]{Rudin} that extreme points of the compact convex set
$\mathit{M}$ of minimizers of $\Gamma \left(h\right) $ over $K$ are generalized minimizers of $h$:
\begin{theorem}[Minimization of real-valued functions -- III] \label{theorem trivial sympa 1 copy(1)}\mbox{ }\newline Let $K$ be any (non--empty) compact convex subset of a locally convex real space $\mathcal{X}$ and $h:K\rightarrow \lbrack \mathrm{k},\infty ]$ be any extended real--valued
function with $\mathrm{k}\in \mathbb{R}$. Then extreme points of the compact convex set $\mathit{M}$ belong to the set of generalized minimizers of $h$, i.e., $\mathcal{E}\left( \mathit{M}\right) \subseteq \overline{ \mathit{\Omega }\left( h,K\right) }$. \end{theorem}
\noindent This last result makes possible a \emph{full characterization} of the closure of the set $\mathit{\Omega }\left( h,K\right) $ in the following sense: Since $\mathit{M}$ is compact and convex, we can study the minimization problem $\inf \,h\left( K_{\mathit{M}}\right) $ for any closed (and hence compact) convex subset $K_{\mathit{M}}\subset \mathit{M}$. Applying Theorem \ref{theorem trivial sympa 1} we get \begin{equation}
\inf \,h\left( K_{\mathit{M}}\right) =\inf \,\Gamma \left( h|_{K_{\mathit{M} }}\right) \left( K_{\mathit{M}}\right) . \label{full characterization1} \end{equation} If \begin{equation*} \inf \,h\left( K_{\mathit{M}}\right) =\inf \,h\left( K\right) \end{equation*} then, by Theorem \ref{theorem trivial sympa 1 copy(1)}, \begin{equation*} \mathcal{E}\left( \mathit{M}_{K_{\mathit{M}}}\right) \subseteq \overline{
\mathit{\Omega }\left( h|_{K_{\mathit{M}}},K_{\mathit{M}}\right) }\subseteq \overline{\mathit{\Omega }\left( h,K\right) }, \end{equation*}
where $\mathit{M}_{K_{\mathit{M}}}$ is the compact convex set of minimizers of $\Gamma \left( h|_{K_{\mathit{M}}}\right) $ over $K_{\mathit{M}}\subset \mathit{M}$. In general, $\mathcal{E}\left( \mathit{M}_{K_{\mathit{M} }}\right) \backslash \mathcal{E}\left( \mathit{M}\right) \neq \emptyset $ because $\mathit{M}_{K_{\mathit{M}}}$ is not necessarily a face of $\mathit{M }$. Thus we discover in this manner new points of $\overline{\mathit{\Omega } \left( h,K\right) }$ not contained in $\mathcal{E}\left( \mathit{M}\right) $ . Choosing a sufficiently large family $\{K_{\mathit{M}}\}$ of closed convex subsets of $\mathit{M}$ we can exhaust the set $\overline{\mathit{\Omega } \left( h,K\right) }$ through the union $\cup $ $\{\mathcal{E}\left( \mathit{M }_{K_{\mathit{M}}}\right) \}$. Note that this construction can be performed in an inductive way: For each set $\mathit{M}_{K_{\mathit{M}}}$ of minimizers consider further closed convex subsets $K_{\mathit{M}}^{\prime }\subset $ $\mathit{M}_{K_{\mathit{M}}}$. The art consists in choosing the family $\{K_{\mathit{M}}\}$ appropriately, i.e., it should be as small as possible and the extreme points of $\mathit{M}_{K_{\mathit{M}}}$ should possess some reasonable characterization. Of course, the latter heavily depends on the function $h$ and on particular properties of the compact convex set $K$ (e.g., density of $\mathcal{E}(K)$, metrizability, etc.).
To close this section we recall that the $\Gamma $--regularization $\Gamma \left( h\right) $ of a function $h$ on $K$ equals its twofold \emph{ Legendre--Fenchel transform} -- also called the \emph{biconjugate } (function) of $h$. See, for instance, \cite[Paragraph 51.3]{Zeidler3}. Indeed, $\Gamma \left( h\right) $ is the largest lower semi--continuous and convex minorant of $h$ (cf. Corollary \ref{Biconjugate} ). However, in contrast to the $\Gamma $--regularization the notion of Legendre--Fenchel transform requires the use of dual pairs (cf. Definition \ref{dual pairs}). Since, for any locally convex real space $\mathcal{X}$ together with the space $\mathcal{X}^{\ast }$ of linear continuous funtionals $\mathcal{X} \to \mathbb{R}$ (dual space) equipped with the weak$^{\ast }$--topology, $(\mathcal{X},\mathcal{X}^{\ast })$ is a dual pair, the Legendre--Fenchel transform can be defined on any locally convex real space $\mathcal{X}$ as follows:
\begin{definition}[The Legendre--Fenchel transform] \label{Legendre--Fenchel transform}\mbox{ }\newline Let $K$ be a (non--empty) compact convex subset of a locally convex real space $\mathcal{X}$. For any extended real--valued
function $h:K\rightarrow \left( -\infty,\infty \right] $, $h\not\equiv \infty$,
its Legendre--Fenchel transform $h^{\ast }$ is the convex weak$^{\ast }$--lower semi--continuous extented function from $\mathcal{X}^{\ast }$ to $\left( -\infty ,\infty \right] $ defined, for any $x^{\ast }\in \mathcal{ X}^{\ast }$, by \begin{equation*} h^{\ast }\left( x^{\ast }\right) :=\underset{x\in K}{\sup }\left\{ x^{\ast }\left( x\right) -h\left( x\right) \right\} . \end{equation*} \end{definition} See also \cite[Definition 51.1]{Zeidler3}. Note that, together with its weak$^{\ast }$--topology, the dual space $\mathcal{X}^{\ast }$ of any locally convex space $\mathcal{X}$ is also a locally convex space, see \cite[Theorems 3.4 (b) and 3.10]{Rudin}. Therefore, in case nothing is further specified, the space $\mathcal{X} ^{\ast }$ is always equipped with its weak$^{\ast }$--topology.
The Legendre--Fenchel transform is strongly related to the notion of Fenchel subdifferentials (see also \cite{Phelps-conv}):
\begin{definition}[Fenchel subdifferentials] \label{tangent functional}\mbox{ }\newline Let $h:\mathcal{X}\rightarrow \mathbb{(-\infty },\infty ]$ be any extended real--valued function on a real topological vector space $\mathcal{X}$. A continuous linear functional $\mathrm{d}h_{x}\in \mathcal{X}^{\ast }$ is said to be a Fenchel subgradient (or tangent) of the function $h$ at $x\in \mathcal{X}$ iff, for all $x^{\prime }\in \mathcal{X}$, $h(x+x^{\prime })\geq h(x)+\mathrm{d} h_{x}(x^{\prime })$. The set $\partial h(x)\subset \mathcal{X}^{\ast }$ of Fenchel subgradients of $h$ at $x$ is called Fenchel
subdifferential of $h$ at $x$. \end{definition}
\noindent Theorem \ref{theorem trivial sympa 1} establishes a link between generalized minimizers and Fenchel subdifferentials:
\begin{theorem}[Subdifferentials of continuous convex functions -- I] \label{theorem trivial sympa 3}Let $K$ be any (non--empty) compact convex subset of a locally convex real space $\mathcal{X}$ and $h:K\rightarrow \lbrack \mathrm{k},\infty ]$ be any extended real--valued
function with $\mathrm{k}\in \mathbb{R}$. Then the Fenchel subdifferential
$\partial h^{\ast }(x^{\ast })\subset \mathcal{X}$ of $h^{\ast }$
at the point $x^{\ast }\in \mathcal{X}^{\ast }$ is the (non--empty) compact convex set \begin{equation*} \partial h^{\ast }(x^{\ast })=\overline{\mathrm{co}}\left( \mathit{\Omega } \left( h-x^{\ast },K\right) \right) . \end{equation*} \end{theorem}
\noindent This last result -- proven in Section \ref{Section Proofs-III} -- generalizes the Lanford III--Robinson theorem \cite[Theorem 1]{LanRob} which has only been proven for separable real Banach spaces $\mathcal{X}$ and continuous convex functions $h:\mathcal{X}\rightarrow \mathbb{R}$, cf. Theorem \ref{Land.Rob}.
Indeed, for any extended real--valued function
$h$ from a compact convex subset $K\subset \mathcal{X}$
of a locally convex real space $\mathcal{X}$ to $\left( -\infty ,\infty \right] $, let \begin{equation*} \mathcal{Y}^{\ast }:=\left\{ x^{\ast }\in \mathcal{X}^{\ast }:h^{\ast }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ has a unique Fenchel subgradient }\mathrm{d}h_{x^{\ast }}^{\ast }\in \mathcal{X} \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ at }x^{\ast }\right\} . \end{equation*} For all $x^{\ast }\in \mathcal{X}^{\ast }$ and any open neighborhood $ \mathcal{V}$ of\ $\{0\}\subset \mathcal{X}^{\ast }$, we also define the set \begin{equation} \mathcal{T}_{x^{\ast },\mathcal{V}}:=\overline{\left\{ \mathrm{d}h_{y^{\ast }}^{\ast }:y^{\ast }\in \mathcal{Y}^{\ast }\cap (x^{\ast }+\mathcal{V)} \right\} }^{\mathcal{X}}\subset \mathcal{X} \label{set t voisinage} \end{equation} and denote by $\mathcal{T}_{x^{\ast }}$ the intersection \begin{equation} \mathcal{T}_{x^{\ast }}:=\bigcap\limits_{\mathcal{V}\ni 0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ open}} \mathcal{T}_{x^{\ast },\mathcal{V}}. \label{set t voinagebis} \end{equation} Here, $\overline{\;\cdot \;}^{\mathcal{X}}$ denotes the closure w.r.t. the topology of $\mathcal{X}$. Then we observe first that Theorem \ref{theorem trivial sympa 3} implies that the set $\partial h^{\ast }(x^{\ast })\subset \mathcal{X}$ of Fenchel subgradients of $h^{\ast }$ at the point $x^{\ast }\in \mathcal{X}^{\ast }$ is included in the closed convex hull of the set $\mathcal{T}_{x^{\ast }}$ provided $\mathcal{Y}^{\ast }$ is dense in $\mathcal{X}^{\ast }$ (cf. Section \ref{Section Proofs-IV}):
\begin{corollary}[Subdifferentials of continuous convex functions -- II] \label{corollary explosion lanford-robinson}Let $K$ be any (non--empty) compact convex subset of a locally convex real space $\mathcal{X}$ and $ h:K\rightarrow \lbrack \mathrm{k},\infty ]$ be any extended real--valued
function with $\mathrm{k}\in \mathbb{R}$. If $\mathcal{Y}^{\ast }$ is dense in $\mathcal{X}^{\ast }$ then, for any $x^{\ast }\in \mathcal{X}^{\ast }$, \begin{equation*} \partial h^{\ast }(x^{\ast })\subseteq \overline{\mathrm{co}}\left( \mathcal{T }_{x^{\ast }}\right) . \end{equation*} \end{corollary}
\noindent This last result applied on separable Banach spaces yields, in turn, the following assertion (cf. Section \ref{Section Proofs-IV copy(1)}):
\begin{corollary}[The Lanford III--Robinson theorem] \label{corollary explosion lanford-robinson copy(1)}\mbox{ }\newline Let $\mathcal{X}$ be a separable Banach space and $h:\mathcal{X}\rightarrow \mathbb{R}$ be any convex function which is globally Lipschitz continuous. If the set \begin{equation*} \mathcal{Y}:=\left\{ x\in \mathcal{X}:h\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ has a unique Fenchel subgradient } \mathrm{d}h_{x}\in \mathcal{X}^{\ast }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ at }x\right\} \end{equation*} is dense in $\mathcal{X}$ then the Fenchel subdifferential $\partial h(x)$ of $h$, at any $x\in \mathcal{X}$, is the weak$^{\ast }$--closed convex hull of the set $\mathcal{Z}_{x}$. Here, at fixed $x\in \mathcal{X}$, $\mathcal{Z}_{x}$ is the set of functionals $x^{\ast }\in \mathcal{X}^{\ast }$ such that there is a net $\{x_{i}\}_{i\in I}$ in $\mathcal{Y}$ converging to $x$ with the property that the unique Fenchel subgradient $\mathrm{d}h_{x_{i}}\in \mathcal{X}^{\ast }$ of $h$ at $x_{i}$ converges towards $x^{\ast }$ in the weak$^{\ast}$--topology. \end{corollary}
\noindent Recall that the Mazur theorem shows that the set $\mathcal{Y}$ on which a continuous convex function $h$ is G\^{a}teaux differentiable, i.e., the set $\mathcal{Y}$ for which $h$ has exactly one Fenchel subgradient \textrm{d}$h_{x}\in \mathcal{X}^{\ast }$ at any $x\in \mathcal{Y}$, is dense in a separable Banach space $\mathcal{X}$, cf. Theorem \ref{Mazur} and Remark \ref{Mazur remark}. Therefore, for globally Lipschitz continuous and convex functions, the Lanford III--Robinson theorem \cite[Theorem 1] {LanRob} (cf. Theorem \ref{Land.Rob}) directly follows from Corollary \ref {corollary explosion lanford-robinson copy(1)}. Observe that, in which concerns Fenchel subdifferentials of convex continuous functions on Banach spaces, the case of global Lipschitz continuous functions is already the most general case: For any continuous convex function $h$ on a Banach space $ \mathcal{X}$ and any $x\in \mathcal{X}$, there are ${\Greekmath 0122} >0$ and a globally Lipschitz continuous convex function $g$ such that $g\left( y\right) =h\left( y\right) $ whenever $\left\Vert x-y\right\Vert <{\Greekmath 0122} $. In particular, $g$ and $h$ have the same Fenchel subgradients at $x$. Remark, indeed, that continuous convex functions $h$ on a Banach space $\mathcal{X}$ are locally Lipschitz continuous and an example of such a global Lipschitz continuous convex function is given by \begin{equation*} g\left( x\right) :=\inf \left\{ z\in \mathbb{R}:\left( x,z\right) \in \left[ \mathrm{epi}\left( h\right) +\mathcal{C}_{{\Greekmath 010B} }\right] \right\} , \end{equation*} for sufficiently small ${\Greekmath 010B} >0$. Here, \begin{equation*} \mathcal{C}_{{\Greekmath 010B} }:=\left\{ \left( x,z\right) \in \mathcal{X}\times \mathbb{R}:z\geq 0,\left\Vert x\right\Vert \leq {\Greekmath 010B} z\right\} \end{equation*} and $\mathrm{epi}\left( h\right) $ is the epigraph of $h$ defined by \begin{equation*} \mathrm{epi}\left( h\right) :=\left\{ \left( x,z\right) \in \mathcal{X} \times \mathbb{R}:z\geq f\left( x\right) \right\} . \end{equation*}
The rest of the paper is structured as follows. Section \ref{Section Proofs} gives the detailed proofs of Lemma \ref{theorem trivial sympa 1 copy(2)}, Theorems \ref{theorem trivial sympa 1}, \ref{theorem trivial sympa 3}, and Corollaries \ref{corollary explosion lanford-robinson}--\ref{corollary explosion lanford-robinson copy(1)}. Then, Section \ref{Concluding remarks} discusses an additional observation which is relevant in the context of minimization of non--convex or non--semi--continuous functions and which does not seem to have been observed before. Indeed, Lemma \ref{Bauer maximum principle bis} gives an extension of the Bauer maximum principle (Lemma \ref {Bauer maximum principle}). Finally, Section \ref{Section appendix} is a concise appendix about dual pairs, barycenters in relation with the $\Gamma $--regularization, the Mazur theorem, and the Lanford III--Robinson theorem.
\section{Proofs\label{Section Proofs}}
This section gives the detailed proofs of Lemma \ref{theorem trivial sympa 1 copy(2)}, Theorems \ref{theorem trivial sympa 1}, \ref{theorem trivial sympa 3}, and Corollaries \ref{corollary explosion lanford-robinson}--\ref {corollary explosion lanford-robinson copy(1)}. Up to Corollary \ref {corollary explosion lanford-robinson copy(1)}, we will always assume that $ K $ is a (non--empty) compact convex subset of a locally convex real space $ \mathcal{X}$ and $h:K\rightarrow \lbrack \mathrm{k},\infty ]$ is any extended real--valued
function with $\mathrm{k}\in \mathbb{R}$. In Lemma \ref{theorem trivial sympa 1 copy(2)} the metrizability of the topology on $K$ is also assumed. In Corollary \ref{corollary explosion lanford-robinson copy(1)} $\mathcal{X}$ is a separable Banach space and $h:\mathcal{X}\rightarrow \mathbb{R}$ is any globally Lipschitz continuous convex function.
\subsection{Proof of Lemma \protect\ref{theorem trivial sympa 1 copy(2)} \label{Section Proofs-I}}
Because the subset $K\subset \mathcal{X}$ is metrizable and compact, it is sequentially compact and we can restrict ourselves to sequences instead of more general nets. Using any metric $d(x,y)$ on $K$ generating the topology we define, at fixed ${\Greekmath 010E} >0$, the extended real--valued function $h_{{\Greekmath 010E} }$ from $K$ to $[\mathrm{k},\infty ]$ by \begin{equation*} h_{{\Greekmath 010E} }\left( x\right) :=\inf \,h(\mathcal{B}_{{\Greekmath 010E} }\left( x\right) ) \end{equation*} for any $x\in K$, where \begin{equation} \mathcal{B}_{{\Greekmath 010E} }\left( x\right) :=\left\{ y\in K:\ d(x,y)<{\Greekmath 010E} \right\} \label{ball} \end{equation} is the ball (in $K$) of radius ${\Greekmath 010E} >0$ centered at $x\in K$. The family $ \{h_{{\Greekmath 010E} }\left( x\right) \}_{{\Greekmath 010E} >0}$ of extended real--valued
functions is clearly increasing as ${\Greekmath 010E} \searrow 0$ and is bounded from above by $h(x)$. Therefore, for any $x\in K$, the limit of $h_{{\Greekmath 010E} }\left( x\right) \geq \mathrm{k}$ as ${\Greekmath 010E} \searrow 0$ exists and defines an extended
real--valued function \begin{equation*} x\mapsto h_{0}\left( x\right) :=\underset{{\Greekmath 010E} \searrow 0}{\lim } \,h_{{\Greekmath 010E} }\left( x\right) \end{equation*} from $K$ to $[\mathrm{k},\infty ]$.
In fact, this construction is well--known and the function $h_{0}$ is called the \emph{lower semi--continuous hull} of $h$ as it is a lower semi--continuous extended real--valued function
from $K$ to $[\mathrm{k},\infty ]$. Indeed, for all ${\Greekmath 010E} >0$ and any sequence $\{x_{n}\}_{n=1}^{\infty }\subset K$ converging to $x\in K$, there is $N_{{\Greekmath 010E} }>0$ such that, for all $ n>N_{{\Greekmath 010E} }$, $x_{n}\in \mathcal{B}_{{\Greekmath 010E} /2}\left( x\right) $ which implies that $\mathcal{B}_{{\Greekmath 010E} /2}\left( x_{n}\right) \subset \mathcal{B} _{{\Greekmath 010E} }\left( x\right) $. In particular, $h_{{\Greekmath 010E} }\left( x\right) \leq h_{{\Greekmath 010E} /2}\left( x_{n}\right) $ for all ${\Greekmath 010E} >0$ and $n>N_{{\Greekmath 010E} }$. Since the family $\{h_{{\Greekmath 010E} }\left( x\right) \}_{{\Greekmath 010E} >0}$ defines an increasing sequence as ${\Greekmath 010E} \searrow 0$, it follows that \begin{equation*} h_{{\Greekmath 010E} }\left( x\right) \leq \ \liminf_{n\rightarrow \infty }h_{0}\left( x_{n}\right) \end{equation*} for any ${\Greekmath 010E} >0$ and $x\in K$. In the limit ${\Greekmath 010E} \searrow 0$ the latter yields the lower semi--continuity of the extended real--valued
function $h_{0}$ on $K $. Moreover, \begin{equation} h_{0}\left( x\right) \geq h_{{\Greekmath 010E} }\left( x\right) \geq \inf \,h(K)\geq \mathrm{k}>-\infty \label{definition de h lower3bis} \end{equation} for any $x\in K$ and ${\Greekmath 010E} >0$.
We observe now that $h$ and $h_{0}$ have the same infimum on $K$: \begin{equation} \inf h_{0}\left( K\right) =\inf h(x). \label{definition de h lower3} \end{equation} This can be seen by observing first that there is $y\in K$ such that \begin{equation} \inf h_{0}\left( K\right) =h_{0}\left( y\right) \label{definition de h lower4} \end{equation} because of the lower semi--continuity of $h_{0}$. Since $h_{{\Greekmath 010E} }\leq h$ on $K$ for any ${\Greekmath 010E} >0$, we have $h_{0}\leq h\ $on $K$, which combined with (\ref{definition de h lower3bis}) and (\ref{definition de h lower4}) yields Equality (\ref{definition de h lower3}).
Additionally, for all ${\Greekmath 010E} >0$ and any minimizer $y\in K$ of $h_{0}$ over $K$, there is a sequence $\{x_{{\Greekmath 010E} ,n}\}_{n=1}^{\infty }\subset \mathcal{B }_{{\Greekmath 010E} }\left( y\right) $ of approximating minimizers of $h$ over $ \mathcal{B}_{{\Greekmath 010E} }\left( y\right) $, that is, \begin{equation*} h_{{\Greekmath 010E} }\left( y\right) :=\inf \,h(\mathcal{B}_{{\Greekmath 010E} }\left( y\right) )= \underset{n\rightarrow \infty }{\lim }h(x_{{\Greekmath 010E} ,n})\leq h(y). \end{equation*} We can assume without loss of generality that \begin{equation*}
d(x_{{\Greekmath 010E} ,n},y)\leq {\Greekmath 010E} \mathrm{\quad and\quad }|h(x_{{\Greekmath 010E}
,n})-h_{{\Greekmath 010E} }\left( y\right) |\leq 2^{-n} \end{equation*} for all $n\in \mathbb{N}$ and all ${\Greekmath 010E} >0$. Note that $h_{{\Greekmath 010E} }\left( y\right) \rightarrow $ $h_{0}\left( y\right) $ as ${\Greekmath 010E} \searrow 0$. Thus, by taking any function $p({\Greekmath 010E} )\in \mathbb{N}$ satisfying $p({\Greekmath 010E} )>{\Greekmath 010E} ^{-1}$ we obtain that $x_{{\Greekmath 010E} ,p({\Greekmath 010E} )}$ converges to $y\in K$ as ${\Greekmath 010E} \searrow 0$ with the property that $h(x_{{\Greekmath 010E} ,p({\Greekmath 010E} )})$ converges to $h_{0}\left( y\right) $. Using Equalities (\ref{definition de h lower3}) and (\ref{definition de h lower4}) we obtain that all minimizers of (\ref{definition de h lower4}) are generalized minimizers of $h$, i.e., \begin{equation*} \mathit{\Omega }\left( h_{0},K\right) \subseteq \mathit{\Omega }\left( h,K\right) . \end{equation*} The converse inclusion \begin{equation*} \mathit{\Omega }\left( h,K\right) \subseteq \mathit{\Omega }\left( h_{0},K\right) \end{equation*} is straightforward because one has the inequality $h_{0}\leq h$ on $K$ as well as Equality (\ref{definition de h lower3}).
\subsection{Proof of Theorem \protect\ref{theorem trivial sympa 1}\label {Section Proofs-II}}
The assertion (i) of Theorem \ref{theorem trivial sympa 1} is a standard result. Indeed, by Definition \ref{gamm regularisation}, $\Gamma \left( h\right) \leq h$ on $K$ and thus \begin{equation*} \inf \,\Gamma \left( h\right) \left( K\right) \leq \inf \,h\left( K\right) . \end{equation*} The converse inequality is derived by restricting the supremum in Definition \ref{gamm regularisation} to constant maps $m$ from $\mathcal{X}$ to $ \mathbb{R}$ with $\mathrm{k}\leq m\leq h$ on $K$.
Observe that the variational problem $\inf \,\Gamma \left( h\right) (K)$ has minimizers and the set $\mathit{M}=\mathit{\Omega }\left( \Gamma \left( h\right) ,K\right) $ of all minimizers of $\Gamma \left( h\right) $ is convex and compact. For any $y\in \mathit{\Omega }\left( h,K\right) $, there is a net $\left\{ x_{i}\right\} _{i\in I}\subset K$ of approximating minimizers of $h$ on $K$ converging to $y$. In particular, since the function $\Gamma \left( h\right) $ is lower semi--continuous and $\Gamma \left( h\right) \leq h$ on $ K$, we have that \begin{equation*} \Gamma \left( h\right) (y)\leq \underset{I}{\liminf }\,\Gamma \left( h\right) (x_{i})\leq \underset{I}{\lim }\,h(x_{i})=\inf \,h(K)=\inf \,\Gamma \left( h\right) (K), \end{equation*} i.e., $y\in \mathit{M}$. Since $\mathit{M}$ is convex and compact, we obtain that \begin{equation} \mathit{M}\supset \overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) . \label{inclusion1} \end{equation} So, we prove now the converse inclusion. We can assume without loss of generality that $\overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) \neq K$ since otherwise there is nothing to prove. We show\ next that, for any $x\in K\backslash \overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) $, we have $x\notin $ $\mathit{M}$ .
As $\overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) $ is a closed set of a locally convex real space $\mathcal{X}$, for any $x\in K\backslash \overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right)$, there is an open and convex neighborhood $\mathcal{V}_{x}\subset $ $\mathcal{X}$ of $\{0\}\subset \mathcal{X}$ which is symmetric, i.e., $ \mathcal{V}_{x}=-\mathcal{V}_{x}$, and which satisfies \begin{equation*} \mathcal{G}_{x}\cap \left[ \{x\}+\mathcal{V}_{x}\right] =\emptyset \end{equation*} with \begin{equation*} \mathcal{G}_{x}:=K\cap \left[ \overline{\mathrm{co}}\left( \mathit{\Omega } \left( h,K\right) \right) +\mathcal{V}_{x}\right] . \end{equation*} This follows from \cite[Theorem 1.10]{Rudin} together with the fact that each neighborhood of $\{0\}\subset \mathcal{X}$ contains some open and convex neighborhood of $\{0\}\subset \mathcal{X}$ because $\mathcal{X}$ is locally convex. Observe also that any one--point set $\{x\}\subset $ $ \mathcal{X}$ is trivially compact.
For any neighborhood $\mathcal{V}_{x}$ of $\{0\}\subset \mathcal{X}$ in a locally convex real space, there is another convex, symmetric, and open neighborhood $\mathcal{V}_{x}^{\prime }$ of $\{0\}\subset \mathcal{X}$ such that $[\mathcal{V}_{x}^{\prime }+\mathcal{V}_{x}^{\prime }]\subset \mathcal{V }_{x}$, see proof of \cite[Theorem 1.10]{Rudin}. Let \begin{equation*} \mathcal{G}_{x}^{\prime }:=K\cap \left[ \overline{\mathrm{co}}\left( \mathit{ \Omega }\left( h,K\right) \right) +\mathcal{V}_{x}^{\prime }\right] . \end{equation*} Then the following inclusions hold: \begin{equation} \overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) \subset \mathcal{G}_{x}^{\prime }\subset \overline{\mathcal{G}_{x}^{\prime }} \subset \mathcal{G}_{x}\subset \overline{\mathcal{G}_{x}}\subset K\backslash \{x\}. \label{eq sup} \end{equation} Since $K$, $\mathcal{V}_{x}$, $\mathcal{V}_{x}^{\prime }$, and $\overline{ \mathrm{co}} \left( \mathit{\Omega }\left( h,K\right) \right) $ are all convex sets, $\mathcal{G}_{x}$ and $\mathcal{G}_{x}^{\prime }$ are also convex. Seen as subsets of $K$ they are open neighborhoods of $\overline{ \mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) $.
The set $\mathcal{X}$ is a Hausdorff space and thus any compact subset $K$ of $\mathcal{X}$ is a normal space. By Urysohn lemma, there is a continuous function \begin{equation*} f_{x}:K\rightarrow \lbrack \inf h(K),\inf h(K\backslash \mathcal{G} _{x}^{\prime })] \end{equation*} satisfying $f_{x}\leq h$ and \begin{equation*} f_{x}\left( y\right) =\left\{ \begin{array}{ll} \inf h(K) & \mathrm{for\ }y\in \overline{\mathcal{G}_{x}^{\prime }}. \\ \inf h(K\backslash \mathcal{G}_{x}^{\prime }) & \mathrm{for\ }y\in K\backslash \mathcal{G}_{x}. \end{array} \right. \end{equation*} By compactness of $K\backslash \mathcal{G}_{x}^{\prime }$ and the inclusion $\mathit{\Omega }\left( h,K\right) \subset \mathcal{G}_{x}^{\prime }$, observe that \begin{equation*} \inf h(K\backslash \mathcal{G}_{x}^{\prime })>\inf h(K). \end{equation*} Then we have per construction that \begin{equation} f_{x}(\overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) )=\{\inf h(K)\} \label{Omega f sympa} \end{equation} and \begin{equation} f_{x}^{-1}(\inf h(K))=\mathit{\Omega }\left( f_{x},K\right) \subset \mathcal{ G}_{x} \label{Omega f sympabis} \end{equation} for any $x\in K\backslash \overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) $.
We use now the $\Gamma $--regularization $\Gamma \left( f_{x}\right) $ of $ f_{x}$ on the set $K$ and denote by $\mathit{M}_{x}=\mathit{\Omega }\left( \Gamma (f_{x}),K\right) $ its non--empty set of minimizers over $K$. Applying Theorem \ref{Thm - Corollary I.3.6}, for any $y\in \mathit{M}_{x}$, we have a probability measure ${\Greekmath 0116} _{y}\in M_{1}^{+}(K)$\ on $K$ with barycenter $y$ such that \begin{equation} \Gamma \left( f_{x}\right) \left( y\right) =\int_{K}\mathrm{d}{\Greekmath 0116} _{y}(z)\;f_{x}\left( z\right) . \label{herve bis} \end{equation} As $y\in \mathit{M}_{x}$, i.e., \begin{equation} \Gamma \left( f_{x}\right) \left( y\right) =\inf \,\Gamma \left( f_{x}\right) (K)=\inf f_{x}(K), \label{herve 2} \end{equation} we deduce from (\ref{herve bis}) that \begin{equation*} {\Greekmath 0116} _{y}(\mathit{\Omega }\left( f_{x},K\right) )=1 \end{equation*} and it follows that $y\in $ $\overline{\mathrm{co}}\left( \mathit{\Omega } \left( f_{x},K\right) \right) $, by Theorem \ref{thm barycenter}. Using ( \ref{Omega f sympabis}) together with the convexity of the open neighborhood $\mathcal{G}_{x}$ of $\overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) $ we thus obtain \begin{equation} \mathit{M}_{x}\subset \overline{\mathrm{co}}\left( \mathit{\Omega }\left( f_{x},K\right) \right) \subset \overline{\mathcal{G}_{x}} \label{herve 3} \end{equation} for any $x\in K\backslash \overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) $.
We remark now that the inequality $f_{x}\leq h$ on $K$ yields $\Gamma \left( f_{x}\right) \leq \Gamma \left( h\right) $ on $K$ because of Corollary \ref {Biconjugate}. As a consequence, it results from (i) and (\ref{Omega f sympa} ) that the set $\mathit{M}$ of minimizers of $\Gamma \left( h\right) $ over $ K$ is included in $\mathit{M}_{x}$, i.e., $\mathit{M}\subset \mathit{M}_{x}$ . Hence, by (\ref{eq sup}) and (\ref{herve 3}), we have the inclusions \begin{equation} \mathit{M}\subset \overline{\mathcal{G}_{x}}\subset K\backslash \{x\}. \label{inclusion2bis} \end{equation} Therefore, we combine (\ref{inclusion1}) with (\ref{inclusion2bis}) for all $ x\in K\backslash \overline{\mathrm{co}}\left( \mathit{\Omega }\left( h,K\right) \right) $ to obtain the desired equality in the assertion (ii) of Theorem \ref{theorem trivial sympa 1}.
\subsection{Proof of Theorem \protect\ref{theorem trivial sympa 3}\label {Section Proofs-III}}
The proof of Theorem \ref{theorem trivial sympa 3} is a simple consequence of Theorem \ref{theorem trivial sympa 1} together with the following well--known result:
\begin{lemma}[Fenchel subgradients as minimizers] \label{theorem trivial sympa 2}\mbox{ }\newline Let $(\mathcal{X},\mathcal{X}^{\ast })$ be a dual pair and $h\not\equiv \infty$
be any extended real--valued function from a (non--empty) convex subset $K\subseteq \mathcal{X}$ to $ (-\infty ,\infty ]$. Then the Fenchel subdifferential $\partial h^{\ast }(x^{\ast })\subset \mathcal{X}$ of $h^{\ast }$ at the point $x^{\ast }\in \mathcal{X} ^{\ast }$ is the (non--empty) set $\mathit{M}_{x^{\ast }}$ of minimizers over $K$ of the map \begin{equation*} y\mapsto \Gamma \left( h\right) \left( y\right) -x^{\ast }\left( y\right) \end{equation*} from $K\subseteq \mathcal{X}$ to $(-\infty ,\infty ]$. \end{lemma}
\textit{Proof. }The proof is standard and simple, see, e.g., \cite[Theorem I.6.6]{Simon}. Indeed, any Fenchel subgradient $x\in \mathcal{X}$ of the Legendre--Fenchel transform $h^{\ast }$ at the point $x^{\ast }\in \mathcal{X }$ satisfies the inequality: \begin{equation} x^{\ast }\left( x\right) +h^{\ast }\left( y^{\ast }\right) -y^{\ast }\left( x\right) \geq h^{\ast }\left( x^{\ast }\right) \label{landford1landford1} \end{equation} for any $y^{\ast }\in \mathcal{X}^{\ast }$, see Definition \ref{tangent functional}. Since $h^{\ast }=h^{\ast \ast \ast }$ and $\Gamma \left( h\right) =h^{\ast \ast }$ (cf. Corollary \ref{Biconjugate} and \cite[ Proposition 51.6]{Zeidler3}), we have (\ref{landford1landford1}) iff \begin{equation*} x^{\ast }\left( x\right) +\underset{y^{\ast }\in \mathcal{X}^{\ast }}{\inf } \left\{ h^{\ast }\left( y^{\ast }\right) -y^{\ast }\left( x\right) \right\} =x^{\ast }\left( x\right) -\Gamma \left( h\right) \left( x\right) \geq \underset{y\in K}{\sup }\left\{ x^{\ast }\left( y\right) -\Gamma \left( h\right) \left( y\right) \right\} , \end{equation*} see Definition \ref{Legendre--Fenchel transform}.
\qed\medbreak
We combine now Theorem \ref{theorem trivial sympa 1} with Lemma \ref{theorem trivial sympa 2} to characterize the Fenchel subdifferential $\partial h^{\ast }(x^{\ast })\subset \mathcal{X}$ of $h^{\ast }$ at the point $x^{\ast }\in \mathcal{X}^{\ast }$ as the closed convex hull of the set $\mathit{\Omega } \left( h-x^{\ast },K\right) $ of generalized minimizers of $h$ over a compact convex subset $K$, see Definition \ref{gamm regularisation copy(4)}. Indeed, for any $x^{\ast }\in \mathcal{X}^{\ast }$, \begin{equation*} \Gamma \left( h-x^{\ast }\right) =\Gamma \left( h\right) -x^{\ast }, \end{equation*} see Definition \ref{gamm regularisation}.
\subsection{Proof of Corollary \protect\ref{corollary explosion lanford-robinson}\label{Section Proofs-IV}}
For $x^{\ast }\in \mathcal{X}^{\ast }$ and any open neighborhood $\mathcal{V} $ of $\{0\}\subset \mathcal{X}^{\ast }$, we define the map $g_{\mathcal{V} ,x^{\ast }}$ from $\mathcal{X}$ to $[\mathrm{k},\infty ]$ with $\mathrm{k} \in \mathbb{R}$ by \begin{equation*} g_{\mathcal{V},x^{\ast }}\left( x\right) :=\left\{ \begin{array}{c} \Gamma \left( h\right) \left( x\right) \\ \infty \end{array} \begin{array}{l} \mathrm{for}\ x=\mathrm{d}h_{y^{\ast }}^{\ast }\mathrm{\ with}\ y^{\ast }\in \mathcal{Y}^{\ast }\cap (x^{\ast }+\mathcal{V)}, \\ \mathrm{otherwise.} \end{array} \right. \end{equation*} For any $y^{\ast }\in \mathcal{Y}^{\ast }\cap \left( x^{\ast }+\mathcal{V} \right) $, one has the equality $g_{\mathcal{V},x^{\ast }}^{\ast }\left( y^{\ast }\right) =h^{\ast }\left( y^{\ast }\right) $. This easily follows from the fact that \begin{align*} h^{\ast }\left( y^{\ast }\right) & =\underset{z\in K}{\sup }\left\{ y^{\ast }\left( z\right) -\Gamma \left( h\right) \left( z\right) \right\} =y^{\ast }\left( x\right) -\Gamma \left( h\right) \left( x\right) \\ & =\underset{z\in K}{\sup }\left\{ y^{\ast }\left( z\right) -g_{\mathcal{V} ,x^{\ast }}\left( z\right) \right\} =g_{\mathcal{V},x^{\ast }}^{\ast }\left( y^{\ast }\right) \end{align*} with $x:=\mathrm{d}h_{y^{\ast }}^{\ast }$, see proof of Lemma \ref{theorem trivial sympa 2}. Let $\mathcal{W}$ be any open neighborhood of $ \{0\}\subset \mathcal{X}^{\ast }$. Then, for any $z\in K$, the set \begin{equation*} \{{\Greekmath 010E} ^{\ast }(z):{\Greekmath 010E} ^{\ast }\in \mathcal{W}\}\subset \mathbb{R} \end{equation*} is bounded, by continuity of the linear map ${\Greekmath 010E} ^{\ast }\mapsto {\Greekmath 010E} ^{\ast }(z)$. From the the principle of uniform boundedness for compact convex sets, i.e., the version of the Banach--Steinhaus theorem stated, for instance, in \cite[Theorem 2.9]{Rudin}, the set \begin{equation*} \{{\Greekmath 010E} ^{\ast }(z):{\Greekmath 010E} ^{\ast }\in \mathcal{W},\,z\in K\}\subset \mathbb{R} \end{equation*} is also bounded. Thus, for any $z^{\ast }\in \mathcal{X}^{\ast }$, \begin{eqnarray*}
\lim_{s\searrow 0}\sup \left\{ |h^{\ast }\left( z^{\ast }\right) -h^{\ast
}\left( z^{\ast }+{\Greekmath 010E} ^{\ast }\right) |\,:\,{\Greekmath 010E} ^{\ast }\in s\mathcal{W }\right\} &=&0, \\
\lim_{s\searrow 0}\sup \left\{ |g_{\mathcal{V},x^{\ast }}^{\ast }\left( z^{\ast }\right) -g_{\mathcal{V},x^{\ast }}^{\ast }\left( z^{\ast }+{\Greekmath 010E}
^{\ast }\right) |\,:\,{\Greekmath 010E} ^{\ast }\in s\mathcal{W}\right\} &=&0. \end{eqnarray*} This implies the continuity of $h^{\ast }$ and $g_{\mathcal{V},x^{\ast }}^{\ast }$. Hence, from the density of $\mathcal{Y}^{\ast }$, $h^{\ast }=g_{ \mathcal{V},x^{\ast }}^{\ast }$ on the open neighborhood $\left( x^{\ast }+ \mathcal{V}\right) $ of $\{x^{\ast }\}\subset \mathcal{X}^{\ast }$. In particular, $h^{\ast }$ and $g_{\mathcal{V},x^{\ast }}^{\ast }$ have the same Fenchel subgradients at the point $x^{\ast }$. From Theorems \ref{theorem trivial sympa 1 copy(1)} and \ref{theorem trivial sympa 3}, for each open neighborhood $\mathcal{V}$ of $\{0\}\subset \mathcal{X}^{\ast }$, the extreme Fenchel subgradients of $h^{\ast }$ at $x^{\ast }$ are all contained in the set $\mathcal{T}_{x^{\ast },\mathcal{V}}$ defined by (\ref{set t voisinage} ). Corollary \ref{corollary explosion lanford-robinson} thus follows.
\subsection{Proof of Corollary \protect\ref{corollary explosion lanford-robinson copy(1)}\label{Section Proofs-IV copy(1)}}
Note that $h^{\ast \ast }=h$ because the function $h$ is continuous and convex. By the global Lipschitz continuity of $h$, \begin{equation*} h\left( x\right) =\underset{x^{\ast }\in \mathcal{X}^{\ast }}{\sup }\left\{ x^{\ast }\left( x\right) -h^{\ast }\left( x^{\ast }\right) \right\} = \underset{x^{\ast }\in K}{\sup }\left\{ x^{\ast }\left( x\right) -h^{\ast }\left( x^{\ast }\right) \right\} \end{equation*} with $K:=\mathcal{B}_{R}\left( 0\right) \subset \mathcal{X}^{\ast }$ being some ball of sufficiently large radius $R>0$ centered at $0$. The set $K$ is weak$^{\ast }$--compact, by the Banach--Alaoglu theorem.
Now, for any fixed $x\in \mathcal{X}$ and all $x^{\ast }\in \mathcal{Z} _{x}\subset \mathcal{X}^{\ast }$, by definition of the set $\mathcal{Z}_{x}$ , there is a net $\{x_{i}\}_{i\in I}$ in $\mathcal{Y}$ converging to $x$ with the property that the unique Fenchel subgradient $x_{i}^{\ast }:=\mathrm{d}h_{x_{i}}\in \mathcal{X}^{\ast }$ of $h$ at $x_{i}$ converges towards $x^{\ast }$ in the weak$^{\ast }$--topology. Therefore, by continuity of $h$, for any fixed $x\in \mathcal{X}$ and all $x^{\ast }\in \mathcal{Z}_{x}$, \begin{equation*} h\left( x\right) =\underset{y^{\ast }\in \mathcal{X}^{\ast }}{\sup }\left\{ y^{\ast }\left( x\right) -h^{\ast }\left( y^{\ast }\right) \right\} = \underset{I}{\lim }\ h\left( x_{i}\right) =\underset{I}{\lim }\left\{ x_{i}^{\ast }\left( x\right) -h^{\ast }\left( x_{i}^{\ast }\right) \right\} , \end{equation*} with $\{x_{i}^{\ast }\}_{i\in I}$ converging to $x^{\ast }$. In other words, \begin{equation*} \mathcal{Z}_{x}\subset \mathit{\Omega }\left( h^{\ast }-x,K\right) , \end{equation*} see Definition \ref{gamm regularisation copy(4)}. Thus, by Theorem \ref {theorem trivial sympa 3} and Corollary \ref{corollary explosion lanford-robinson}, it suffices to prove that $\mathcal{T}_{x}\subset \mathcal{Z}_{x}$.
By density of $\mathcal{Y}$ in $\mathcal{X}$, observe that the set \begin{equation*} \mathcal{T}_{x,\mathcal{V}}:=\overline{\left\{ \mathrm{d}h_{y}:y\in \mathcal{ Y}\cap (x+\mathcal{V)}\right\} }^{\mathcal{X}^{\ast }}\subset \mathcal{X}^{\ast } \end{equation*} is non--empty for any open neighborhood $\mathcal{V}$ of $\{0\}\subset \mathcal{X}$. Meanwhile, the weak$^{\ast }$--compact set $K$ is metrizable with respect to (w.r.t.) the weak$^{\ast }$--topology, by separability of $ \mathcal{X}$, see \cite[Theorem 3.16]{Rudin}. In particular, $K$ is sequentially compact and we can restrict ourselves to sequences instead of more general nets. In particular, by (\ref{set t voisinage})--(\ref{set t voinagebis}), one has \begin{equation} \mathcal{T}_{x}=\bigcap\limits_{n\in \mathbb{N}}\mathcal{T}_{x,\mathcal{B} _{1/n}\left( 0\right) } \label{inclusion sup} \end{equation} with $\mathcal{B}_{{\Greekmath 010E} }\left( x\right) $ being the ball (in $K$) of radius ${\Greekmath 010E} >0$ centered at $x\in K$. Here, $\mathcal{B}_{{\Greekmath 010E} }\left( x\right) $ is defined by (\ref{ball}) for any metric $d$ on $K$ generating its weak$^{\ast }$--topology. For any $x^{\ast }\in \mathcal{T}_{x}\subset K$ and any $n\in \mathbb{N}$, there are per definition a sequence $ \{x_{n,m}^{\ast }\}_{m=1}^{\infty }$ converging to $x^{\ast }$ in $K$ as $ m\rightarrow \infty $ and an integer $N_{n}>0$ such that, for all $m\geq N_{n}$, $d(x^{\ast },x_{n,m}^{\ast })\leq 2^{-n}$ and $x_{n,m}^{\ast }= \mathrm{d}h_{x_{n,m}}$ for some $x_{n,m}\in \mathcal{Y}\cap \lbrack x+ \mathcal{B}_{1/n}\left( 0\right) ]$. Taking any function $p(n)\in \mathbb{N}$ satisfying $p(n)>N_{n}$ and converging to $\infty $ as $n\rightarrow \infty $ we obtain a sequence $\{x_{n,p(n)}^{\ast }\}_{n=1}^{\infty }$ converging to $ x^{\ast }\in \mathcal{Z}_{x}$ as $n\rightarrow \infty $. This yields the inclusion $\mathcal{T}_{x}\subset \mathcal{Z}_{x}$.
\section{Further Remarks\label{Concluding remarks}}
We give here an additional observation which is\ not necessarily directly related to the main results of the paper. It concerns an extension of the Bauer maximum principle \cite[Theorem I.5.3.]{Alfsen}. See \cite{BruPedra2} for an application to statistical mechanics.
First, recall that the $\Gamma $--regularization $\Gamma \left( h\right) $ of an extended real--valued function $h$ is a convex and lower semi--continuous function on a compact convex subset $K$. Moreover, every convex and lower semi--continuous function on $K$ equals its own $\Gamma $--regularization on $K$ (see, e.g., \cite[Proposition I.1.2.]{Alfsen}):
\begin{proposition}[$\Gamma $--regularization of lower semi--cont. conv. maps ] \label{lemma gamma regularisation}Let $h$ be any extended function from a (non--empty) compact convex subset $K\subset \mathcal{X}$ of a locally convex real space $\mathcal{X}$ to $\left( -\infty ,\infty \right] $. Then the following statements are equivalent:\newline \emph{(i)} $\Gamma \left( h\right) =h$ on $K$.\newline \emph{(ii)} $h$ is a lower semi--continuous convex function on $K$. \end{proposition}
\noindent This proposition is a standard result. The compactness of $K$ is in fact not necessary but $K$ should be a closed convex set. This result can directly be proven without using the fact that the $\Gamma $--regularization $\Gamma \left( h\right) $ of a function $h$ on $K$ equals its twofold \emph{Legendre--Fenchel transform} -- also called the \emph{biconjugate } (function) of $h$. Indeed, $\Gamma \left( h\right) $ is the largest lower semi--continuous and convex minorant of $h$:
\begin{corollary}[Largest lower semi--cont. convex minorant of $h$] \label{Biconjugate}\mbox{ }\newline Let $h$ be any extended function from a (non--empty) compact convex subset $ K\subset \mathcal{X}$ of a locally convex real space $\mathcal{X}$ to $ \left( -\infty ,\infty \right] $. Then its $\Gamma $--regularization $\Gamma \left( h\right) $ is its largest lower semi--continuous convex minorant on $ K $. \end{corollary}
\textit{Proof. }For any lower semi--continuous convex extended real--valued
function $f$ defined on $K$ satisfying $f\leq h$, we have, by Proposition \ref{lemma gamma regularisation}, that \begin{equation*} f\left( x\right) =\sup \left\{ m(x):m\in \mathrm{A}\left( \mathcal{X}\right)
\;\RIfM@\expandafter\text@\else\expandafter\mbox\fi{and }m|_{K}\leq f\leq h\right\} \leq \Gamma \left( h\right) \left( x\right) \end{equation*} for any $x\in K$.
\qed\medbreak
\noindent In particular, if $(\mathcal{X},\mathcal{X}^{\ast })$ is a dual pair and $h \not\equiv \infty$ is any extended function from $K$ to $(-\infty ,\infty ]$ then $\Gamma \left( h\right) =h^{\ast \ast }$, see \cite[Proposition 51.6]{Zeidler3}.
Proposition \ref{lemma gamma regularisation} has another interesting consequence: An extension of the Bauer maximum principle \cite[Theorem I.5.3. ]{Alfsen} which, in the case of convex functions, is:
\begin{lemma}[Bauer maximum principle] \label{Bauer maximum principle}\mbox{ }\newline Let $\mathcal{X}$ be a locally convex real space. An upper semi--continuous convex real--valued
function $h$ over a compact convex subset $K\subset \mathcal{X} $ attains its maximum at an extreme point of $K$, i.e., \begin{equation*} \sup \,h\left( K\right) =\max \,h\left( \mathcal{E}(K)\right) . \end{equation*} Here, $\mathcal{E}(K)$ is the (non--empty) set of extreme points of $K$. \end{lemma}
\noindent Indeed, by combining Proposition \ref{lemma gamma regularisation} with Lemma \ref{Bauer maximum principle} it is straightforward to check the following statement which does not seem to have been observed before:
\begin{lemma}[Extension of the Bauer maximum principle] \label{Bauer maximum principle bis}\mbox{ }\newline Let $h_{\pm }$ be two convex real--valued
functions from a locally convex real space $\mathcal{X}$ to $\left( -\infty ,\infty \right] $ such that $h_{-}$ and $h_{+}$ are respectively lower and upper semi--continuous. Then the supremum of the sum $h:=h_{-}+h_{+}$ over a compact convex subset $K\subset \mathcal{X}$ can be reduced to the (non--empty) set $\mathcal{E}(K)$ of extreme points of $K$, i.e., \begin{equation*} \sup \,h\left( K\right) =\sup \,h\left( \mathcal{E}(K)\right) . \end{equation*} \end{lemma}
\textit{Proof. }We first use Proposition \ref{lemma gamma regularisation} in order to write $h_{-}=\Gamma \left( h_{-}\right) $ as a supremum over affine and continuous functions. Then we commute this supremum with the one over $ K$ and apply the Bauer maximum principle to obtain that \begin{equation*} \sup \,h\left( K\right) =\sup \left\{ \sup \,\left[ m+h_{+}\right] (\mathcal{
E}(K)):m\in \mathrm{A}\left( \mathcal{X}\right) \;\RIfM@\expandafter\text@\else\expandafter\mbox\fi{and }m|_{K}\leq h_{-}|_{K}\right\} . \end{equation*} The lemma follows by commuting again both suprema and by using $h_{-}=\Gamma \left( h_{-}\right) $.
\qed\medbreak
\noindent Observe, however, that under the conditions of the lemma above, the supremum of $h=h_{-}+h_{+}$ is generally not attained on $\mathcal{E}(K)$ .
\section{Appendix\label{Section appendix}}
\setcounter{equation}{0}
For the reader's convenience we give here a short review on the following subjects:
\begin{itemize} \item Dual pairs of locally convex real spaces, see, e.g., \cite{Rudin};
\item Barycenters and $\Gamma $--regularization of real--valued functions, see, e.g., \cite{Alfsen};
\item The Mazur and Lanford III--Robinson theorems, see \cite{LanRob,Mazur}. \end{itemize}
\noindent These subjects are rather standard. Therefore, we keep the exposition as short as possible and only concentrate on results used in this paper.
\subsection{Dual Pairs of Locally Convex Real Spaces}
The notion of \emph{dual pairs} is defined as follow:
\begin{definition}[Dual pairs] \label{dual pairs}\mbox{ }\newline For any locally convex space $(\mathcal{X},{\Greekmath 011C} )$, let $\mathcal{X}^{\ast }$ be its dual space, i.e., the set of all continuous linear functionals on $ \mathcal{X}$. Let ${\Greekmath 011C} ^{\ast }$ be any locally convex topology on $ \mathcal{X}^{\ast }$. $(\mathcal{X},\mathcal{X}^{\ast })$ is called a dual pair iff, for all $x\in \mathcal{X}$, the functional $x^{\ast }\mapsto x^{\ast }(x)$ on $\mathcal{X}^{\ast }$ is continuous w.r.t. ${\Greekmath 011C} ^{\ast }$, and all linear functionals which are continuous w.r.t. ${\Greekmath 011C} ^{\ast }$ have this form. \end{definition}
\noindent By \cite[Theorems 3.4 (b) and 3.10]{Rudin}, a typical example of a dual pair $(\mathcal{X},\mathcal{X}^{\ast })$ is given by any locally convex real space $\mathcal{X}$ equipped with a topology ${\Greekmath 011C} $ and $\mathcal{X} ^{\ast }$ equipped with the ${\Greekmath 011B} (X^{\ast },X)$--topology ${\Greekmath 011C} ^{\ast }$ , i.e., the weak$^{\ast }$--topology. We also observe that if $(\mathcal{X}, \mathcal{X}^{\ast })$ is a dual pair w.r.t. ${\Greekmath 011C} $ and ${\Greekmath 011C} ^{\ast }$ then $(\mathcal{X}^{\ast },\mathcal{X})$ is a dual pair w.r.t. ${\Greekmath 011C} ^{\ast }$ and ${\Greekmath 011C} $.
\subsection{Barycenters and $\Gamma $--regularization}
The theory of compact convex subsets of a locally convex real (topological vector) space $\mathcal{X}$ is standard. For more details, see, e.g., \cite {Alfsen}. An important observation is the Krein--Milman theorem (see, e.g., \cite[Theorems 3.4 (b) and 3.23]{Rudin}) which states that any compact convex subset $K\subset \mathcal{X}$ is the closure of the convex hull of the (non--empty) set $\mathcal{E}(K)$ of its extreme points. Restricted to finite dimensions this theorem corresponds to a classical result of Minkowski which, for any $x\in K$ in a (non--empty) compact convex subset $ K\subset \mathcal{X}$, states the existence of a finite number of extreme points $\hat{x}_{1},\ldots ,\hat{x}_{k}\in \mathcal{E}(K)$ and positive numbers ${\Greekmath 0116} _{1},\ldots ,{\Greekmath 0116} _{k}\geq 0$ with $\Sigma _{j=1}^{k}{\Greekmath 0116} _{j}=1$ such that \begin{equation} x=\overset{k}{\sum\limits_{j=1}}{\Greekmath 0116} _{j}\hat{x}_{j}. \label{barycenter1} \end{equation} To this\ simple decomposition we can associate a probability measure, i.e., a \emph{normalized positive Borel regular measure}, ${\Greekmath 0116} $\ on $K$.
Borel sets of any set $K$ are elements of the ${\Greekmath 011B} $--algebra $\mathfrak{B }$ generated by closed -- or open -- subsets of $K$. Positive Borel regular measures are the positive countably additive set functions ${\Greekmath 0116} $ over $ \mathfrak{B}$ satisfying \begin{equation*} {\Greekmath 0116} \left( B\right) =\sup \left\{ {\Greekmath 0116} \left( C\right) :C\subset B,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }C \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ closed}\right\} =\inf \left\{ {\Greekmath 0116} \left( O\right) :B\subset O,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ } O\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ open}\right\} \end{equation*} for any Borel subset $B\in \mathfrak{B}$ of $K$. If $K$ is compact then any positive Borel regular measure ${\Greekmath 0116} $ (one--to--one) corresponds to an element of the set $M^{+}(K)$ of Radon measures with ${\Greekmath 0116} \left( K\right) =\left\Vert {\Greekmath 0116} \right\Vert $, and we write \begin{equation} {\Greekmath 0116} \left( h\right) =\int_{K}\mathrm{d}{\Greekmath 0116} (\hat{x})\;h\left( \hat{x}\right) \label{barycenter1bis} \end{equation} for any continuous function $h$ on $K$. A probability measure ${\Greekmath 0116} \in M_{1}^{+}(K)$ is per definition a positive Borel regular measure ${\Greekmath 0116} \in M^{+}(K)$ which is \emph{normalized}: $\left\Vert {\Greekmath 0116} \right\Vert =1$.
Therefore, using the probability measure ${\Greekmath 0116} _{x}\in M_{1}^{+}(K)$\ on $K$ defined by \begin{equation*} {\Greekmath 0116} _{x}=\overset{k}{\sum\limits_{j=1}}{\Greekmath 0116} _{j}{\Greekmath 010E} _{\hat{x}_{j}} \end{equation*} with ${\Greekmath 010E} _{y}$ being the Dirac -- or point -- mass\footnote{${\Greekmath 010E} _{y}$ is the Borel measure such that, for any Borel subset $B\in \mathfrak{B}$ of $ K$, ${\Greekmath 010E} _{y}(B)=1$ if $y\in B$ and ${\Greekmath 010E} _{y}(B)=0$ if $y\notin B$.} at $y$, Equation (\ref{barycenter1}) can be seen as an integral defined by ( \ref{barycenter1bis}) for the probability measure ${\Greekmath 0116} _{x}\in M_{1}^{+}(K)$ : \begin{equation} x=\int_{K}\mathrm{d}{\Greekmath 0116} _{x}(\hat{x})\;\hat{x}\ . \label{barycenter2} \end{equation} The point $x$ is in fact the \emph{barycenter} of the probability measure $ {\Greekmath 0116} _{x}$. This notion is defined in the general case as follows (cf. \cite[ Eq. (2.7) in Chapter I]{Alfsen}):
\begin{definition}[Barycenters of probability measures in convex sets] \label{def barycenter}Let $K\subset \mathcal{X}$ be any (non--empty) compact convex subset of a locally convex real space $\mathcal{X}$ and let ${\Greekmath 0116} \in M_{1}^{+}(K)$ be a probability measure on $K$. We say that $x\in K$ is the barycenter\footnote{ Other terminologies existing in the literature: \textquotedblleft $x$ is represented by ${\Greekmath 0116} $\textquotedblright , \textquotedblleft $x$ is the resultant of ${\Greekmath 0116} $\textquotedblright .} of ${\Greekmath 0116} $ if, for all $z^{\ast }\in \mathcal{X}^{\ast }$, \begin{equation*} z^{\ast }\left( x\right) =\int_{K}\mathrm{d}{\Greekmath 0116} (\hat{x})\;z^{\ast }\left( \hat{x}\right) . \end{equation*} \end{definition}
\noindent Barycenters are well--defined for \emph{all} probability measures in convex compact subsets of locally convex real spaces (cf. \cite[Theorems 3.4 (b) and 3.28]{Rudin}):
\begin{theorem}[Well-definiteness and uniqueness of barycenters] \label{thm barycenter}\mbox{ }\newline Let $K\subset \mathcal{X}$ be any (non--empty) compact subset of a locally convex real space $\mathcal{X}$ such that $\overline{\mathrm{co}}\left( K\right) $ is also compact. Then, for any probability measure ${\Greekmath 0116} \in M_{1}^{+}(K)$ on $K$,$\ $there is a unique barycenter $x_{{\Greekmath 0116} }\in \overline{ \mathrm{co}}\left( K\right) $. \end{theorem}
\noindent Note that Barycenters can also be defined in the same way via affine continuous functions instead of continuous linear functionals, see, e.g., \cite[Proposition I.2.2.]{Alfsen} together with \cite[Theorem 1.12] {Rudin}.
It is natural to ask whether, for any $x\in K$ in the compact convex set $K$ , there is a (possibly not unique) probability measure ${\Greekmath 0116} _{x}$ on $K$ (pseudo--) supported on $\mathcal{E}(K)$ with barycenter $x$. Equation (\ref {barycenter2}) already gives a first positive answer to that problem in the finite dimensional case. The general case, which is a remarkable refinement of the Krein--Milman theorem, has been proven by Choquet--Bishop--de Leeuw (see, e.g., \cite[Theorem I.4.8.]{Alfsen}).
We conclude now by a crucial property concerning the $\Gamma $ --regularization of extended real--valued
functions in relation with the concept of barycenters (cf. \cite[Corollary I.3.6.]{Alfsen}):
\begin{theorem}[Barycenters and $\Gamma $--regularization] \label{Thm - Corollary I.3.6}\mbox{ }\newline Let $K\subset \mathcal{X}$ be any (non--empty) compact convex subset of a locally convex real space $\mathcal{X}$ and $h:K \to \mathbb{R}$
be a continuous real--valued function. Then, for any $x\in K$, there is a probability measure ${\Greekmath 0116} _{x}\in M_{1}^{+}(K)$\ on $K$ with barycenter $x$ such that \begin{equation*} \Gamma \left( h\right) \left( x\right) =\int_{K}\mathrm{d}{\Greekmath 0116} _{x}(\hat{x} )\;h\left( \hat{x}\right) . \end{equation*} \end{theorem}
\noindent This theorem is a very important statement used to prove Theorem \ref{theorem trivial sympa 1}.
\subsection{The Mazur and Lanford III--Robinson Theorems}
If $\mathcal{X}$ is a separable real Banach space and $h$ is a continuous convex real--valued
function on $\mathcal{X}$ then it is well--known that $h$ has, on each point $x\in \mathcal{X}$, at least one Fenchel subgradient \textrm{d}$h\in \mathcal{X}^{\ast }$. The Mazur theorem describes the set $\mathcal{Y}$ on which a continuous convex function $h$ is G\^{a}teaux differentiable, more precisely, the set $\mathcal{Y}$ for which $h$ has exactly one Fenchel subgradient \textrm{d}$h_{x}\in \mathcal{X}^{\ast }$ at any $x\in \mathcal{Y}$:
\begin{theorem}[Mazur] \label{Mazur}\mbox{ }\newline Let $\mathcal{X}$ be a separable real Banach space and let $h:\mathcal{X} \rightarrow \mathbb{R}$ be a continuous convex function. The set $\mathcal{ Y}\subset \mathcal{X}$ of elements where $h$ has exactly one Fenchel subgradient \textrm{d}$h_{x}\in \mathcal{X}^{\ast }$ at any $x\in \mathcal{Y}$ is residual, i.e., a countable intersection of dense open sets. \end{theorem}
\begin{remark} \label{Mazur remark}By Baire category theorem, the set $\mathcal{Y}$ is dense in $\mathcal{X}$. \end{remark}
\noindent The Lanford III--Robinson theorem \cite[Theorem 1]{LanRob} completes the Mazur theorem by characterizing the Fenchel subdifferential $\partial h(x)\subset \mathcal{X}^{\ast }$ at any $x\in \mathcal{X}$:
\begin{theorem}[Lanford III -- Robinson] \label{Land.Rob}\mbox{ }\newline Let $\mathcal{X}$ be a separable real Banach space and let $h:\mathcal{X} \rightarrow \mathbb{R}$ be a continuous convex function. Then the Fenchel subdifferential $\partial h(x)\subset \mathcal{X}^{\ast }$ of $h$, at any $ x\in \mathcal{X}$, is the weak$^{\ast }$--closed convex hull of the set $ \mathcal{Z}_{x}$. Here, at fixed $x\in \mathcal{X}$, $\mathcal{Z}_{x}$ is the set of functionals $x^{\ast }\in \mathcal{X}^{\ast }$ such that there is a net $\{x_{i}\}_{i\in I}$ in $\mathcal{Y}$ converging to $x$ with the property that the unique Fenchel subgradient $\mathrm{d}h_{x_{i}}\in \mathcal{X}^{\ast }$ of $h$ at $x_{i}$ converges towards $x^{\ast }$ in the weak$^{\ast}$--topology. \end{theorem}
\addcontentsline{toc}{section}{References}
\end{document} | arXiv |
Mat. Sb.:
Mat. Sb., 2010, Volume 201, Number 2, Pages 29–78 (Mi msb7515)
This article is cited in 51 scientific papers (total in 52 papers)
Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Padé approximants
A. I. Aptekarev†, V. G. Lysov
M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences
Abstract: The paper considers Hermite-Padé approximants to systems of Markov functions defined by means of directed graphs. The minimization problem for the energy functional is investigated for a vector measure whose components are related by a given interaction matrix and supported in some fixed system of intervals. The weak asymptotics of the approximants are obtained in terms of the solution of this problem. The defining graph is allowed to contain undirected cycles, so the minimization problem in question is considered within the class of measures whose masses are not fixed, but allowed to 'flow' between intervals. Strong asymptotic formulae are also obtained. The basic tool that is used is an algebraic Riemann surface defined by means of the supports of the components of the extremal measure. The strong asymptotic formulae involve standard functions on this Riemann surface and solutions of some boundary value problems on it. The proof depends upon an asymptotic solution of the corresponding matrix Riemann-Hilbert problem.
Bibliography: 40 titles.
Keywords: Hermite-Padé approximants, multiple orthogonal polynomials, weak and strong asymptotics, extremal equilibrium problems for a system of measures, matrix Riemann-Hilbert problem.
† Author to whom correspondence should be addressed
DOI: https://doi.org/10.4213/sm7515
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Sbornik: Mathematics, 2010, 201:2, 183–234
MSC: Primary 42C05, 41A21; Secondary 30E25
Received: 22.12.2008 and 03.09.2009
Citation: A. I. Aptekarev, V. G. Lysov, "Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Padé approximants", Mat. Sb., 201:2 (2010), 29–78; Sb. Math., 201:2 (2010), 183–234
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Sarah is planning to visit relatives in England and Spain. On the day that she wants to buy the currencies for her trip, one euro costs \$1.50 and one British pound costs \$2.00. Which of the equations represent combinations of these currencies can Sheila buy for \$700?
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After a fundraiser, the treasurer for a minor soccer league invested some of the money in a savings account that paid 2.5\%/year and the rest in a government bond that paid 3.5\%/year. After one year, the money earned \$140 in interest. Define two variables, write an equation.
Gary drove his pickup truck from Cornwall to Chatham. He left Cornwall at 8:15 a.m. and drove at a steady 100 km/h along Highway 401. The graph below shows how the fuel in the tank varied over time.
a) What do the coordinates of the point (5, 25.5) tell you about the amount of fuel?
b) How much fuel was in the tank at 11:45 a.m.?
c) The low fuel warning light came on when 6 L of fuel remained. At what time did this light come on?
Ready Car charges \$59/day plus \$0.14/km to rent a car. Best Car charges \$69/day plus \$0.11/km. Set up an equation which you can use to compare these two rental rates. What advice would you give someone who wants to rent a car from one of these companies?
Solve the linear system graphically.
\displaystyle \begin{array}{lllll} & x + y = 2 \\ & x = 2y + 2 \\ \end{array}
Q5a
\displaystyle \begin{array}{lllll} & y -x = 1 \\ & 2x - y = 1 \\ \end{array}
Q5b
Tools-R-Us rents snow blowers for a base fee of \$20 plus \$8/h. Randy's Rentals rents snowblowers for a base fee of \$12 plus \$10/h.
a) Create an equation that represents the cost of renting a snowblower from Tools-R-Us.
b) Create the corresponding equation for Randy?s Rentals.
c) Solve the system of equations graphically.
d) What does the point of intersection mean in this situation?
Use substitution to solve each system.
\displaystyle \begin{array}{lllll} & 2x + 3y = 7 \\ & -2x - 1 = y \\ \end{array}
Use substitution to solve the system.
\displaystyle \begin{array}{lllll} & 3x - 4y = 5 \\ & x -y = 5 \\ \end{array}
\displaystyle \begin{array}{lllll} & 5x + 2y = 18 \\ & 2x + 3y = 16 \\ \end{array}
Q7c
\displaystyle \begin{array}{lllll} & 9 = 6x - 3y \\ & 4x - 3y = 5 \\ \end{array}
Q7d
Courtney paid a one-time registration fee to join a fitness club. She also pays a monthly fee. After three months, she had paid \$315. After seven months, she had paid \$535. Determine the registration fee and the monthly fee.
A rectangle has a perimeter of 40 m. Its length is 2 m greater than its width.
a) Represent this situation with a linear system.
b) Solve the linear system using substitution.
c) What do the numbers in the solution represent? Explain.
Which linear system below is equivalent to the system that is shown in the graph?
\displaystyle \begin{array}{llllll} &\text{A}. &2x - 5y =4 &\text{B}. &x - 3y = -1\\ &&-x + y = 1 && 2x + y = 4 \\ \end{array}
a) Which of the following is an equivalent system of linear equation?
\displaystyle \begin{array}{lllll} & -2x -3y = 5\\ & 3x - y = 9 \\ \end{array}
b) What is the solution for above?
Use elimination to solve the linear system.
\displaystyle \begin{array}{lllll} & 2x -3y = 13\\ & 5x - y = 13 \\ \end{array}
Q12a
\displaystyle \begin{array}{lllll} & x - 3y = 0\\ & 3x - 2y = -7 \\ \end{array}
\displaystyle \begin{array}{lllll} & 3x + 21 = 5y\\ & 4y + 6 = - 9x\\ \end{array}
Q12c
\displaystyle \begin{array}{lllll} & x - \frac{1}{3}y = - 1\\ & \frac{2}{3}x -\frac{1}{4}y = - 1\\ \end{array}
Q12d
Lily needs 200 g of chocolate that is 86\% cocoa for a cake recipe. He has one kind of chocolate that is 99\% cocoa and another kind that is 70\% cocoa. How much of each kind of chocolate does he need to make the cake? Round your answer to the nearest gram.
A Grade 10 class is raising money for a school— building project in Uganda. To buy 35 desks and 3 chalkboards, the students need to raise $2082. To buy 40 desks and 2 chalkboards, they need to raise $2238. Determine the cost of a desk and the cost of a chalkboard.
Solve the linear system.
\displaystyle \begin{array}{llll} &2(2x -1) - (y - 4) =11 \\ &3(1 - x) - 2(y - 3) =-7 \\ \end{array}
Juan is a cashier at a variety store. He has a total of \$580 in bills. He has 76 bills, consisting of \$5 bills and \$10 bills. How many of each type does he have?
Sketch a linear system that has no solution.
The linear system 6x + 5y= 10 and ax+ 2y = b has an infinite number of solutions. Determine a and b.
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\begin{document}
\title[Fractional Choquard equations with decaying potentials] {Semi-classical states for fractional Choquard equations with decaying potentials}
\thanks {The research was supported by the Natural Science Foundation of China (No. 12271196, 11931012).} \author[ Y. Deng, S. Peng, X. Yang, ]{Yinbin Deng \textsuperscript{1}, Shuangjie Peng\textsuperscript{2} and Xian Yang \textsuperscript{3}}
{
\footnotetext[1]{School of Mathematics and Statistics \& Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, P. R. China. Email: [email protected].}
\footnotetext[2]{School of Mathematics and Statistics \& Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, P. R. China. Email: [email protected]}
\footnotetext[3]{ School of Mathematics and Statistics \& Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, P. R. China. Email: [email protected].}
}
\begin{abstract}
This paper deals with the following fractional Choquard equation
$$\varepsilon^{2s}(-\Delta)^su +Vu=\varepsilon^{-\alpha}(I_\alpha*|u|^p)|u|^{p-2}u\ \ \ \mathrm{in}\ \mathbb R^N,$$ where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$ is the fractional Laplacian, $N>2s$, $s\in(0,1)$, $\alpha\in\big((N-4s)_{+}, N\big)$, $p\in[2, \frac{N+\alpha}{N-2s})$, $I_\alpha$ is a Riesz potential, $V\in C\big(\mathbb R^N, [0, +\infty)\big)$ is an electric potential. Under some assumptions on the decay rate of $V$ and the corresponding range of $p$, we prove that the problem has a family of solutions $\{u_\varepsilon\}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$. Since the potential $V$ decays at infinity, we need to employ a type of penalized argument and implement delicate analysis on the both nonlocal terms to establish regularity, positivity and asymptotic behaviour of $u_\varepsilon$, which is totally different from the local case. As a contrast, we also develop some nonexistence results, which imply that the assumptions on $V$ and $p$ for the existence of $u_\varepsilon$ are almost optimal. To prove our main results, a general strong maximum principle and comparison function for the weak solutions of fractional Laplacian equations are established.
The main methods in this paper are variational methods, penalized technique and some comparison principle developed in this paper.
{\bf Key words:} Fractional Choquard; penalized method; variational methods; decaying potentials; comparison principle
{\bf AMS Subject Classifications:} 35J15, 35A15, 35J10. \end{abstract}
\maketitle \section{Introduction}\label{s1} In this paper, we study the following nonlinear fractional Choquard equation \begin{equation}\label{eqs1.1}
\varepsilon^{2s}(-\Delta)^su +Vu=\varepsilon^{-\alpha}(I_\alpha*|u|^p)|u|^{p-2}u \ \ \ \mathrm{in}\ \mathbb R^N, \end{equation}
where~$\varepsilon>0$~is a parameter, $N>2s$, $s\in(0,1)$, $\alpha\in(0, N)$, $p\in[2, \frac{N+\alpha}{N-2s})$, $V\in C\big({\mathbb R}^N, [0, \infty)\big)$ is an external potential, $I_{\alpha}=A_{N,\alpha}|x|^{\alpha-N}$ is the Riesz potential with $A_{N,\alpha}=\frac{\Gamma(\frac{N-\alpha}{2})}{2^\alpha\pi^{N/2}\Gamma(\frac{\alpha}{2})}$ (see \cite{rie}) and could be interpreted as the Green function of $(-\Delta)^{\frac{\alpha}{2}}$ in $\mathbb R^N$ satisfying the semigroup property $I_{\alpha+\beta}=I_\alpha*I_\beta$ for $\alpha,\beta>0$ such that $\alpha+\beta<N$, $(-\Delta)^s$ is the fractional Laplacian defined as \begin{eqnarray*}
(-\Delta)^su(x)&:=&C(N,s)P.V.\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}~\mathrm{d} y\\
&=&C(N,s)\lim\limits_{r\to 0}\int_{\mathbb{R}^N\backslash B_r(x)}\frac{u(x)-u(y)}{|x-y|^{N+2s}}~\mathrm{d} y \end{eqnarray*}
with $C(N,s)=\big(\int_{\mathbb{R}^N}\frac{1-\cos(\zeta_1)}{|\zeta|^{N+2s}}\mathrm{d}\zeta\big)^{-1}$ (see \cite{ege2012}). In view of a path integral over the L\'{e}vy flights paths, the fractional Laplacian was introduced by Laskin (\cite{3}) to model fractional quantum mechanics.
When $s=\frac{1}{2}$, $N=3$ and $\alpha=2$, problem \eqref{eqs1.1} is related to the following well-known boson stars equation (see \cite{es,on,bos,fro,len,lel}) \begin{equation}\label{q1r}
i\partial_t\psi=\sqrt{-\Delta+m^2}\psi+(V(x)-E)\psi-(I_2*|\psi|^2)\psi,\quad \psi:[0,T)\times\mathbb R^3\to\mathbb C, \end{equation} which can effectively describe the dynamics and gravitational collapse of relativistic boson stars, where $m\ge0$ is a mass parameter and $\sqrt{-\Delta+m^2}$ is the kinetic energy operator defined via its symbol $\sqrt{\xi^2 + m^2}$ in Fourier space. In the massless case ($m=0$), a standing wave $\psi(x,t):=e^{iEt}u(x)$ of \eqref{q1r} leads to a solution $u$ of \begin{align*}
\sqrt{-\Delta}u+Vu=(I_2*|u|^2)u\quad \mathrm{in}\ \mathbb R^3. \end{align*}
When $s=1$, equation \eqref{eqs1.1} boils down to the following classical Choquard equation: \begin{equation}\label{la} -\varepsilon^2\Delta u+Vu=\varepsilon^{-\alpha}(I_\alpha*|u|^p)|u|^{p-2}u\ \ \ \mathrm{in}\ \mathbb{R}^N, \end{equation} which was introduced by Choquard in $1976$ in the modeling of a one-component plasma (\cite{lie}). The equation can also be derived from the Einstein-Klein-Gordon and Einstein-Dirac system (\cite{giu}). Equation \eqref{la} can be seen as a stationary nonlinear Schr\"odinger equation with an attractive long range interaction (represented by the nonlocal term) coupled with a repulsive short range interaction (represented by the local nonlinearity). While for the most of the relevant physical applications $p=2$, the case $p\neq 2$ may appear in several relativistic models of the density functional theory. When $V\in L^1_{\mathrm{loc}}(\mathbb R^N)$ is a non-constant electric potential, \eqref{la} can model the physical phenomenon in which particles are under the influence of an external electric field.
When $\varepsilon>0$ is a small parameter, which is typically related to the Planck constant, from the physical prospective \eqref{eqs1.1} is particularly important, since its solutions as $\varepsilon\to 0$ are called semi-classical bound states. Physically, it is expected that in the semi-classical limit $\varepsilon\to 0$ there should be a correspondence between solutions of the equation \eqref{eqs1.1} and critical points of the potential $V$, which governs the classical dynamics.
For fixed $\varepsilon>0$, for instance $\varepsilon=1$, problem \eqref{eqs1.1} becomes \begin{equation}\label{fldd} (-\Delta)^s u+ V u=(I_\alpha*|u|^p)|u|^{p-2}u\ \ \ \mathrm{in}\ \mathbb{R}^N. \end{equation} In the case that $V(x)$ is a constant $\lambda>0$, $N\ge 3$ and $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2s})$, it was verified in \cite{dss} that problem \eqref{fldd}
has a positive radial decreasing ground state $U_{\lambda}$. Moreover, if $p\ge2$, it holds that $U_{\lambda}$ decays as follows: \begin{equation}\label{sjg}
U_\lambda=\frac{C}{|x|^{N+2s}}+o(|x|^{-N-2s})\ \ \ \mathrm{as}\ |x|\to\infty \end{equation} for some $C>0$.
Noting that $I_\alpha*|u|^p\to |u|^p$ as $\alpha\to0$ for all $u\in C_0^\infty(\mathbb{R}^N)$, we see that equation \eqref{eqs1.1} is formally associated to the following well-known fractional Schr\"{o}dinger equation: \begin{equation}\label{flc}
\varepsilon^{2s}(-\Delta)^s u+Vu=|u|^{2p-2}u\ \ \ \mathrm{in}\ \mathbb{R}^N, \end{equation}
which has been widely studied in recent years. For example, when $\varepsilon=1$ and $V\equiv \lambda>0$, by Fourier analysis and extending \eqref{flc} into a local problem in $\mathbb R^{N+1}_{+}$(see \cite{24}), Frank et al. in \cite{Frank.L-CPAM-2016} proved that the ground state of \eqref{flc} is unique up to translation. In \cite{paj2012}, it was proved that \eqref{flc} has a positive radial ground state when the nonlinear term is replaced by general nonlinear term. When $\varepsilon\to0$, it was shown in \cite{co2016} and \cite{xsc2019} that $\eqref{flc}$ has a family of solutions concentrating at a local minimum of $V$ in the nonvanishing case $\inf_{x\in\mathbb{R}^N}V(x)>0$ and the vanishing case $\inf_{x\in\mathbb{R}^N}V(x)|x|^{2s}<\infty$ respectively. For more results about \eqref{flc}, we would like to refer the readers to \cite{vab2017,jmj2014,ss2013,dmv,sil} and the references therein.
Inspired by the penalization method in \cite{mp1996} for \eqref{flc} with $s=1$,
Moroz et al. in \cite{Ms} introduced a novel penalized technique and obtained a family of single-peak solutions for \eqref{la} under various assumptions on the decay of $V$.
However, for the double nonlocal case, i.e., $s\in(0,1)$ and $\alpha\in (0,N)$, there seems no result on the study of semi-classical solutions
for \eqref{eqs1.1} with vanishing potentials (particularly the potentials with compact support). If $V$ tends to zero at infinity,
the action functional corresponding to \eqref{eqs1.1} is typically not well defined nor Fr\'echet differentiable on $H_{V, \varepsilon}^s(\mathbb R^N)$ (which is defined later). Even in the local case $s=1$, this difficulty is not only technical. As was pointed out in \cite{mor1}, the local Choquard equations with fast decaying potentials indeed may not have positive solutions or even positive super-solutions for certain ranges of parameters. Hence the existence of semi-classical bound states to $\eqref{eqs1.1}$ in the case $\liminf_{|x|\to\infty}V(x) = 0$ is an interesting but hard problem. In this paper, we will focus on the type of problems with the potential $V$ decaying arbitrarily or even being compactly supported. It is worth pointing out that, compared with the local case $s=1$, the nonlocal effects from both $(-\Delta)^s\ (0<s<1)$ and the nonlocal nonlinear term will cause some new difficulties different from \cite{Ms,xsc2019}. For instance, the double nonlocal effects make it quite difficult to derive the uniform regular estimates and construct the penalized function and sup-solution.
In order to state our main results, we first introduce some notations.
For~$0<s<1$~,~the usual fractional Sobolev space is defined as $$ H^s(\mathbb R^N)=\big\{u\in L^2(\mathbb R^N):[u]_s<\infty\big\}, $$ endowed with the norm
$\|u\|_{H^s(\mathbb{R}^N)}=\big(\|u\|_{L^2(\mathbb{R}^N)}^2+[u]_s^2\big)^{\frac{1}{2}}$, where $[u]_s$ is defined as $$
[u]_s^2:=\iint_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}=\int_{\mathbb R^N}|(-\Delta)^{s/2}u|^2. $$
For $N>2s$, we define the space $\dot H^s(\mathbb R^N)$ as $$\dot H^s(\mathbb R^N)=\Big\{u\in L^{2_s^{\ast}}(\mathbb R^N):[u]_s^2<\infty\Big\},$$ which is the completion of $C_c^{\infty}(\mathbb R^N)$ under the norm $[u]_s$, where $2_s^{\ast}:=\frac{2N}{N-2s}$ is the fractional Sobolev critical exponent.
Without loss of generality, hereafter, we define $I_{\alpha}=\frac{1}{|x|^{N-\alpha}}$ and \begin{equation*}
(-\Delta)^su(x):=2\lim\limits_{r\to 0}\int_{\mathbb{R}^N\backslash B_r(x)}\frac{u(x)-u(y)}{|x-y|^{N+2s}}~\mathrm{d} y. \end{equation*}
Our study will rely on the following weighted Hilbert space $$
H_{V, \varepsilon}^s(\mathbb R^N):=\Big\{u\in \dot{H}^s(\mathbb R^N):\int_{\mathbb R^N}V(x)|u|^2<\infty \Big\},$$ with the inner product $$
\langle u,v\rangle_\varepsilon=\varepsilon^{2s}\iint_{\mathbb R^{2N}}\frac{\big(u(x)-u(y)\big)\big(v(x)-v(y)\big)}{|x-y|^{N+2s}} +\int_{\mathbb R^N} V(x)uv$$ and the corresponding norm
$$\|u\|_\varepsilon=\Big(\varepsilon^{2s}[u]_s^2+\int_{\mathbb R^N}V(x)|u|^2\Big)^{\frac{1}{2}}.$$
We assume that $V$ satisfies the following assumption:
($\mathcal{V}$) $V\in C(\mathbb{R}^N,[0,+\infty))$, and there exists a bounded open set~$\Lambda \subset \mathbb R^N$~such that $$ 0<V_0=\underset{x\in \Lambda}{\rm inf}V(x)<\underset{x\in \partial\Lambda}{\rm min} V(x). $$ Moreover, we assume without loss of generality that $0\in\Lambda$ and $\partial \Lambda$ is smooth. From the assumption $(\mathcal{V})$, we choose a smooth bounded open set $U\subset\mathbb R^N$ such that $\Lambda\subset\subset U$ and $\inf_{x\in U\setminus\Lambda}V(x)>V_0$.~
We say that $u$ is a weak solution to equation $\eqref{eqs1.1}$ if $u\in{H_{V,\varepsilon}^s(\mathbb{R}^N)}$ satisfies \begin{equation*}\label{445} \langle u,\varphi\rangle_\varepsilon=\varepsilon^{-\alpha}\int_{\mathbb{R}^N}(I_\alpha*|u|^p)|u|^{p-2}u\varphi \end{equation*} for any $\varphi\in {H_{V,\varepsilon}^s(\mathbb{R}^N)}$.
For convenience, hereafter, given $\Omega\subset\mathbb{R}^N$ and $\tau>0$, we denote $C^\tau(\Omega)=C^{[\tau],\tau-[\tau]}(\Omega)$ with $[\tau]$ denoting the largest integer no larger than $\tau$.
Now we state our main results. \begin{theorem}\label{thm1.1} Let $V$ satisfy ($\mathcal{V}$), $N>2s$, $\alpha\in \big((N-4s)_+,N\big)$, $p\in [2,\frac{N+\alpha}{N-2s})$ satisfying one of the following two assumptions:
$(\mathrm{\mathcal{Q}1})$ $p> p_*:=1+\frac{\max\{s+\frac{\alpha}{2},\alpha\}}{N-2s}$;
$(\mathrm{\mathcal{Q}2})$ $p> p_\omega:=1+\frac{\alpha+2s}{N+2s-\omega}$ if $\inf_{x\in\mathbb{R}^N}(1+|x|^{\omega})V(x)>0$ for some $\omega\in(0,2s]$.
\noindent Then there exists an $\varepsilon_0>0$ such that for all $\varepsilon\in(0,\varepsilon_0)$, problem $(\ref{eqs1.1})$ admits a positive weak solution $u_{\varepsilon}\in C^\sigma_{\mathrm{loc}}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ with $\sigma\in (0,\min\{2s,1\})$, which owns the following two properties:
i) $u_{\varepsilon}$ has a global maximum point $x_{\varepsilon}\in~\bar{\Lambda}$ such that
$$\lim\limits_{\varepsilon\to 0}V(x_{\varepsilon})=V_0~$$
and
\begin{equation*}\label{yyy}
u_{\varepsilon}(x)\leq\frac{C\varepsilon^{\gamma}}{\varepsilon^{\gamma}+|x-x_{\varepsilon}|^{\gamma}}
\end{equation*}
for a positive constant $C$ independent of $\varepsilon$, where $\gamma>0$ is a positive constant close to $N-2s$ from below if $(\mathrm{\mathcal{Q}1})$ holds and close to $N+2s-\omega$ from below if $(\mathrm{\mathcal{Q}2})$ holds;
ii) $u_\varepsilon$ is a classical solution to \eqref {eqs1.1} and $u_\varepsilon \in C_{\mathrm{loc}}^{2s+\vartheta}(\mathbb{R}^N)$ for some $\vartheta\in (0,1)$ if $V\in C_{\mathrm{loc}}^\varrho(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ for some $\varrho\in (0,1)$. \end{theorem}
We also have the following nonexistence result, which implies that the assumptions $(\mathrm{\mathcal{Q}1})$-$(\mathrm{\mathcal{Q}2})$ on $p$ and $V$ in Theorem \ref {thm1.1} are almost optimal.
\begin{theorem}\label{thm1.2'} Let $N>2s$ and $V\in C(\mathbb{R}^N,[0,+\infty))$. Then \eqref{eqs1.1} has no nonnegative nontrivial continuous weak solutions if
$p\in (1,1+\frac{s+\frac{\alpha}{2}}{N-2s})\cup[2, 1+\frac{\alpha}{N-2s})$ and $\limsup_{|x|\to\infty}(1+|x|^{2s})V(x)=0$.
\end{theorem}
\begin{remark}
We do not need any extra assumptions on $V$ out of $\Lambda$ in $(\mathrm{\mathcal{Q}1})$, which means that $V$ can decay arbitrarily even have compact support. The restriction $p\ge2$ in Theorem \ref{thm1.1} is crucially required since $u_\varepsilon^{p-2}$ will be unbounded if $p<2$. Noting that $ (p_*,+\infty)\cap [2,\frac{N+\alpha}{N-2s})\subset(p_{2s},+\infty)\cap [2,\frac{N+\alpha}{N-2s})$ and $p_\omega$ is decreasing on $\omega\in (0,2s]$, one can see from Theorem \ref{thm1.1} that the restriction on $p$ is weaker when $V$ decays slower. Specially, when $\omega<\min\{2s,N-\alpha\}$, the restriction on $p$ in $(\mathrm{\mathcal{Q}2})$ holds naturally since $p_\omega<2$.
The proof of our main results depends strongly on Proposition \ref{tb}, which is a basis of applying comparison principle. We use a tremendous amount of delicate analysis to check Proposition \ref{tb}.
\end{remark}
Let us now elaborate the main difficulties and novelties in our proof.
We will use the variational sketch to prove our results, hence it is natural to consider the following functional corresponding to $\eqref{eqs1.1}$ \begin{equation}\label{zr}
E_\varepsilon(v):=\frac{1}{2}\|u\|_\varepsilon^2-\frac{1}{2p\varepsilon^\alpha}\int_{\mathbb{R}^N}|I_\frac{\alpha}{2}*|u|^p|^2,\ v\in {H_{V,\varepsilon}^s(\mathbb{R}^N)}, \end{equation}
whose critical points are weak solutions of \eqref{eqs1.1}. However, $E_\varepsilon$ is not well defined when $V$ decays very fast. For example, the function $\omega_\mu:=(1+|x|^2)^{-\frac{\mu}{2}}\in {H_{V,\varepsilon}^s(\mathbb{R}^N)}$ but $\int_{\mathbb{R}^N}|I_\frac{\alpha}{2}* w_\mu^p|^2=+\infty$ for any $\mu\in(\frac{N-2s}{2},\frac{N+\alpha}{2p})$ if $V\le C(1+|x|^{2s})^{-1}$. In addition, it is hard to verify directly the (P.S.) condition only under the local assumption $(\mathcal{V})$ on $V$. Furthermore, due to the nonlocal effect of the Choquard term, if $V$ decays to 0 at infinity, it is very tricky to obtain a priori regular estimate desired for a weak solution $u$ of \eqref{eqs1.1} because we neither know whether $u\in L^\infty(\mathbb{R}^N)$ nor know whether $I_\alpha*u^p\in L^\infty(\mathbb{R}^N)$. To overcome these difficulties, we employ a type of penalized idea to modify the nonlinearity. We will introduce the following penalized problem (see \eqref{gp} and \eqref{eqs3.2}) \begin{align}\label{Aeq1.7} \quad\varepsilon^{2s} (-\Delta)^s u+V(x)u\nonumber &=p\varepsilon^{-\alpha}\Big(I_\alpha*\int_0^{u_+}\big(\chi_\Lambda t_+^{p-1}+\chi_{\mathbb{R}^N\setminus\Lambda}\min\{t^{p-1}_+,\mathcal P_\varepsilon(x)\}\big)\Big)\\ & \ \ \quad\quad\quad \times\Big(\chi_\Lambda u_+^{p-1}+\chi_{\mathbb{R}^N\setminus\Lambda}\min\{u^{p-1}_+,\mathcal P_\varepsilon(x)\}\Big). \end{align}
Under better pre-assumptions (see \eqref{eqs2.1}, $(\mathcal{P}_1), (\mathcal{P}_2)$ in Section \ref{sec2}) on the penalized function $\mathcal{P}_{\varepsilon}$, the functional corresponding to \eqref{Aeq1.7} is $C^1$ in ${H_{V,\varepsilon}^s(\mathbb{R}^N)}$ and satisfies the (P.S.) condition. Hence the standard min-max procedure results in a critical point $u_{\varepsilon}$ which solves equation \eqref{Aeq1.7}. To prove that $u_\varepsilon$ is indeed a solution to the original problem $(\ref{eqs1.1})$, a crucial step is to show that \begin{equation}\label{Aeq1.8} u^{p-1}_{\varepsilon}\le \mathcal{P}_{\varepsilon}\ \ \text{in}\ \mathbb R^N\backslash\Lambda, \end{equation} in which some new difficulties caused by the nonlocal term $(-\Delta)^s$($0<s<1$) and the nonlocal nonlinear term will be involved.
Firstly, we need to prove the concentration of $u_{\varepsilon}$ (see Lemma \ref{jz}). This step relies on the uniform regularity of $u_{\varepsilon}$.
However, under the double nonlocal effect of $(-\Delta)^s$ and the Choquard term, the regularity estimates here are non-trivial after the truncation of the nonlinear term (see \eqref{gp}). In \cite{dss}, using essentially the fact that week solutions of \eqref{la} belongs to $L^2(\mathbb{R}^N)$, some regularity results for solutions of \eqref{la} were obtained. But in our case, the solutions $u_{\varepsilon}$ may not be $L^2$-integrable if especially $V$ is compactly supported. To overcome this difficulty, we first use directly the Moser iteration to get the uniform $L^\infty$-estimates (see Lemma \ref{w}) and then apply a standard convolution argument (see \cite[Proposition 5]{re}) to get the uniform H\"{o}lder estimates. Our proof is quite different from that of \cite{dss}, since the $L^2$-norm of $u_{\varepsilon}$ here is unknown for fast decay $V$.
We emphasize here that the upper bound on the energy (see Lemma \ref{lem4.5}) and the construction of the penalized function play a key role in the regularity estimates since we expect not only the sufficient regularity
estimates for fixed $\varepsilon>0$ but also the uniform regularity estimates for all $\varepsilon\in (0,\varepsilon_0)$.
Secondly, the double nonlocal effects from the Choquard term and the operator $(-\Delta)^s$ make the construction of penalized function and sup-solution to the linearized equation (see \eqref{q1}) derived from the concentration of $u_{\varepsilon}$ more difficult than that in \cite{Ms,xsc2019}. By large amounts of delicate nonlocal analysis, we find a sup-solution
$$w_\mu=\frac{1}{(1+|x|^2)^{\frac{\mu}{2}}},$$
where $\mu>0$ is a constant depending on different decay rates of $V$ (see the assumptions $(\mathcal{Q}_1)-(\mathcal{Q}_2)$ in Theorem \ref{thm1.1} above). We would like to emphasize that the sup-solutions above imply that the solutions $u_{\varepsilon}$ can decay fast than $|x|^{2s-N}$ or even $|x|^{-N}$ if $V$ decays slowly, which is quite different from \cite{xsc2019}. Moreover, the different behavior of $(-\Delta)^sw_\mu$ and $-\Delta w_\mu$, for instance
$(-\Delta)^sw_\mu\sim |x|^{-N-2s}$
and
$-\Delta w_\mu\sim |x|^{-\mu-2}$
as $|x|\to\infty$ for any $\mu>N$, makes our proof quite different from that of \cite{Ms}.
Using the decay properties of $(-\Delta)^sw_\mu$ (see Proposition \ref{tb}), we indeed provide a specific comparison function $w_\mu$ to derive decay estimates from above and below for solutions of general fractional equations. As an application, Proposition \ref{tb} is used to the full in the proof of Theorem \ref{thm1.2'} by carrying out a skillful iteration procedure. We point out that it is interesting that Proposition \ref{tb} can also be applied to the case $\inf_{\mathbb{R}^N}V(x)>0$. For instance, for constant $\kappa>0$, instead of the comparison functions constructed by the Bessel Kernel (see \cite[Lemmas 4.2 and 4.3]{paj2012}), function $w_{N+2s}(\lambda x)$ can be taken as a super-solution ($\lambda$ small) or a sub-solution ($\lambda$ large) to
$$(-\Delta)^s u+\kappa u=0,\quad |x|\ge R_\lambda$$ for some suitable $R_\lambda>0$.
The proof of Theorem \ref{thm1.2'} depends strongly on the positivity of solutions. To this end, we establish a general strong maximum principle for weak super-solutions (see \eqref{wq9}).
It should be mentioned that the potential $V$ affects the decay properties of solutions. On one hand, assume that $c<(1+|x|^{2s})V(x)<C$ for $C,c>0$, then by Remark \ref{rma}, $u_\varepsilon$ given by Theorem \ref{thm1.1} satisfies
$ u_\varepsilon\ge \frac{C_\varepsilon}{1+|x|^N}$ for some $C_\varepsilon>0$, and thereby
$$\limsup_{|x|\to\infty}u_\varepsilon(x)(1+|x|)^{N+2s}=+\infty.$$
On the other hand, we can check by the same way as that in \cite{dss}, that any nonnegative weak solution ${u}_\varepsilon$ to \eqref{eqs1.1} must satisfy
$$\limsup_{|x|\to\infty}{u}_\varepsilon(x)(1+|x|)^{N+2s}<\infty,$$ for $p\in [2,\frac{N+\alpha}{N-2s})$ if $\inf_{x\in\mathbb{R}^N}V(x)>0$.
Hence, the solution $u_\varepsilon$ has different decay behavior at infinity between the nonvanishing case ($\inf_{x\in\mathbb{R}^N}V(x)>0$) and the vanishing case ($\lim_{|x|\to\infty}V(x)=0$). In fact, we believe that solutions decay faster if $V$ decays slower (see the choice of $\gamma$ in Theorem \ref{thm1.1}) .
This paper will be organized as follows: In Section \ref{sec2}, we modify the nonlinear term of \eqref{eqs1.1} and get a new well-defined penalized functional whose critical point $u_{\varepsilon}$ can be obtained by min-max procedure in \cite{wm}. In Section \ref{sec3}, we give the essential energy estimates and regularity estimates of $u_{\varepsilon}$ and prove the concentration property of $u_{\varepsilon}$. In Section \ref{sec4}, the concentration of $u_\varepsilon$ will be used to linearize the penalized equation for which we construct a suitable super-solution and the penalized function. We also prove the decay estimates on $u_{\varepsilon}$ by comparison principle, which shows that $u_{\varepsilon}$ solves indeed the origin problem \eqref{eqs1.1}. In Section \ref{s6}, we present some nonexistence results and verify Theorem \ref{thm1.2'}.
Throughout this paper, fixed constants are frequently denoted by $C>0$ or $c>0$, which may change from line to line if necessary, but are always independent of the variable under consideration. What's more, $\varepsilon\in (0,\varepsilon_0)$ and $\varepsilon_0$ can be taken smaller depending on the specific needs.
\section{The penalized problem}\label{sec2}
In this section, we introduce a penalized functional which satisfies all the assumptions of Mountain Pass Theorem by truncating the nonlinear term outside $\Lambda$, and obtain a nontrivial Mountain-Pass solution $u_{\varepsilon}$ to the modified problem.
We first list the following inequalities which are essential in this paper.
\begin{proposition}\label{prop2.1}{\rm(\cite{ {lr2008}} Sharp\ fractional\ Hardy\ inequality)}
Let~$N> 2s, s\in (0,1)$. Then for any $u\in~\dot{H}^s(\mathbb R^N)$, there exists a constant $\mathcal{C}_{N,s}>0$ depending only on $N$ and $s$ such that $$
\mathcal{C}_{N,s}\int_{\mathbb R^N}\frac{|u(x)|^2}{|x|^{2s}}\le [u]_s^2. $$ \end{proposition}
\begin{proposition}\label{prop2.2}{\rm(\cite{{ege2012}} Fractional\ embedding\ theorem)} Let $N>2s$, then the embeddings $\dot{H}^s(\mathbb R^N)\subset L^{2_s^*}(\mathbb R^N)$ and $H^s(\mathbb R^N)\subset L^q(\mathbb R^N)$ are continuous for any $q\in [2, 2_s^*]$. Moreover, the following embeddings are compact $$H^s(\mathbb{R}^N)\subset L_{\mathrm{loc}}^q(\mathbb{R}^N),\ \dot{H}^s(\mathbb{R}^N)\subset L_{\mathrm{loc}}^q(\mathbb{R}^N),\quad q\in[1,2_s^{\ast}).$$ \end{proposition}
\begin{proposition}\label{prop2.3}{\rm(Rescaled\ Sobolev\ inequality)} Assume $N>2s$ and $q\in [2, 2_s^{*}]$. Then for every $u\in H^s_{V,\varepsilon}(\mathbb R^N)$, it holds
$$\int_{\Lambda}|u|^q\leq\frac{C}{\varepsilon^{N(\frac{q}{2}-1)}}\Big(\int_{\mathbb R^N}\varepsilon^{2s}|(-\Delta)^{s/2}u|^2+V|u|^2 \Big)^{\frac{q}{2}},$$ where $C>0$ depends only on $N$, $q$ and $V_0$. \end{proposition} \begin{proof} Actually, by H\"{o}lder inequality, Young's inequality and Proposition \ref{prop2.2}, we have \begin{align*} \begin{split}
\|u\|_{L^q(\Lambda)}&\leq\|u\|_{L^2(\Lambda)}^{\theta}\|u\|_{L^{2_s^*}(\Lambda)}^{1-\theta}\leq C\varepsilon^{-\beta}\|u\|_{L^2(\Lambda)}^{\theta} \varepsilon^{\beta}[u]_s^{1-\theta}\\
&\leq C\theta\varepsilon^{-\frac{\beta}{\theta}}\|u\|_{L^2(\Lambda)}+C(1-\theta)\varepsilon^{\frac{\beta}{1-\theta}}[u]_s\\
&\leq \frac{C}{\varepsilon^{N(\frac{1}{2}-\frac{1}{q})}}\Big(\varepsilon^s[u]_s+\Big(\int_{\mathbb R^N}V|u|^2\Big)^{\frac{1}{2}}\Big), \end{split} \end{align*} where $\frac{1}{q}=\frac{\theta}{2}+\frac{1-\theta}{2_s^*}$, $\beta=\theta N(\frac{1}{2}-\frac{1}{q})$ and $\inf_{\Lambda}V=V_0>0$. \end{proof}
\begin{proposition}\label{prop2.4}{\rm(\cite{lflm} Hardy-Littlewood-Sobolev\ inequality)} Let $N\in \mathbb{N}$, $\alpha\in (0, N)$ and $q\in (1, \frac{N}{\alpha})$. If $u\in L^q(\mathbb R^N)$, then $I_{\alpha}*u\in L^{\frac{Nq}{N-\alpha q}}$ and $$
\Big(\int_{\mathbb{R}^{N}}\left|I_{\alpha} * u\right|^{\frac{N q}{N-\alpha q}}\Big)^{\frac{N-\alpha q}{Nq}} \leq C\Big(\int_{\mathbb{R}^{N}}|u|^{q}\Big)^{\frac{1}{q}} ,$$ where $C>0$ depends only on $\alpha$, $N$ and $q$. \end{proposition}
\begin{proposition}\label{prop2.5}{\rm(\cite{sewg} Weighted Hardy-Littlewood-Sobolev\ inequality)} Let $N\in \mathbb{N}$, $\alpha\in (0, N)$. If
$u \in L^{2}\left(\mathbb{R}^{N},|x|^{\alpha} \mathrm{d} x\right)$, then $I_{\frac{\alpha}{2}} * u \in L^{2}\left(\mathbb{R}^{N}\right)$ and $$
\int_{\mathbb{R}^{N}}|I_{\frac{\alpha}{2}} * u|^{2} \leq C_{\alpha} \int_{\mathbb{R}^{N}}|u(x)|^{2}|x|^{\alpha}, $$ where $C_{\alpha}=\frac{1}{2^{\alpha}}\Big(\frac{\Gamma\left(\frac{N-\alpha}{4}\right)}{\Gamma\left(\frac{N+\alpha}{4}\right)}\Big)^{2}$. \end{proposition}
By the assumption $(\mathcal{V})$, we choose a family of nonnegative penalized functions $\mathcal{P}_{\varepsilon}\in L^\infty(\mathbb R^N)$ for $\varepsilon>0$ small in such a way that \begin{equation}\label{eqs2.1}
\mathcal{P}_{\varepsilon}(x)=0~ \text{for}~ x\in~\Lambda\ \text{and}\ \lim\limits_{\varepsilon\to 0} \| \mathcal{P}_{\varepsilon}\|_{L^\infty (\mathbb R^N)}=0. \end{equation} The explicit construction of $\mathcal{P}_{\varepsilon}$ will be described later in Section \ref{sec4}. Before that, we only need the following two embedding assumptions on $\mathcal{P}_{\varepsilon}$:
$\left(\mathcal{P}_{1}\right)$ the space $H_{V,\varepsilon}^s\left(\mathbb{R}^{N}\right)$ is compactly embedded into $L^{2}\left(\mathbb{R}^{N},\mathcal{P}_{\varepsilon}(x)^{2}|x|^{\alpha} \mathrm{d} x\right)$,
$\left(\mathcal{P}_{2}\right)$ there exists $\kappa\in (0,1/2)$ such that \begin{equation}\label{a}
\frac{p}{\varepsilon^{\alpha}} \int_{\mathbb{R}^{N}}\left|I_{\frac{\alpha}{2}} *\left(\mathcal{P}_{\varepsilon} u\right)\right|^{2} \leq
\frac{pC_\alpha}{\varepsilon^{\alpha}} \int_{\mathbb{R}^{N}}|\mathcal{P}_{\varepsilon} u|^2|x|^\alpha
\le\kappa \int_{\mathbb{R}^{N}} \varepsilon^{2s}|(-\Delta)^{\frac{s}{2}} u|^{2}+V(x)|u|^{2} \end{equation} for $u \in {H_{V,\varepsilon}^s(\mathbb{R}^N)}$, where $C_\alpha$ is given by Proposition \ref {prop2.5}.
Basing on the two assumptions above, we define the penalized nonlinearity $g_{\varepsilon}:\mathbb R^N\times\mathbb R\to \mathbb R$ as \begin{equation}\label{gp} g_{\varepsilon}(x, t):=\chi_{\Lambda}(x) t_{+}^{p-1}+\chi_{\mathbb{R}^{N} \backslash \Lambda}(x) \min \big\{t_{+}^{p-1}, \mathcal{P}_{\varepsilon}(x)\big\}, \end{equation} where $\chi_{\Omega}$ is the characteristic function corresponding to $\Omega\subset\mathbb{R}^N$. Set $G_{\varepsilon}(x,t)=\int_0^tg_{\varepsilon}(x,r)~dr.$ One can check that $g_{\varepsilon}(x, t) \leq t_{+}^{p-1}$ in $\mathbb{R}^N\times\mathbb R$ and
\begin{align}\label{3y7} &0 \leq G_{\varepsilon}(x, t)\le g_{\varepsilon}(x, t) t \leq t_{+}^{p} \chi_{\Lambda}+\mathcal{P}_{\varepsilon}(x) t_{+}\chi_{\Lambda^c}\quad \mathrm{in}\ \mathbb{R}^N\times\mathbb R,\nonumber\\ &0 \leq G_{\varepsilon}(x, t) \leq \frac{1}{p} t_{+}^{p} \chi_{\Lambda}+\mathcal{P}_{\varepsilon}(x) t_{+}\chi_{\Lambda^c}\quad \mathrm{in}\ \mathbb{R}^N\times\mathbb R,\\ &0 \leq pG_{\varepsilon}(x, t)=g_{\varepsilon}(x, t) t=t_+^p\quad \mathrm{in}\ \Lambda\times\mathbb R.\nonumber \end{align}
We consider the following penalized problem \begin{equation}\label{eqs3.2} \varepsilon^{2s} (-\Delta)^s u+V u=p \varepsilon^{-\alpha}\big(I_{\alpha} * G_{\varepsilon}(x,u)\big) g_{\varepsilon}(x,u) \quad \text { in } \mathbb{R}^{N}, \end{equation} whose Euler-Lagrange functional $J_{\varepsilon}~:~H_{V,\varepsilon}^s(\mathbb R^N)\to \mathbb R$ is defined as $$
J_{\varepsilon}(u)=\frac{1}{2}\|u\|_\varepsilon^2-\frac{p}{2 \varepsilon^{\alpha}} \int_{\mathbb{R}^{N}}|I_{\frac{\alpha}{2}} * G_{\varepsilon}(x,u)|^{2}. $$ For $u\in H_{V,\varepsilon}^s(\mathbb R^N)$, if $p\in [\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2s}]$, by ($\mathcal{V}$), Propositions \ref{prop2.3} and \ref{prop2.4}, we have \begin{equation}\label{b}
\frac{p}{\varepsilon^{\alpha}}\int_{\mathbb R^N}|I_\alpha*(\chi_\Lambda |u|^p)|^2\leq \frac{C}{\varepsilon^{\alpha}}\Big(\int_{\Lambda}|u|^{\frac{2Np}{N+\alpha}}\Big)^{\frac{N+\alpha}{N}}
\leq \frac{C}{\varepsilon^{(p-1)N}}\Big(\int_{\mathbb{R}^{N}} \varepsilon^{2s}|(-\Delta)^{\frac{s}{2}} u|^{2}+V|u|^{2}\Big)^p, \end{equation} where $\frac{2Np}{N+\alpha}\in[2, 2_s^*]$.
From \eqref{a}, \eqref{3y7} and \eqref{b}, we conclude that \begin{equation*}\label{c}
\frac{p}{2 \varepsilon^{\alpha}} \int_{\mathbb{R}^{N}}\left|I_{\frac{\alpha}{2}} * G_{\varepsilon}(x, u)\right|^{2}\leq C\|u\|_\varepsilon^2
+\frac{C}{\varepsilon^{(p-1)N}}\|u\|_\varepsilon^{2p}, \end{equation*} which implies that $J_\varepsilon$ is well defined in ${H_{V,\varepsilon}^s(\mathbb{R}^N)}$ if $\left(\mathcal{P}_{2}\right)$ holds.
Next, we prove that the functional $J_{\varepsilon}$ is $C^1$ in ${H_{V,\varepsilon}^s(\mathbb{R}^N)}$.
\begin{lemma}\label{lem3.1} If $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2s})$ and $(\mathcal{P}_{1})$-$(\mathcal{P}_{2})$ hold, then $J_{\varepsilon} \in C^{1}(H_{V,\varepsilon}^{s}\left(\mathbb{R}^{N})\right)$ and $$ \left\langle J_{\varepsilon}^{\prime}(u), \varphi\right\rangle=\left\langle u, \varphi\right\rangle_\varepsilon-\frac{p}{\varepsilon^{\alpha}} \int_{\mathbb{R}^{N}}\big(I_{\alpha} * G_{\varepsilon}(x,u)\big) g_{\varepsilon}(x,u) \varphi,\,\,\forall\,u \in H_{V,\varepsilon}^{s}\left(\mathbb{R}^{N}\right),\,\varphi \in H_{V,\varepsilon}^{s}\left(\mathbb{R}^{N}\right). $$ \end{lemma}
\begin{proof} In fact, it suffices to show that the nonlinear term
$$\mathcal{J}_{\varepsilon}:=\int_{\mathbb{R}^{N}}\left|I_{\frac{\alpha}{2}} * G_{\varepsilon}(x, u)\right|^{2}$$ is $C^1$ in ${H_{V,\varepsilon}^s(\mathbb{R}^N)}$. Let $u_n\to u $ in $H_{V,\varepsilon}^s(\mathbb R^N)$. Noting that $\frac{2Np}{N+\alpha}<2_s^*$, from \eqref{3y7}, $(\mathcal{P}_1)$, Propositions \ref{prop2.2}, \ref{prop2.4} and \ref{prop2.5}, we deduce that \begin{align}\label{g}
&\int_{\mathbb R^N}|I_{\frac{\alpha}{2}}*\big(G(x,u_n)-G(x,u)\big)|^2\nonumber\\
\leq&2\int_{\mathbb R^N}|I_{\frac{\alpha}{2}}*\big(\chi_\Lambda(|u_n|^p-|u|^p)\big)|^2+2\int_{\mathbb R^N}|I_{\frac{\alpha}{2}}*(\mathcal{P}_\varepsilon|u_n-u|)|^2\nonumber\\
\leq&C\Big(\int_{\Lambda}(|u|_n^p-|u|^p)^{\frac{2 N}{N+\alpha}}\Big)^{\frac{N+\alpha}{N}}+C\int_{\mathbb{R}^{N}} |u_n-u|^{2}\mathcal{P}_{\varepsilon}^2|x|^{\alpha}\nonumber\\ =&o_n(1), \end{align} which yields that $\mathcal{J}_{\varepsilon}$ is continuous.
For any $\varphi \in {H_{V,\varepsilon}^s(\mathbb{R}^N)}$ and $0<|t|<1$, by \eqref{3y7}, it holds \begin{align*}
&\quad\left||I_{\frac{\alpha}{2}} * G_{\varepsilon}(x, u+t\varphi)|^{2}-|I_{\frac{\alpha}{2}} * G_{\varepsilon}(x, u)|^{2}\right|/t\\
&\leq C\Big(\big|I_{\frac{\alpha}{2}}*\big((|u|^p+|\varphi|^p)\chi_{\Lambda}\big)\big|^2+
\big|I_{\frac{\alpha}{2}}*\big(\mathcal{P}_{\varepsilon}(|u|+|\varphi|)\big)\big|^2\Big)\in L^1(\mathbb R^N). \end{align*} Then by Dominated Convergence Theorem, we get \begin{align*}
\langle\mathcal{J}_{\varepsilon}'(u),\varphi\rangle&=\lim\limits_{t\to 0}\int_{\mathbb R^N}\frac{|I_{\frac{\alpha}{2}} * G_{\varepsilon}(x, u+t\varphi)|^{2}-|I_{\frac{\alpha}{2}} * G_{\varepsilon}(x, u)|^{2}}{t}\\
&=\int_{\mathbb R^N}\lim\limits_{t\to 0}\frac{|I_{\frac{\alpha}{2}} * G_{\varepsilon}(x, u+t\varphi)|^{2}-|I_{\frac{\alpha}{2}} * G_{\varepsilon}(x, u)|^{2}}{t}\\
&=2\int_{\mathbb R^N}\Big(I_{\frac{\alpha}{2}} * G_{\varepsilon}(x, u)\Big)\Big(I_{\frac{\alpha}{2}} * \big(g_{\varepsilon}(x, u)\varphi\big)\Big),\\
&=2\int_{\mathbb R^N}\big(I_\alpha * G_{\varepsilon}(x, u)\big)g_{\varepsilon}(x, u)\varphi, \end{align*} which indicates the existence of Gateaux derivative.
For the continuity of $\mathcal{J}_{\varepsilon}'$, we observe that \begin{align*} \langle\mathcal{J}_{\varepsilon}'(u_n)-\mathcal{J}_{\varepsilon}'(u),\varphi\rangle =&2\int_{\mathbb R^N}\big(I_{\frac{\alpha}{2}} * G_{\varepsilon}(x, u_n)\big)\Big(I_{\frac{\alpha}{2}} *\Big(\big(g_{\varepsilon}(x, u_n)-g_{\varepsilon}(x, u)\big)\varphi\Big)\Big)\\ &+2\int_{\mathbb R^N}\Big(I_{\frac{\alpha}{2}} * \big(G_{\varepsilon}(x, u_n)-G_{\varepsilon}(x, u)\big)\Big)\Big(I_{\frac{\alpha}{2}} *\big(g_{\varepsilon}(x, u)\varphi\big)\Big). \end{align*} Then, by H\"{o}lder inequality and calculations similar to \eqref{g}, we deduce that \begin{align*}
&\big|\langle\mathcal{J}_{\varepsilon}'(u_n)-\mathcal{J}_{\varepsilon}'(u),\varphi\rangle\big|
= o_n(1)\|\varphi\|_\varepsilon. \end{align*} Hence $\mathcal{J}_{\varepsilon}'(u)$ is continuous and the proof is completed. \end{proof}
Furthermore, we deduce that $J_{\varepsilon}$ satisfies the (P.S.) condition. \begin{lemma}\label{lem3.3} If $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2s})$ and $\left(\mathcal{P}_1\right)$-$\left(\mathcal{P}_2\right)$ hold, then $J_{\varepsilon}$ satisfies the (P.S.) condition. \end{lemma}
\begin{proof} By Lemma \ref{lem3.1}, $J_{\varepsilon} \in C^{1}(H_{V,\varepsilon}^{s}\left(\mathbb{R}^{N})\right)$. Let $\{u_n\}\subset {H_{V,\varepsilon}^s(\mathbb{R}^N)}$ satisfy $J_{\varepsilon}(u_n)\leq c$ and $J_{\varepsilon}'(u_n)\to 0.$ We claim that $\{u_n\}$ is bounded in ${H_{V,\varepsilon}^s(\mathbb{R}^N)}$. Indeed, by \eqref{3y7}, we have \begin{align}\label{d} \begin{split} J_{\varepsilon}(u_n)-\frac{1}{2}\langle J_{\varepsilon}'(u_n),u_n\rangle =&\frac{p}{2\varepsilon^\alpha}\int_{\mathbb R^N}\big(I_\alpha* G(x,u_n)\big)\big(g_\varepsilon(u_n)u_n-G_\varepsilon(x,u_n)\big)\\[3mm] \ge&\frac{p-1}{2p\varepsilon^\alpha}\int_{\Lambda}\big(I_\alpha* (\chi_\Lambda u_{n+}^p)\big)u_{n+}^p. \end{split} \end{align} On the other hand, in view of \eqref{3y7}, Young's inequality and \eqref{a}, we see that \begin{align}\label{e}
\frac{1}{2}\|u_n\|_\varepsilon^2=&\frac{p}{2\varepsilon^\alpha}\int_{\mathbb{R}^N}|I_\frac{\alpha}{2}* G(x,u_n)|^2\mathrm{d} x+J_\varepsilon(u_n)\nonumber\\
\le&\frac{p}{2\varepsilon^\alpha}\int_{\mathbb{R}^N}\big|\frac{1}{p}I_\frac{\alpha}{2}* (\chi_\Lambda u_{n+}^p)+I_\frac{\alpha}{2}* (\mathcal{P}_\varepsilon |u_n|)\big|^2\mathrm{d} x+J_\varepsilon(u_n)\nonumber\\
\le&\kappa\|u_n\|_{\varepsilon}^{2}+\frac{1}{p\varepsilon^\alpha}\int_{\mathbb{R}^{N}}\big|I_{\frac{\alpha}{2}} *\left(\chi_{\Lambda} u_{n+}^{p}\right)\big|^{2}+J_\varepsilon(u_n), \end{align} Then it holds from $\kappa<1/2$ and \eqref{d}--\eqref{e} that \begin{align}\label{f}
\|u_n\|_\varepsilon^2\le C'_1J_\varepsilon(u_n)+C'_2|\langleJ_\varepsilon'(u_n),u_n\rangle|, \end{align}
where $C'_1, C'_2>0$ are constants independent of $\varepsilon$. Then $\|u_n\|_\varepsilon\le C$. Up to a subsequence, we have $u_n\rightharpoonup u$ in ${H_{V,\varepsilon}^s(\mathbb{R}^N)}$.
By the same proof as \eqref{g}, we have \begin{align}\label{d1} \begin{split} \int_{\mathbb{R}^{N}}\big(I_\alpha * G_{\varepsilon}(x,u_{n})\big)g_{\varepsilon}(x,u_{n}) u_{n}\to \int_{\mathbb{R}^{N}}\big(I_\alpha * G_{\varepsilon}(x,u)\big)g_{\varepsilon}(x,u) u \end{split} \end{align} and \begin{align*} \int_{\mathbb{R}^{N}}\big(I_\alpha * G_{\varepsilon}(x,u_{n})\big)g_{\varepsilon}(x,u_{n}) u\to \int_{\mathbb{R}^{N}}\big(I_\alpha * G_{\varepsilon}(x,u)\big)g_{\varepsilon}(x,u) u. \end{align*} It follows from $u_n\rightharpoonup u$ in ${H_{V,\varepsilon}^s(\mathbb{R}^N)}$ that \begin{align}\label{ddh}
0=\lim_{n\to\infty}\langleJ_\varepsilon'(u_n),u\rangle=\|u\|_\varepsilon^2-\frac{p}{\varepsilon^\alpha}\int_{\mathbb{R}^{N}}\big(I_\alpha * G_{\varepsilon}(x,u)\big)g_{\varepsilon}(x,u) u. \end{align} Combining $\eqref{d1}$ with \eqref{ddh}, we get \begin{align*}\label{d3}
\lim_{n\to\infty}\|u_n-u\|_\varepsilon^2=&\lim_{n\to\infty}(\|u_n\|_\varepsilon^2-\|u\|_\varepsilon^2)\\ =&\lim_{n\to\infty}\frac{p}{\varepsilon^\alpha}\Big(\int_{\mathbb{R}^{N}}\big(I_{\alpha} * G_{\varepsilon}(x, u_{n})\big) g_{\varepsilon}(x, u_{n}) u_{n}-\int_{\mathbb{R}^{N}}\big(I_{\alpha} * G_{\varepsilon}(x, u)\big) g_{\varepsilon}(x, u) u\Big)\\ &+\lim_{n\to\infty}\langleJ_\varepsilon'(u_n),u_n\rangle=0, \end{align*} which completes the proof. \end{proof}
Finally, it is easy to check that $J_{\varepsilon}$ owns the Mountain Pass Geometry, so by Lemma \ref {lem3.1} and Lemma \ref {lem3.3}, we can find a critical point for $J_{\varepsilon}$ via min-max theorem (\cite{wm}).
Define the Mountain-Pass value $c_{\varepsilon}$ as \begin{equation}\label{Ade2.11}
c_{\varepsilon}:=\inf_{\gamma\in \Gamma_{\varepsilon}}\max_{t\in[0,1]}J_{\varepsilon}(\gamma(t)), \end{equation} where $$\Gamma_{\varepsilon}:=\Big\{\gamma\in C\big([0,1],{H_{V,\varepsilon}^s(\mathbb{R}^N)}\big)\mid\gamma(0)=0,\ J_{\varepsilon}\big(\gamma(1)\big)<0\Big\}.$$
We have the following lemma immediately. \begin{lemma}\label{lem3.6}Let $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2s})$ and $\left(\mathcal{P}_1\right)$-$\left(\mathcal{P}_2\right)$ hold. Then $c_{\varepsilon}$ can be achieved by a $u_{\varepsilon}\in {H_{V,\varepsilon}^s(\mathbb{R}^N)}\setminus\{0\}$, which is a nonnegative weak solution of the penalized equation $(\ref{eqs3.2})$. \end{lemma}
\begin{proof} The existence is trivial by Lemmas \ref{lem3.1}, \ref{lem3.3} and the min-max procedure in \cite{wm}.
Letting $u_{\varepsilon,-}$ be a test function in $\eqref{eqs3.2}$, we obtain \begin{equation}\label{d31} \begin{aligned}
&\varepsilon^{2s}\iint_{\mathbb{R}^{2 N}} \frac{|u_{\varepsilon,-}(x)-u_{\varepsilon,-}(y)|^{2}}{|x-y|^{N+2 s}}+\int_{\mathbb{R}^{N}} V|u_{\varepsilon,-}|^{2}\\
\le&\varepsilon^{2s}\iint_{\mathbb{R}^{2 N}} \frac{\big(u_{\varepsilon,-}(x)-u_{\varepsilon,-}(y)\big)\big(u_{\varepsilon,+}(x)-u_{\varepsilon,+}(y)\big)}{|x-y|^{N+2 s}}\le0, \end{aligned} \end{equation} which leads to $u_{\varepsilon,-}=0$ and thereby $u_{\varepsilon}$ is nonnegative. \end{proof}
To expect the positivity of $u_\varepsilon$, we give the following strong maximum principle. \begin{lemma}\label{w61} Let $c(x)\in L^\infty_{\mathrm{loc}}(\mathbb{R}^N)$ and $u\in \dot{H}^s(\mathbb{R}^N)$ be a weak supersolution to \begin{align}\label{wq9}
(-\Delta)^su+c(x)u=0,\quad x\in\mathbb{R}^N. \end{align}
If $u\in C(\mathbb{R}^N)$ and $u\ge0$ in $\mathbb{R}^N$, then either $u\equiv0$ in $\mathbb{R}^N$ or $u>0$ in $\mathbb{R}^N$. \end{lemma} \begin{proof} Suppose by contradiction that there exist $x_0,y_0\in\mathbb{R}^N$ such that $u(x_0)=0$ and $u(y_0)>0$. Denote
$$r:=\frac{|x_0-y_0|}{2},\ R:=2\max\{|x_0|,|y_0|\},\ \sigma:=\|c(x)\|_{L^\infty(B_r(x_0))},\ M:=\max_{B_R(0)}u(x).$$ Clearly, $B_r(x_0)\subset B_R(0)$, $y_0\in B_R(0)\backslash B_r(x_0)$ and $u$ weakly satisfies \begin{align}\label{qgs}
(-\Delta)^su+\sigma u\ge(\sigma-c(x))u\ge0,\quad x\in B_r(x_0). \end{align}
Define $\bar{u}=\min\{M,u(x)\}$. We see that $\bar{u}\in C(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, $0\le \bar{u}\le u(x)$ in $\mathbb{R}^N$ and $\bar{u}=u(x)$ in $B_R(0)$. Moreover, since $|\bar{u}(x)-\bar{u}(y)|\le|u(x)-u(y)|$, we deduce that $\bar{u}\in \dot{H}^s(\mathbb{R}^N)$.
We claim that the following problem \begin{equation}\label{w18}\left\{
\begin{aligned}
(-\Delta)^sv+\sigma v=&0,\quad x\in B_r(x_0),\\
v=&\bar{u},\quad x\in \mathbb{R}^N\backslash B_r(x_0)
\end{aligned}\right. \end{equation} has a weak solution $v\in \dot{H}^s(\mathbb{R}^N)$.
Indeed, define the following Hilbert space $$ \mathcal{H}_0^s(B_r(x_0)):=\big\{\phi\in H^s(\mathbb{R}^N): \phi\equiv0\ \mathrm{on}\ \mathbb{R}^N\backslash B_r(x_0)\big\}.$$ Since $(-\Delta)^s\bar{u}+\sigma \bar{u}\in \big(\mathcal{H}_0^s(B_r(x_0))\big)^{-1}$ in the sense of \begin{align*}
\big \langle (-\Delta)^s\bar{u}+\sigma \bar{u}, \phi\big\rangle:=\int_{\mathbb{R}^N}(-\Delta)^{s/2}\bar{u}(-\Delta)^{s/2}\phi+ \int_{B_r(x_0)}\sigma \bar{u}\phi,\quad \phi\in \mathcal{H}_0^s(B_r(x_0)), \end{align*}
it follows from Riesz representation theorem that there exists $w\in \mathcal{H}_0^s(B_r(x_0))$ satisfying weakly \begin{equation*}\left\{
\begin{aligned}
(-\Delta)^sw+\sigma w=&-(-\Delta)^s\bar{u}-\sigma \bar{u},\quad& x\in B_r(x_0),\\
w=&0,\quad& x\in \mathbb{R}^N\backslash B_r(x_0).
\end{aligned}\right. \end{equation*} Consequently, $v=\bar{u}+w$ solves \eqref{w18} in the weak sense.
Let $v\in \dot{H}^s(\mathbb{R}^N)$ be a weak solution of \eqref{w18}, using \eqref{qgs}-\eqref{w18} and comparison principle we deduce \begin{align}\label{xye}
v(x)\le u(x),\quad x\in B_r(x_0). \end{align} Since $\bar{u}=u$ in $B_r(x_0)$, it follows that $v(x)\le \bar{u}$ in $\mathbb{R}^N$. On the other hand, taking $v_-$ as a test function in \eqref{w18}, we have $v\ge0$ in $\mathbb{R}^N$. As a result, $0\le v\le \bar{u}$ and $v\in L^{\infty}(\mathbb{R}^N)$.
By the regularity theory in \cite[Proposition 5]{re} and \cite[Theorem 12.2.5]{clp}, there holds $v\in C_{\mathrm{loc}}^{2s+\gamma}(B_r(x_0))$ for some $\gamma>0$, which implies $v$ is a classical solution to \eqref{w18}. If $v(x_0)=0$, then we have \begin{equation*}
C(N,s)P.V.\int_{\mathbb{R}^N}\frac{0-v(y)}{|x_0-y|^{N+2s}}=(-\Delta)^sv(x_0)+\sigma v(x_0)=0, \end{equation*} which and $v(y)\ge0$ implies that $v\equiv0$ in $\mathbb{R}^N$. This contradicts to $v(y_0)=\bar{u}(y_0)=u(y_0)>0$. Therefore, $v(x_0)>0$ and thereby $u(x_0)\ge v(x_0)>0$,
which contradicts to $u(x_0)=0$. \end{proof}
\begin{remark} The proof of Lemma \ref{w61} will be much easier if $u$ is a classical solution to \eqref{wq9}. Indeed, if there exists $x_0\in \mathbb{R}^N$ such that $u(x_0)=0$, then \begin{equation*}
C(N,s)P.V.\int_{\mathbb{R}^N}\frac{0-u(y)}{|x_0-y|^{N+2s}}=(-\Delta)^su(x_0)+c(x_0) u(x_0)\ge0, \end{equation*} which and $u\ge0$ imply $u\equiv0$. \end{remark}
\section{Concentration phenomena of penalized solutions}\label{sec3} In this section, we aim to prove the concentration of $u_{\varepsilon}$ given in Lemma \ref{lem3.6}. We prove that $u_{\varepsilon}$ has a maximum point concentrating at a local minimum of $V$ in $\Lambda$ as $\varepsilon\to 0$. This concentration phenomenon is crucial in linearizing the penalized equation $(\ref{eqs3.2})$. We prove the concentration through comparing energy, in which more regularity results on $u_{\varepsilon}$ will be needed.
Before studying asymptotic behavior of $u_\varepsilon$ as $\varepsilon\to 0$, we first give some knowledge about the limiting problem of \eqref{eqs3.2}: \begin{equation}\label{eqs4.1}
(-\Delta)^s u +\lambda u=(I_\alpha*|u|^p)|u|^{p-2}u, \quad x\in \mathbb R^N, \end{equation} where $\lambda>0$ is a constant and $u\in {H^s(\mathbb{R}^N)}$. The limiting functional $\mathcal{I}_\lambda:{H^s(\mathbb{R}^N)}\to\mathbb R$ corresponding to equation $(\ref{eqs4.1})$ is
$$\mathcal{I}_{\lambda}(u)=\frac{1}{2}\iint_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}+\frac{\lambda}{2}\int_{\mathbb R^N}|u|^2-\frac{1}{2p}\int_{\mathbb R^N}\big|I_\frac{\alpha}{2}*|u|^p\big|^2.$$ By Proposition \ref{prop2.4}, $\mathcal{I}_\lambda$ is well-defined in ${H^s(\mathbb{R}^N)}$ if $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2s})$. We denote the limiting energy by \begin{equation}\label{c1} \mathcal{C}(\lambda):=\inf_{u\in {H^s(\mathbb{R}^N)}\setminus\{0\}}\sup_{t\geq 0}\mathcal{I}_\lambda(tu). \end{equation}
Since $\mathcal{I}_\lambda(|u|)\le \mathcal{I}_\lambda(u)$ for $u\in {H^s(\mathbb{R}^N)}$, $\mathcal{I}_\lambda$ is continuous and $C_c^{\infty}(\mathbb{R}^N)$ is dense in ${H^s(\mathbb{R}^N)}$, we deduce that \begin{equation}\label{c2} \mathcal{C}(\lambda)=\mathop{\inf}_{u\in C_c^\infty(\mathbb{R}^N)\setminus\{0\}\atop{u\ge0}}\sup_{t\geq 0}\mathcal{I}_\lambda(tu). \end{equation}
The following lemma implies the homogeneity of $\mathcal{I}_\lambda$. \begin{lemma}\label{lem4.4}
Let $\lambda>0$, $p\in (\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2s})$ and $u\in {H^s(\mathbb{R}^N)}$, then $$\mathcal{C}(\lambda)=\lambda^{\frac{\alpha+2s}{2s(p-1)}-\frac{N-2s}{2s}}\mathcal{C}(1).$$ \end{lemma} In particular, since $p< \frac{N+\alpha}{N-2s}$, $\mathcal C(\lambda)$ is strictly increasing with respect to $\lambda$. \begin{proof} For any $u\in {H^s(\mathbb{R}^N)}$, we define $u_\lambda(x)=\lambda^{\frac{\alpha+2s}{4s(p-1)}}u(\lambda^{\frac{1}{2s}}x).$ A trivial verification shows that $u$ is a critical point of $\mathcal{I}_1$ if and only if $u_\lambda$ is a critical point of $\mathcal{I}_\lambda$, then the assertion follows by the definition of $\mathcal C(\lambda)$. \end{proof}
In this section, we always assume that ($\mathcal{P}_1$) and ($\mathcal{P}_2$) hold. By the analysis above, we now give the upper bound of the Mountain-Pass energy $c_{\varepsilon}$. \begin{lemma}\label{lem4.5}
It holds
\begin{equation*}\label{eqs4.4} \limsup\limits_{\varepsilon\to0}\frac{c_{\varepsilon}}{\varepsilon^N}\leq\mathcal{C}(V_0). \end{equation*} Moreover, there exists a constant $C>0$ independent of $\varepsilon\in(0,\varepsilon_0)$ such that \begin{equation}\label{eqs4.5}
\|u_{\varepsilon}\|_\varepsilon^2\leq C\varepsilon^N, \end{equation} where $u_{\varepsilon}$ is given by Lemma $\ref{lem3.6}$. \end{lemma} \begin{proof} For a nonnegative function $\psi\in C_c^{\infty}(\mathbb R^N)\setminus\{0\}$ and $a\in\Lambda$ with $V(a)=V_0$, we define $$\psi_\epsilon(x):=\psi\Big(\frac{x-a}{\varepsilon}\Big).$$
Clearly, $\mathrm{supp}(\psi_{\varepsilon})\subset\Lambda$ for $\varepsilon$ small, then $G_\varepsilon(x,\psi_\varepsilon)=\frac{1}{p}|\psi_\varepsilon|^p$. Since \begin{equation*} \begin{aligned}
&\lim _{\varepsilon \rightarrow 0} \int_{\mathbb{R}^{N}}V\left(\varepsilon x+a)|\psi\right|^{2}
=\int_{\mathbb{R}^{N}}V(a)|\psi|^{2}, \end{aligned} \end{equation*} we can select $T_0>0$ so large that $\gamma_{\varepsilon}(t):=tT_0\psi_{\varepsilon}\in\Gamma_{\varepsilon}$ and \begin{align*} c_{\varepsilon}\leq \max\limits_{t\in[0,1]}J_{\varepsilon}\big(\gamma_{\varepsilon}(t)\big)
=&\varepsilon^N\max\limits_{t\in[0,1]}\Big(\frac{1}{2}\iint_{\mathbb R^{2N}}\frac{|tT_0\psi(x)-tT_0\psi(y)|^2}{|x-y|^{N+2s}}\\
&+\frac{1}{2}\int_{\mathbb R^N}V(\varepsilon x+a)|tT_0\psi|^2-\frac{1}{2p}\int_{\mathbb R^N}|I_\frac{\alpha}{2}* |tT_0\psi|^p|^2\Big)\\ =&\varepsilon^N\big(\max\limits_{t\in[0,T_0]}\mathcal{I}_{V(a)}(t\psi)+o_{\varepsilon}(1)\big)\leq\varepsilon^N(\sup\limits_{t>0}\mathcal{I}_{V(a)}\big(t\psi)+o_{\varepsilon}(1)\big). \end{align*} By $(\ref{c2})$ and the arbitrariness of $\psi$, we deduce that $$\limsup\limits_{\varepsilon\to0}\frac{c_{\varepsilon}}{\varepsilon^N}\leq\mathop{\inf}_{\psi\in C_c^\infty(\mathbb{R}^N)\setminus\{0\}\atop{\psi\ge0}}\sup_{t> 0}\mathcal{I}_{V(a)}(t\psi)=\mathcal{C}\big(V(a)\big)=\mathcal{C}(V_0).$$
Besides, it follows from $\eqref{f}$ that $\|u_\varepsilon\|_\varepsilon^2\le C\varepsilon^N$ for a constant $C>0$ independent of $\varepsilon$. \end{proof}
The concentration phenomenon of $u_\varepsilon$ will be proved by comparing the Mountain-Pass energy $c_{\varepsilon}$ with the limiting energy $\mathcal{C}(V_0)$. One key step is to verify that the rescaled function of $u_\varepsilon$ does not vanish as $\varepsilon\to 0$, which needs some further regularity estimates on $u_{\varepsilon}$. To this end, we first use Moser iteration to get the uniform global $L^{\infty}$-estimate.
\begin{lemma}\label{w} Let $\alpha\in ((N-4s)_+,N)$, $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2s})$ and $u_\varepsilon$ be given by Lemma \ref{lem3.6}, then it holds \begin{equation*}\label{eqs4.6}
\|u_\varepsilon\|_{L^\infty(\mathbb{R}^N)}\le C, \end{equation*} where $C>0$ is a constant independent of $\varepsilon$. \end{lemma} \begin{proof} Since $u_\varepsilon\ge0$ satisfies $\eqref{eqs3.2}$ and $G_\varepsilon(x,u_\varepsilon)\le g_\varepsilon(x,u_\varepsilon)u_\varepsilon$, it follows from $\eqref{eqs4.5}$ that \begin{equation}\label{d8} \begin{aligned}
\frac{p}{\varepsilon^\alpha}\int_{\mathbb{R}^{N}}|I_{\frac{\alpha}{2}} * G_{\varepsilon}(x, u_\varepsilon)|^{2} \le& \frac{p}{\varepsilon^\alpha}\int_{\mathbb{R}^{N}}\left(I_{\alpha} * G_{\varepsilon}\left(x, u_\varepsilon\right)\right) g_{\varepsilon}\left(x, u_{\varepsilon}\right) u_{\varepsilon}\\
=&\|u_\epsilon \|_\epsilon ^2 \le C\varepsilon^N. \end{aligned} \end{equation}
Fix any sequence $\{y_\varepsilon\}_{\varepsilon>0}\subset\mathbb{R}^N$ and define $v_\varepsilon(y)=u_\varepsilon(y_\varepsilon+\varepsilon y)$ for $y\in\mathbb{R}^N$. It is easy to check that $v_\varepsilon \in H_{V_\varepsilon}^s(\mathbb{R}^N):=\{u\in{\dot{H}^s(\mathbb{R}^N)} \ | \ \int_{\mathbb{R}^N}V_\varepsilon |u|^2<\infty\}$ is a weak solution to the rescaled equation \begin{equation}\label{d9} \begin{aligned} (-\Delta)^s v_\varepsilon+V_\varepsilon v_\varepsilon=p\big(I_\alpha* \mathcal G_\varepsilon(x,v_\varepsilon)\big)\mathfrak g_\varepsilon(x,v_\varepsilon), \end{aligned} \end{equation} where $V_\varepsilon(x)=V(y_\varepsilon+\varepsilon x)$ and \begin{align}\label{G} \mathcal G_\varepsilon(x,s)=G_\varepsilon(y_\varepsilon+\varepsilon x,s),\ \ \mathfrak g_\varepsilon(x,s)=g_\varepsilon(y_\varepsilon+\varepsilon x,s).\nonumber \end{align} Since $V_\varepsilon,v_\varepsilon\ge0$ and $\mathfrak g_\varepsilon(x,s)\le s_+^{p-1}$, we deduce that $v_\varepsilon$ weakly satisfies \begin{equation}\label{dd} \begin{aligned} (-\Delta)^s v_\varepsilon\le C\big(I_\alpha* \mathcal G_\varepsilon(x,v_\varepsilon)\big)v_\varepsilon^{p-1}. \end{aligned} \end{equation} From $\eqref{eqs4.5}$, $\eqref{d8}$ and Proposition \ref{prop2.2}, by a change of variable, we have \begin{equation}\label{de} \begin{aligned}
\int_{\mathbb{R}^{N}}|I_{\frac{\alpha}{2}} * \mathcal G_{\varepsilon}\left(x, v_{\varepsilon}\right)|^{2}=\frac{1}{\varepsilon^{N+\alpha}}\int_{\mathbb{R}^{N}}|I_{\frac{\alpha}{2}} * G_{\varepsilon}\left(x, u_{\varepsilon}\right)|^{2}\le C \end{aligned} \end{equation} and \begin{equation}\label{dff} \begin{aligned}
&\|v_\varepsilon\|^2_{L^{2_s^*}(\mathbb{R}^N)}\le C\Big([v_\varepsilon]^2_s+\int_{\mathbb{R}^N}V_\varepsilon v_\varepsilon^2\Big)=\frac{C}{\varepsilon^N} \|u_\varepsilon \|_\varepsilon ^2 \le C. \end{aligned} \end{equation} Let $\beta\ge1$ and $T>0$. Define \begin{equation}\label{qdj} \varphi_{\beta,T}(t)=\left\{\begin{array}{l} 0, \text { if } t \leqslant 0, \\ t^{\beta}, \text { if } 0<t<T, \\ \beta T^{\beta-1}(t-T)+T^{\beta}, \text { if } t \geqslant T. \end{array}\right. \end{equation} Since $\varphi_{\beta,T}$ is convex and Lipschitz, we see that \begin{align}\label{wql}
\varphi_{\beta,T}(v_\varepsilon), \varphi_{\beta,T}'(v_\varepsilon)\ge0\ \mathrm{and}\ \varphi_{\beta,T}(v_\varepsilon), \varphi_{\beta,T}(v_\varepsilon)\varphi_{\beta,T}'(v_\varepsilon)\in H_{V_\varepsilon}^s(\mathbb{R}^N). \end{align} Moreover, $\varphi_{\beta,T}(v_\varepsilon)$ satisfies the following inequality \begin{equation}\label{dg} \begin{aligned} (-\Delta)^s\varphi_{\beta,T}(v_\varepsilon)\le\varphi_{\beta,T}'(v_\varepsilon)(-\Delta)^s v_\varepsilon \end{aligned} \end{equation} in the weak sense. It follows from Proposition \ref{prop2.2} that \begin{align}\label{dk}
\|\varphi_{\beta,T}(v_\varepsilon)\|^2_{L^{2_s^*}(\mathbb{R}^N)}&\le C\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}\varphi_{\beta,T}(v_\varepsilon)|^2\nonumber\\ &=C\int_{\mathbb{R}^N}\varphi_{\beta,T}(v_\varepsilon)(-\Delta)^s\varphi_{\beta,T}(v_\varepsilon)\nonumber\\ &\le C\int_{\mathbb{R}^N}\varphi_{\beta,T}(v_\varepsilon)\varphi_{\beta,T}'(v_\varepsilon)(-\Delta)^s v_\varepsilon. \end{align} Noting the fact that $v_\varepsilon\varphi_{\beta,T}'(v_\varepsilon)\le\beta\varphi_{\beta,T}(v_\varepsilon)$, by $\eqref{dd}$, \eqref{wql} and \eqref{dk}, we obtain that \begin{align}\label{dl}
&\|\varphi_{\beta,T}(v_\varepsilon)\|^2_{L^{2_s^*}(\mathbb{R}^N)} \le C\beta\int_{\mathbb{R}^N}\big(\varphi_{\beta,T}(v_\varepsilon)\big)^2\big(I_\alpha*\mathcal G_\varepsilon(x,v_\varepsilon)\big)v_\varepsilon^{p-2}:=L_1. \end{align} By H\"{o}lder inequality, $\eqref{de}$ and Proposition \ref{prop2.4}, we have the following estimate on $L_1$: \begin{align}\label{dp}
L_1\le&C\beta\Big(\int_{\mathbb{R}^N}\Big|I_\frac{\alpha}{2}* \Big(\big(\varphi_{\beta,T}(v_\varepsilon)\big)^2v_\varepsilon^{p-2}\Big)\Big|^2\Big)^{\frac{1}{2}}\Big(\int_{\mathbb{R}^N}\left|I_\frac{\alpha}{2}* \mathcal G_\varepsilon(x,v_\varepsilon)\right|^2\Big)^{\frac{1}{2}}\nonumber\\ \le& C\beta\Big(\int_{\mathbb{R}^N}\big(\varphi_{\beta,T}(v_\varepsilon)\big)^{\frac{4N}{N+\alpha}}v_\varepsilon^{(p-2)\frac{2N}{N+\alpha}}\Big)^{\frac{N+\alpha}{2N}}. \end{align} Substituting \eqref{dp} into \eqref{dl}, we conclude that \begin{align}
\|\varphi_{\beta,T}(v_\varepsilon)\|^2_{L^{2_s^*}(\mathbb{R}^N)} \le C\beta\Big(\int_{\mathbb{R}^N}\big(\varphi_{\beta,T}(v_\varepsilon)\big)^{\frac{4N}{N+\alpha}}v_\varepsilon^{(p-2)\frac{2N}{N+\alpha}}\Big)^{\frac{N+\alpha}{2N}}.\nonumber \end{align} Letting $T\to\infty$, by Monotone Convergence Theorem, we get \begin{align}\label{ttog}
\Big(\int_{\mathbb{R}^N}v_\varepsilon^{\be2_s^*}\Big)^{\frac{2}{2_s^*}}\le C\beta\Big(\int_{\mathbb{R}^N}v_\varepsilon^{\beta\frac{4N}{N+\alpha}+(p-2)\frac{2N}{N+\alpha}}\Big)^{\frac{N+\alpha}{2N}}. \end{align} Choosing $\{\beta_{i}\}_{i\ge1}$ so that $$\beta_{i+1}\frac{4N}{N+\alpha}+(p-2)\frac{2N}{N+\alpha}=\beta_i2_s^*,\ \ \beta_0=1,$$ we have $$ \beta_{i+1}+d=\frac{N+\alpha}{2(N-2s)}(\beta_i+d),\quad d=\frac{\frac{p}{2}-1}{1-\frac{1}{2}\frac{N+\alpha}{N-2s}}> -1, $$ and $\frac{N+\alpha}{2(N-2s)}>1$ by $\alpha>(N-4s)_+$.
Letting $\beta=\beta_{i+1}$ in $\eqref{ttog}$, we obtain \begin{equation*}\label{eg} \begin{aligned} \Big(\int_{\mathbb{R}^N}v_\varepsilon^{2_s^*\beta_{i+1}}\Big)^\frac{1}{2_s^*(\beta_{i+1}+d)} \le(C\beta_{i+1})^\frac{1}{2(\beta_{i+1}+d)}\Big(\int_{\mathbb{R}^N}v_\varepsilon^{2_s^*\beta_{i}}\Big)^\frac{1}{2_s^*(\beta_{i}+d)}. \end{aligned} \end{equation*} Therefore, by iteration, one gets that \begin{align}
\Big(\int_{\mathbb{R}^N}v_\varepsilon^{2_s^*\beta_{i}}\Big)^\frac{1}{2_s^*(\beta_{i}+d)}\le\prod_{i=1}^{\infty}(C\beta_i)^{\frac{1}{2(\beta_i+d)}} \Big(\int_{\mathbb{R}^N}v_\varepsilon^{2_s^*}\Big)^\frac{1}{2_s^*(1+d)}\le C,\nonumber \end{align} which implies $(\int_{\mathbb{R}^N}v_\varepsilon^{2_s^*\beta_{i}})^\frac{1}{2_s^*\beta_{i}}\le C$ too, where $C>0$ is some constant independent of $i$ and $\varepsilon$. Letting $i\to\infty$, we conclude that
$\|v_\varepsilon\|_{L^\infty(\mathbb{R}^N)}\le C$ uniformly for $\varepsilon$.
By the definition of $v_\varepsilon$, we complete the proof. \end{proof}
\begin{remark} As shown in \cite[Proposition 5]{re} and \cite[Theorem 12.2.1]{clp}, because of the nonlocal nature of
$(-\Delta)^s$ ($0<s<1$), the H\"{o}lder estimate and Schauder estimate for solutions of fractional equations demand the global $L^\infty$ information instead of local $L^\infty$ information, which is quite different from the classical case $(s=1)$. To ensure a uniform upper bound of $\|u_\varepsilon\|_{L^\infty(\mathbb{R}^N)}$ for $\varepsilon\in(0,\varepsilon_0)$, Lemma \ref{lem4.5} plays a key role, see \eqref{de}-\eqref{dff}. \end{remark}
Now we are going to give the $L^{\infty}$-estimate for the Choquard term. \begin{lemma}\label{I} Let $\alpha\in\big((N-4s)_+,N\big)$, $p\in(\frac{N+\alpha}{N},\frac{N+\alpha}{N-2s})$ and $u_\varepsilon$ be given by Lemma \ref{lem3.6}, then for any sequence $\{y_\varepsilon\}_{\varepsilon>0}\in\mathbb{R}^N$, it holds
$$\|I_\alpha*\big(\mathcal G_\varepsilon(x,v_\varepsilon)\big)\|_{L^\infty(\mathbb{R}^N)}\le C,$$ where $v_\varepsilon(y)=u_\varepsilon(y_\varepsilon+\varepsilon y)$, $\mathcal G_{\varepsilon}(y,s)=G_\varepsilon(y_\varepsilon+\varepsilon y,s)$, $C>0$ is a constant independent of $\varepsilon$ and $\{y_\varepsilon\}_{\varepsilon>0}$. \end{lemma}
\begin{proof}
From $\eqref{dff}$ and Lemma \ref{w}, i.e., $\|v_\varepsilon\|_{L^{2_s^*}(\mathbb{R}^N)}\le C$ and $\|v_\varepsilon\|_{L^{\infty}(\mathbb{R}^N)}\le C$, we get $\|v_\varepsilon\|_{L^{q}(\mathbb{R}^N)}\le C$ uniformly for $\varepsilon>0$ and $q\ge2_s^*$. By \eqref{3y7}, we have \begin{align}\label{i} I_\alpha*\big(\mathcal G_\varepsilon(x,v_\varepsilon)\big)\le &I_\alpha*\big(\mathcal P_\varepsilon(y_\varepsilon+\varepsilon y)v_\varepsilon\big)+\frac{1}{p}I_\alpha*\big(\chi_\Lambda(y_\varepsilon+\varepsilon y)v_\varepsilon^p\big)\nonumber\\ :=&D_1+D_2. \end{align}
We first estimate $D_1$. By a change of variable, H\"{o}lder inequality, ($\mathcal{P}_2$) and $\eqref{eqs4.5}$, we have \begin{align}\label{D1}
D_1=&\int_{|x-y|\le1}\frac{1}{|x-y|^{N-\alpha}}\mathcal P_\varepsilon(y_\varepsilon+\varepsilon y)v_\varepsilon~\mathrm{d} y+\int_{|x-y|>1}\frac{1}{|x-y|^{N-\alpha}}\mathcal P_\varepsilon(y_\varepsilon+\varepsilon y)v_\varepsilon~\mathrm{d} y\nonumber\\
\le&\|\mathcal P_\varepsilon\|_{L^\infty(\mathbb{R}^N)}\|v_\varepsilon\|_{L^\infty(\mathbb{R}^N)}\int_{|x-y|\le1}\frac{1}{|x-y|^{N-\alpha}}~\mathrm{d} y\nonumber\\
&+\Big(\int_{|x-y|>1}\frac{1}{|x-y|^{2N-2\alpha}|y_\varepsilon+\varepsilon y|^\alpha}~\mathrm{d} y\Big)^{\frac{1}{2}}
\Big(\int_{|x-y|>1}\mathcal P_\varepsilon^2(y_\varepsilon+\varepsilon y)v_\varepsilon^2|y_\varepsilon+\varepsilon y|^\alpha~\mathrm{d} y\Big)^{\frac{1}{2}}\nonumber\\
\le&C_1+\Big(\frac{1}{\varepsilon^\alpha}\int_{|y|>1}\frac{1}{|(\frac{y_\varepsilon}{\varepsilon}+x)-y|^\alpha|y|^{2N-2\alpha}}~\mathrm{d} y\Big)^{\frac{1}{2}}\Big(\frac{1}{\varepsilon^N}\int_{\mathbb{R}^N}\mathcal P_\varepsilon^2 u_\varepsilon^2|y|^\alpha\Big)^{\frac{1}{2}}\le C, \end{align}
where we have used the fact that $\sup_{z\in\mathbb{R}^N}\int_{|y|>1}\frac{1}{|z-y|^\alpha|y|^{2N-2\alpha}}~\mathrm{d} y\le C.$
Next we estimate $D_2$. By a change of variable, Proposition \ref{prop2.3} and $\eqref{eqs4.5}$, it holds \begin{align}\label{D2}
D_2=&\int_{|x-y|\le1}\frac{1}{|x-y|^{N-\alpha}}\chi_\Lambda(y_\varepsilon+\varepsilon y)v_\varepsilon^p~\mathrm{d} y+\int_{|x-y|>1}\frac{1}{|x-y|^{N-\alpha}}\chi_\Lambda(y_\varepsilon+\varepsilon y)v_\varepsilon^p~\mathrm{d} y\nonumber\\
&\le\|v_\varepsilon\|_{L^\infty(\mathbb{R}^N)}^p\int_{|x-y|\le1}\frac{1}{|x-y|^{N-\alpha}}~\mathrm{d} y+\frac{1}{\varepsilon^N}\int_{\Lambda}|u_\varepsilon|^p\le C. \end{align}
Substituting \eqref{D1} and \eqref{D2} into \eqref{i}, we see that $\|I_\alpha*\big(\mathcal G_\varepsilon(x,v_\varepsilon)\big)\|_{L^\infty(\mathbb{R}^N)}\le C$ uniformly for $\varepsilon$. \end{proof}
\begin{remark}\label{qt0} The upper energy estimates (Lemma \ref{lem4.5}) and the properties of penalization play a very important role in Lemma \ref{I} (see \eqref{D1}-\eqref{D2}). On the other hand, the regularity helps us to check Lemma \ref{lem4.7} (see \eqref{jx1}), which is a significant step to make it possible to realize the desired penalization. This indicates that the regularity and the construction of penalization are not mutually independent but interrelated.
\end{remark}
In terms of Lemma \ref{w} and Lemma \ref{I}, we continue to prove the locally H\"{o}lder estimate of $u_{\varepsilon}$, where the fact $\|u_\varepsilon\|_{L^\infty(\mathbb{R}^N)}\le C$ in Lemma \ref{w} is essential. \begin{lemma}\label{s} Let $\alpha\in\big((N-4s)_+,N\big)$, $p\in(\frac{N+\alpha}{N},\frac{N+\alpha}{N-2s})$ and $u_\varepsilon$ be given by Lemma \ref{lem3.6}, then for any $R>0$ and $\varepsilon\in(0,\varepsilon_0)$, we have $v_\varepsilon\in C^\sigma(B_R(0))$ for any $\sigma\in (0,\min\{2s,1\})$ and
$$\|v_\varepsilon\|_{C^\sigma(B_R(0))}\le C(\sigma, N,s,\alpha, R, y_0),$$ where $C>0$ is independent of $\varepsilon$, $v_\varepsilon=u_\varepsilon(y_\varepsilon+\varepsilon y)$ such that $y_\varepsilon\to y_0$ for some $y_0\in \mathbb{R}^N$ as $\varepsilon \to 0$.
If we assume additionally that $V\in L^\infty(\mathbb{R}^N)$, then the estimate above is global, i.e., $v_\varepsilon\in C^\sigma(\mathbb{R}^N)$ and \begin{equation}\label{g1} \begin{aligned}
\|v_\varepsilon\|_{C^\sigma(\mathbb{R}^N)}\le C(\sigma,N,s,\alpha). \end{aligned} \end{equation}
\end{lemma}
\begin{proof}
Fix $R>0$ and any $y_*\in B_R(0)$, we have $B_3(y_*)\subset B_{R+3}(0)$. Since $y_\varepsilon\to y_0$ as $\varepsilon \to 0$, there exists $R_0>0$ such that $y_\varepsilon\in B_{R_0}(y_0)$ for $\varepsilon\in(0,\varepsilon_0)$. Denote
$C_{R,y_0}=\sup_{y\in B_{\tilde{R}}(0)}V(y)$, where $\tilde{R}=R+3+R_0+|y_0|$, we have $y_\varepsilon+B_3(y_*)\subset B_{\tilde{R}}(0)$.
Recalling $\eqref{d9}$ and Lemma \ref{w}, we see that $v_\varepsilon\in \dot{H}^s(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ solves weakly the following equation \begin{equation}\label{k1} \begin{aligned} (-\Delta)^s v_\varepsilon=f_\varepsilon,\ \ x\in B_3(y_*), \end{aligned} \end{equation}
where $f_\varepsilon:=p\big(I_\alpha* \mathcal G_\varepsilon(x,v_\varepsilon)\big)\mathfrak g_\varepsilon(x,v_\varepsilon)-V_\varepsilon v_\varepsilon$. By Lemmas \ref{w}, \ref{I} and the above analysis, it holds that
$f_\varepsilon\in L_{\mathrm{loc}}^\infty(\mathbb{R}^N)$ and $\|f_\varepsilon\|_{L^\infty(B_1(y_*))}\le C+CC_{R,y_0}$. From Proposition 5 in \cite{re}, it follows that $v_\varepsilon\in C^\sigma\big(B_{1/4}(y_*)\big)$ for any $\sigma\in (0,\min\{2s,1\})$ and \begin{equation}\label{k2} \begin{aligned}
\|v_\varepsilon\|_{C^\sigma(B_{1/4}(y_*))}\le C(\|v_\varepsilon\|_{L^\infty(\mathbb{R}^N)}+\|f_\varepsilon\|_{L^\infty(B_1(y_*))})\le C+CC_{R,y_0}, \end{aligned} \end{equation} where $C$ and $C_{R,y_0}$ are independent of $y_*\in B_R(0)$. For any $y_1,y_2\in B_R(0)$ and $y_1\neq y_2$, we have
$y_1,y_2\in B_{1/4}(y_1)$ if $|y_1-y_2|<\frac{1}{4}$. It follows from $\eqref{k2}$ that \begin{equation}\label{k3} \begin{aligned}
\frac{|v_\varepsilon(y_1)-v_\varepsilon(y_2)|}{|y_1-y_2|^\sigma}\le C+CC_{R,y_0}. \end{aligned} \end{equation}
If $|y_1-y_2|\ge\frac{1}{4}$, we deduce that \begin{equation}\label{k4} \begin{aligned}
\frac{|v_\varepsilon(y_1)-v_\varepsilon(y_2)|}{|y_1-y_2|^\sigma}\le 8\|v_\varepsilon\|_{L^\infty(\mathbb{R}^N)}\le C. \end{aligned} \end{equation} Therefore, by $\eqref{k3}$ and $\eqref{k4}$, we have \begin{equation*}\label{k5} \begin{aligned}
&[v_\varepsilon]_{C^\sigma(B_R(0))}=\mathop{\sup}_{y_1,y_2\in B_R(0)\atop{y_1\neq y_2}}\frac{|v_\varepsilon(y_1)-v_\varepsilon(y_2)|}{|x-y|^\sigma} \le C+CC_{R,y_0}. \end{aligned} \end{equation*}
Furthermore, if $V\in L^\infty(\mathbb{R}^N)$, then $C_{R,y_0}\le \|V\|_{L^\infty(\mathbb{R}^N)}$ and thereby $\|v_\varepsilon\|_{C^\sigma(\mathbb{R}^N)}\le C$. Thus the assertion holds.
\end{proof}
By the regularity above, now we can give a lower bound on the energy of $u_{\varepsilon}$ by blow-up analysis.
\begin{lemma}\label{lem4.7} Let $\alpha\in ((N-4s)_+,N)$, $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2s})$, $\{\varepsilon_n\}\subset\mathbb R_+$ with $\lim\limits_{n\to\infty}\varepsilon_n=0$, $u_n:=u_{\varepsilon_n}$ be given by Lemma \ref{lem3.6} and $\{(x_n^j)_{n\ge 1}\subset\mathbb R^N:1\le j\le k\}$ be $k$ families of points satisfying $\lim\limits_{n\to\infty}x_n^j=x_{\ast}^j$. If the following statements hold \begin{equation}\label{eqs1b}
V(x_\ast^j)>0,\quad\lim_{n\to\infty}\frac{|x_n^i-x_n^j|}{\varepsilon_n}=\infty\ \ \text{for\ every}\ 1\le i\neq j\le k \end{equation} and \begin{equation}\label{eqs4.17}
\liminf_{n\to\infty}\|u_{n}\|_{L^{\infty}(B_{\varepsilon_n\rho}(x_n^j))}+\varepsilon_n^{-\alpha}\|I_\alpha* G_{\varepsilon_n}(x,u_{n})\|_{L^\infty(B_{\varepsilon_n\rho}(x_n^j))}>0 \end{equation} for $1\le j\le k$ and some $\rho>0$, then $x_*^j\in\bar\Lambda$ and \begin{equation*}\label{eqs4.18} \liminf_{n\to\infty}\frac{J_{\varepsilon_n}(u_{\varepsilon_n})}{\varepsilon_n^N}\geq\sum_{j=1}^k\mathcal{C}\big(V(x_{\ast}^j)\big), \end{equation*} where $\mathcal{C}\big(V(x_{\ast}^j)\big)$ is given by \eqref{c1}. \end{lemma}
\begin{proof} The rescaled function $v_n^j$ defined as $v_n^j(x)=u_{n}(x_n^j+\varepsilon_nx)$ satisfies \begin{equation}\label{eqs4.20} (-\Delta)^sv_n^j+V_n^jv_n^j=p\big(I_\alpha* \mathcal{G}_n^j(v_n^j)\big)\mathfrak{g}_n^j(v_n^j), \end{equation} where $V_n^j(x)=V(x_n^j+\varepsilon_nx),\ \mathcal{G}_n^j(v_n^j)=G_{\varepsilon_n}(x_n^j+\varepsilon_nx,v_n^j),\ \mathfrak{g}_n^j(v_n^j)=g_{\varepsilon_n}(x_n^j+\varepsilon_nx,v_n^j)$. We also denote the rescaled set $\Lambda_n^j=\{y\in\mathbb{R}^N:x_n^j+\varepsilon_ny\in\Lambda\}$. Since $\Lambda$ is smooth, up to a subsequence, we can assume that $\chi_{\Lambda_n^j}\to\chi_{\Lambda_*^j}$ a.e. as $n\to\infty$, where $\Lambda_*^j\in\{\mathbb{R}^N, H, \emptyset\}$ and $H$ is a half-space in $\mathbb{R}^N$.
By Lemma \ref{lem4.5}, we have $\|u_n\|_{\varepsilon_n}^2\le C\varepsilon_n^N$. A change of variable and Proposition \ref{prop2.3} implies that \begin{eqnarray}\label{h1}
[v_n^j]_s^2+\int_{\mathbb{R}^N}V_n^j(v_n^j)^2=\frac{1}{\varepsilon_n^N}\|u_n\|_{\varepsilon_n}^2\leq C, \end{eqnarray} and \begin{align}\label{wan}
\int_{\mathbb{R}^N}\chi_{\Lambda_n^j}(v_n^j)^p~\mathrm{d} y=\frac{1}{\varepsilon_n^N}\int_{\mathbb{R}^N}\chi_{\Lambda}u_n^p~\mathrm{d} y\le C. \end{align} Moreover, since $\mathcal{G}_n^j(v_n^j)\le \mathfrak{g}_n^j(v_n^j)v_n^j$, by \eqref{eqs4.20} and \eqref{h1}, we have \begin{align}\label{qjl}
p\int_{\mathbb{R}^N}\big(I_\alpha* \mathcal{G}_n^j(v_n^j)\big)\mathcal{G}_n^j(v_n^j)\le[v_n^j]_s^2+\int_{\mathbb{R}^N}V_n^j(v_n^j)^2\le C. \end{align} Taking a subsequence if necessary, there exists $v_*^j\in{\dot{H}^s(\mathbb{R}^N)}$ such that $v_n^j\rightharpoonup v_*^j$ weakly in ${\dot{H}^s(\mathbb{R}^N)}$, $v_n^j\to v_*^j$ strongly in $L_{\mathrm{loc}}^q(\mathbb{R}^N)$ for $q\in [1,2_s^*)$ and $v_n^j\to v_*^j$ a.e. as $n\to\infty$. Besides, $p\mathcal{G}_n^j(v_n^j)\to \chi_{\Lambda_*^j}(v_*^j)^p$ a.e. as $n\to\infty$.
By the weak lower semicontinuity of the norms and Fatou's lemma, we have \begin{align}\label{pol}
\int_{\mathbb{R}^N}\big(I_\alpha*(\chi_{\Lambda_*^j}(v_*^j)^p)\big)\chi_{\Lambda_*^j}(v_*^j)^p\le \liminf_{n\to\infty}\int_{\mathbb{R}^N}p^2\big(I_\alpha* \mathcal{G}_n^j(v_n^j)\big)\mathcal{G}_n^j(v_n^j)\leq C, \end{align} and $$[v_*^j]_s^2+\int_{\mathbb{R}^N}V(x_*^j)(v_*^j)^2\le \liminf_{n\to\infty}\left([v_n^j]_s^2+\int_{\mathbb{R}^N}V_n^j(v_n^j)^2\right)\leq C,$$
which implies that $v_*^j\in {H^s(\mathbb{R}^N)}$ since $V(x_*^j)>0$. In addition, $v_*^j\ge0$ a.e. in $\mathbb{R}^N$ since $v_n^j\ge0$ a.e. in $\mathbb{R}^N$. Moreover, by Proposition \ref{prop2.2}, Lemma \ref{w} and Lemma \ref{s}, we deduce that $v_n^j\to v_*^j$ in $L_{\mathrm{loc}}^q(\mathbb{R}^N)$ for any $q\in[1,+\infty]$ as $n\to\infty$ and $\|v_n^j\|_{L^q(\mathbb{R}^N)}\le C$ for any $q\in[2_s^*,+\infty]$.
We claim that \begin{equation}\label{4oo} p\big(I_\alpha* \mathcal{G}_n^j(v_n^j)\big)\toI_\alpha* \big(\chi_{\Lambda_*^j}(v_*^j)^p\big) \ \ in \ \ L_{\mathrm{loc}}^\infty(\mathbb{R}^N)\ \ as \ \ n\to\infty. \end{equation}
Indeed, by Fatou's lemma and Lemma \ref{I}, we have \begin{align}\label{4o}
&\|I_\alpha* \big(\chi_{\Lambda_*^j}(v_*^j)^p\big)\|_{L^\infty(\mathbb{R}^N)}\le \sup_{n\in\mathbb N}\|p\big(I_\alpha* \mathcal{G}_n^j(v_n^j)\big)\|_{L^\infty(\mathbb{R}^N)}\le C. \end{align} For any given $R>1$ and $x\in B_R(0)$, it holds \begin{equation}\label{4i} \begin{aligned}
&\big|p\big(I_\alpha* \mathcal{G}_n^j(v_n^j)\big)-I_\alpha* \big(\chi_{\Lambda_*^j}(v_*^j)^p\big)\big|\\
\le&\int_{\mathbb{R}^N}\frac{1}{|x-y|^{N-\alpha}}\big|\chi_{\Lambda_n^j}(v_n^j)^p-\chi_{\Lambda_*^j}(v_*^j)^p\big|~\mathrm{d} y+
p\int_{\mathbb{R}^N}\frac{1}{|x-y|^{N-\alpha}}\mathcal{P}_{\varepsilon_n}(x_n^j+\varepsilon_ny)v_n^j~\mathrm{d} y. \end{aligned} \end{equation} By H\"{o}lder inequality, \eqref{eqs2.1}, ($\mathcal{P}_2$) and $\eqref{h1}$, letting $M>2$, we have \begin{align}\label{4s}
&\int_{\mathbb{R}^N}\frac{1}{|x-y|^{N-\alpha}}\mathcal{P}_{\varepsilon_n}(x_n^j+\varepsilon_ny)v_n^j~\mathrm{d} y\nonumber\\
\le&\|\mathcal{P}_{\varepsilon_n}\|_{L^\infty(\mathbb{R}^N)}\|v_n^j\|_{L^\infty(\mathbb{R}^N)}\int_{|x-y|\le MR}\frac{1}{|x-y|^{N-\alpha}}~\mathrm{d} y\nonumber\\
&+\Big(\int_{\{|x-y|>MR\}\cap(\Lambda_n^j)^c}\frac{1}{|x-y|^{2N-2\alpha}|x_n^j+\varepsilon_ny|^\alpha}~\mathrm{d} y\Big)^{\frac{1}{2}}
\Big(\frac{1}{\varepsilon^N}\int_{\mathbb{R}^N}\mathcal{P}_{\varepsilon_n}^2u_n^2|y|^\alpha~\mathrm{d} y\Big)^{\frac{1}{2}}\nonumber\\
\le& C\|\mathcal{P}_{\varepsilon_n}\|_{L^\infty(\mathbb{R}^N)}M^\alpha R^\alpha+\frac{C}{(MR)^{\frac{N-\alpha}{4}}}\Big(\int_{\{|x-y|>MR\}\cap(\Lambda_n^j)^c}\frac{1}{|x-y|^{\frac{3}{2}N-\frac{3}{2}\alpha}|\frac{x_n^j}{\varepsilon_n}+y|^\alpha}~\mathrm{d} y\Big)^{\frac{1}{2}}\nonumber\\
\le&C\|\mathcal{P}_{\varepsilon_n}\|_{L^\infty(\mathbb{R}^N)}M^\alpha R^\alpha+\frac{C}{(MR)^{\frac{N-\alpha}{4}}}. \end{align} On the other hand, by H\"{o}lder inequality, \eqref{wan} and $v_*^j\in H^s(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, it follows that \begin{align}\label{4j}
&\int_{\mathbb{R}^N}\frac{1}{|x-y|^{N-\alpha}}\big|\chi_{\Lambda_n^j}(v_n^j)^p-\chi_{\Lambda_*^j}(v_*^j)^p\big|~\mathrm{d} y\nonumber\\
\le&\Big(\int_{|y|\le MR}\frac{1}{|x-y|^{N-\beta}}~\mathrm{d} y\Big)^{\frac{N-\alpha}{N-\beta}}
\Big(\int_{|y|\le MR}\big|\chi_{\Lambda_n^j}(v_n^j)^p-\chi_{\Lambda_*^j}(v_*^j)^p\big|^{\frac{N-\beta}{\alpha-\beta}}\Big)^{\frac{\alpha-\beta}{N-\beta}}\nonumber\\
&+\Big(\int_{|y|> MR}\frac{1}{|x-y|^{\gamma}}~\mathrm{d} y\Big)^{\frac{N-\alpha}{\gamma}}
\Big(\int_{|y|> MR}(v_*^j)^{p\frac{\gamma}{\gamma+\alpha-N}}\Big)^{\frac{\gamma+\alpha-N}{\gamma}}\nonumber\\ &+\frac{1}{[(M-1)R]^{N-\alpha}}\int_{\mathbb{R}^N}\chi_{\Lambda_n^j}(v_n^j)^p~\mathrm{d} y\nonumber\\
\le&\Big(\int_{|y|\le (M+1)R}\frac{1}{|y|^{N-\beta}}\Big)^{\frac{N-\alpha}{N-\beta}}
\Big(\int_{|y|\le MR}\big|\chi_{\Lambda_n^j}(v_n^j)^p-\chi_{\Lambda_*^j}(v_*^j)^p\big|^{\frac{N-\beta}{\alpha-\beta}}\Big)^{\frac{\alpha-\beta}{N-\beta}}\nonumber\\ &+\frac{C}{[(M-1)R]^{(\gamma-N)\frac{N-\alpha}{\gamma}}}+\frac{C}{[(M-1)R]^{N-\alpha}}, \end{align} where $0<\beta<\alpha$ and $\gamma>N$ such that $p\frac{\gamma}{\gamma+\alpha-N}\ge2$. Since $v_n^j\to v_*^j$ in $L_{\mathrm{loc}}^q(\mathbb{R}^N)$ as $n\to\infty$ for $q\in[1,+\infty]$, by Dominated Convergence Theorem, we have \begin{align}\label{4k}
&\lim_{n\to\infty}\int_{|y|\le MR}\big|\chi_{\Lambda_n^j}(v_n^j)^p-\chi_{\Lambda_*^j}(v_*^j)^p\big|^{\frac{N-\beta}{\alpha-\beta}}=0 \end{align} and \begin{align}\label{4f} &p\mathcal{G}_n^j(v_n^j)\to\chi_{\Lambda_*^j}(v_*^j)^{p},\ \mathfrak{g}_n^j(v_n^j)\to \chi_{\Lambda_*^j}(v_*^j)^{p-1}\ \mathrm{in}\ L^q(B_R(0))\ \mathrm{for\ any}\ q\ge1. \end{align} From $\eqref{4i}$--$\eqref{4k}$ and \eqref{eqs2.1}, we conclude that \begin{align}\label{4h}
&\lim_{n\to\infty}\|p\big(I_\alpha* \mathcal{G}_n^j(v_n^j)\big)-I_\alpha* \big(\chi_{\Lambda_*^j}(v_*^j)^p\big)\|_{L^\infty(B_R(0))}=0, \end{align} which gives (\ref {4oo}).
Taking any $\varphi\in C_c^\infty(\mathbb{R}^N)$ as a test function in $\eqref{eqs4.20}$ and letting $n\to\infty$, from $\eqref{4f}$, $\eqref{4h}$ and $v_n^j\rightharpoonup v_*^j$ in ${\dot{H}^s(\mathbb{R}^N)}$, we deduce that $v_*^j$ satisfies
\begin{align}\label{jx} (-\Delta)^s v_*^j+V(x_*^j)v_*^j=\big(I_\alpha*\chi_{\Lambda_*^j}(v_*^j)^p\big)\chi_{\Lambda_*^j}(v_*^j)^{p-1}. \end{align} Since $v_n^j\to v_*^j$ and $pI_\alpha*\big(\mathcal{G}_n^j(v_n^j)\big)\toI_\alpha*\big(\chi_{\Lambda_*^j}(v_*^j)^p\big)$ in $L^\infty_{\mathrm{loc}}(\mathbb{R}^N)$, from assumption $\eqref{eqs4.17}$, we have \begin{align}\label{jx1}
&\|v_*^j\|_{L^\infty(B_\rho(0))}+\|I_\alpha*\big(\chi_{\Lambda_*^j}(v_*^j)^p\big)\|_{L^\infty(B_\rho(0))}\nonumber\\
=&\lim_{n\to\infty}\left(\|v_n^j\|_{L^\infty(B_\rho(0))}+p\|I_\alpha*\big(\mathcal{G}_n^j(v_n^j)\big)\|_{L^\infty(B_\rho(0))}\right)\nonumber\\
=&\lim_{n\to\infty}\big(\|u_{n}\|_{L^{\infty}(B_{\varepsilon_n\rho}(x_n^j))}+p\varepsilon_n^{-\alpha}\|I_\alpha* G_{\varepsilon_n}(x,u_{n})\|_{L^\infty(B_{\varepsilon_n\rho}(x_n^j))}\big)>0. \end{align} Consequently, $v_*^j\neq0$ and $\Lambda_*^j\neq\emptyset$. In particular, $x_*^j\in \bar\Lambda$.
Define the functional $T^j:H^s(\mathbb R^N)\to\mathbb R$ associated with equation $\eqref{jx}$ as
\begin{align*}T^j(u)&=\frac{1}{2}\iint_{\mathbb R^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}+\frac{V(x_\ast^j)}{2}\int_{\mathbb R^N}|u|^2-\frac{1}{2p}\int_{\mathbb R^N}\big|I_\frac{\alpha}{2}*(\chi_{\Lambda_*^j}|u|^p)\big|^2. \end{align*} Since $ \chi_{\Lambda_\ast^j}\leq1$ and $v_*^j$ is a nontrivial nonnegative solution to equation $\eqref{jx}$, it holds \begin{equation}\label{jxn} T^j(v_*^j)=\max_{t>0}T^j(tv_*^j)\geq \sup_{t>0}\mathcal{I}_{V(x_\ast^j)}(tv_*^j)\geq\inf_{u\in H^s(\mathbb{R}^N)\setminus\{0\}}\sup_{t>0}\mathcal{I}_{V(x_\ast^j)}(tu)=\mathcal{C}\big(V(x_\ast^j)\big). \end{equation}
Now we begin estimating the energy of $u_n$. Fixing $R>0$, by the assumption $(\ref{eqs1b})$, we have $B_{2\varepsilon_nR}(x_n^j)\cap B_{2\varepsilon_nR}(x_n^l)=\emptyset$ if $j\neq l$ for $n$ large enough. Then by Fatou's lemma, $v_n^j\rightharpoonup v_*^j$ in ${\dot{H}^s(\mathbb{R}^N)}$, $\eqref{4f}$, $\eqref{4h}$ and \eqref{jxn}, we have \begin{align}\label{eqs48}
&\liminf_{n\to\infty}\frac{1}{\varepsilon_n^N}\Big(\frac{1}{2}\int_{\cup_{j=1}^kB_{\varepsilon_nR}(x_n^j)}\Big(\int_{\mathbb R^N}\varepsilon_n^{2s}\frac{|u_{n}(x)-u_{n}(y)|^2}{|x-y|^{N+2s}}\mathrm{d} y\Big)\mathrm{d} x\nonumber\\ &\quad+\frac{1}{2}\int_{\cup_{j=1}^kB_{\varepsilon_nR}(x_n^j)}V(x)u_{n}^2-\frac{p}{2\varepsilon_n^\alpha}\int_{\cup_{j=1}^kB_{\varepsilon_nR}(x_n^j)}\big(I_\alpha* G_{\varepsilon_n}(x,u_{n})\big)G_{\varepsilon_n}(x,u_{n})\Big)\nonumber\\
=&\liminf_{n\to\infty}\sum_{j=1}^k\Big(\frac{1}{2}\int_{B_R(0)}\Big(\int_{\mathbb R^N}\frac{|v_n^j(x)-v_n^j(y)|^2}{|x-y|^{N+2s}}\mathrm{d} y\Big)\mathrm{d} x\nonumber\\ &\quad+\frac{1}{2}\int_{B_R(0)}V_n^j(v_n^j)^2-\frac{1}{2p}\int_{B_R(0)}\big(I_\alpha* \mathcal{G}_n^j(v_n^j)\big)\mathcal{G}_n^j(v_n^j)\Big)\nonumber\\
\geq&\sum_{j=1}^k\Big(\frac{1}{2}\int_{B_R(0)}\Big(\int_{\mathbb R^N}\frac{|v_*^j(x)-v_*^j(y)|^2}{|x-y|^{N+2s}}\mathrm{d} y\Big)\mathrm{d} x\nonumber\\ &\quad+\frac{1}{2}\int_{B_R(0)}V(x_\ast^j)(v_*^j)^2-\frac{1}{2p}\int_{B_R(0)}\big(I_\alpha*(\chi_{\Lambda_*^j}(v_*^j)^p)\big)\chi_{\Lambda_*^j}(v_*^j)^p\Big)\nonumber\\
\geq&\sum_{j=1}^k\Big(T^j(v_*^j)-\frac{1}{2}\int_{\mathbb R^N\setminus B_R(0)}\Big(\int_{\mathbb R^N}\frac{|v_*^j(x)-v_*^j(y)|^2}{|x-y|^{N+2s}}\mathrm{d} y\Big)\mathrm{d} x
-\frac{1}{2}\int_{\mathbb R^N\setminus B_R(0)}V(x_\ast^j)|v_*^j|^2\Big)\nonumber\\ \ge&\sum_{j=1}^k\mathcal{C}\big(V(x_\ast^j)\big)+o_R(1). \end{align}
Next we estimate the integral outside the balls above. Let $\eta\in C^{\infty}(\mathbb R^N)$ be such that $0\leq\eta\leq1$, $\eta=0$ on $B_1(0)$ and $\eta=1$ on $\mathbb R^N\setminus B_2(0)$. Define $$\psi_{n,R}(x)=\prod_{j=1}^k\eta(\frac{x-x_n^j}{\varepsilon_nR}).$$ Taking $\psi_{n,R}u_{n}$ as a test function to the penalized equation $(\ref{eqs3.2})$, we get \begin{align}\label{uu}
&\iint_{\mathbb R^{2N}}\varepsilon_n^{2s}\psi_{n,R}(x)\frac{|u_{n}(x)-u_{n}(y)|^2}{|x-y|^{N+2s}} +\int_{\mathbb R^N}V\psi_{n,R}u_{n}^2\nonumber\\ &=-\varepsilon_n^{2s}\mathcal{R}_n+\frac{p}{\varepsilon_n^\alpha}\int_{\mathbb{R}^N}\big(I_\alpha* G_{\varepsilon_n}(x,u_n)\big)g_{\varepsilon_n}(x,u_n)\psi_{n,R}u_n, \end{align} where \begin{align}
&\mathcal{R}_n=\iint_{\mathbb R^{2N}}\frac{u_n(y)\big(u_n(x)-u_n(y)\big)\big(\psi_{n,R}(x)-\psi_{n,R}(y)\big)}{|x-y|^{N+2s}}.\nonumber \end{align} Noting $G_{\varepsilon_n}(x,u_n)\le g_{\varepsilon_n}(x,u_n)u_n$, it follows from $\eqref{uu}$ that \begin{align}\label{ut}
&\frac{1}{\varepsilon_n^N}\Big(\frac{1}{2}\int_{\mathbb R^N\setminus\cup_{j=1}^kB_{\varepsilon_nR}(x_n^j)}\Big(\int_{\mathbb R^N}\varepsilon_n^{2s}\frac{|u_{n}(x)-u_{n}(y)|^2}{|x-y|^{N+2s}}\mathrm{d} y\Big)\mathrm{d} x+\frac{1}{2}\int_{\mathbb R^N\setminus\cup_{j=1}^kB_{\varepsilon_nR}(x_n^j)}V(x)|u_{n}|^2\nonumber\\ &\ \ -\frac{p}{2\varepsilon_n^\alpha}\int_{\mathbb R^N\setminus\cup_{j=1}^kB_{\varepsilon_nR}(x_n^j)}\big(I_\alpha* G_{\varepsilon_n}(x,u_n)\big)G_{\varepsilon_n}(x,u_n)\Big)\nonumber\\ \ge&-\frac{\varepsilon_n^{2s-N}}{2}\mathcal{R}_n+\frac{p}{2\varepsilon_n^{N+\alpha}}\int_{\mathbb R^N\setminus\cup_{j=1}^kB_{\varepsilon_nR}(x_n^j)}\big(I_\alpha* G_{\varepsilon_n}(x,u_n)\big)g_{\varepsilon_n}(x,u_n)u_n(\psi_{n,R}-1). \end{align} From \eqref{4o}, $\eqref{4f}$, $\eqref{4h}$ and \eqref{pol}, we obtain \begin{align}\label{eqs51}
&\limsup_{n\to\infty}\Big|\frac{p}{\varepsilon_n^{N+\alpha}}\int_{\mathbb R^N\setminus\cup_{j=1}^kB_{\varepsilon_nR}(x_n^j)}\big(I_\alpha* G_{\varepsilon_n}(x,u_n)\big)g_{\varepsilon_n}(x,u_n)u_n(\psi_{n,R}-1)\Big|\nonumber\\
\leq&\limsup_{n\to\infty}\sum_{j=1}^k\Big|\frac{p}{\varepsilon_n^{N+\alpha}}\int_{B_{2\varepsilon_nR}(x_n^j)\setminus B_{\varepsilon_nR}(x_n^j)}\big(I_\alpha* G_{\varepsilon_n}(x,u_n)\big)g_{\varepsilon_n}(x,u_n)u_n\Big|\nonumber\\ =&\limsup_{n\to\infty}\sum_{j=1}^kp\int_{B_{2R}\setminus B_{R}}\big(I_\alpha* \mathcal{G}_n(v_n^j)\big)\mathfrak{g}_n(v_n^j)v_n^j\nonumber\\ =&\sum_{j=1}^k\int_{B_{2R}\setminus B_{R}}\big(I_\alpha*(\chi_{\Lambda_*^j}(v_*^j)^p)\big)\chi_{\Lambda_*^j}(v_*^j)^p=o_R(1). \end{align}
It remains to estimate $\mathcal{R}_n$. Noticing
$$|\psi_{n,R}(x)-\psi_{n,R}(y)|\le\sum_{j=1}^k\Big|\eta\Big(\frac{x-x_n^j}{\varepsilon_nR}\Big)-\eta\Big(\frac{y-x_n^j}{\varepsilon_nR}\Big)\Big|,$$ by H\"{o}lder inequality and scaling, from $\eqref{h1}$ we have \begin{align}\label{ug}
|\mathcal{R}_n|\le&\sqrt{N}\Big(\iint_{\mathbb R^{2N}}\frac{|u_n(x)-u_n(y)|^2}{|x-y|^{N+2s}}\Big)^{\frac{1}{2}}
\Big(\iint_{\mathbb R^{2N}}\sum_{j=1}^k\frac{|u_n(y)|^2|\eta\big(\frac{x-x_n^j}{\varepsilon_nR}\big)-\eta\big(\frac{y-x_n^j}{\varepsilon_nR}\big)|^2}{|x-y|^{N+2s}}\Big)^{\frac{1}{2}}\nonumber\\
\le&C\varepsilon_n^{N-2s}\sum_{j=1}^k\Big(\iint_{\mathbb R^{2N}}\frac{|v_n^j(y)|^2|\eta\big(\frac{x}{R}\big)-\eta\big(\frac{y}{R}\big)|^2}{|x-y|^{N+2s}}\Big)^{\frac{1}{2}}. \end{align}
Next we estimate the last integral in $\eqref{ug}$, which can be divided into four parts. In the region $B_{2R}(0)\times B_{4R}(0)$, since $|\eta\big(\frac{x}{R}\big)-\eta\big(\frac{y}{R}\big)|\le \frac{C|x-y|}{R}$ and $v_n^j\to v_*^j$ in $L_{\mathrm{loc}}^2(\mathbb{R}^N)$, we get \begin{align}\label{uk}
&\limsup_{n\to\infty}\int_{B_{2R}(0)}|v_n^j(y)|^2~\mathrm{d} y\int_{B_{4R}(0)}\frac{\left|\eta\big(\frac{x}{R}\big)-\eta\big(\frac{y}{R}\big)\right|^2}{|x-y|^{N+2s}}~\mathrm{d} x\nonumber\\
\le& \limsup_{n\to\infty}\frac{C}{R^2}\int_{B_{2R}(0)}|v_n^j(y)|^2~\mathrm{d} y\int_{B_{6R}(0)}\frac{1}{|x|^{N+2s-2}}~\mathrm{d} x\nonumber\\
=&\frac{C}{R^{2s}}\int_{B_{2R}(0)}|v_*^j(y)|^2~\mathrm{d} y=o_R(1). \end{align} Similarly, in the region $B_{4R}(0)\times B_{2R}(0)$, \begin{align}\label{utt}
&\limsup_{n\to\infty}\int_{B_{4R}(0)}|v_n^j(y)|^2~\mathrm{d} y\int_{B_{2R}(0)}\frac{\left|\eta\big(\frac{x}{R}\big)-\eta\big(\frac{y}{R}\big)\right|^2}{|x-y|^{N+2s}}~\mathrm{d} x =o_R(1). \end{align}
In the region $B_{2R}(0)\times \big(\mathbb{R}^N\setminus B_{4R}(0)\big)$, since $\left|\eta\big(\frac{x}{R}\big)-\eta\big(\frac{y}{R}\big)\right|\le 2$, \begin{align}\label{ul}
&\limsup_{n\to\infty}\int_{B_{2R}(0)}|v_n^j(y)|^2~\mathrm{d} y\int_{\mathbb{R}^N\setminus B_{4R}(0)}\frac{\left|\eta\big(\frac{x}{R}\big)-\eta\big(\frac{y}{R}\big)\right|^2}{|x-y|^{N+2s}}~\mathrm{d} x\nonumber\\
\le& \limsup_{n\to\infty}C\int_{B_{2R}(0)}|v_n^j(y)|^2~\mathrm{d} y\int_{\mathbb{R}^N\setminus B_{2R}(0)}\frac{1}{|x|^{N+2s}}~\mathrm{d} x\nonumber\\
=&\frac{C}{R^{2s}}\int_{B_{2R}(0)}|v_*^j(y)|^2~\mathrm{d} y=o_R(1). \end{align} In the region $(\mathbb{R}^N\setminus B_{4R}(0))\times B_{2R}(0)$, by H\"{o}lder inequality and $\eqref{h1}$, we have \begin{align}\label{ux}
&\limsup_{n\to\infty}\int_{\mathbb{R}^N\setminus B_{4R}(0)}|v_n^j(y)|^2~\mathrm{d} y\int_{ B_{2R}(0)}\frac{\left|\eta\big(\frac{x}{R}\big)-\eta\big(\frac{y}{R}\big)\right|^2}{|x-y|^{N+2s}}~\mathrm{d} x\nonumber\\
\le&\limsup_{n\to\infty} CR^N\int_{\mathbb{R}^N\setminus B_{4R}(0)}(v_n^j)^2\frac{1}{|y|^{N+2s}}~\mathrm{d} y\nonumber\\
\le&\limsup_{n\to\infty}\Big(CR^N\Big(\int_{\mathbb{R}^N}(v_n^j)^{2_s^*}~\mathrm{d} y\Big)^{\frac{N-2s}{N}}\Big(\int_{\mathbb{R}^N\setminus B_{4R^{2}}(0)}\frac{1}{|y|^{(N+2s)\frac{N}{2s}}}~\mathrm{d} y\Big)^{\frac{2s}{N}}\nonumber\\ &\qquad+\frac{C}{R^{2s}}\int_{B_{4R^{2}}(0)\setminus B_{4R}(0)}(v_n^j)^2~\mathrm{d} y\Big)\nonumber\\ \le& \frac{C}{R^N}+\frac{C}{R^{2s}}\int_{B_{4R^{2}}(0)\setminus B_{4R}(0)}(v_*^j)^2~\mathrm{d} y=o_R(1). \end{align} Thus we conclude from $\eqref{ug}$--$\eqref{ux}$ that \begin{align}\label{uy}
|\mathcal{R}_n|=\varepsilon_n^{N-2s}o_R(1). \end{align} Putting $\eqref{eqs48}$, $\eqref{ut}$, $\eqref{eqs51}$ and $\eqref{uy}$ together and letting $R\to\infty$, we conclude that \begin{equation*}\label{tt} \liminf_{n\to\infty}\frac{J_{\varepsilon_n}(u_{n})}{\varepsilon_n^N}\geq\sum_{j=1}^k\mathcal{C}\big(V(x_{\ast}^j)\big). \end{equation*} Hence we complete the proof. \end{proof}
At the end of this section, by comparing the Mountain-Pass energy $c_{\varepsilon}$ in (\ref{Ade2.11}) and the limiting energy in \eqref{c2}, we apply Lemma \ref{lem4.7} to prove that the penalized solution $u_{\varepsilon}$ concentrates at a local minimum of $V$ in $\Lambda$ as $\varepsilon\to0$.
\begin{lemma}\label{jz}Let $\alpha\in\big((N-4s)_+,N\big)$, $p\in(\frac{N+\alpha}{N},\frac{N+\alpha}{N-2s})$ and $u_\varepsilon$ be given by Lemma \ref{lem3.6}. Then there exists a family of points $\{x_{\varepsilon}\}_{\varepsilon>0}\subset\Lambda$ and $\rho>0$ such that
$({\rm\romannumeral1})~~\liminf\limits_{\varepsilon\to0}\|u_{\varepsilon}\|_{L^{\infty}(B_{\varepsilon\rho}(x_{\varepsilon}))}>0;$
$({\rm\romannumeral2})~\lim\limits_{\varepsilon\to0}V(x_{\varepsilon})=V_0;$
$({\rm\romannumeral3})\liminf\limits_{\varepsilon\to0}{\rm dist}(x_{\varepsilon}, \Lambda^c)>0;$
$({\rm\romannumeral4})\limsup\limits_{R\to\infty}\limsup \limits_{\varepsilon\to0}\|u_{\varepsilon}\|_{L^{\infty}(U\setminus B_{\varepsilon R}(x_{\varepsilon}))}+\frac{1}{\varepsilon^\alpha}\|I_\alpha* G_{\varepsilon}(x,u_\varepsilon)\|_{L^\infty(U\setminus B_{\varepsilon R}(x_\varepsilon))}=0.$ \end{lemma} \begin{proof} Testing the equation $\eqref{eqs3.2}$ by $u_\varepsilon$ and applying \eqref{3y7} and Young's inequality, we have \begin{align}\label{y1}
&\int_{\mathbb{R}^N}(\varepsilon^{2s}|(-\Delta)^{\frac{s}{2}} u_\varepsilon|^2+Vu_\varepsilon^2)=\frac{p}{\varepsilon^\alpha}\int_{\mathbb{R}^N}\big(I_\alpha* G_\varepsilon(x,u_\varepsilon)\big)g_\varepsilon(x,u_\varepsilon)u_\varepsilon\nonumber\\
\le&\frac{2p}{\varepsilon^{\alpha}} \int_{\mathbb{R}^{N}}|I_{\frac{\alpha}{2}} *(\chi_{\Lambda} u_\varepsilon^{p})|^{2}+\frac{2p}{ \varepsilon^{\alpha}} \int_{\mathbb{R}^{N}}|I_{\frac{\alpha}{2}} *(\mathcal{P}_{\varepsilon} u_\varepsilon)|^{2}, \end{align} By Proposition \ref{prop2.5} and the assumption ($\mathcal{P}_2$), it holds \begin{align}\label{y2}
&\frac{p}{ \varepsilon^{\alpha}}\int_{\mathbb{R}^{N}}|I_{\frac{\alpha}{2}} *(\mathcal{P}_{\varepsilon} u_\varepsilon)|^{2}
\le \kappa\int_{\mathbb{R}^N}\varepsilon^{2s}|(-\Delta)^{\frac{s}{2}} u_\varepsilon|^2+Vu_\varepsilon^2. \end{align} Since $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-2s}$, we choose $1<p'<p$ such that $2<\frac{2Np'}{N+\alpha}<2_s^*$. By Proposition \ref{prop2.3} and Proposition \ref{prop2.4}, \begin{align}\label{y3}
\frac{1}{\varepsilon^{\alpha}} \int_{\mathbb{R}^{N}}|I_{\frac{\alpha}{2}} *(\chi_{\Lambda} u_\varepsilon^{p})|^{2} \le &\frac{C}{\varepsilon^\alpha}\Big(\int_{\Lambda}u_\varepsilon^{\frac{2Np}{N+\alpha}}\Big)^\frac{N+\alpha}{N}
\le\frac{C}{\varepsilon^\alpha}\|u_\varepsilon\|_{L^\infty(\Lambda)}^{2p-2p'}\Big(\int_{\Lambda}u_\varepsilon^{\frac{2Np'}{N+\alpha}}\Big)^\frac{N+\alpha}{N}\nonumber\\
\le&\frac{C}{\varepsilon^{(p'-1)N}}\|u_\varepsilon\|_{L^\infty(\Lambda)}^{2p-2p'}\Big(\int_{\mathbb{R}^N}\varepsilon^{2s}|(-\Delta)^{\frac{s}{2}} u_\varepsilon|^2+Vu_\varepsilon^2\Big)^{p'}. \end{align}
Substituting \eqref{y2}-\eqref{y3} into \eqref{y1}, by $u_\varepsilon\not\equiv0$ and $\eqref{eqs4.5}$, we get \begin{align}\label{y4}
1-2\kappa\le& \frac{C}{\varepsilon^{(p'-1)N}}\|u_\varepsilon\|_{L^\infty(\Lambda)}^{2p-2p'}\|u_\varepsilon\|_\varepsilon^{2(p'-1)}
\le C\|u_\varepsilon\|_{L^\infty(\Lambda)}^{2p-2p'}. \end{align} Lemma \ref{s} means that $u_{\varepsilon}$ is continuous on $\bar\Lambda$, so we can choose $x_{\varepsilon}\in\bar\Lambda$ as a maximum point of $u_{\varepsilon}$ in $\bar{\Lambda}$. It follows from $\kappa<1/2$ and $\eqref{y4}$ that
$$\liminf_{\varepsilon\to0}\|u_{\varepsilon}\|_{L^{\infty}(B_{\varepsilon\rho}(x_{\varepsilon}))}\geq\liminf_{\varepsilon\to0}\|u_{\varepsilon}\|_{L^{\infty}(\Lambda)}>0.$$ Taking any subsequence $\{x_{\varepsilon_n}\}\subset\{x_{\varepsilon}\}$ such that $\lim\limits_{n\to\infty}x_{\varepsilon_n}=x_{\ast}$, by Lemmas $\ref{lem4.5}$ and $\ref{lem4.7}$ we obtain $$\mathcal{C}\big(V_0\big)\geq\liminf_{n\to\infty}\frac{J_{\varepsilon_n}(u_{\varepsilon_n})}{\varepsilon_n^N}\geq\mathcal{C}\big(V(x_{\ast})\big).$$ From the assumption $(\mathcal{V})$ and Lemma $\ref{lem4.4}$, there hold $V(x_{\ast})=V_0$ and $x_{\ast}\in \Lambda$. By the arbitrariness of $ \{x_{\varepsilon_n}\}$, we have $\lim\limits_{\varepsilon\to 0}V(x_{\varepsilon})=V_0$ and then $\liminf\limits_{\varepsilon\to 0}{\rm dist}(x_\varepsilon, \Lambda^c)>0$.
Finally we prove $(\rm\romannumeral4)$ by contradiction. If (iv) does not hold, then there exist $\{\varepsilon_n\}\subset\mathbb R^+$~with ~$\varepsilon_n\to0$ and $\{z_{\varepsilon_n}\}\subset U$ such that
$$\liminf_{n\to\infty}\|u_{\varepsilon_n}\|_{L^{\infty}(B_{\varepsilon_n\rho}(z_{\varepsilon_n}))}+\frac{1}{\varepsilon_n^\alpha}\|I_\alpha* G_{\varepsilon_n}(x,u_{\varepsilon_n})\|_{L^\infty(B_{\varepsilon_n\rho}(z_{\varepsilon_n}))}>0 $$ and
$$\lim\limits_{n\to\infty}\frac{|x_{\varepsilon_n}-z_{\varepsilon_n}|}{\varepsilon_n}=\infty.$$ Since $\bar U$ is compact, we can assume $z_{\varepsilon_n}\to z_\ast\in\bar{U}$, then $V(z_{\ast})\geq V_0>0$. By Lemmas $\ref{lem4.5}$ and $\ref{lem4.7}$ again, we have $$\mathcal{C}\big(V(x_{\ast})\big)\geq\liminf_{n\to\infty}\frac{J_{\varepsilon_n}(u_{\varepsilon_n})}{\varepsilon_n^N} \geq\mathcal{C}\big(V(x_{\ast})\big)+\mathcal{C}\big(V(z_{\ast})\big),$$ which is impossible and hence the proof is completed. \end{proof}
\
\section{Recover the original problem}\label{sec4} In this section, we show that $u_\varepsilon$ given by Lemma \ref{lem3.6} is indeed a solution to the original problem \eqref{eqs1.1} by comparison principle. To do this, the first step is to linearize the penalized problem.
{Beforehand, we state some facts and notations used frequently in this section. Let $\{x_{\varepsilon}\}$ be the points given by Lemma \ref{jz}. By Lemma $\ref {jz}$ (iii), we have \begin{align}\label{ss}
c_\Lambda\varepsilon |x|\le|x_\varepsilon+\varepsilon x|\le C_\Lambda\varepsilon|x|,\ \ x\in\mathbb{R}^N\setminus\Lambda_\varepsilon, \end{align} where $\Lambda_\varepsilon:=\{x\mid x_\varepsilon+\varepsilon x\in\Lambda\}$, $c_\Lambda,C_\Lambda>0$ are some constants depending on $\Lambda$ but independent of $x$ and $\varepsilon$.
Define the rescaled space \begin{align}
H_{V_\varepsilon}^s(\mathbb{R}^N):=\Big\{\psi\in {\dot{H}^s(\mathbb{R}^N)} \ \Big | \ \int_{\mathbb{R}^N}V_\varepsilon\psi^2<\infty\Big\},\nonumber \end{align} where $V_{\varepsilon}(x)=V(\varepsilon x+x_{\varepsilon})$. From ($\mathcal{P}_2$), by rescaling, we have \begin{align}\label{sg}
p\int_{\mathbb{R}^N}|I_{\frac{\alpha}{2}}*(\tilde{\mathcal{P}}_{\varepsilon}\varphi)|^2\le \kappa\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} \varphi|^2+V_\varepsilon|\varphi|^2,\quad \forall\ \varphi\in H_{V_\varepsilon}^s(\mathbb{R}^N), \end{align} where $\tilde{\mathcal{P}}_{\varepsilon}(x)=\mathcal{P}_{\varepsilon}(\varepsilon x+x_{\varepsilon})$.
We also define the set of test functions for the weak sub(super)-solutions outside a ball \begin{align} H_{c,R}^s(\mathbb{R}^N):=\left\{\psi\in {\dot{H}^s(\mathbb{R}^N)}, \psi\ge 0\mid \mathrm{supp} \psi\ \mathrm{is\ compact},\ \psi=0\ \mathrm{in}\ B_R(0)\right\}.\nonumber \end{align}
\begin{proposition}\label{prop4.1} Let $\alpha\in\big((N-4s)_+,N\big)$, $p\in[2,\frac{N+\alpha}{N-2s})$, ($\mathcal{P}_1$)-($\mathcal{P}_2$) hold, $u_\varepsilon$ be given by Lemma \ref{lem3.6}, $\{x_{\varepsilon}\}_\varepsilon$ be the family of points given by Lemma \ref{jz}. Denote $v_{\varepsilon}(\cdot)=u_{\varepsilon}(\varepsilon\cdot + x_{\varepsilon})$, then there exist $\nu>0$, $R_*>0$ and $\varepsilon_{R}>0$ such that for any given $R>R_*$ and $\varepsilon\in (0,\varepsilon_{R})$, $v_{\varepsilon}$ is a weak sub-solution to the following equation \begin{equation}\label{q1} (-\Delta)^s v +\frac{1}{2}V_{\varepsilon}v= \big(pI_\alpha*(\tilde{\mathcal P_\varepsilon}v)+\nu\varepsilon^{N-\alpha}I_{\alpha,\varepsilon}\big)\tilde{\mathcal P_\varepsilon},\ x\in\mathbb R^N\setminus B_R(0), \end{equation} i.e., \begin{align}\label{hk} \int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} v_\varepsilon(-\Delta)^{\frac{s}{2}}\varphi+\frac{1}{2}V_\varepsilon v_\varepsilon\varphi\le\int_{\mathbb{R}^N}\big(pI_\alpha*(\tilde{\mathcal P_\varepsilon}v_\varepsilon)+\nu\varepsilon^{N-\alpha}I_{\alpha,\varepsilon}\big)\tilde{\mathcal P_\varepsilon}\varphi, \end{align} for all $\varphi\in H_{c,R}^s(\mathbb{R}^N)$, where~$V_{\varepsilon}(x)=V(\varepsilon x+x_{\varepsilon})$,~$\tilde{\mathcal{P}}_{\varepsilon}(x)=\mathcal{P}_{\varepsilon}(\varepsilon x+x_{\varepsilon})$, $I_{\alpha,\varepsilon}=I_\alpha(\varepsilon x+x_{\varepsilon})$. \end{proposition}
\begin{proof} By Lemma \ref{jz}, since $p\ge2$, there exists $R_*>0$ and $\varepsilon_{R}>0$ such that \begin{align}\label{edig}
&p\varepsilon^{-\alpha}\big(I_\alpha* G_\varepsilon(x,u_\varepsilon)\big)u_\varepsilon^{p-2}\le \frac{1}{2} V_0\quad \mathrm{in}\ \ U\setminus B_{\varepsilon R}(x_\varepsilon) \end{align} for any $R>R_*$ and $0<\varepsilon<\varepsilon_{R}$.
Fix $\varphi\in H_{c,R}^s(\mathbb{R}^N)$. Taking $\varphi_{\varepsilon}(\cdot)=\varphi (\frac{\cdot - x_{\varepsilon}}{\varepsilon})$ as a test function in $\eqref{eqs3.2}$ for $u_\varepsilon$, namely \begin{align}\label{ephj}
\varepsilon^{2s}\int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} u_\varepsilon(-\Delta)^{\frac{s}{2}}\varphi_{\varepsilon}+Vu_\varepsilon\varphi_{\varepsilon}=\int_{\mathbb{R}^N}p\varepsilon^{-\alpha}\left(I_{\alpha} * G_{\varepsilon}(x, u_\varepsilon)\right) g_{\varepsilon}(x, u_\varepsilon)\varphi_{\varepsilon}. \end{align} By $g_\varepsilon(x,u_\varepsilon)\le u_\varepsilon^{p-1}$, \eqref{edig} and $\inf_{U}V=V_0$, we have \begin{align}\label{z1} p\varepsilon^{-\alpha}\big(I_\alpha* G_\varepsilon(x,u_\varepsilon)\big)g_\varepsilon(x,u_\varepsilon)\le p\varepsilon^{-\alpha}\big(I_\alpha* G_\varepsilon(x,u_\varepsilon)\big)u_\varepsilon^{p-1}\le\frac{1}{2} Vu_\varepsilon\ \mathrm{in}\ U\setminus B_{\varepsilon R}(x_\varepsilon). \end{align} Moreover, by \eqref{3y7}, we have \begin{align}\label{z2} p\varepsilon^{-\alpha}\big(I_\alpha* G_\varepsilon(x,u_\varepsilon)\big)g_\varepsilon(x,u_\varepsilon)\le p\varepsilon^{-\alpha}\Big(I_\alpha* \big(\mathcal P_\varepsilon u_\varepsilon+\frac{1}{p}\chi_\Lambda u_\varepsilon^p\big)\Big)\mathcal P_\varepsilon\ \mathrm{in}\ \mathbb{R}^N\setminus U. \end{align} Since ${\rm dist}(\Lambda,\partial U)>0$, $p\ge2$, by Proposition \ref{prop2.3} and $\eqref{eqs4.5}$, we have \begin{align}\label{z3}
\varepsilon^{-\alpha}I_\alpha*\big(\chi_\Lambda u_\varepsilon^p\big)\le C\frac{I_\alpha}{\varepsilon^\alpha}\int_{\Lambda}u_\varepsilon^p\le C'I_\alpha\varepsilon^{N(1-\frac{p}{2})-\alpha}\|u_\varepsilon\|_\varepsilon^{\frac{p}{2}}\le \nu I_\alpha\varepsilon^{N-\alpha}\ \mathrm{in}\ \mathbb{R}^N\setminus U, \end{align} where $\nu>0$ is independent of $R$ and $\varepsilon$.
Note that $\varphi_\varepsilon=0\ \mathrm{in}\ B_{\varepsilon R}(x_\varepsilon)$. Substituting \eqref{z1}--\eqref{z3} into \eqref{ephj}, we get \begin{align}
\varepsilon^{2s}\int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} u_\varepsilon(-\Delta)^{\frac{s}{2}}\varphi_{\varepsilon}+\frac{1}{2}Vu_\varepsilon\varphi_{\varepsilon}\le\int_{\mathbb{R}^N}\left(p \varepsilon^{-\alpha} I_{\alpha} *\left(\mathcal{P}_{\varepsilon} u_{\varepsilon}\right)+\nu \varepsilon^{N-\alpha} I_{\alpha}\right) \mathcal{P}_{\varepsilon}\varphi_{\varepsilon}.\nonumber \end{align} Therefore, it follows by scaling that \begin{align} \int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} v_\varepsilon(-\Delta)^{\frac{s}{2}}\varphi+\frac{1}{2}V_\varepsilon v_\varepsilon\varphi\le\int_{\mathbb{R}^N}\big(pI_\alpha*(\tilde{\mathcal P_\varepsilon}v_\varepsilon)+\nu\varepsilon^{N-\alpha}I_{\alpha,\varepsilon}\big)\tilde{\mathcal P_\varepsilon}\varphi.\nonumber \end{align} The conclusion then follows by the arbitrariness of $\varphi$. \end{proof}
Next, we establish the comparison principle: \begin{proposition}\label{4z}(Comparison principle) Let $(\mathcal{P}_2)$ hold and $v\in \dot{H}^s(\mathbb{R}^N)$ with $\int_{\mathbb{R}^N}V_\varepsilon v_+^2<\infty$. If $v$ satisfies weakly \begin{align}\label{n1} (-\Delta)^s v +\frac{1}{2}V_{\varepsilon}v\leq p\big(I_\alpha*(\tilde{\mathcal P_\varepsilon}v)\big)\tilde{\mathcal P_\varepsilon}\ \mathrm{in}\ \mathbb R^N\setminus B_R(0), \end{align} and $v\le0$ in $B_R(0)$, then $v\le0$ in $\mathbb{R}^N$. \end{proposition} \begin{proof} Clearly, $v_+=0$ in $B_R(0)$ and $v_+\in \dot{H}^s(\mathbb{R}^N)$. Then there exists $\{\varphi_n\}_{n\ge1}\subset H_{c,R}^s(\mathbb{R}^N)$ such that $\varphi_n\to v_+$ in $\dot{H}^s(\mathbb{R}^N)$ as $n\to\infty$. Indeed, by \cite[Lemma 5]{pap}, we can choose $\varphi_n=\eta(\frac{x}{n})v_+$ where $\eta\in C_c^\infty(\mathbb{R}^N,[0,1])$ satisfying $\eta\equiv1$ in $B_R(0)$ and $\mathrm{supp}\eta\subset B_{2R}(0)$.
Taking $\varphi_n$ as a test function into $\eqref{n1}$, since $\tilde{\mathcal P_\varepsilon}v\le \tilde{\mathcal P_\varepsilon}v_+$, we see that \begin{align}\label{wbj} &\int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} v_+(-\Delta)^{\frac{s}{2}}\varphi_n+\frac{1}{2}V_\varepsilon v_+\varphi_n\le p\int_{\mathbb{R}^N}\big(I_\alpha*(\tilde{\mathcal P_\varepsilon}v_+)\big)\tilde{\mathcal P_\varepsilon}\varphi_n, \end{align} where we have used that \begin{align}\label{23j}
\int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} v_+(-\Delta)^{\frac{s}{2}}\varphi_n\le \int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} v(-\Delta)^{\frac{s}{2}}\varphi_n. \end{align} Since $\varphi_n\to v_+$ in $\dot{H}^s(\mathbb{R}^N)$ as $n\to\infty$, it follows that \begin{align}\label{24j}
\lim_{n\to\infty}\int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} v_+(-\Delta)^{\frac{s}{2}}\varphi_n=\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} v_+|^2. \end{align} Clearly, since $\varphi_n\le v_+$,
$$\int_{\mathbb{R}^N}\big(I_\alpha*(\tilde{\mathcal P_\varepsilon}v_+)\big)\tilde{\mathcal P_\varepsilon}\varphi_n\le\int_{\mathbb{R}^N}\big(I_\alpha*(\tilde{\mathcal P_\varepsilon}v_+)\big)\tilde{\mathcal P_\varepsilon}v_+= \int_{\mathbb{R}^N}|I_\frac{\alpha}{2}*(\tilde{\mathcal P_\varepsilon}v_+)|^2.$$ Moreover, by Fatou's Lemma, \begin{align}\label{25j}
\int_{\mathbb{R}^N}V_\varepsilon |v_+|^2\le\liminf_{n\to\infty} \int_{\mathbb{R}^N}V_\varepsilon v_+\varphi_n. \end{align} Therefore, recalling \eqref{wbj} and letting $n\to\infty$, from Proposition \ref{prop2.5} and \eqref{sg}, we get \begin{align}
\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} v_+|^2+\frac{1}{2}V_\varepsilon v_+^2\le& p\int_{\mathbb{R}^N}|I_\frac{\alpha}{2}*(\tilde{\mathcal P_\varepsilon}v_+)|^2\nonumber\\
\le&\kappa\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} v_+|^2+V_\varepsilon v_+^2,\nonumber \end{align} which implies $v_+=0$ since $v_+\in H_{V_\varepsilon}^s(\mathbb{R}^N)$ and $\kappa<1/2$. \end{proof}
Now we construct the super-solutions for the linear penalized problem $\eqref{q1}$. The sup-solutions are selected as \begin{align}\label{sn}
w_\mu=\frac{1}{(1+|x|^2)^\frac{\mu}{2}}, \end{align} which belongs to $C^{k,\beta}(\mathbb{R}^N)$ for any $k\in \mathbb N$ and $\beta\in(0,1)$. Particularly, $(-\Delta)^s w_\mu$ is well-defined pointwise.
The following two propositions for estimating the nonlocal term $(-\Delta)^s w_\mu$ are given by our other paper \cite {dpy2023}.
\begin{proposition}\label{tb} For any $\mu\in(0,+\infty)$, there exists constants $R_\mu, C_\mu, \tilde{C}_\mu>0$ depending only on $\mu$, $N$ and $s$ such that \begin{align} \left\{
\begin{array}{ll}
0<C_\mu\displaystyle\frac{1}{|x|^{\mu+2s}}\le(-\Delta)^s w_\mu\le3C_\mu\frac{1}{|x|^{\mu+2s}}, & \mathrm{if }\ |x|>R_\mu\ \mathrm{and}\ \mu\in(0,N-2s);
\\
(-\Delta)^s w_\mu=C_{N-2s}w_\mu^{2_s^*-1},\ x\in \mathbb{R}^N,& \mathrm{if }\ \mu=N-2s;
\\
-3C_\mu\displaystyle\frac{1}{|x|^{\mu+2s}}\le(-\Delta)^s w_\mu\le -C_\mu\frac{1}{|x|^{\mu+2s}}<0,& \mathrm{if }\ |x|>R_\mu\ \mathrm{and}\ \mu\in(N-2s,N);
\\
-\displaystyle\frac{\tilde{C}_{N}\ln|x|}{|x|^{N+2s}}\le(-\Delta)^s w_\mu\le-\frac{C_{N}\ln|x|}{|x|^{N+2s}}<0,& \mathrm{if }\ |x|>R_\mu\ \mathrm{and}\ \mu=N,
\\
-\displaystyle\frac{\tilde{C}_\mu}{|x|^{N+2s}}\le(-\Delta)^s w_\mu\le -\frac{C_\mu}{|x|^{N+2s}}<0, & \mathrm{if }\ |x|>R_\mu\ \mathrm{and}\ \mu>N. \end{array} \right.\nonumber \end{align}
\end{proposition}
\begin{proposition}\label{cc5} $w_\mu\in \dot{H}^s(\mathbb{R}^N)$ for $\mu>\frac{N-2s}{2}$ and $w_\mu\notin \dot{H}^s(\mathbb{R}^N)$ for $0<\mu\le\frac{N-2s}{2}$. Moreover, for any $\mu>\frac{N-2s}{2}$, \begin{align*}
\int_{\mathbb{R}^N}(-\Delta)^{s/2}w_\mu(-\Delta)^{s/2}\phi=\int_{\mathbb{R}^N}(-\Delta)^{s}w_\mu\phi,\quad \forall \phi\in \dot{H}^s(\mathbb{R}^N). \end{align*} \end{proposition}
Now we are in a position to construct the super-solutions of (\ref {q1}). We assume the prescribed form of the penalization: \begin{align}\label{ph}
\mathcal P_\varepsilon(x)=\frac{\varepsilon^\theta}{|x|^\tau}\chi_{\Lambda^c}, \end{align} where $\theta, \tau>0$ are two parameters which will be determined later. Moreover, in order to described the following proof conveniently, we give some notations as follows: \begin{equation}\label{fd}F^\varepsilon_{\theta,\tau, \mu}(x):=
\frac{\varepsilon^{2\theta-2\tau}\chi_{\Lambda_\varepsilon^c}}{|x|^{\mu+2\tau-\alpha}}+\frac{(\varepsilon^{2\theta-2\tau}+\varepsilon^{\theta-\tau})\ln(|x|+e)\chi_{\Lambda_\varepsilon^c}}{|x|^{N-\alpha+\tau}}, \quad \mu+\tau>\alpha.
\end{equation} and \begin{equation}\label{fdd}G^\varepsilon_{\mu}(x):=\left\{
\begin{aligned}
&\frac{\chi_{\Lambda_\varepsilon^c}}{|x|^{\mu+2s}},\ \mathrm{if}\ \mu\in\big(\frac{N-2s}{2},N-2s\big),\\
&\frac{\varepsilon^{-2s}\chi_{\Lambda_\varepsilon^c}}{|x|^{\mu+2s}},\ \mathrm{if}\ \mu\in(N-2s,N),\ \inf_{\mathbb{R}^N}V(x)(1+|x|^{\omega})>0\ \mathrm{with}\ \omega=2s,\\
&\frac{\varepsilon^{-\omega}\chi_{\Lambda_\varepsilon^c}}{|x|^{\mu+\omega}},\ \mathrm{if}\ \mu\in(N,N+2s-\omega),\ \inf_{\mathbb{R}^N}V(x)(1+|x|^{\omega})>0, \ \omega\in (0,2s).
\end{aligned}\right.
\end{equation}
\begin{proposition}\label{5b}(Construction of sup-solutions) Let $$ \mu\in\Big(\frac{N-2s}{2},N+2s\Big)\Big\backslash\{N,N-2s\},\,\,\mu+\tau>\alpha, $$ and $\{x_{\varepsilon}\}_\varepsilon$ be the family of points given by Lemma \ref{jz}.
If $F^\varepsilon_{\theta,\tau, \mu}\le \lambda G^\varepsilon_{\mu}$ for given $\lambda>0$ and $\varepsilon$ small depending on $\lambda$ , then $w_\mu$ is a supper-solution of (\ref {q1}) in the classical sense, i.e. \begin{align}\label{wd} (-\Delta)^s w_\mu+\frac{1}{2}V_\varepsilon w_\mu\ge\big(pI_\alpha*(\tilde{\mathcal P_\varepsilon}w_\mu)+\nu\varepsilon^{N-\alpha}I_{\alpha,\varepsilon}\big)\tilde{\mathcal P_\varepsilon},\ x\in\mathbb R^N\setminus B_R(0) \end{align} for given $R>0$ large enough, where $\tilde{\mathcal P_\varepsilon}(x)=\mathcal P_\varepsilon(x_\varepsilon+\varepsilon x)$, $I_{\alpha,\varepsilon}(x)=I_\alpha(x_\varepsilon+\varepsilon x)$, $V_\varepsilon(x)=V(x_\varepsilon+\varepsilon x)$. \end{proposition} \begin{proof} We first consider the right hand side of $\eqref{wd}$. For given $R>\max\{R_\mu,1\}$, since $\liminf_{\varepsilon\to0}{\rm dist}(x_\varepsilon, \Lambda^c)>0$, we have $B_R(0)\subset \Lambda_\varepsilon:=\{x\mid x_\varepsilon+\varepsilon x\in \Lambda\}$ for small $\varepsilon$. Reviewing $\eqref{ss}$, we have \begin{align}\label{al} &\big(pI_\alpha*(\tilde{\mathcal P_\varepsilon}w_\mu)+\nu\varepsilon^{N-\alpha}I_{\alpha,\varepsilon}\big)\tilde{\mathcal P_\varepsilon}\nonumber\\
\le&\frac{p}{c_\Lambda^{2\tau}}\varepsilon^{2\theta-2\tau}\Big(I_\alpha*\Big(\frac{\chi_{B^c_1(0)}}{|x|^{\mu+\tau}}\Big)\Big)\frac{\chi_{\Lambda_\varepsilon^c}}{|x|^\tau}
+\frac{\nu}{c_\Lambda^{N-\alpha+\tau}}\varepsilon^{\theta-\tau}\frac{\chi_{\Lambda_\varepsilon^c}}{|x|^{N-\alpha+\tau}}. \end{align} There exists a constant $C>0$ such that for $\mu+\tau>\alpha$,
\begin{equation}\label{df}\Big(I_\alpha*\big(\frac{\chi_{B^c_1(0)}}{|x|^{\mu+\tau}}\big)\Big)(x)\le \frac{C}{|x|^{\mu+\tau-\alpha}}+\frac{C\ln(|x|+e)}{|x|^{N-\alpha}},\quad x\in\mathbb{R}^N\backslash\{0\}.
\end{equation} Indeed, for any $x\in\mathbb{R}^N\backslash\{0\}$, we have \begin{align*}
&\int_{\mathbb{R}^N}\frac{1}{|x-y|^{N-\alpha}}\frac{\chi_{B^c_1(0)}(y)}{|y|^{\mu+\tau}}dy\\
=&\int_{B_{|x|/2}(x)}\frac{1}{|x-y|^{N-\alpha}}\frac{\chi_{B^c_1(0)}(y)}{|y|^{\mu+\tau}}dy+
\int_{B_{|x|/2}(0)}\frac{1}{|x-y|^{N-\alpha}}\frac{\chi_{B^c_1(0)}(y)}{|y|^{\mu+\tau}}dy\\
&+\int_{B^c_{|x|/2}(x)\cap B^c_{|x|/2}(0)}\frac{1}{|x-y|^{N-\alpha}}\frac{\chi_{B^c_1(0)}(y)}{|y|^{\mu+\tau}}dy\\
\le&\frac{C}{|x|^{\mu+\tau}}\int_{B_{|x|/2}(x)}\frac{1}{|x-y|^{N-\alpha}}dy+\frac{C}{|x|^{N-\alpha}}\int_{1\le |y|\le \frac{|x|}{2}}\frac{1}{|y|^{\mu+\tau}}dy\\
&+C\int_{B^c_{|x|/2}(0)}\frac{1}{|y|^{N-\alpha+\mu+\tau}}dy\\
\le&\frac{C}{|x|^{\mu+\tau-\alpha}}+\frac{C}{|x|^{N-\alpha}}\Big(1+\frac{1}{|x|^{\mu+\tau-N}}+\ln(|x|+e)\Big), \end{align*}
where we use that $|x-y|\ge \frac{1}{3}|y|$ if $y\in B^c_{|x|/2}(x)\cap B^c_{|x|/2}(0)$. Then \eqref{df} holds.
Recalling the definition of $F^\varepsilon_{\theta,\tau, \mu}$ in \eqref{fd}, we infer from $\eqref{al}$ and $\eqref{df}$ that \begin{align} \big(pI_\alpha*(\tilde{\mathcal P_\varepsilon}w_\mu)+\nu\varepsilon^{N-\alpha}I_{\alpha,\varepsilon}\big)\tilde{\mathcal P_\varepsilon}\le CF^\varepsilon_{\theta,\tau, \mu}.\nonumber
\end{align}
Now we consider the left hand side of $\eqref{wd}$ in different decay rates of $V$ stated in \eqref{fdd}.
\noindent\textbf{Case 1.} $\mu\in (\frac{N-2s}{2}, N-2s)$.
From Proposition \ref{tb}, we have $(-\Delta)^s w_\mu\ge\frac{C}{|x|^{\mu+2s}}$ for $|x|>R$.
\noindent\textbf{Case 2.} $\inf_{x\in\mathbb{R}^N}V(x)(1+|x|^{2s})>0$ and $N-2s<\mu<N$.
From Proposition \ref{tb}, for $R$ large, we have
$$(-\Delta)^s w_\mu+\frac{1}{2}V_{\varepsilon} w_\mu\ge -\frac{3C_\mu}{|x|^{2s}}\frac{1}{|x|^\mu}+\frac{1}{2}V_0\frac{1}{(1+|x|^2)^{\mu/2}}\ge0,\ \ x\in \Lambda_\varepsilon\setminus B_R(0).$$
Since $\inf_{x\in\mathbb{R}^N}V(x) (1+|x|^{2s})>0$, there exists $C>0$ such that $V(x)\ge \frac{C}{|x|^{2s}}$ for $|x|\ge1$. By $\eqref{ss}$, for $\varepsilon>0$ small,
we have \begin{eqnarray*}
(-\Delta)^s w_\mu+\frac{1}{2}V_{\varepsilon} w_\mu&\ge&-\frac{3C_\mu}{|x|^{2s+\mu}}+\frac{C}{2C_\Lambda^{2s}\varepsilon^{2s}}
\frac{1}{|x|^{2s}}\frac{1}{(1+|x|^2)^{\mu/2}}\\
&\ge&\frac{C\varepsilon^{-2s}}{|x|^{\mu+2s}},\quad x\in\mathbb R^N\backslash\Lambda_\varepsilon. \end{eqnarray*}
\noindent\textbf{Case 3.} $\inf_{x\in\mathbb{R}^N}V(x)(1+|x|^{\omega})>0$ for some $\omega\in (0,2s)$ and $N<\mu< N+2s-\omega$.
From Proposition \ref{tb}, we get for $R$ large and $\varepsilon$ small that
$$(-\Delta)^s w_\mu+\frac{1}{2}V_{\varepsilon} w_\mu\ge -\frac{\tilde{C}_\mu}{|x|^{N+2s}}+\frac{1}{2}V_0\frac{1}{(1+|x|^2)^{\mu/2}}\ge0,\quad x\in\Lambda_\varepsilon\setminus B_R(0).$$
Since $\inf_{x\in\mathbb{R}^N}V(x) (1+|x|^{\omega})>0$, there exists $C_\omega>0$ such that $V(x)\ge \frac{C_\omega}{|x|^{\omega}}$ for $|x|\ge1$. Thus for $\varepsilon>0$ small, it follows by $\eqref{ss}$ and Proposition \ref{tb} that $$(-\Delta)^s w_\mu+\frac{1}{2}V_{\varepsilon} w_\mu\ge-
\frac{\tilde{C}_\mu}{|x|^{N+2s}}+\frac{C_\omega}{2C_\Lambda^{\omega}\varepsilon^{\omega}}\frac{1}{|x|^{\omega}}\frac{1}{(1+|x|^2)^{\mu/2}}\ge\frac{C\varepsilon^{-\omega}}{|x|^{\mu+\omega}}$$ for all $x\in\mathbb{R}^N\setminus \Lambda_\varepsilon$.
Summarizing the three cases above, the conclusion follows by the assumption $F^\varepsilon_{\theta,\tau, \mu}\le \lambda G^\varepsilon_{\mu}$ for $\lambda$ small. \end{proof}
\begin{remark}
Note that there is no restrictions on $V$ out set of $\Lambda$ in case 1, which indicates that $V$ will not have influence outside $\Lambda$ during the construction in case 1. However, if $V$ further satisfies $\inf_{x\in\mathbb{R}^N}V(x)(1+|x|^{\omega})>0$ for $\omega\in(0,2s]$, we are able to take $\mu> N-2s$ due to the effect of $V$. More precisely, $(-\Delta)^s w_\mu$ can be absorbed by $V_{\varepsilon} w_\mu$ outside $\Lambda_\varepsilon$. \end{remark}
Next, by means of the sup-solutions above, we are going to apply the comparison principle in Proposition \ref{4z} to prove Theorem $\ref{thm1.1}$. We need to verify firstly that the two pre-assumptions ($\mathcal{P}_1$)-($\mathcal{P}_2$) in Section \ref{sec2} hold under some choices of the parameters $\tau,\theta$.
\begin{proposition}\label{48} Assume that one of the following two conditions holds:
($\mathcal{S}_1$) $\alpha+2s<2\tau$, $\alpha+2s<2\theta$;
($\mathcal{S}_2$) $\alpha<2\theta$ and $\alpha+\omega<2\tau$ when
$\inf_{x\in\mathbb{R}^N}V(x)(1+|x|^\omega)>0$ with $\omega\in(0,2s]$.
\noindent Then the penalized function $\mathcal{P}_{\varepsilon}$ defined by (\ref {ph}) satisfies ($\mathcal{P}_1$) and ($\mathcal{P}_2$) in Section \ref{sec2}. \end{proposition}
\begin{proof} We first verify ($\mathcal{P}_2$).
{\bf The case under the assumption ($\mathcal{S}_1$):} For any $\varphi\in {\dot{H}^s(\mathbb{R}^N)}$, by the assumption $\alpha+2s<2\tau$, $\alpha+2s<2\theta$ and Hardy inequality (Proposition \ref{prop2.1}), for $\varepsilon$ small we have, \begin{align}\label{hp}
\frac{pC_\alpha}{\varepsilon^\alpha}\int_{\mathbb{R}^N}\mathcal P_\varepsilon^2|\varphi|^2|x|^\alpha=&pC_\alpha\varepsilon^{2\theta-\alpha}\int_{\Lambda^c}\frac{1}{|x|^{2\tau-\alpha}}|\varphi|^2\nonumber\\
\le& C\varepsilon^{2\theta-\alpha}\int_{\Lambda^c}\frac{|\varphi|^2}{|x|^{2s}}\le\kappa\int_{\mathbb{R}^N}\varepsilon^{2s}|(-\Delta)^{\frac{s}{2}}\varphi|^2, \end{align} which implies $(\mathcal{P}_2)$.
{\bf The case under the assumption ($\mathcal{S}_2$):} Clearly, there exists a $C_\omega>0$ such that $V\ge\frac{C_\omega}{|x|^\omega}$ in $\mathbb{R}^N\setminus\Lambda$. By the assumptions $\alpha<2\theta$ and $\alpha+\omega<2\tau$, for $\varepsilon$ small, we have \begin{align}
\frac{pC_\alpha}{\varepsilon^\alpha}\int_{\mathbb{R}^N}\mathcal P_\varepsilon^2|\varphi|^2|x|^\alpha\le&pC_\alpha\varepsilon^{2\theta-\alpha}\int_{\Lambda^c}\frac{1}{|x|^{2\tau-\alpha}}|\varphi|^2\nonumber\\
\le&\kappa C_\omega\int_{\Lambda^c}\frac{1}{|x|^\omega}|\varphi|^2\le\kappa\int_{\mathbb{R}^N} V|\varphi|^2,\nonumber \end{align} which also implies $(\mathcal{P}_2)$.
Next we turn to check ($\mathcal{P}_1$) . Let $\{v_n\}_{n\in\mathbb N}$ is a bounded sequence in ${H_{V,\varepsilon}^s(\mathbb{R}^N)}$. Up to a subsequence, there exists some $v\in{H_{V,\varepsilon}^s(\mathbb{R}^N)}$ such that $v_n\rightharpoonup v$ in ${H_{V,\varepsilon}^s(\mathbb{R}^N)}$ and $v_n\to v$ in $L_{\mathrm{loc}}^q(\mathbb{R}^N)$ for $q\in [1,2_s^*)$. Let $\varepsilon<1$ and $M>1$ such that $\Lambda\subset B_M(0)$.
{\bf The case under the assumption ($\mathcal{S}_1$):} By the assumption $\alpha+2s<2\tau$, $\alpha+2s<2\theta$ and Hardy inequality, \begin{align}\label{zr}
&\int_{\mathbb{R}^N}|v_n-v|^2\mathcal P_\varepsilon^2|x|^\alpha\nonumber\\
=&\varepsilon^{2\theta}\int_{\mathbb{R}^N\setminus B_M(0)}\frac{|v_n-v|^2}{|x|^{2\tau-\alpha}}+\varepsilon^{2\theta}\int_{B_M(0)\setminus\Lambda}\frac{|v_n-v|^2}{|x|^{2\tau-\alpha}}\nonumber\\
\le&\frac{\varepsilon^{2s}}{M^{2\tau-\alpha-2s}}\int_{\mathbb{R}^N\setminus B_M(0)}\frac{|v_n-v|^2}{|x|^{2s}}+C\int_{B_M(0)\setminus\Lambda}|v_n-v|^2,\nonumber\\
\le&\frac{C}{M^{2\tau-\alpha-2s}}\big(\sup_{n\in\mathbb N}\varepsilon^{2s}[v_n]_s^2+\varepsilon^{2s}[v]_s^2\big)+C\int_{B_M(0)\setminus\Lambda}|v_n-v|^2, \end{align}
which implies $v_n\to v$ in $L^2\big(\mathbb{R}^N, \mathcal P_\varepsilon^2|x|^\alpha\mathrm{d} x\big)$ as $n\to\infty$ and thereby ($\mathcal{P}_1$) holds.
{\bf The case under the assumption ($\mathcal{S}_2$):} Noting $V\ge\frac{C_\omega}{|x|^\omega}$ in $\mathbb{R}^N\setminus\Lambda$, by the assumption $\alpha+\omega<2\tau$, \begin{align}\label{zr}
&\int_{\mathbb{R}^N}|v_n-v|^2\mathcal P_\varepsilon^2|x|^\alpha\nonumber\\
=&\varepsilon^{2\theta}\int_{\mathbb{R}^N\setminus B_M(0)}|v_n-v|^2\frac{1}{|x|^{2\tau-\alpha}}+\varepsilon^{2\theta}\int_{B_M(0)\setminus\Lambda}|v_n-v|^2\frac{1}{|x|^{2\tau-\alpha}}\nonumber\\
\le&\frac{1}{M^{2\tau-\alpha-\omega}}\int_{\mathbb{R}^N\setminus B_M(0)}\frac{|v_n-v|^2}{|x|^{\omega}}+C\int_{B_M(0)\setminus\Lambda}|v_n-v|^2,\nonumber\\
\le&\frac{C}{M^{2\tau-\alpha-\omega}}(\sup_{n\in\mathbb N}\int_{\mathbb{R}^N}V|v_n|^2+\int_{\mathbb{R}^N}V|v|^2)+C\int_{B_M(0)\setminus\Lambda}|v_n-v|^2, \end{align}
which indicates $v_n\to v$ in $L^2\big(\mathbb{R}^N, \mathcal P_\varepsilon^2|x|^\alpha\mathrm{d} x\big)$ as $n\to\infty$ and so ($\mathcal{P}_1$) holds.
Then we complete the proof. \end{proof}
Secondly, we use the comparison principle in Proposition \ref{4z} to get the upper decay estimates of $u_{\varepsilon}$.
\begin{proposition}\label{49} Let $\alpha\in \big((N-4s)_+,N\big)$, $p\in [2,\frac{N+\alpha}{N-2s})$. Assume that one of the following three conditions holds:
($\mathcal{U}_1$) $2s<2\tau-\alpha$ and $\alpha<\tau<\theta$, $\mu\in (\frac{N-2s}{2},N-2s)$;
($\mathcal{U}_2$) $\alpha+2s<\tau$ and $\tau<\theta$, $\mu\in (N-2s,N)$, when
$\inf_{x\in\mathbb{R}^N}V(x)(1+|x|^{2s})>0$;
($\mathcal{U}_3$) $\alpha+2s<\tau$ and $\tau<\theta$, $\mu\in(N,N+2s-\omega)$, when
$\inf_{x\in\mathbb{R}^N}V(x)(1+|x|^\omega)>0$ with $\omega\in(0,2s)$.
\noindent Then ($\mathcal{P}_1$)-($\mathcal{P}_2$) hold and there exists $C>0$ independent of small $\varepsilon$ such that $v_\varepsilon:=u_\varepsilon(x_\varepsilon+\varepsilon x)\le Cw_\mu$. In particular, \begin{align}\label{wsq}
u_\varepsilon\le\frac{C\varepsilon^\mu}{|x|^\mu}\ \mathrm{in}\ \mathbb{R}^N\setminus\Lambda, \end{align} where $u_\varepsilon$ is given by Lemma \ref{lem3.6} and $\{x_{\varepsilon}\}_\varepsilon$ is given by Lemma \ref{jz}. \end{proposition} \begin{proof}
It is easy to check that ($\mathcal{S}_1$) holds under the assumption ($\mathcal{U}_1$), and ($\mathcal{S}_2$) holds under one of ($\mathcal{U}_2$) and ($\mathcal{U}_3$). Moreover, we can verify that
$F^\varepsilon_{\theta,\tau, \mu}\le \varepsilon^{\theta-\tau} G^\varepsilon_{\mu}$ for $\theta-\tau>0$. Thus ($\mathcal{P}_1$)-($\mathcal{P}_2$) hold by Proposition \ref{48} and \eqref{wd} holds by Proposition \ref{5b}.
Fix $R$ large enough and let
$$
\bar{w}_\mu=2\sup_{\varepsilon\in(0,\varepsilon_0)}\|v_\varepsilon\|_{L^\infty(\mathbb{R}^N)}R^\mu w_\mu,\,\,\bar{v}_\varepsilon=v_\varepsilon-\bar{w}_\mu.
$$
Clearly, $\bar{v}_\varepsilon\le0$ in $B_R(0)$, $\bar{v}_\varepsilon\in \dot{H}^s(\mathbb{R}^N)$ and $\int_{\mathbb{R}^N}V_\varepsilon(\bar{v}_{\varepsilon,+})^2\le\int_{\mathbb{R}^N}V_\varepsilon v^2_\varepsilon<\infty$. Moreover, from Proposition \ref{prop4.1}, \eqref{wd} and Proposition \ref{cc5}, $\bar{v}_\varepsilon$ satisfies weakly $$(-\Delta)^s\bar{v}_\varepsilon+\frac{1}{2}V_{\varepsilon}\bar{v}_\varepsilon\le p\big(I_\alpha*(\tilde{\mathcal P_\varepsilon}\bar{v}_\varepsilon)\big)\tilde{\mathcal P_\varepsilon}\ \ \mathrm{in}\ \mathbb R^N\backslash B_R(0).$$ It follows from Proposition \ref{4z} that $\bar{v}_\varepsilon\le0$ in $\mathbb{R}^N$. Then $v_\varepsilon\le Cw_\mu$. In particular, if $x\in \mathbb{R}^N\setminus\Lambda$, noting that $\liminf_{\varepsilon\to0}{\rm dist}(x_\varepsilon,\mathbb{R}^N\setminus\Lambda)>0$, it holds \begin{align}\label{sjj}
u_\varepsilon(x)=v_\varepsilon\Big(\frac{x-x_\varepsilon}{\varepsilon}\Big)\le&C\Big(1+\Big|\frac{x-x_\varepsilon}{\varepsilon}\Big|^2\Big)^{-\frac{\mu}{2}}\nonumber\\
\le& \frac{C\varepsilon^\mu}{\varepsilon^\mu+|x-x_\varepsilon|^\mu}\le\frac{C\varepsilon^\mu}{|x|^\mu}. \end{align} This completes the proof. \end{proof}
Finally, we prove Theorem $\ref{thm1.1}$.
\
\noindent\textbf{Proof of Theorem \ref{thm1.1}}:
{\bf The case under the assumption ($\mathcal{Q}_1$), i.e. $p>1+\frac{\max\{s+\frac{\alpha}{2},\alpha\}}{N-2s}$.}
Let $\mu\in (\frac{N-2s}{2},N-2s)$ be sufficiently close to $N-2s$ from below, $\tau$ and $\theta$ be such that
\begin{align}\label{lll}
\max\left\{s+\frac{\alpha}{2},\alpha\right\}<\tau<\theta<\mu(p-1)<(N-2s)(p-1).
\end{align}
By \eqref{lll} and Proposition \ref{49}, ($\mathcal{P}_1$) and ($\mathcal{P}_2$) hold. Then we can find a nonnegative nontrivial weak solution $u_\varepsilon$ to \eqref{eqs3.2} by Lemma \ref{lem3.6}. Moreover, by \eqref{lll} and \eqref{wsq}, $$
u_\varepsilon^{p-1}\le\frac{C\varepsilon^{\mu(p-1)}}{|x|^{\mu(p-1)}}\le\frac{\varepsilon^\theta}{|x|^\tau}=\mathcal P_\varepsilon\ \ \mathrm{in}\ \mathbb{R}^N\setminus\Lambda $$ for $\varepsilon$ small enough. Hence $u_\varepsilon$ is indeed a solution to the original problem $\eqref{eqs1.1}$.
Letting $\{x_{\varepsilon}\}_\varepsilon$ be given by Lemma \ref{jz}. $\eqref{sjj}$ says \begin{align}\label{sxs}
u_\varepsilon\le\frac{C\varepsilon^\mu}{\varepsilon^\mu+|x-x_\varepsilon|^\mu}. \end{align} Moreover, by Lemmas \ref{w} and \ref{s}, we know that $u_\varepsilon\in L^\infty(\mathbb{R}^N)\cap C_{\mathrm{loc}}^\sigma(\mathbb{R}^N)$ for any $\sigma\in (0,\min\{2s,1\})$. It follows by Lemma \ref{w61} that $u_\varepsilon>0$ in $\mathbb{R}^N$.
Next, we derive a higher regular estimate of $u_\varepsilon$ if additionally $V\in C_{\mathrm{loc}}^\varrho(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ for some $\varrho\in (0,1)$.
Since $u_\varepsilon$ is a solution to $\eqref{eqs1.1}$, we see that $v_\varepsilon(y):=u_\varepsilon(x_\varepsilon+\varepsilon y)$ solves \begin{align}\label{2b} (-\Delta)^sv_\varepsilon = h_\varepsilon\ \ \text{in}\ \mathbb R^N, \end{align} where $h_\varepsilon(y)=-V(x_\varepsilon+\varepsilon y)v_\varepsilon+\big(I_\alpha* (v_\varepsilon^p)\big)v_\varepsilon^{p-1}$.
It suffices to prove that $I_\alpha* v_\varepsilon^p\in C^\delta(\mathbb{R}^N)$ for any $\delta\in (0,\min\{1,2s\})$.
In fact, if $I_\alpha* v_\varepsilon^p\in C^\delta(\mathbb{R}^N)$, it follows from Lemmas \ref{w}, \ref{s} and the assumption $V\in C_{\mathrm{loc}}^\varrho(\mathbb R^N)$ that $h_\varepsilon\in C_{\mathrm{loc}}^\vartheta(\mathbb R^N)$ for some $\vartheta\in\left(0, \min\{1,2s,\varrho\}\right)$. Thus, for any given $R>1$, from \cite[Theorem 12.2.5]{clp}, we know $v_\varepsilon\in C^{2s+\vartheta}(B_R(0))$ satisfying
$$\|v_\varepsilon\|_{C^{2s+\vartheta}(B_R(0))}\le C\Big(\|h_\varepsilon\|_{C^\vartheta(B_{3R}(0))}+\|v_\varepsilon\|_{L^\infty(\mathbb{R}^N)}\Big).$$ Since $R>1$ is arbitrary, by rescaling, we deduce that $u_\varepsilon\in C_{\mathrm{loc}}^{2s+\vartheta}(\mathbb{R}^N)$.
In the following, we verify $I_\alpha* v_\varepsilon^p\in C^\delta(\mathbb{R}^N)$ for any $\delta\in (0,\min\{1,2s\})$. Actually, fix any $\delta\in (0,\min\{1,2s\})$, from $\eqref{g1}$, we have $v_\varepsilon\in C^\delta(\mathbb{R}^N)$. By lemma \ref{I}, we find $I_\alpha* v_\varepsilon^p\in L^\infty(\mathbb{R}^N)$. Besides, for any given $x_1,x_2\in \mathbb{R}^N$, $x_1\neq x_2$, since $\mu(p-1)>\alpha$, we have \begin{align}
&\frac{|I_\alpha* (v_\varepsilon^p)(x_1)-I_\alpha* (v_\varepsilon^p)(x_2)|}{|x_1-x_2|^\delta}\nonumber\\
\le&\int_{\mathbb{R}^N}\frac{1}{|y|^{N-\alpha}}\frac{|v_\varepsilon^p(x_1-y)-v_\varepsilon^p(x_2-y)|}{|x_1-x_2|^\delta}\mathrm{d} y\nonumber\\
\le& C\|v_\varepsilon\|_{C^\delta(\mathbb{R}^N)}\int_{\mathbb{R}^N}\frac{1}{|y|^{N-\alpha}}\left(v_\varepsilon^{p-1}(x_1-y)+v_\varepsilon^{p-1}(x_2-y)\right)\mathrm{d} y\nonumber\\
\le&C\int_{\mathbb{R}^N}\frac{1}{|x_1-y|^{N-\alpha}}\frac{1}{1+|y|^{\mu(p-1)}}\mathrm{d} y+C\int_{\mathbb{R}^N}\frac{1}{|x_2-y|^{N-\alpha}}\frac{1}{1+|y|^{\mu(p-1)}}\mathrm{d} y\nonumber\\
\le&2C\sup_{x\in\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{1}{|x-y|^{N-\alpha}}\frac{1}{1+|y|^{\mu(p-1)}}\mathrm{d} y\le C,\nonumber \end{align} where we use the fact that
$v_\varepsilon^{p-1}(y)= u^{p-1}_\varepsilon(x_\varepsilon+\varepsilon y)\le\frac{C}{1+|y|^{\mu (p-1)}}$ by \eqref{sxs}.
Therefore, $u_\varepsilon\in C_{\mathrm{loc}}^{2s+\vartheta}(\mathbb{R}^N)$, and hence $u_\varepsilon$ is a classical solution to \eqref{eqs1.1}.
The proofs for the other cases are similar, so we only give the corresponding choice of $p$ and parameters.
{\bf The case under the assumption ($\mathcal{Q}_2$) with $\omega =2s$, i.e. $p>1+\frac{\alpha+2s}{N}$, $\inf_{x\in\mathbb{R}^N}V(x)(1+|x|^{2s})>0$.}
Let $\mu\in (N-2s,N)$ be sufficiently close to $N$ from below, $\tau$ and $\theta$ satisfy
\begin{align}
\alpha+2s<\tau<\theta<\mu(p-1)<N(p-1).\nonumber
\end{align}
{\bf The case under the assumption ($\mathcal{Q}_2$) with $\omega\in(0,2s)$, i.e. $p>1+\frac{\alpha+2s}{N+2s-\omega}$, $\inf_{x\in\mathbb{R}^N}V(1+|x|^{\omega})>0$ for $\omega\in(0,2s)$.}
Let $\mu$ be sufficiently close to $N+2s-\omega$ from below, $\tau$ and $\theta$ satisfy
\begin{align}
\alpha+2s<\tau<\theta<\mu(p-1)<(N+2s-\omega)(p-1).\nonumber
\end{align} The proof of Theorem \ref{thm1.1} is then completed.
Under specific decay assumptions on $V$, we can also get the lower decay estimates of $u_{\varepsilon}$. For example, taking $w_N:=\frac{1}{(1+|x|^2)^{N/2}}$, by \eqref{wqb} and Proposition \ref{tb}, we can verify that \begin{align*}
\varepsilon^{2s}(-\Delta)^sw_N+Vw_N\le -\frac{\varepsilon^{2s}C_N\ln|x|}{|x|^{N+2s}}+\frac{C}{1+|x|^{2s}}\frac{1}{(1+|x|^2)^{N/2}}<0,\quad |x|>R_\varepsilon, \end{align*} for some $R_\varepsilon>0$ large enough. On the other hand, letting $u_\varepsilon$ be a positive weak solution of \eqref{eqs1.1}, it is clear that \begin{align*}
\varepsilon^{2s}(-\Delta)^su_\varepsilon+Vu_\varepsilon>0,\quad x\in\mathbb{R}^N. \end{align*} It follows from comparison principle that \begin{align*}
w_N\le \frac{1}{\inf_{x\in B_{R_\varepsilon}(0)}u_\varepsilon}u_\varepsilon,\quad x\in \mathbb{R}^N, \end{align*} i.e., \begin{align*}
u_\varepsilon\ge w_N\inf_{x\in B_{R_\varepsilon}(0)}u_\varepsilon\ge \frac{C_\varepsilon}{1+|x|^N} \end{align*} for some $C_\varepsilon>0$ since $u_\varepsilon>0$ in $\overline{B_{R_\varepsilon}(0)}$. Thus we obtain the following remark:
\begin{remark}\label{rma} Assume $p\in [2,\frac{N+\alpha}{N-2s})$, $p> 1+\frac{\alpha+2s}{N}$ and \begin{equation}\label{wqb}
c\le V(x)(1+|x|^{2s})\le C,\quad x\in\mathbb{R}^N, \end{equation} for constants $C,c>0$. Let $u_\varepsilon$ be given by Theorem \ref{thm1.1}. Then \begin{align*}
u_\varepsilon\ge \frac{C_\varepsilon}{1+|x|^N}, \end{align*} for a constant $C_\varepsilon>0$ depending on $\varepsilon$. \end{remark}
\section{Nonexistence results}\label{s6} In this section, we aim to obtain some nonexistence results for \eqref{eqs1.1}. Before that, we present the following comparison principle. \begin{lemma}\label{ws8}(Comparison principle) Let $f(x)\in L^1_{\mathrm{loc}}(\mathbb{R}^N\backslash\{0\})$ with $f(x)\ge0$. Suppose $\tilde v\in \dot{H}(\mathbb{R}^N)\cap C(\mathbb{R}^N)$ with $\tilde v>0$ being a weak supersolution to \begin{align*}
(-\Delta)^s v+Vv= f(x),\quad x\in \mathbb{R}^N\backslash B_{R}(0), \end{align*} and $\underline{v}_\lambda \in \dot{H}(\mathbb{R}^N)\cap C(\mathbb{R}^N)$ with $\underline{v}_\lambda >0$ being a weak subsolution to \begin{align*}
(-\Delta)^s {v}+V{v}= \lambda f(x),\quad x\in \mathbb{R}^N\backslash B_{R'}(0), \end{align*} where $R,R',\lambda>0$ are constants. Then there holds \begin{align*}
\tilde v\ge C \underline {v}_\lambda,\quad x\in\mathbb{R}^N, \end{align*} where $C>0$ is a constant depending only on $\lambda$, $\tilde{R}:=\max\{R,R'\}$, $\min_{B_{\tilde{R}}(0)} \tilde v$ and $\max_{B_{\tilde{R}}(0)}\underline{v}_\lambda$. \end{lemma} \begin{proof}
Define $$\bar{v}:=\min\Big\{1,\frac{\min_{B_{\tilde{R}}(0)}\tilde v}{\max_{B_{\tilde{R}}(0)}\underline{v}_\lambda}\Big\}\frac{1}{\max\{1,\lambda\}}\underline{v}_\lambda,\,\,w:=\bar{v}-\tilde v. $$
Clearly, $w\le0$ in $B_{\tilde{R}}(0)$ and $w$ weakly satisfies \begin{align}\label{wq7}
(-\Delta)^s w+Vw\le 0,\quad x\in \mathbb{R}^N\backslash B_{\tilde{R}}(0). \end{align} Then by the same arguments as \eqref{23j}, \eqref{24j} and \eqref{25j}, we get $w_+\le0$ in $\mathbb{R}^N$, which completes the proof. \end{proof}
To prove Theorem \ref{thm1.2'}, we need to give the following decay properties for the nonlocal Choquard term. \begin{lemma}\label{l46}It holds that \begin{equation}\label{qg8}
I_\alpha*w^p_\mu\ge\frac{C}{|x|^{N-\alpha}}+\frac{C}{|x|^{\mu p-\alpha}},\quad |x|\ge2, \end{equation} where $C>0$ is a constant depending only on $N$, $\alpha$, $\mu$ and $p$. \end{lemma}
\begin{proof}Let $|x|\ge2$. \begin{align}\label{wh0}
(I_\alpha*w^p_\mu)(x)\ge& \int_{B_{|x|/2}(x)}\frac{C}{|x-y|^{N-\alpha}|y|^{\mu p}}dy+\int_{B_{|x|/2}(0)}\frac{1}{|x-y|^{N-\alpha}(1+|x|^2)^{\frac{\mu p}{2}}}dy\nonumber\\
&+\int_{B^c_{2|x|}(0)}\frac{C}{|x-y|^{N-\alpha}|y|^{\mu p}}dy\nonumber\\
\ge&\frac{C}{|x|^{\mu p}}\int_{B_{|x|/2}(x)}\frac{C}{|x-y|^{N-\alpha}}dy+\frac{C}{|x|^{N-\alpha}}\int_{B_{|x|/2}(0)}\frac{1}{(1+|x|^2)^{\frac{\mu p}{2}}}dy\nonumber\\
&+\int_{B^c_{2|x|}(0)}\frac{C}{|y|^{N-\alpha+\mu p}}dy\nonumber\\
\ge&\frac{C}{|x|^{\mu p-\alpha}}+\frac{C}{|x|^{N-\alpha}}+\int_{B^c_{2|x|}(0)}\frac{C}{|y|^{N-\alpha+\mu p}}dy. \end{align} Note that
\begin{equation*}\int_{B^c_{2|x|}(0)}\frac{C}{|y|^{N-\alpha+\mu p}}dy=\left\{
\begin{aligned}
\frac{C}{|x|^{\mu p-\alpha}}&,\quad \mu p>\alpha,\\
+\infty&,\quad \mu p\le \alpha,
\end{aligned}\right. \end{equation*}
The the conclusion follows immediately by \eqref{wh0}. \end{proof}
Now we are going to prove Theorem \ref{thm1.2'}.
Without of loss generality, we may assume $\varepsilon =1$. It suffices to consider the following equation \begin{align}\label{wqs}
(-\Delta)^su+V(x)u=(I_\alpha*|u|^{p})|u|^{p-2}u, \quad x\in\mathbb{R}^N. \end{align}
\begin{proof}[{\bf Proof of Theorem \ref{thm1.2'}}]
Assume that $p\in (1,1+\frac{s+\frac{\alpha}{2}}{N-2s})\cup[2, 1+\frac{\alpha}{N-2s})$ and $\limsup_{|x|\to\infty}(1+|x|^{2s})V(x)=0$. Then for given $\epsilon>0$, $V(x)\le \frac{\epsilon}{1+|x|^{2s}}$ in $\mathbb{R}^N\backslash B_{R_\epsilon}(0)$ for some $R_\epsilon>0$. Afterwards, $\epsilon>0$ can be taken smaller if necessary.
Suppose by contradiction that $u\in H^s_{V,1}(\mathbb{R}^N)\cap C(\mathbb{R}^N)$ is a nonnegative nontrivial weak solution to \eqref{wqs}. There holds \begin{align}\label{wq81}
\int_{\mathbb{R}^N}(I_\alpha*u^p)u^p=[u]_s^2+\int_{\mathbb{R}^N}Vu^2<\infty. \end{align} Moreover, by Lemma \ref{w61}, $u>0$ in $\mathbb{R}^N$.
Let $\mu_1\in (N-2s,N)$ be a parameter. By Propositions \ref{tb} and \ref{cc5}, $w_{\mu_1}$ weakly satisfies \begin{align}\label{qy9}
(-\Delta)^s w_{\mu_1}+V(x)w_{\mu_1}\le-\frac{C_{\mu_1}}{|x|^{\mu_1+2s}}+\frac{\epsilon}{|x|^{\mu_1+2s}}\le0,\quad x\in \mathbb{R}^N\backslash B_{R_1}(0) \end{align} for some $R_1>0$. It follows by \eqref{qy9} and Lemma \ref{ws8} that \begin{align}\label{24f}
u\ge C_1w_{\mu_1} \end{align} for a constant $C_1>0$.
Now we divide the proof into the following two cases.
{\bf Case 1: $1< p<1+\frac{s+\frac{\alpha}{2}}{N-2s}$}.
By Lemma \ref{l46}, we have \begin{align}\label{pq9}
I_\alpha*w_\mu^p\ge \frac{C}{|x|^{\mu p-\alpha}},\quad |x|\ge2. \end{align}
Choose $\mu_2\in (\frac{N-2s}{2},N-2s)$ and $\mu_1\in (N-2s,N)$ such that \begin{equation}\label{w60}
N>\mu_2+2s>\mu_1(2p-1)-\alpha. \end{equation} From \eqref{wqs}, \eqref{24f} and \eqref{pq9}, we get \begin{align*}
(-\Delta)^su+Vu\ge\frac{C}{|x|^{\mu_1(2p-1)-\alpha}},\quad |x|\ge2. \end{align*} In addition, Proposition \ref{tb}, Proposition \ref{cc5} and \eqref{w60} indicate that $w_{\mu_2}$ weakly satisfies \begin{align}\label{ro9}
(-\Delta)^s w_{\mu_2}+V(x)w_{\mu_2}\le \frac{C_{\mu_2}}{|x|^{\mu_2+2s}}+\frac{\epsilon}{|x|^{\mu_2+2s}}\le \frac{C}{|x|^{\mu_1(2p-1)-\alpha}},\quad x\in \mathbb{R}^N\backslash B_{R_2}(0) \end{align} for some $R_2>0$. As a consequence of Lemma \ref{ws8}, there exists $C_2>0$ such that \begin{align*}
u\ge C_2w_{\mu_2}. \end{align*} It follows from \eqref{pq9} that \begin{align*}
(-\Delta)^su+Vu\ge\frac{C}{|x|^{\mu_2(2p-1)-\alpha}},\quad |x|\ge2. \end{align*} Set $\mu_{i+1}:=\mu_i(2p-1)-\alpha-2s$, $i\ge 2$, i.e., $$\mu_{i}=(2p-1)^{i-2}\Big(\mu_{2}-\frac{\alpha+2s}{2p-2}\Big)+\frac{\alpha+2s}{2p-2},\quad i\ge2.$$ Due to $2p-1>1$ and $\mu_2<N-2s<\frac{\alpha+2s}{2p-2}$, it follows that $\mu_{i+1}<\mu_{i}<N-2s$ for $i\ge2$ and $\mu_i\to-\infty$ as $i\to\infty$.
Fix $i\ge2$ such that $\mu_i>\frac{N-2s}{2}, \mu_{i+1}>\frac{N-2s}{2}$. We claim that there exists constants $C_{i},C_{i+1}>0$ such that \begin{align}\label{d0n}
u\ge C_{i+1}w_{\mu_{i+1}}\ \mathrm{if}\ u\ge C_{i}w_{\mu_{i}}. \end{align} In fact, if $u\ge C_iw_{\mu_i}$, then by \eqref{pq9}, \begin{align*}
(-\Delta)^s u+V(x)u\ge \frac{C}{|x|^{\mu_{i}(2p-1)-\alpha}},\quad x\in \mathbb{R}^N\backslash B_1(0). \end{align*} On the other hand, thanks to Proposition \ref{tb} and Proposition \ref{cc5}, $w_{\mu_{i+1}}$ weakly satisfies \begin{align*}
(-\Delta)^s w_{\mu_{i+1}}+V(x)w_{\mu_{i+1}}\le & \frac{C_{\mu_{i+1}}}{|x|^{\mu_{i+1}+2s}}+\frac{\epsilon}{|x|^{\mu_{i+1}+2s}}\\
\le & \frac{C}{|x|^{\mu_i(2p-1)-\alpha}},\quad x\in \mathbb{R}^N\backslash B_{R_i}(0) \end{align*} for some $R_i>0$. As a consequence of Lemma \ref{ws8}, the claim \eqref{d0n} holds immediately.
Therefore, for any $\mu>\frac{N-2s}{2}$, by finite iteration from \eqref{d0n}, we obtain \begin{align*}
u\ge d_\mu w_{\mu},\quad x\in\mathbb{R}^N \end{align*} for some constant $d_\mu>0$. Choosing $\mu>\frac{N-2s}{2}$ such that $2\mu p-\alpha<N$, we get \begin{align*}
\int_{\mathbb{R}^N}(I_\alpha*u^p)u^p\ge C\int_{\mathbb{R}^N\backslash B_1(0)}\frac{1}{|x|^{2\mu p-\alpha}}=+\infty, \end{align*} which contradicts to \eqref{wq81}.
{\bf Case 2: $2\le p<1+\frac{\alpha}{N-2s}$}.
Reviewing Lemma \ref{l46}, in this case, we will apply the following estimate instead of \eqref{pq9} in Case 1, \begin{align}\label{pq9'}
I_\alpha*w_\mu^p\ge \frac{C}{|x|^{N-\alpha}},\quad |x|\ge2. \end{align}
Since $2<1+\frac{\alpha}{N-2s}$, we have $\alpha>N-2s$. Pick $\mu_2\in (\frac{N-2s}{2},N-2s)$ and $\mu_1\in (N-2s,N)$ such that \begin{equation}\label{w6o}
N>\mu_2+2s>\mu_1(p-1)+N-\alpha. \end{equation} Through \eqref{wqs}, \eqref{24f} and \eqref{pq9'}, we get \begin{align*}
(-\Delta)^su+Vu\ge\frac{C}{|x|^{N-\alpha+\mu_1(p-1)}},\quad |x|\ge2. \end{align*} On the other hand, Proposition \ref{tb}, Proposition \ref{cc5} and \eqref{w6o} imply that $w_{\mu_2}$ weakly satisfies \begin{align*}
(-\Delta)^s w_{\mu_2}+V(x)w_{\mu_2}&\le \frac{C_{\mu_2}}{|x|^{\mu_2+2s}}+\frac{\epsilon}{|x|^{\mu_2+2s}}\\
&\le \frac{C}{|x|^{N-\alpha+\mu_1(p-1)}},\quad x\in \mathbb{R}^N\backslash B_{R_2}(0) \end{align*} for some $R_2>0$. Hence, by Lemma \ref{ws8}, there exists $C_2>0$ such that \begin{align*}
u\ge C_2w_{\mu_2}. \end{align*}
It follows from \eqref{pq9'} that \begin{align*}
(-\Delta)^su+Vu\ge\frac{C}{|x|^{N-\alpha+\mu_2(p-1)}},\quad |x|\ge2. \end{align*} Set $\mu_{i+1}:=\mu_i(p-1)+N-\alpha-2s$, $i\ge 2$, i.e., \begin{equation*}
\begin{aligned}
&\mu_{i}=\mu_2+(i-2)(N-\alpha-2s),\quad i\ge2,\quad&\ \mathrm{if}\ p=2;\\
&\mu_{i}=(p-1)^{i-2}\Big(\mu_2+\frac{N-\alpha-2s}{p-2}\Big)+\frac{N-\alpha-2s}{2-p},\quad i\ge2,\quad&\ \mathrm{if}\ p>2.
\end{aligned} \end{equation*} Since $\alpha>N-2s$ and $\mu_2+\frac{N-\alpha-2s}{p-2}<N-2s+\frac{N-\alpha-2s}{p-2}<0$ for $p<1+\frac{\alpha}{N-2s}$, it follows that $\mu_{i}<N-2s$ for $i\ge2$ and $\mu_i\to-\infty$ as $i\to\infty$.
By finite iterations similar to those in Case 1, for any $\mu>\frac{N-2s}{2}$, we can find a constant $d_\mu>0$ satisfying \begin{align*}
u\ge d_\mu w_{\mu},\quad x\in\mathbb{R}^N. \end{align*} Setting $\mu>\frac{N-2s}{2}$ such that $\mu p+N-\alpha<N$, we derive \begin{align*}
\int_{\mathbb{R}^N}(I_\alpha*u^p)u^p\ge C\int_{\mathbb{R}^N\backslash B_1(0)}\frac{1}{|x|^{\mu p+N-\alpha}}=+\infty, \end{align*} which contradicts to \eqref{wq81}.
As a result, we complete the proof of Theorem \ref{thm1.2'}. \end{proof}
\end{document} | arXiv |
Stacked generative adversarial networks for image compositing
Bing Yu ORCID: orcid.org/0000-0002-1697-60891,2,
Youdong Ding1,2,
Zhifeng Xie1,2 &
Dongjin Huang1,2
Perfect image compositing can harmonize the appearance between the foreground and background effectively so that the composite result looks seamless and natural. However, the traditional convolutional neural network (CNN)-based methods often fail to yield highly realistic composite results due to overdependence on scene parsing while ignoring the coherence of semantic and structural between foreground and background. In this paper, we propose a framework to solve this problem by training a stacked generative adversarial network with attention guidance, which can efficiently create a high-resolution, realistic-looking composite. To this end, we develop a diverse adversarial loss in addition to perceptual and guidance loss to train the proposed generative network. Moreover, we construct a multi-scenario dataset for high-resolution image compositing, which contains high-quality images with different styles and object masks. Experiments on the synthesized and real images demonstrate the efficiency and effectiveness of our network in producing seamless, natural, and realistic results. Ablation studies show that our proposed network can improve the visual performance of composite results compared with the application of existing methods.
Image compositing. is a fundamental technique in image editing that focuses on seamlessly integrating the foreground region of the source image into another target background. Ideally, a seamless composite result can trick humans into believing that it is not a fake image. However, as shown in Fig. 1a, some differences in appearance between the foreground and background, including illumination, lighting, white balance, and shading, severely reduce the fidelity of image composition. Therefore, to achieve highly realistic compositing, it is necessary to eliminate differences in appearance between the original foreground region and the target background as much as possible.
Comparison of compositing methods on real cut-and-paste image. a Cut-and-paste. b MVCC [1]. c RC [2]. d DIH [3]. e DPH [4]. f Our proposed network
Early techniques performed gradient-domain blending [1, 5] or alpha matting [6] operations to refine the foreground region for seamless compositing. However, as shown in Fig. 1b, they ignored some essential consistency constraints; thus, their composite results often appear unrealistic. Subsequently, some harmonization methods [7, 8] attempted to yield seamless and realistic results by transferring the visual appearance, texture, and even noise patterns. between images before gradient-domain compositing [5]. Unfortunately, they did not take into account global semantic and structure information and produced unrealistic composite results when the foreground region and target background were very different.
As a powerful learning method, the deep neural network has been successfully applied to various fields of image processing, including image compositing. However, traditional convolutional neural network (CNN)-based methods [2–4, 9] are still tentative and imperfect for high-fidelity compositing. As shown in Fig. 1c, the realism CNN method [2] generates image composite results with unsatisfactory appearance through simple color parameter optimization. Deep image harmonization [3] was subsequently able to capture both the context and semantic information from images through a joint training scheme in which the scene parsing decoders can control semantic organization and generate sharp content efficiently in the process of compositing. However, if scene understanding fails, this method cannot produce a realistic composite result. As shown in Fig. 1d, due to some semantic errors, the composite effects of the deep image harmonization method are not sufficiently harmonized between the foreground and background. In addition, as shown in Fig. 1e, the recent deep painterly harmonization method [4] does not seem to work well for the adjustment of the appearance of nature images.
Recently, several generative adversarial networks (GANs) [10, 11] have been introduced to achieve image compositing. Although these GAN models have the ability to harmonize composite images, they cannot solve all compositing issues, including appearance artifacts and high resolution. Especially for high-resolution compositing, the GAN models that take context encoders as the generative network can only output composite results with a low resolution of 64×64. Thus, they cannot directly generate high-resolution composites, and gradient-based optimization is needed as a post-processing step to create high-resolution images.
In this paper, we propose a stacked generative adversarial network that can create realistic-looking image composites through an end-to-end network. As shown in Fig. 2, our model includes two generators, three discriminators, and multiple loss terms. The inputs to this network are a cut-and-paste composite image and its corresponding mask, and the output is a harmonized high-resolution composite result. Our new model can construct stacked generators and discriminators to harmonize the composite image and determine whether a composite image looks realistic and natural. The generators are essential components for training and testing, while the discriminators are auxiliary components used only for training. Furthermore, after building a multi-scenario high-resolution dataset, our new network can achieve stable training and faster convergence solving in three steps: (1) train generator G1 and all discriminators; (2) fix the parameters of generator G1, and then train generator G2; and (3) jointly fine-tune the whole network.
The overview of the proposed stacked generative adversarial network. It consists of two generators and three discriminators: the output feature map of G1 is concatenated with the input image and serves as the input of G2. The discriminators D1 and D2 have an identical network architecture but operate at different image scales. D2 and D3 operate at the same image scales but have distinct architecture
Briefly, to reduce appearance differences between the foreground and background, our end-to-end network can fully consider the texture and structure information of the background while effectively preserving the semantic consistency of the composite result. As shown in Fig. 1f, some appearance artifacts (e.g., illumination, contrast, noise, and texture) can be effectively eliminated by our new model; thus, the composite result is seamless and realistic. This paper makes four main contributions, summarized as follows:
We propose a novel stacked generative adversarial network for high-resolution image compositing. It explores the cascade attention guidance generation strategy and aims to achieve a realistic-looking composite result. Unlike the state-of-the-art GAN-based methods, our network generates a harmonized image in an end-to-end manner.
We introduce the shift-connection layer [12] to the image compositing task. The layer can utilize long-range and multilevel dependencies across different features to guide generation, improving the structure and texture consistency of the composite image. By doing so, we can take into account the advantages of learning-based and exemplar-based methods and obtain a more realistic composite result compared with the state-of-the-art methods.
We propose a specialized discriminator for high-resolution image compositing that can employ diverse adversarial strategies at different scales to strengthen the ability of detail discrimination.
We build a multi-scenario dataset for high-resolution image compositing that mainly contains indoor and outdoor scenes with different styles. To our knowledge, this is the first high-resolution publicly available dataset for image compositing.
The organization of this paper is as follows. Section 2 briefly reviews the existing relevant works. Section 3 describes the proposed network and implementation details. Section 4 verifies the proposed method through a number of comparisons and describes ablation studies through experiments. Section 5 briefly summarizes this work and discusses possible future work.
In this section, we briefly introduce three subdomains, namely, image compositing, learning-based image editing, and image synthesis using GANs, with particular attention to related works.
Image compositing
Gradient-domain compositing [1, 5] can adjust the foreground region and the background region to be consistent in terms of illumination by blending the transition region between them. To make the composite image look more realistic, Sunkavalli et al. [7] proposed transferring the appearance of the target image to the source image before blending them. Darabi et al. [8] proposed combining Poisson blending [5] with patch-based synthesis in a unified framework (image melding) to produce a realistic composite. To avoid inconsistent colors and sacrificing texture sharpness, Darabi et al.'s work introduced an extra gradual transition operation between the foreground and background. Xue et al. [13] proposed using statistics and machine learning to study the realism of composites. Recently, deep neural networks have further improved image realism by learning context and semantic information. Zhu et al. [2] proposed a CNN-based model to distinguish composite images from realistic photographs. Tsai et al. [9] proposed using a scene parsing deep network to replace the sky background in a given photograph. These authors further proposed an end-to-end CNN method [3] for image appearance harmonization that could automatically learn both the context and semantic information of the input image and could be trained for both compositing and scene parsing tasks. Wu et al. [11] proposed a two-step method for high-resolution image compositing by combining Wasserstein GAN with multiscale gradient-based methods. Tan et al. [14] proposed a model that learns to predict foreground objects from source images before dealing with appearance compatibility. In contrast to the abovementioned methods, our GAN-based model can take into account the advantages of both exemplar-based and learning-based methods for high-fidelity image compositing.
Learning-based image editing
Many researchers have leveraged deep learning for image editing with the goal of modifying an image using given image pairs as training data. Zhang et al. [15] proposed a CNN-based image colorization method in which a color recommender system was used to help users interactively use the trained model to translate a gray image to a image. Wang et al. [16] proposed a learning-based image super-resolution method that uses an improved deep CNN to reconstruct a high-resolution image from a given low-resolution image. A deep reinforcement learning-based image enhancement method was proposed by Park et al. [17] that used the MIT-Adobe FiveK dataset [18] to model the stepwise nature of the human retouching process. Yan et al. [12] introduced a novel image inpainting model that uses attention-guided U-net [19] as the generator that fills in marked missing regions with suitable structure and texture. Our method shares a similar concept with learning-based methods and incorporates the advantages of multiple editing models to propose a novel trainable GAN architecture for image compositing.
Image synthesis using GANs
While GANs [20] can generate photorealistic images from random noise, the generated results might not be in accordance with the user's requirements. It is worth emphasizing some recent works on deep image synthesis using GANs. Conditional GANs [21, 22] are new models that generate images based on particular inputs other than simple noise, thus providing user-controllable results. Isola et al. [23] proposed a pix2pix method that explores conditional GANs to translate semantic label maps into photorealistic images. To solve the pix2pix model's unstable performance during adversarial training for high-resolution synthesis tasks, Wang et al. [24] synthesized 2048×1024 resolution realistic-looking photos through a robust training objective together with coarse-to-fine generators and multiscale discriminators. Recently, Xian et al. [25] introduced local texture loss to train a generative adversarial network that can take the texture patches and sketches as inputs and output a shoe or bag. Our method is inspired by the above successful work and is within the framework of image-to-image translation GANs. With our adversarial training objective as well as stacked generators and diverse discriminators, we can not only realize automatic image compositing but also achieve better results compared to existing methods.
Proposed method
In this section, we first introduce the attention-guided cascaded generative network and multiple losses. We then describe the training scheme that jointly fine-tunes all the networks together after two separate training processes. Finally, we introduce the multi-scenario synthesized dataset collection method.
Stacked generators
Given a source image ysrc and a target image ytrg, the cut-and-paste composite image y can be given as follows:
$$ y = y_{src}\odot M + y_{trg}\odot (1 - M) $$
where ⊙ is element-wise multiplication. M is a binary mask corresponding to the foreground region with a value of 1 and 0 for the background region. Our goal is to generate a natural-looking composite result \(\hat {y}\) in which the contents are the same as the cut-and-paste input but the appearance is more natural and realistic.
Similar to the pix2pix network [23], our generator is based on the U-net architecture and leverages the property of skip connections between each layer of the encoder and those of the corresponding layer of the decoder. This architecture maintains the texture and details of the image that are lost during the compression process in the encoder, which is important for image compositing [3] and other image editing tasks [26, 27]. Given a U-net of n layers, we denote Φl(y) as the encoder feature of the lth layer and Φn−l(y) as the decoder feature of the (n−l)th layer. In addition, we denote Ψl(M) as a binary mask corresponding to the foreground region in both the encoder feature Φl(y) and the decoder feature Φn−l(y). Ψl(M) is computed by an extra network that has the same architecture as the U-net encoder but with a network width of 1.
The pix2pix framework is designed to generate low-resolution images if applied directly to 512×512 resolution image synthesis. We find that the training is unstable and the generated results are unsatisfactory. Since stacked networks can be competent for high-resolution image synthesis because of their progressive refinement capability [26, 28, 29], we introduce this concept to our compositing task. Our network consists of two generators in which the second one is stacked upon the first. We call the first generator G1 and the second generator G2. Given a cut-and-paste image y, generator G1 is trained to produce a first feature map G1(y). Then, G1(y) is concatenated with the original image y and serves as the input for the second generator G2. The generator is given by the tuple G={G1,G2}, as showed in Fig. 2. The detailed architecture of the stacked generators is listed in Table 1.
Table 1 The architecture of G1/G2 network. "IN" represents InstanceNorm, "LReLU" represents Leaky ReLU activation, "Conv."/"DeConv." denotes convolutional/transposed convolutional layer with kernel size of 4, "st" means stride, "Concat" explains the skip connections, "Guidance" means guidance loss operation, and "Shift" means shift-connection operation. The different layers of G1 and G2 are listed separately
Attention guidance compositing
As a state-of-art appearance compositing method, the deep image harmonization method [3] adjusts the masked parts conditioned on their surroundings. However, we have found that this method can produce a distorted appearance or structural inconsistency between the foreground and background when the appearance of particular scenes is improperly remembered due to the limitation of training samples. In contrast, as a traditional compositing method, the image melding method [8] uses exemplar-based synthesis to smoothly transform from the source region to the target region, which avoids obviously inconsistent appearance. This suggests that matching by patches might lead to a more harmonious result. Motivated by these observations, our network takes into account the advantages of learning-based and exemplar-based methods for image compositing. We introduce the shift-connection attention layer [12] in our generators, which can guide the generator to obtain global semantic and structural information, improving the structure and texture consistency of the result.
Formally, let Ω be the foreground region and \(\overline {\Omega } \) be the background region. For each (Φn−l(y))p with location p∈Ω, its nearest neighbor searching in (Φl(y))q (location \(q \in \overline {\Omega }\)) can be independently defined as [12]:
$$ q^{*}(p)= \text{arg} \mathop{\text{max}}\limits_{q \in \overline{\Omega}}\frac{\left < (\Phi_{n-l}(y))_{p}, (\Phi_{l}(y))_{q}\right>}{\left\| (\Phi_{n-l}(y))_{p} \right\|_{2} \left\|(\Phi_{l}(y))_{q}\right\|_{2}} $$
and the shift vector is obtained by [12]:
$$ u_{p} = q^{*}(p)-p $$
Then, we spatially rearrange the encoder feature (Φl(y))q according to the shift vector to obtain a new estimate [12]:
$$ (\Phi^{\text{shift}}_{n-l}(y))_{p} = (\Phi_{l}(y))_{p+u_{p}} $$
The shift-connection layer takes Φl(y), Φn−l(y), and Ψl(M) as inputs and outputs a new shift-connection feature \(\Phi ^{\text {shift}}_{n-l}(y)\). The layer is embedded in the decoders of both G1 and G2 to guide generation. On the one hand, the layer can thus use the information from the background region of the feature to generate new appearances in the foreground region. On the other hand, the layer also helps to model global dependencies across generated regions, ensuring that the details at each location are carefully coordinated with the details at a distance.
Training losses
The choice of GAN discriminator is especially important for learning-based high-resolution image editing tasks. To obtain realistic-looking generated results, multiple discriminators at different image scales [24] or different image patches [30] have been proposed. Considering that the shape and size of the foreground region in the cut-and-paste image are arbitrary and the resolution of the generating task is high, our compositing network constructs three diverse PatchGAN discriminators [22, 23]. The discriminators receive the generated composite or the ground truth at different scales and attempt to classify the content as either "real" or ''fake." We denote the discriminators as D1, D2, and D3. The discriminator is given by the tuple D={D1,D2,D3}, as shown in Fig. 2. Specifically, the generated and real high-resolution images are downsampled by a factor of 2 to obtain image pyramids of 2 scales. Then, D1 is trained to differentiate real and generated images at the finest scale, and D2 and D3 are both trained to differentiate images at the coarsest scale. The detailed architecture of the discriminators is presented in Table 2. The discriminators D1 and D2 have identical network architectures, while D3 differs from them. With the discriminators, our adversarial loss is defined as:
$$ \mathcal{L}_{\text{adv}} = \underset{G}{\text{min}}\underset{D_{1},D_{2},D_{3}}{\text{max}}\sum_{k=1,2,3}\mathcal{L}_{GAN}(G,D_{k}) $$
Table 2 The architecture of D1/D2/D3 network. Annotations are the same as Table 1. The different layers of D1, D2, and D3 are listed separately
where k is the number of discriminators. The objective function \(\mathcal {L}_{GAN}(G,D_{k})\) is given by:
$$ \mathcal{L}_{GAN}(G,D_{k})=E_{x_{k} \sim p_{\text{data}}(x_{k})}[logD_{k}(x_{k})]+E_{y_{k} \sim p_{\text{data}}(y_{k})}[log(1-D_{k}(G(y_{k})))] $$
where yk is a cut-and-paste image and xk is the corresponding ground truth image. Specifically, y1 and x1 correspond to the finest scale, and y2, y3 and x2, x3 correspond to the coarsest scale. \(E_{x_{k} \sim p_{\text {data}}(x_{k})}\) represents the mathematical expectation of logDk(xk), where xk follows the probability distribution pdata(xk). \(E_{y_{k} \sim p_{\text {data}}(y_{k})}\) represents the mathematical expectation of log(1−Dk(G(yk))), where yk follows the probability distribution pdata(yk).
Recent GAN methods [25, 27] have found it effective to combine the adversarial loss with other additional multiple loss terms. First, we choose to use the traditional L2 pixel loss to stabilize the training. It is defined as the mean squared error (MSE) between a generated image and its reference image:
$$ \mathcal{L}_{L2} = \left\|G(y)-x\right\|^{2}_{2} $$
where G(y) is the output of a given cut-and-paste composite using a generator and x is the corresponding ground truth.
Next, we further include the perceptual loss term, which is used in various editing tasks, such as image inpainting [27] and image super-resolution [31]. Given a cut-and-paste input, we would like the composite result to look realistic and the foreground and background regions to be compatible. The features extracted from the middle layers of the pretrained very deep network represent high-level semantic perception. We defined the perceptual loss using the active layer of the pretrained VGG-19 [32] network on the ImageNet dataset [33]. The loss is defined as the MSE between the feature representations of a generated image and its ground truth:
$$ \mathcal{L}_{\text{per}} = \left\|\phi(G(y))-\phi(x)\right\|^{2}_{2} $$
where ϕ(·) is the activation map of the selected layer.
Our final loss term is used to encourage the compositing network to focus on the masked foreground region. We use the guidance loss on the decoder feature of U-net proposed by Yan et al. [12]. It is defined as the MSE between the masked feature representations:
$$ \mathcal{L}_{\text{gui}} = \sum_{j=1,2}\left\|(\Psi_{l}(M)\odot \Phi^{j}_{n-l}(y))-(\Psi_{l}(M)\odot \Phi^{j}_{l}(x))\right\|^{2}_{2} $$
where j is the generator number, \(\Phi ^{j}_{n-l}(y)\) is the decoder feature of cut-and-paste input on the (n−l)th layer for G1 or G2, and \(\Phi ^{j}_{l}(x)\) is the encoder feature of ground truth on the lth layer. Note that the guidance loss is only deployed to the decoder feature maps of the (n−3)th layer for G1 and G2 in our method.
Our combined loss is defined as the sum of all the above loss functions:
$$ \mathcal{L}_{\text{total}} = w_{\text{adv}}\mathcal{L}_{\text{adv}} + w_{L2}\mathcal{L}_{L2} + w_{\text{per}}\mathcal{L}_{\text{per}} + w_{\text{gui}}\mathcal{L}_{\text{gui}} $$
where wadv, wL2, wper, and wgui are the weight parameters for the adversarial, L2, perceptual, and guidance losses, respectively.
During the training, three discriminators are trained to distinguish the generated results from the ground truth, while the stacked compositing networks are trained to fake the discriminators. Since the high-resolution image compositing task itself is very challenging, we need to train the network carefully to make it converge. The training procedure is divided into three phases. First, generator G1 and discriminator D are trained for \(T_{G_{1}}\) epochs. Then, generator G1 is fixed, and generator G2 is trained from scratch jointly with discriminator D for \(T_{G_{2}}\) epochs. Finally, generator G1, generator G2, and discriminator D are trained jointly until the end of the training. An overview of the training procedure is shown in Algorithm 1.
In all experiments, we set the weight wadv=0.002, wL2=1, wper=0.01, and wgui=1. Our network is optimized using the Adam algorithm [34] with a learning rate of 0.0002. We train our models at an input resolution of 512×512, and the batch size is 1. Data augmentation, such as cropping, is also adopted during training.
Synthetic datasets
Data acquisition is the foundation of a successful training network. In our experiment, a masked image pair containing the cut-and-paste and composite result is required as the input and ground truth for the network. However, there are currently no public datasets for our task. To solve this problem, we selected two public datasets (MIT-Adobe FiveK [18] and Archive of Many Outdoor Scenes (AMOS) [35]) to create our multi-scenario training dataset for compositing through appearance editing. Two different processes are described in Fig. 3.
Data acquisition methods for our multi-scenario dataset. a MIT-Adobe FiveK. b AMOS
MIT-Adobe FiveK consists of 5000 raw images, each of which is paired with five retouched images using Adobe Lightroom by 5 trained photographers, A/B/C/D/E. The 6 editions of the same image have different styles. We randomly select one of the 6 versions of the image as the target image and then randomly select one of the remaining 5 versions as the source image. Therefore, there are 30 sets of 5000 target-source paired images (i.e., 150,000 paired images). To create more foreground objects or scenes from the source image, we manually annotate multiple object-level masks for each image in the dataset using the LabelMe annotation tool [36]. When generating input data, we first randomly select a mask and manually segment a region from the source image. Then, we crop this segmented region and overlay on the target image (i.e., ground truth image) to generate the cut-and-paste composite. We reserve 109 images (i.e., 3270 masked paired images) for testing, and the model is trained on the remaining 4891 (i.e., 146,730 paired images) images.
To cover richer object categories and scene styles, we use images from outdoor webcams, which contain images that are captured at the same location but change dramatically with lighting, weather, and season. We construct the compositing dataset using sequences from 92 webcams (the webcam numbers are the same as the famous Transient Attributes Database [37]) selected from AMOS by color transfer. First, given a target image from the camera sequence, we pick 20–30 other images of the same camera taking pictures at other times as transfer reference images. Second, instead of using the simple color and illumination histogram statistics method in Tsai et al. [3], we use a patch-based matching method [38] to transfer the appearance between two images with similar content. In this way, we produce 20–30 images of different styles from the given target image while maintaining the same content and scene. Third, for each camera sequence, we repeat the above steps to select 3–10 target images and produce multiple images of different styles. Fourth, all original targets and color transfer results are manually reviewed to ensure that there will be no artifacts or noise. Fifth, we obtain multiple object-level masks for each target image using the LabelMe tool. We use the original target image as the ground truth and crop a segmented foreground from its corresponding produced image in a different style to overlay on the original image. We reserve 1365 masked paired images from 7 webcams for testing and train the model on the remaining 21,658 paired images from another 85 webcams. To distinguish them from the original datasets, we call our compositing datasets FiveK and AMOS in the following experimental discussion.
In this section, we first describe the experimental setup. We then provide comparisons of synthesized images and real images with several metric methods, including user studies. Finally, we conduct five ablation studies on our network design.
Our model is implemented on PyTorch v0.3.1, CUDNN v7.0.5, and CUDA v9.0 and run on hardware with an NVIDIA TITAN X GPU (12GB). We separately train and test on our two synthesized datasets. Since the GAN loss curve does not reveal much information in training image-to-image translation GANs [23], we check whether the training has converged by observing L2 and perceptual loss curves. On the one hand, the L2 term can reflect how close the results are to ground truth images at the pixel level. On the other hand, the perceptual term can reflect the perceptual similarity between generated images and ground truth images. Figures 4 and 5 show the L2 and perceptual loss convergence curves of different training phases on the two datasets, respectively. For FiveK, we set \(T_{G_{1}}=6\) (880,380 iterations), \(T_{G_{2}}=1\) (146,730 iterations), and TG=3 (440,190 iterations). For AMOS, we set \(T_{G_{1}}=16\) (346,528 iterations), \(T_{G_{2}}=10\) (216,580 iterations), and TG=30 (649,740 iterations). For each dataset, the training takes approximately 3 weeks. Compositing a single cut-and-paste image of 512×512 takes less than 0.7 s.
Training convergence curves of L2 loss. a FiveK. b AMOS
Training convergence curves of perceptual loss. a FiveK. b AMOS
Comparison with existing methods
For synthesized images, we compare our results with MVCC [1], IM [8], DIH [3], and GP [11] at 512×512 resolution. For DIH [3] and GP [11], we use the pretrained models provided by the authors. Note that DIH [3] uses a combination of three public datasets, including MIT-Adobe FiveK, to train the model, and GP [11] uses the transient attributes database as the training dataset.
The images shown in Figs. 6 and 7 are taken from the FiveK and AMOS test datasets. Although the foreground appearances of the MVCC results are well blended using mean-value coordinates, some obvious artifacts can be found, as shown in Figs. 6c and 7c. IM results showed no significant improvement in visual appearance. DIH is effective in semantic compositing, and the visual appearance of the results shows better performance than MVCC and IM. However, the boundary between the foreground and background of the DIH results is not seamless enough, and there are obvious jagged edges. In addition, DIH models the dependencies between the scene semantics and its surface appearance, but these kinds of semantic priors do not always work well; for example, the yellow flower foreground in the composite of the three rows of Fig. 6e is adjusted to green, and the result is far from the ground truth (GT). GP adopts a multistage scheme to combine a deep network and Poisson blending, while its GAN model generates poor results at low resolution and leads to incorrect enlargement in the subsequent high-resolution optimization step, resulting in unrealistic images, as shown in Fig. 7e. Overall, the proposed method performs favorably in generating realistic, seamless, and harmonious images. The foreground appearance of our results is most consistent with the corresponding background.
Example results on synthesized FiveK dataset. a GT. b Cut-and-paste. c MVCC [1]. d IM [8]. e DIH [3]. f Our proposed network. Our composite results obtained the highest PSNR value scores
Example results on synthesized AMOS dataset. a GT. b Cut-and-paste. c MVCC [1]. d IM [8]. e GP [11]. f Our proposed network. Our composite results obtained the highest PSNR value scores
In addition, we use three quantitative criteria to evaluate the proposed and other methods. First, the peak signal-to-noise ratio (PSNR), which is used by Tsai et al. [3], can reflect how close the result is to GT. Second, the structural similarity index (SSIM) attempts to quantify the visibility of structural differences between the result and GT. Third, the learned perceptual image patch similarity (LPIPS) [39], which agrees surprisingly well with human judgment, is used to assess the perceptual similarity between two images. Note that unlike PSNR and SSIM, smaller values mean greater perceptual similarity for LPIPS. Tables 3 and 4 show the quantitative scores between GT and composite results for FiveK and AMOS, respectively. The scores are calculated based on the mean values of a random subset of 300 images selected from each of the two test datasets. Our proposed image compositing network performs better than other methods in terms of PSNR, SSIM, and LPIPS metrics.
Table 3 Comparisons of methods on the FiveK test dataset
Table 4 Comparisons of methods on the AMOS test dataset
For real images, we compare our results with MVCC [1], IM [8], RC [2], GP [11], and DIH [3]. To demonstrate that the models trained on our multi-scenario dataset can be generalized to real cut-and-paste composite images, we created a test set of 30 high-resolution real composite images and combined 50 public high-quality images collected by Xue et al. [13] and Tsai et al. [3], resulting in a real cut-and-paste composite set that contains 80 images. Since Xue et al.'s statistical method has no public code, our results are not compared with it.
Figure 8 shows some experimental comparisons selected from the real composite set. The MVCC and IM can solve some of the inconsistencies between two parts of inputs, but the results are not satisfactory. RC's realism prediction model is effective in handling easily distinguishable cut-and-paste composite input; nevertheless, it generates unsatisfactory results, especially the transition region between foreground and background (e.g., there are distinct jagged outlines at the boundary of the foreground in the results of the second, third, and fifth rows). For GP, it generates visually poor results. For DIH, because the model utilizes semantic information to adjust cut-and-paste input, it is limited by the training dataset. If the scene semantics are incorrectly judged, this will lead to unrealistic outputs (e.g., the fourth, sixth, and eighth rows). Compared with others, the proposed model can better predict the global structure and thus maintain the consistency of the context, resulting in realistic-looking composites.
Example results on real images. a Cut-and-paste. b Mask. c MVCC [1]. d IM [8]. e RC [2]. f GP [11]. g DIH [3]. h Our proposed network
Figure 9 illustrates one example where the same foreground (i.e., the zebra) is copied to different backgrounds (i.e., a street in dim light and a zebra herd in the sun). For RC, the discriminative model cannot correctly predict the degree of perceived visual realism of the given inputs, so the appearance of the foreground is almost never adjusted. For DIH, regardless of the scene, the context-aware encoder-decoder recovers the fur color constrained by the trained prior knowledge to almost invariable results. In contrast to the two methods mentioned above, with the proposed network, the foregrounds can be adjusted according to the surrounding scene and luminance.
Real example with same foreground and different backgrounds. a Cut-and-paste. b Partial enlarged details in a. c RC [2]. d Partial enlarged details in c. e DIH [3]. f Partial enlarged details in e. g Our proposed network. h Partial enlarged details in g
To better understand the performance of our methods, we conducted quantitative assessment studies with users, similar to Tsai et al. [9]. Participants were shown an input cut-and-paste composite and six results from MVCC, IM, RC, GP, DIH, and the proposed method. Each participant was asked to rate each group according to the realistic nature of the images using a 5-point Likert scale (1 for worst, 5 for best). We asked 20 users to provide feedback by giving users 30 tuples of images selected from our real cut-and-paste composite set. The average scores of individual images in the evaluation set are shown in Fig. 10. Most of our scores are above 3.0. Our scores outperform MVCC in 80%, IM in 80%, RC in 80%, GP in 100%, and DIH in 73%.
Average evaluation scores for each image, sorted by our score. The proposed method performs better than others in most cases
Ablation studies
The main differences between our compositing method and other methods are the stacked generative adversarial network architecture and the combined loss function. Thus, five groups of experiments in the FiveK dataset were conducted to analyze the effect of stacked generators, diverse discriminators, shift-connection operations, perceptual loss, and guidance loss on composite results. Table 3 shows that the proposed network achieved better scores in terms of PSNR, SSIM, and LPIPS metrics compared to the other five strategies.
To evaluate the effectiveness of stacked generators for high-resolution compositing, we trained our network without using generator G2. The number of training epochs was constrained to be the same as the original model. As shown in Fig. 11, the results generated by a single (non-stacked) generator may not be satisfactory and have obvious artifacts. In addition, the consistent improvement in the quantitative assessment scores of our models clearly demonstrates the benefits of the cascaded refinement approach.
Effect of stacked generators. a GT. b Cut-and-paste [PSNR = 26.19 db, LPIPS = 0.0750]. c Ours (w/o G2) [PSNR = 30.91 db, LPIPS = 0.0986]. d Ours [PSNR = 32.48 db, LPIPS = 0.0387]. e GT. f Cut-and-paste [PSNR = 19.00 db, LPIPS = 0.06326]. g Ours (w/o G2) [PSNR = 33.30 db, LPIPS = 0.0428]. h Ours [PSNR = 35.00 db, LPIPS = 0.0274]
To evaluate the effectiveness of our specialized discriminator for high-resolution compositing, we trained our network only with discriminator D1. Visually, as shown in Fig. 12, we observed that the model using the combination of three diverse PatchGAN discriminators could reduce artifacts and improve appearance in terms of realism.
Effect of diverse discriminators. a GT. b Cut-and-paste [PSNR = 23.06 db, LPIPS = 0.0669]. c Ours (w/ D1 only) [PSNR = 24.75 db, LPIPS = 0.0518]. d Ours [PSNR = 28.15 db, LPIPS = 0.0444]. e GT. f Cut-and-paste [PSNR = 20.27 db, LPIPS = 0.0873]. g Ours (w/ D1 only) [PSNR = 28.66 db, LPIPS = 0.0685]. h Ours [PSNR = 32.10 db, LPIPS = 0.0495]
We trained a model without using the shift-connection layer. As shown in Fig. 13, the operation helps to obtain representation for the foreground (i.e., the man or the flower) from the background region, resulting in composites with consistent regions.
Effect of shift-connection layer. a GT. b Cut-and-paste [PSNR = 27.17 db, LPIPS = 0.0592]. c Ours (w/o Shift) [PSNR = 29.59 db, LPIPS = 0.0548]. d Ours [PSNR = 33.82 db, LPIPS = 0.03167]. e GT. f Cut-and-paste [PSNR = 28.40 db, LPIPS = 0.0422]. g Ours (w/o Shift) [PSNR = 29.90 db, LPIPS = 0.0402]. h Ours [PSNR = 32.71 db, LPIPS = 0.0257]
We trained a model without perceptual loss. As shown in Fig. 14, the composites generated by the model without \(\mathcal {L}_{\text {per}}\) have ghosting. In addition, the significant advantage in LPIPS scores for the model with \(\mathcal {L}_{\text {per}}\) shows that the perceptual loss can greatly improve visual perception.
Effect of perceptual loss. a GT. b Cut-and-paste [PSNR = 33.82 db, LPIPS = 0.0196]. c Ours (w/o \(\mathcal {L}_{\text {per}}\)) [PSNR = 31.64 db, LPIPS = 0.0580]. d Ours [PSNR = 35.83 db, LPIPS = 0.0167]. e GT. f Cut-and-paste [PSNR = 31.66 db, LPIPS = 0.0100]. g Ours (w/o \(\mathcal {L}_{\text {per}}\)) [PSNR = 34.33 db, LPIPS = 0.0464]. h Ours [PSNR = 36.37db, LPIPS = 0.009]
We trained a model without guidance loss. As shown in Fig. 15, guidance loss is helpful in preserving better visual appearance. We observed that the color and luminance of foregrounds with \(\mathcal {L}_{\text {gui}}\) were closer to GT.
Effect of guidance loss. a GT. b Cut-and-paste [PSNR = 28.65 db, LPIPS = 0.0257]. c Ours (w/o \(\mathcal {L}_{\text {gui}}\)) [PSNR = 27.98 db, LPIPS = 0.0281]. d Ours [PSNR = 29.86 db, LPIPS = 0.0248]. e GT. f Cut-and-paste [PSNR = 17.12 db, LPIPS = 0.0921]. g Ours (w/o \(\mathcal {L}_{\text {gui}}\)) [PSNR = 21.72 db, LPIPS = 0.0424]. h Ours [PSNR = 33.19 db, LPIPS = 0.0247]
Our model trained on the proposed multi-scenario dataset can handle the composition of high-resolution real cut-and-paste images in most cases. However, if the input image is significantly different from the training data, it may still fail. Figure 16 shows two examples of our failure case, where the appearance of foregrounds and backgrounds is not sufficiently natural and harmonious.
Failure cases. a Cut-and-paste. b Ours. c Cut-and-paste. d Ours
In this paper, we proposed a stacked GAN method for high-resolution image compositing. Given a cut-and-paste composite, the proposed network can adjust the foreground appearance and output a harmonized image that looks realistic. We have shown that by using stacked generators, diverse discriminators, and multiple loss constraints, it is possible to train a good performance model. In addition, we demonstrated that our network can be implemented in three steps to achieve stable training and faster convergence. Our method utilizes a cascade attention guidance generation strategy and generates more harmonious and consistent results than state-of-the-art methods. Future studies will focus on improving the speed of high-resolution compositing of the proposed network and expanding the training dataset.
The datasets for high-resolution image compositing generated during the current study are available in the Baidu Cloud repository, https://pan.baidu.com/s/1WmJ5P7ToSeA9FS4vgmMfaA (download password: 1111).
AMOS:
Archive of many outdoor scenes
PSNR:
Peak signal to noise ratio
SSIM:
Structural similarity index
LPIPS:
Learned perceptual image patch similarity
MVCC:
Mean-value coordinates cloning
RC:
Realism CNN
DIH:
Deep image harmonization
Deep painterly harmonization
Image melding
GP:
Gaussian-Poisson
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The authors thank the editor and anonymous reviewers.
This work was supported by the National Natural Science Foundation of China (61303093, 61402278) and the Shanghai Natural Science Foundation (19ZR1419100).
Shanghai Film Academy, Shanghai University, Yanchang Road, Shanghai, 200072, China
Bing Yu, Youdong Ding, Zhifeng Xie & Dongjin Huang
Shanghai Engineering Research Center of Motion Picture Special Effects, Shanghai University, Yanchang Road, Shanghai, 200072, China
Bing Yu
Youdong Ding
Zhifeng Xie
Dongjin Huang
All authors take part in the discussion of the work described in this manuscript. YB wrote the first version of the manuscript. DY, XZ, and HD did part experiments of the paper. All authors read and approved the final manuscript.
Bing Yu received his Ph.D. degree in digital media technology from Shanghai University, Shanghai, China, in 2020. He received his BS degrees in computer science and technology from Zhengzhou University, Zhengzhou, China, in 2011, and his MS degree in computer science and technology from North Minzu University, Yinchuan, China, in 2015. He is now a lecturer with Shanghai University, Shanghai, China. His current research interests include deep learning and image processing.
Youdong Ding received his Ph.D. degree in mathematics from University of Science and Technology of China, Hefei, China, in 1997. He was a post-doctor at the Department of Mathematics of Fudan University, Shanghai, China, from 1997 to 1999. He is now a professor with Shanghai University, Shanghai, China. His research interests are computer graphics, image processing, and digital media technology.
Zhifeng Xie received his Ph.D. degree in computer application technology from Shanghai Jiao Tong University, Shanghai, China, in 2013. He was a research assistant at the Department of Computer Science, City University of Hong Kong, Hong Kong, China, in 2011. He is now an associate professor with Shanghai University, Shanghai, China. His research interests include image/video editing, computer graphics, and digital media technology.
Dongjin Huang received his Ph.D. degree in computer application technology from Shanghai University, Shanghai, China, in 2011. He was a post-doctor with University of Bradford, UK, from 2012 to 2013. He is now an assistant professor with Shanghai University, Shanghai, China. His research interests are augmented reality, computer vision, and computer graphics.
Correspondence to Bing Yu.
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Yu, B., Ding, Y., Xie, Z. et al. Stacked generative adversarial networks for image compositing. J Image Video Proc. 2021, 10 (2021). https://doi.org/10.1186/s13640-021-00550-w
DOI: https://doi.org/10.1186/s13640-021-00550-w
Generative adversarial networks
Deep neural network | CommonCrawl |
Analysis of statistical parametric maps
Conclusions and future research
Optimized statistical parametric mapping procedure for NIRS data contaminated by motion artifacts
Neurometric analysis of body schema extension
Satoshi Suzuki1Email author
Brain Informatics20174:70
https://doi.org/10.1007/s40708-017-0070-x
Accepted: 20 July 2017
This study investigated the spatial distribution of brain activity on body schema (BS) modification induced by natural body motion using two versions of a hand-tracing task. In Task 1, participants traced Japanese Hiragana characters using the right forefinger, requiring no BS expansion. In Task 2, participants performed the tracing task with a long stick, requiring BS expansion. Spatial distribution was analyzed using general linear model (GLM)-based statistical parametric mapping of near-infrared spectroscopy data contaminated with motion artifacts caused by the hand-tracing task. Three methods were utilized in series to counter the artifacts, and optimal conditions and modifications were investigated: a model-free method (Step 1), a convolution matrix method (Step 2), and a boxcar-function-based Gaussian convolution method (Step 3). The results revealed four methodological findings: (1) Deoxyhemoglobin was suitable for the GLM because both Akaike information criterion and the variance against the averaged hemodynamic response function were smaller than for other signals, (2) a high-pass filter with a cutoff frequency of .014 Hz was effective, (3) the hemodynamic response function computed from a Gaussian kernel function and its first- and second-derivative terms should be included in the GLM model, and (4) correction of non-autocorrelation and use of effective degrees of freedom were critical. Investigating z-maps computed according to these guidelines revealed that contiguous areas of BA7–BA40–BA21 in the right hemisphere became significantly activated (\(t(15); p<.001\), \(p<.01\), and \(p<.001\), respectively) during BS modification while performing the hand-tracing task.
Body schema
Near-infrared spectroscopy
General linear model
Statistical parametric mapping
Hand-tracing task
Motion artifacts
In the human brain, peri-personal space [1] is represented by embedding the spatial volume of external objects, such as a hat (clothing) or a stick (tool), into an internal body map [2]. In this process, people typically feel the object as an extension of their own body [3]. The mechanism underlying this sophisticated cognitive process is known as body schema (BS) modification [4]. This mechanism is considered a form of homuncular flexibility, involving constant changes to the shape of the homunculus, which is an approximate internal map of the human body in the cortex that is often visualized as a distorted human body [5]. The concept of the BS was initially proposed by Head and Holmes [6], who defined it as a postural model of the body that actively organizes and modifies the impressions produced by incoming sensory impulses [6]. The existence of the BS was confirmed in the 1990s by analyzing brain function in macaque monkeys [7]. The BS is considered to be vital for spatial cognitive function and is associated with various brain areas, including the sensorimotor cortex [8], Broca's area (BA44), the inferior parietal lobule (BA40) [9], the primary motor cortex (BA4) [10], and the mirror neuron system [11]. One experimental approach to examining the BS involves the induction of a "confused" brain state by presenting mismatching visual and haptic stimuli, as in the rubber hand illusion (RHI) [12, 13]. Similar variations, such as the visual–proprioceptive synchrony judgment task [14] and the visual–proprioceptive mismatch task [15], have also been examined. Other studies utilized motion illusions to examine the BS more directly. Motion illusions arise when somatic sensations are confused by physically vibrating the muscle spindle that provides axial and limb position information to the central nervous system [16]. Examples include illusory arm movement [17], the Pinocchio illusion [18], and the waist-shrinking illusion [19]. Electrical stimuli applied to the skin can also induce similar motion illusions [20].
In addition, the BS plays a significant role in the ability to drive a vehicle [21]. For example, a person's sense of car width is a form of BS modification [22]. For a skilled driver, the whole spatial volume of the vehicle body is perceived as an extension of the driver's peri-personal space [23]. Such spatial cognitive function is also involved in teleoperation systems that require the operator to manipulate a machine remotely [24]. In both driving a car and remote operation of a robot, the machine (car or robot) must be manipulated like one's own body. This sensation of body ownership is a type of BS modification [25, 26].
Additionally, the BS is heavily involved in some cognitive disorders [27]. Alice in Wonderland syndrome (involving distorted awareness of body size, mass, or its position in space), autotopagnosia (involving mislocalization of body parts and bodily sensations), and phantom sensation (awareness of an amputated limb) are examples of such disorders. Because the BS is related to such varied human functions, a quantitative method for evaluating the strength of BS modification may be useful both for rehabilitation of spatial cognition disorders and for the estimation of spatial cognitive skill during vehicle operation. Possible BS measurement methods such as functional magnetic resonance imaging (fMRI), positron emission tomography (PET), and magnetoencephalography (MEG) are, however, not adequate for evaluation, because they all require the participant's head to be fixed to a stationary measurement unit mounted on the floor. Natural BS modification is difficult to induce using such stationary measurement devices because participants cannot move their bodies freely. Near-infrared spectroscopy (NIRS) is an alternative measurement method in which the measurement unit can be attached to the participant's head while still permitting head movement. NIRS may thus have useful applications in daily life, and portable NIRS systems have recently been marketed commercially.
While NIRS may be an appropriate measurement technique for measuring brain activity during daily tasks, the relationship between NIRS activity and modifications to the BS is not currently understood. While a brain map of BS modification would be useful, no analysis procedure for constructing a map from NIRS data contaminated by motion artifacts has been established to date. Even mild motion, such as an arm movement, causes strong artifacts in NIRS data. As such, there are several experimental limitations involved in current NIRS methods: the need for participants to maintain a sitting posture, the restriction of movement to the right upper arm only, the inability to twist one's head, and the need to avoid conversation, all of which may induce cognition-related brain activity that contaminates NIRS data.
Statistical parametric mapping (SPM) has recently become a popular method for investigating the spatial distribution of brain activity [28] in studies using fMRI and PET. Several studies describing the application of SPM to NIRS data have been reported [14, 29–31]. According to the SPM procedure, characteristics of brain activity are identified statistically using a general linear model (GLM) [32] to evaluate the accuracy of fit of brain activity against a canonical response pattern of cerebral blood flow. Random effects are then analyzed using the accuracy of fit. Because of the effort expended in recent decades to develop the SPM software package [33], this approach has been established as a standard analysis method for fMRI and PET interpretation. SPM is thus becoming the de facto standard for examining common features in human brain activity. It is widely used for investigating brain functions that relate to a wide brain area.
However, unlike fMRI and PET, various adjustments of experimental design analysis are required in NIRS studies, because of the following issues:
Issue 1 :
To be analyzed with the GLM, signals must satisfy the assumption of normal distribution [32]; however, the actual responses of regional cerebral blood flow (rCBF) are not necessarily normally distributed.
It is difficult to satisfy the GLM assumption of non-autocorrelation of errors, since rCBF is time dependent [14].
It is challenging to distinguish meaningful low-frequency components in rCBF from true noise, such as drift and bias.
These issues have often been implicitly ignored in previous studies because the default parameters of the SPM software were applied without careful consideration [31].
In addition, motion artifacts strongly affect NIRS data when analyzing brain activity that accompanies body motion. As such, consideration of body motion is inevitable because natural body motion is required to combine the visual and haptic senses that are involved in BS. Importantly, ill-conditioned data arising from motion artifact contamination cannot satisfy the requirements of the GLM–SPM because well-conditioned data are implicitly required for comparison with the canonical waveform. For this reason, most previous studies of NIRS–SPM have utilized experimental paradigms that prohibit body motion (as in fMRI and MEG studies). As such, these methods cannot be directly used to analyze BS modification accompanying motion artifacts. Overall, the brain areas associated with BS modification induced by spontaneous body motion have not yet been comprehensively examined in humans, although these mechanisms have been identified in the monkey brain [7], and fragmentary human evidence has been reported [9, 11].
Consequently, existing studies of NIRS–SPM have been unable to use the unique benefit of the NIRS method, which permits movement of the head and body. Therefore, the current study sought to establish a practical NIRS–SPM procedure using a task that requires the control of hand and arm movements. The main aims of this study were as follows:
Creating guidelines for using NIRS–SPM to analyze rCBF accompanied by body motion artifacts.
Quantifying the spatial distribution of human brain activity during BS modification induced by natural and spontaneous body motion.
Concerning the first aim, several conditions and modifications were investigated using the following three steps: Step (1) a model-free method analyzing cerebral blood volume (CBV), Step (2) a convolution matrix method known as the orthodox GLM, and Step (3) a boxcar-function-based Gaussian convolution method.
2.1 Hand-tracing task
A hand-tracing task was devised to examine differences in brain activity related to BS modification. In this task, participants were instructed to trace the curve of Japanese Hiragana characters that were printed on paper (Task 1) or projected onto a screen (Task 2). In Task 1, participants used the right forefinger to trace the characters. In Task 2, participants used a 1.5-m stick held in the right hand to trace characters that were projected 2.0 m in front of them. Importantly, Task 1 entails the use of the BS of participants' own body only, whereas Task 2 involves an extension of the BS to the tip of the long stick. During both tasks, participants sat on a chair, and the sitting position was adjusted so that the participant could touch the characters with the tip of the finger or stick. The size of projected characters was enlarged in proportion to the distance to the screen to keep the perturbation of hand motion similar in both tasks. To avoid inducing unnecessary brain activation from environmental light and sound, participants performed the tasks while wearing noise-canceling headphones inside a tent covered with a curtain. Thirty-second rests were given after each task, as shown in Fig. 1a. The investigator touched the shoulder of the participant to signal the start and end of each task.
a Experimental time sequence, b design matrix \({\mathbf {X}}\) for the GLM: b shows the elements in the design matrix \({\mathbf {X}}\) in black (\({\hbox {value}}=1\)) and white (\({\hbox {value}}=0\)) in the gray image. Refer to Sect. 3.2 for details of the design matrix
2.2 NIRS measurement
Previous evidence suggests that performing a hand-tracing task would be likely to involve activation in the primary motor area (M1), the primary somatosensory area (S1), and the premotor area (PM) [34–37], because the action involved in hand tracing requires the control of hand and arm movements. The activation of these brain areas alone, however, does not provide sufficient evidence to identify BS modification. Hence, the present study also examined the inferior parietal lobule (BA40). This brain region is not thought to directly relate to hand motion, but is one of the cortical areas associated with BS [9]. Therefore, in the current experiment, areas around BA40 were monitored using ETG-4000 (Hitachi Medico, Tokyo, Japan) using two \(3 \times 3\) holders, as shown in Fig. 2. A total of 24 data channels were measured. The NIRS probes were attached to participants' heads using the international 10–20 system so that C3(4) was located at the forefront of the upper array of the holder, and the second vertical line of the holder was perpendicular to the nasion–inion line, as shown in Fig. 2a. Thus, C3(C4) and T3(T4) on the left (right) hemispheres corresponded to Ch.17(Ch.3) and Ch.15(Ch.5), respectively.
a Locations of probes, b channel layout
The content and procedure of the experiment were approved by the Tokyo Denki University Human Bioethics Review Committee, and experiments were conducted after explaining the experiment to the participants in full and receiving written consent. Sixteen healthy university students (20–23 years old) participated (\(N=16\)) in this experiment.
3 Analyses
To determine the optimal conditions for NIRS–SPM analysis dealing with rCBF contaminated by motion artifacts, Steps 1–3 were applied to the rCBF data, in sequence. Step 1 examined the rCBF waveform to determine a hemodynamic response function (HRF) candidate and tentatively select the NIRS hemoglobin type for the GLM analysis. In Step 2, a low-frequency noise that causes adverse effects on fitting time-sequential rCBF data to the GLM was eliminated, and the degrees of freedom of the SPM computation were modified in order to obtain correct statistical results. In Step 3, an adequate canonical model in GLM was found, to enhance the accuracy of fit, and an optimal condition for the NIRS–SPM was derived after autocorrelation modification and the final choice of hemoglobin type were identified. These steps fundamentally adhere to the following basic stages of a general SPM approach [28]:
First-level analysis :
A statistical test ascertains whether the rCBF shows significantly different responses according to the task condition with respect to each measurement channel (one-sample t test).
Second-level analysis :
After statistics obtained in the first-level analysis are converted into z-values, an average of the population to which the z-values of all participants belong is tested (random-effects analysis) [38].
The details of these steps and the analysis results are explained below, in sequence.
3.1 Step 1: Model-free method
The increase of total hemoglobin (Hb) has often been used to investigate brain activity in previous studies. However, other wave patterns such as "both oxy-Hb and total-Hb decrease" and "oxy-Hb, deoxy-Hb, and total-Hb increase" have also been used [39]. In the case of young adults, total-Hb concentration tends to reflect both oxy and deoxy changes, since the increase in oxy-Hb tends to be larger than the decrease in deoxy-Hb [39].
It is, however, difficult to select a consistent pattern because these Hb-based parameters also differ according to the age of the participant, the purpose of the research, measurement conditions, and the researchers' preferences. In practice, oxy-Hb is considered by some researchers to be adequate for detecting local brain activity [14, 40, 41], while other studies [29, 42, 43] support the use of deoxy-Hb. Therefore, the first step in this study was to investigate the most appropriate type of hemoglobin using a model-free approach unrelated to the HRF.
Close examination of rCBF responses revealed that regional cerebral blood volume (rCBV) in several channels decreased at the beginning of Task 1 and increased for several seconds at the beginning of Task 2. Based on this observation, a null hypothesis of no difference was tested using a paired t test against two values of rCBV, referred to as \(S_{1 \cdot \tau }\) and \(S_{2 \cdot \tau }\). These values were computed by integrating the rCBF data for \(\tau\) seconds from the beginning of each task.
$$\begin{aligned} S_{l \cdot \tau }= \sum _{i=0}^{\tau /\Delta } (y_l(i)-b_l) / (\tau /\Delta ) \ \ (l=1,2) \end{aligned}$$
$$\begin{aligned} b_l&:= \sum _{i=-\lfloor 5/\Delta \rfloor }^{0} y_l(i) / \left( \lfloor 5/\Delta \rfloor \right) , \end{aligned}$$
where \(y_l(i)\) is the rCBF data at the sampling count i on Task l from the beginning of the task, the sampling interval \(\Delta\) is .1 s, \(b_l\) is a bias computed by averaging 5 s of data just before the beginning of task, and an operator \(\lfloor * \rfloor\) is a floor function. The z-values converted from statistics computed in this paired t test are shown in Table 1. The z-values were computed for each channel using all participants' data (\(N=16\)), and other results obtained using different \(\tau\) values are summarized in the same table. This table shows an existence of significant differences in channels 4–10, 12, and 15 for all types of Hb. This result demonstrates that Tasks 1 and 2 induced significantly different brain activity responses.
Results of statistical tests using a model-free method in Step 1: z-values obtained using a paired t test, \(df=15\)
Blue cells show \(z>2.33\, (p<.01)\), and red cells show \(z>3.09\, (p<.001)\). (Color table online)
The current method was a relatively simple process, and the results shown in Table 1 may possess lower reliability because the parameters [an integral interval \(\tau\) in Eq. (1) and a duration to compute the bias \(b_l\) in Eq. (2)] were determined subjectively without a theoretical guarantee of optimality, despite consideration of the task condition and the actual NIRS response data. This method, however, contributes to an improved understanding of a tendency in a wide area of common brain activity from multi-channel measurement data. Thus, we assumed the results shown in Table 1 were provisional, and used them to judge the validity of the assumption of the existence of a certain HRF wave pattern for the subsequent Steps 2 and 3, which were processed from objective criteria. Because the data in this table indicate that the channels responding to the difference between Tasks 1 and 2 were ch. 4–10, 12, and 15, a canonical waveform of rCBF was computed using the rCBF data measured in these channels for the HRF. That is, we averaged all rCBF data measured in these channels during each task period into one waveform for each task. The results of the canonical forms are shown in Fig. 3. All graphs in the figure show rounded trapezoidal waveforms similar to the convolved response of a boxcar and a Gaussian function. The results revealed that rCBF during Task 1 decreased [see graphs (a1) (b1)] while rCBF during Task 2 increased [see graph (a2) (b2)] for both oxy- and deoxy-Hb. These findings indicated that a waveform similar to that used in previous studies, involving the convolved response of a boxcar and a Gaussian function, was appropriate for the current analysis. In addition, the averaged rCBF waveform in all cases demonstrated substantial variance because of motion artifacts, as indicated by the long error bars in each graph. Notably, the variance in deoxy-Hb was half that of oxy-Hb [as indicated by the difference in the length of the error bars between graph (a1) and graph (b1), and between graph (a2) and graph (b2)]. This result indicates that deoxy-Hb was less affected by individual differences than oxy-Hb.
Taken together, these findings indicate that deoxy-Hb may be better suited for GLM analysis using the convolved response of a boxcar and a Gaussian function for analysis of rCBF affected by motion artifacts. Therefore, the deoxy-Hb NIRS data were mainly analyzed provisionally in the following analyses. The validity of the selection of deoxy-Hb was judged in Steps 2 and 3.
Averaged waveform of hemodynamic responses at channels showing significant differences in Task 1 (left graphs) and Task 2 (right graphs). Error bars in each graph indicate \(.5 \sigma\) of the measured waveforms every 10 s
3.2 Step 2: Convolution matrix method
After Step 1, a GLM–SPM method presented in [44] called the convolution matrix method was applied to the deoxy-Hb responses because the validity of assuming a HRF was confirmed in Step 1, as the procedure presented in [44] is considered a basic version of various extended GLM–SPM methods. To examine differences between the two versions of the hand-tracing task, the following GLM equation was assumed using independent variables \(x_k\ (k=1,2)\) and a response variable y.
$$\begin{aligned} {}^{j}y(i) = {}^{j}b_1\cdot x_1(i)+{}^{j}b_2\cdot x_2(i) +{}^{j}d+{}^{j}e(i), \end{aligned}$$
where \(j\ (=1,\ldots ,24)\) is an index of channel, \(b_k\) are unknown coefficient parameters (const.), d is a drift term (const.), and e is a residual that is assumed to be independently and identically distributed normally with a mean of zero. The maximum of the sampling counter i and the total number of independent variables \(x_k\) are denoted as I and K, respectively. Equations defined by Eq. (3) for all i are summarized in the following matrix form:
$$\begin{aligned} {}^{j}{\mathbf {Y}}= {\mathbf {X}}\cdot {}^{j} {\mathbf {B}}+ {}^{j}{\mathbf {E}}, \end{aligned}$$
where \({\mathbf {X}}\in {{\mathbb {R}}}^{I \times (K+1)}\) is a design matrix, \({\mathbf {Y}}\in {{\mathbb {R}}}^{I}\) is a response vector, \({\mathbf {B}}\in {\mathbb {R}}^{(K+1)}\) is a parameter vector, \({\mathbf {E}}\in {\mathbb {R}}^{I}\) is a residual vector, and \({\mathbf {X}}\) and \({\mathbf {B}}\) are defined as
$$\begin{aligned} {\mathbf {X}} := \left[ \begin{array}{ccc} x_1(1) &{} \quad x_2(1) &{} \quad 1 \\ \vdots &{}\quad \vdots &{}\quad \vdots \\ x_1(I) &{} \quad x_2(I) &{} \quad 1 \\ \end{array} \right] , \quad {}^j {\mathbf {B}} := \left[ \begin{array}{c} {}^j b_1 \\ {}^j b_2 \\ {}^j d \end{array} \right] . \end{aligned}$$
First-level analysis This step investigated whether the hemodynamic responses differed depending on the task conditions by statistically investigating the magnitude of estimations of coefficients in Eq. (3) with respect to each channel for each participant. The details of this technique are explained below.
First, the design matrix \({\mathbf {X}}\) was defined using a time-series signal of a boxcar function with a value of 1 during the task period and a value of 0 otherwise. Second, a convolution matrix \({\mathbf {H}}\in {\mathbb {R}}^{(I+M) \times I}\) was defined using a Gaussian function to approximate the response of the rCBF. Estimates \(\hat{{\mathbf {B}}}\) for \({\mathbf {B}}\) were computed as follows, using the ordinary least squares (OLS) method [44].
$$\begin{aligned} {}^{j}\hat{{\mathbf {B}}}&= \left( {\mathbf {X}}_{a}^{{\mathrm{T}}} {\mathbf {X}}_{a}\right) ^{-1} {\mathbf {X}}_{a}^{{\mathrm{T}}} {\mathbf {H}}\cdot {}^{j} {\mathbf {Y}}\nonumber \\{\mathbf {X}}_{a}&:= {\mathbf {H}}{\mathbf {X}}\end{aligned}$$
Specifically, the convolution matrix \({\mathbf {H}}\) was designed using a Gaussian function with a 4.0-s full width half maximum \(({ FWHM}=4.0)\) [29], and the total length of \({\mathbf {Y}}\) was specified as 135 s (i.e., \(I=1350\)) because the data range including the 15- and 30-s rests at the beginning of Task 1 and the end of Task 2, respectively, was examined, as shown in previous Fig. 1b. Finally, for a statistical test of the estimates \(\hat{{\mathbf {B}}}\), a contrast matrix was chosen as \({\mathbf {C}}=[\ -1\ 1\ 0\ ]\), and the Wald statistic (\(={\mathbf {C}}\hat{{\mathbf {B}}}\)/standard error of slope coefficient) [45] was computed for each channel and tested with a one-sample t test. Importantly, the usual degrees of freedom (DoF) used in common GLM methods computed by \((I-{\mathrm {rank}}({\mathbf {X}}))\) [46] are likely to overestimate the statistic [44] because large statistical values are computed inaccurately when long time-series data are analyzed. To avoid this issue, we used the following alternative Wald statistic t, which was modified using an effective DoF [47]
$$\begin{aligned} t = \frac{{\mathbf {C}}\cdot {}^j \hat{{\mathbf {B}}}}{\sqrt{\epsilon ^2 {\mathbf {C}}\left( {\mathbf {X}}_{a}^{{\mathrm{T}}} {\mathbf {X}}_{a}\right) ^{-1} {\mathbf {X}}_{a}^{{\mathrm{T}}} {\mathbf {V}}{\mathbf {X}}_{a}\left( {\mathbf {X}}_{a}^{{\mathrm{T}}} {\mathbf {X}}_{a}\right) ^{-1} {\mathbf {C}}^{{\mathrm{T}}}},} \end{aligned}$$
where \({\mathbf {V}}:={\mathbf {H}}{\mathbf {H}}^{{\mathrm{T}}}\) and \(\epsilon ^2 \in {\mathbb {R}}^1\) is an unbiased estimator. Here \(\epsilon ^2\) is computed from a residual-forming matrix \({\mathbf {R}}\in {\mathbb {R}}^{(I+M)\times (I+M)}\) and a vector of residuals \({\mathbf {r}}\in {\mathbb {R}}^{(I+M)}\) as
$$\begin{aligned} \epsilon ^2&= {\mathbf {r}}^{{\mathrm{T}}} {\mathbf {r}}/ {\mathrm {trace}}({\mathbf {R}}{\mathbf {V}})\nonumber \\{\mathbf {r}}&:= {\mathbf {R}}{\mathbf {H}}\cdot {}^j {\mathbf {Y}}\nonumber \\{\mathbf {R}}&:= {\mathbf {I}}- {\mathbf {X}}_{a}\left( {\mathbf {X}}_{a}^{{\mathrm{T}}} {\mathbf {X}}_{a}\right) ^{-1} {\mathbf {X}}_{a}^{{\mathrm{T}}} . \end{aligned}$$
Although Eq. (7) can be computed without the direct use of the effective DoF v, the value of v was required to convert the t into a z-value during the second-level analysis. Hence, v was computed [47] by
$$\begin{aligned} v = {\mathrm {trace}}\left( {\mathbf {R}}{\mathbf {V}}\right) ^2 / {\mathrm {trace}}\left( {\mathbf {R}}{\mathbf {V}}{\mathbf {R}}{\mathbf {V}}\right) . \end{aligned}$$
In the present analysis, the effective DoF was approximately 30, while a normal DoF may have been as large as 1350. This example shows that modification using the effective DoF was indispensable for the NIRS–SPM analysis to avoid over-estimation in statistical computation.
Second-level analysis In this analysis, the statistic t computed by Eq. (7) was converted into a z-value using the effective DoF v and tested whether all z-values from all participants were statistically larger than zero for each channel (random-effects analysis).
The general form of the GLM described in Eq. (3) assumes neither a drift nor a trend effect. To fulfill this assumption, the signal is typically passed through a high-pass filter (HPF) to eliminate these effects before applying the OLS. Accurate statistical results cannot be obtained when an inadequate HPF is used. Therefore, in the following analysis, an HPF was tuned based on the Akaike information criterion (AIC). Specifically, the following three types of filters were applied based on a past NIRS study [29]: using no filter and HPFs with cutoff frequencies of .008 and .014 Hz. Table 2 shows the means of all AIC values (at the first-level analysis) and z-values for all 24 channels (at the second-level analysis). Each mean (and standard deviation) was computed by averaging all AICs in all channels for all participants. Similar results obtained by applying this method to oxy- and total-Hb data are also shown in gray in the same table to illustrate the trends caused by different HPFs. These findings revealed that the AIC was improved with a larger cutoff frequency. That is, the AIC obtained using a .014-Hz HPF was improved by approximately 8% compared with the other AICs calculated without the HPF. Therefore, an HPF with a cutoff frequency of .014 Hz with an effective DoF was used for the subsequent step.
Results of statistical test using a convolution matrix method in Step 2: mean values of AIC (at a first-level analysis) and z-values (at a second-level analysis, df = 15)
Blue cells show \(z>2.33\, (p<.01)\) and red cells show \(z>3.09\, (p<.001)\). (Color table online)
3.3 Step 3: Boxcar-function-based Gaussian convolution method
It is possible that the method used in Step 2 overestimated the actual differences because the number of channels showing significant differences in Step 2 (shown in Table 2) was roughly twice that of Step 1 (shown in Table 1). Therefore, we tested another well-used GLM approach that is also implemented in the popular SPM12 software [33]. Furthermore, we tested two modifications intended to deal with problems specific to NIRS.
In Step 3, we first assumed a GLM similar to Eq. (3) using the same boxcar function \(u_k(i)\) [29]. Importantly, this differed from Step 1 in terms of the definition of an independent variable \(x_k(i)\) that was convoluted by \(u_k(i)\) with a Gaussian kernel function g(i) as
$$\begin{aligned} x_k(i)&= (g*u_k)(i)=\sum _{n}^{all} g(n) \cdot u_k(i-n) \nonumber \\g(i) &= \exp \left( -\frac{(\Delta \cdot i)^2}{2 \sigma ^2} \right) \nonumber \\\sigma &=\frac{FWHM}{2\sqrt{2ln2}}. \end{aligned}$$
The \(x_k\) computed by Eq. (10) was used after being normalized with its maximum amplitude, and \(x_k(i)\) for all k was assigned in a design matrix \({\mathbf {X}}\) as a column vector. The gray image of \({\mathbf {X}}\) is shown in Fig. 1b. To examine the response variable signal y(i), the response vector \({\mathbf {Y}}\) was composed from time-series rCBF data that were filtered using an HPF with a cutoff frequency of .014 Hz, as described in Sect. 3.2. Summarizing \({\mathbf {X}}\) and \({\mathbf {Y}}\) into a matrix equation described in Eq. (4), an estimate \(\hat{{\mathbf {B}}}\) was computed using the OLS method by
$$\begin{aligned} \hat{{\mathbf {B}}}=\left( {\mathbf {X}}^{{\mathrm{T}}} {\mathbf {X}}\right) ^{-1} \cdot {\mathbf {X}}^{{\mathrm{T}}} \cdot {\mathbf {Y}} . \end{aligned}$$
Modification 1: Correction of autocorrelation Although there was no autocorrelation for the error e assumed in Eq. (3), this assumption was not satisfied by the actual measured data (described as Issue 1 in Introduction) [14]. For this reason, the OLS estimate is an unbiased estimator, but it is not the best linear unbiased estimator (BLUE). Hence, the statistical evaluation becomes inaccurate [30] and a type I error ("false" brain activation) is more likely to occur. Non-autocorrelation was thus recovered in Step 3 using the Cochrane–Orcutt method [48].
First, the following residual error \({\mathbf {E}}\in {\mathbf {R}} ^I\) was computed using an estimated parameter \(\hat{{\mathbf {B}}}\) obtained by the OLS method without correction of non-autocorrelation.
$$\begin{aligned} {\mathbf {E}}= {\mathbf {Y}}- {\mathbf {X}}\hat{{\mathbf {B}}} \end{aligned}$$
Using elements \([\bar{e}(1), \bar{e}(2), \ldots , \bar{e}(I)]^{{\mathrm{T}}} := {\mathbf {E}}\) in the vector \({\mathbf {E}}\), the Durbin–Watson ratio (DW) was computed by
$$\begin{aligned} DW := \frac{\sum _{i=2}^{I} (\bar{e}(i) - (\bar{e}(i-1))^2}{\sum _{i=2}^{I} (\bar{e}(i))^2} \ \ \in [0,4]. \end{aligned}$$
Next, to examine the original e(i), a first-order autocorrelation model described by
$$\begin{aligned} e(i) = \rho \cdot e(i-1) + w(i) \end{aligned}$$
was assumed using a constant \(\rho\) and a new signal w with a mean of zero and no autocorrelation. An alternative value \(\hat{\rho }\) for \(\rho\) was specified as
$$\begin{aligned} \hat{\rho } = 1 - DW/2 \end{aligned}$$
using the value of DW computed by Eq. (13). Finally, we performed a new OLS estimate using the following autocorrelation-corrected \(x^*\) and \(y^*\) from Eqs. (16) and (17). Then, the newly estimated \(\hat{{\mathbf {B}}}_{2nd}\) was used in a first-level analysis of the SPM.
$$x^*(i)= {\left\{ \begin{array}{ll} \sqrt{1 - {\hat{\rho }}^2} \cdot x(1) &{} i = 1 \\ x(i) - \hat{\rho } \cdot x(i-1) &{} i = 2, \ldots , I \end{array}\right. }$$
$$y^*(i)= {\left\{ \begin{array}{ll} \sqrt{1 - {\hat{\rho }}^2} \cdot y(1) &{} \quad i = 1 \\ y(i) - \hat{\rho } \cdot y(i-1) &{} \quad i = 2, \ldots , I \end{array}\right. }$$
Modification 2: Consideration of higher-order element Previous research proposed that the use of the derivative component of an HRF can be effective for NIRS–GLM analysis [31]. This suggestion corresponds well with the current data because an overshoot-like waveform around the rising and falling edges appeared to be approximated by a derivative term of the canonical HRF, as shown in Fig. 3. Therefore, two new GLM models were considered: (a) a fourth-order model using a new design matrix \({\mathbf {X}}_{1d} \in {\mathbb {R}}^{I \times (4+1)}\) that included the first-derivative terms of \(x_k\) described by Eq. (10); and (b) a sixth-order model using another new design matrix \({\mathbf {X}}_{2d} \in {\mathbb {R}}^{I \times (6+1)}\) that included the first- and second-derivative terms. The three models using \({\mathbf {X}}\), \({\mathbf {X}}_{1d}\), and \({\mathbf {X}}_{2d}\) were called HRF-solo (only HRF), HRF+1.d. (HRF with its first-derivative term), and HRF+2.d. (HRF with its first- and second-derivative terms), respectively.
Total accuracy verification and optimal condition The results obtained from the first- and second-level analyses using Modifications 1 and 2 are summarized in Table 3. This table shows the means of DW and AIC values relevant to all data channels from all participants. Additional results of the analyses of oxy- and total-Hb data are shown in gray to demonstrate the effects obtained by Modifications 1 and 2. Regarding AIC, the other two cases of "no correction of non-autocorrelation (No AR)" and "correction by Cochrane–Orcutt method (AR(1)), i.e., Modification 1" are shown in the same table.
Durbin–Watson ratios and AICs when a boxcar-function-based Gaussian convolution method was applied at Step 3. (Color table online)
Examination of the values of DW related to Modification 1 revealed that they were relatively close to zero in all cases; hence, the assumption of non-autocorrelation of errors e was not satisfied. This finding supports the notion that the method used in Step 2 overestimated the statistic, because it did not involve a correction process for non-autocorrelation. The AIC indices also support potential overestimation at Step 2, since the AIC values in AR(1) were two or three times smaller than in No AR. Therefore, the current results demonstrate that correction for non-autocorrelation is indispensable for NIRS–SPM analysis, whereas several previous studies [31, 46, 47] have not taken this point into consideration. In the analysis of Modification 2, Table 3 reveals that the AIC value improved slightly when the GLM contained higher-order derivative terms. In summary, we found that a GLM model using elements of HRF+2.d. with correction for non-autocorrelation was the most appropriate for analysis of the rCBF measured during the hand-tracing tasks.
Next the appropriateness of the selection of deoxy-Hb was tested after Modifications 1 and 2 were applied. For the AIC values under AR(1) conditions shown in Table 3, differences between the deoxy-Hb and oxy-Hb groups, and between the deoxy-Hb and total-Hb groups, were investigated by Welch's t test, respectively. Because each group included all AIC values computed using HRF-solo, HRF+1.d., and HRF+2.d. models, the number of samples in each group was \(N= 3\) models \({\times } 24\) channels \({\times } 16\) participants \(=1152\). The test revealed that the average of the AIC values computed using deoxy-Hb was significantly smaller than the average of the values computed using oxy-Hb and total Hb, as shown on the right side of the table (the deoxy-Hb vs the oxy-Hb groups: \(t(1700.4)=-24.3, p<.0001\), the deoxy-Hb vs the total-Hb groups: \(t(2299.9)=-4.52, p<.0001\)). Therefore, it can be concluded that deoxy-Hb is better suited for analysis of rCBF since the accuracy of fit to GLM was higher when deoxy-Hb NIRS data were used. Thus, only deoxy-Hb data were used in the subsequent analyses.
We repeated the first- and second-level analyses using these modifications with the optimal conditions. The results are shown in Table 4. The results computed under conditions other than HRF+2.d., i.e., HRF-solo and HRF+1.d., are also shown in gray in the table. Contrast matrices were chosen as \({\mathbf {C}}= [\ -1\ 1\ -1\ 1\ 0\ ](K=4)\) and \({\mathbf {C}}= [\ -1\ 1\ -1\ 1\ -1\ 1\ 0\ ](K=6)\) for HRF+1.d. and HRF+2.d., respectively. Table 4 demonstrates that HRF+2.d. and HRF+1.d. were able to detect more channels showing statistical significance than HRF-solo, and there was a small difference in AIC values. Although the results in Table 4 show similar tendencies for both HRF+1.d. and HRF+2.d., we speculate that HRF+2.d. was more desirable because the AIC value for HRF+2.d. was smaller than that for the HRF+1.d. model.
Results of statistical test using a boxcar-function-based Gaussian convolution method in Step 3: z-values in a second-level analysis for three kinds of design matrices, \(df=15\)
Taking the results of Sects. 3.1–3.3 together, the optimal guidelines for analyzing NIRS–SPM data for rCBF contaminated with motion artifacts can be summarized as follows:
Type of rCBF for SPM analysis: deoxy-Hb.
Prefilter for rCBF: an HPF with cutoff frequencies of .014 Hz.
Method: boxcar-function-based Gaussian convolution method.
GLM: a linear model consisting of an HRF computed using a Gaussian kernel function and its first- and second-derivative terms.
Modifications: correction for non-autocorrelation by the Cochrane–Orcutt method and the use of effective DoF.
Note that these guidelines were obtained by considering a range of issues involved in other NIRS–SPM methods, as described in Introduction. Issue 1 (the unsatisfied assumption of normal distribution in measured data) was attenuated by optimization of statistical procedure using AIC and DW indices in Steps 1–3. Issue 2 (the non-autocorrelation problem) was resolved with Modification 1, using the Cochrane–Orcutt method in Step 3. Finally, Issue 3 (related to noise reduction) was resolved in Step 2.
4 Analysis of statistical parametric maps
The spatial distribution of brain activity associated with BS modification was investigated by focusing on the NIRS–SPM results obtained using the optimal conditions derived in Sect. 3.3. Figure 4 shows z-map images based on the results shown in Table 4. A z-map computed using the optimal conditions obtained in Step 3 is shown in Fig. 4c. This map is a visualization of the z-values that were described by blue numbers in Table 4. Images (a) and (b) are other z-maps based on results obtained in Steps 1 and 2 using the corresponding quasi-optimal conditions.1
Statistical parametric maps (z-map images, \(df=15\)): a model-free method (Step 1); b convolution matrix method (Step 2); c a boxcar-function-based Gaussian convolution method (Step 3). (Color figure online)
The images shown in Fig. 4 are montages of a brain surface image created with the BrainBrowser Surface Viewer (v2.3.0) [49] and the colored z-maps. The colored z-map was drawn by interpolating z-values with a scattered data interpolation function (in MATLAB R2015a) after deforming the positions of the NIRS channel grid with reference to C3(4) and T3(4) positions on the MNI coordinate system [50]. Circles and numbers drawn on the z-map image indicate the position and index of the measurement channels, respectively. Labels of Brodmann's area numbers are provided to indicate several channels where strong significant differences were confirmed. First, Fig. 4c shows that the right hemisphere was dominant. Specifically, significant differences were confirmed in channels 1, 4, 8–10, and 12 (\(z(15)>2.33, p<.01\)). Channel 19, which solely indicates significant differences in the left hemisphere, is close to an area near S1(BA1), which corresponds to the tip of the finger in the cortical homunculus. This finding is in accord with the experimental circumstances, since participants used their right hand in Task 2 (while holding a long stick) more strongly than in Task 1 (which only involved one finger). Examining the positions of these significant channels revealed that the following four contiguous areas of the brain were significantly activated during Task 2: Ar1) somatosensory association cortex (BA7: \(z=3.49, p<.001\)); Ar2) supramarginal gyrus (BA40: \(z=2.93, p<.01\)); Ar3) associative visual cortex (BA19: \(z=3.79, p<.0001\)); and Ar4) middle temporal gyrus (BA21: \(z=3.46, p<.001\)). Although a similar activation pattern at Ar1–Ar4 can be seen in Fig. 4b which was obtained in Step 2, the effects in Step 2 were likely to be overestimated because the z-values obtained in Step 2 were larger than that in Step 3, despite the larger AIC values in Step 3. Therefore, it seems reasonable to conclude that the pattern in the z-map image (c) obtained by Step 3 is a feature of the BS modification related to the hand-tracing task.
Areas BA7 and BA40 related to Ar1 and Ar2 are part of the parietal association cortex. It has been found that damage to these areas causes spatial perception impairment [51]. The parietal association cortex forms the parietal lobe in combination with the S1 area, and the right parietal lobe has been closely linked to spatial skills [52, 53]. Specifically, a bimodal neuron responding to both visual and somatic senses has been reported to exist in the intraparietal sulcus (which is located near BA7) in the right parietal lobe [54]. In addition, the intraparietal sulcus was reported to be associated with the BS in a study of the RHI [55]. These previous findings regarding the right-hemisphere dominance of the parietal lobe are consistent with the current SPM results shown in Fig. 4c. Taken together, these results suggest that the brain areas involved in spatial perception may have been activated when BS extension was required during the use of a long stick in the current study. In addition, the inferior parietal lobule consists of BA40 and the angular gyrus (BA39),2 and the right inferior parietal cortex is related to own-body perception and the illusion of motion [56, 57]. Interestingly, it has been reported that out-of-body experiences [58, 59] and phantom sensations [60] can be induced by stimulating the angular gyrus, one of the areas associated with the BS.
BA19 (in Ar3) is involved in the recognition of the shape and color of objects [61]. In the present study, we speculate that this area may have become active when participants visually examined the characters traced by their fingers. This characteristic may be a feature of the BS extension because the BS is visually dominant [62]. Activation in this area, however, does not necessarily indicate general BS extension because the cognitive processing involved in recognizing Hiragana characters might have also caused neural responses in this region. Moreover, BA21 (in Ar4) is reported to be activated when subjects are engaged in contemplating distance [63]. In the current study, this brain area may have been activated during the estimation of the distance from their own body to the tip of the stick, which would be required to extend the BS spatially.
Taken together, this evidence suggests that the brain activation we observed in contiguous areas in BA7–BA40–BA21 may be related to BS extension during a hand-tracing task.
5 Conclusions and future research
This study described an optimized statistical analysis procedure for NIRS–SPM analysis that involves dealing with rCBF data contaminated by motion artifacts. In addition, we identified the spatial distribution of brain activity associated with BS modification using a hand-tracing task that involved an extension of the BS. Three methodological options were evaluated in turn to determine the optimal conditions for NIRS–SPM analysis: a model-free method in Step 1, a convolution matrix method in Step 2, and a boxcar-function-based Gaussian convolution method in Step 3.
In Step 1, it was found that the actual rCBF waveform during this task could be approximated by a rounded trapezoidal waveform similar to the convolved response of a boxcar with a Gaussian function. Moreover, deoxy-Hb was found to be appropriate for the NIRS–GLM in this experiment, as indicated by the results of diagnostic screening indices concerning individual variance and AIC, which was confirmed in Step 3. In Step 2, to enhance statistical accuracy, conditions for eliminating low-frequency noise and modifying the DoF for statistical testing were investigated using the AIC. In Step 3, correction of non-autocorrelation with derivative components of HRF was applied to a GLM for SPM, by calculating the DW ratio and AIC values. Finally, credible SPM guidelines for NIRS data were obtained. Examination of the best SPM results confirmed that contiguous areas in BA7–BA40–BA21 (BA7: somatosensory association cortex; BA40: supramarginal gyrus; BA21: middle temporal gyrus) in the right hemisphere became significantly active (\(p<.001\), \(p<.01\), and \(p<.001\), respectively) during the hand-tracing tasks, potentially representing BS modification.
Future research could incorporate the NIRS–SPM method described here to exogenously enhance the ability of BS extension using electrical stimuli.
The quasi-optimal condition at Step 1 was "rCBF type: deoxy, time interval of integral: 10 s" because the z-values under this condition were largest, as shown in Table 1. The quasi-optimal condition found in Step 2 was "rCBF type: deoxy, HPF: .014 Hz" because the AIC values under this condition were smallest, as shown in Table 3.
The BA39 area was out of the measurable range of this NIRS setting.
The experiments were performed by Akira Ichinose, Takumi Matsumura, Noriko Kimura, Kei Kawahara, and Toshifumi Chikaraishi. This research would not be possible without the many participants who kindly took part in this study. This research was in part supported by a JSPS Grant-in-Aid for Exploratory Research (Grant No. 25630179) and Grant-in-Aid for Scientific Research (C) (Grant No. 15K06153), Japan.
The author declares that they have no conflict of interest.
Department of Robotics and Mechatronics, Tokyo Denki University, 5 Asahi-Chou, Senju, Adachi-ku, Tokyo 120-8551, Japan
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Standard translation
In modal logic, standard translation is a logic translation that transforms formulas of modal logic into formulas of first-order logic which capture the meaning of the modal formulas. Standard translation is defined inductively on the structure of the formula. In short, atomic formulas are mapped onto unary predicates and the objects in the first-order language are the accessible worlds. The logical connectives from propositional logic remain untouched and the modal operators are transformed into first-order formulas according to their semantics.
Definition
Standard translation is defined as follows:
• $ST_{x}(p)\equiv P(x)$, where $p$ is an atomic formula; P(x) is true when $p$ holds in world $x$.
• $ST_{x}(\top )\equiv \top $
• $ST_{x}(\bot )\equiv \bot $
• $ST_{x}(\neg \varphi )\equiv \neg ST_{x}(\varphi )$
• $ST_{x}(\varphi \wedge \psi )\equiv ST_{x}(\varphi )\wedge ST_{x}(\psi )$
• $ST_{x}(\varphi \vee \psi )\equiv ST_{x}(\varphi )\vee ST_{x}(\psi )$
• $ST_{x}(\varphi \rightarrow \psi )\equiv ST_{x}(\varphi )\rightarrow ST_{x}(\psi )$
• $ST_{x}(\Diamond _{m}\varphi )\equiv \exists y(R_{m}(x,y)\wedge ST_{y}(\varphi ))$
• $ST_{x}(\Box _{m}\varphi )\equiv \forall y(R_{m}(x,y)\rightarrow ST_{y}(\varphi ))$
In the above, $x$ is the world from which the formula is evaluated. Initially, a free variable $x$ is used and whenever a modal operator needs to be translated, a fresh variable is introduced to indicate that the remainder of the formula needs to be evaluated from that world. Here, the subscript $m$ refers to the accessibility relation that should be used: normally, $\Box $ and $\Diamond $ refer to a relation $R$ of the Kripke model but more than one accessibility relation can exist (a multimodal logic) in which case subscripts are used. For example, $\Box _{a}$ and $\Diamond _{a}$ refer to an accessibility relation $R_{a}$ and $\Box _{b}$ and $\Diamond _{b}$ to $R_{b}$ in the model. Alternatively, it can also be placed inside the modal symbol.
Example
As an example, when standard translation is applied to $\Box \Box p$, we expand the outer box to get
$\forall y(R(x,y)\rightarrow ST_{y}(\Box p))$
meaning that we have now moved from $x$ to an accessible world $y$ and we now evaluate the remainder of the formula, $\Box p$, in each of those accessible worlds.
The whole standard translation of this example becomes
$\forall y(R(x,y)\rightarrow (\forall z(R(y,z)\rightarrow P(z))))$
which precisely captures the semantics of two boxes in modal logic. The formula $\Box \Box p$ holds in $x$ when for all accessible worlds $y$ from $x$ and for all accessible worlds $z$ from $y$, the predicate $P$ is true for $z$. Note that the formula is also true when no such accessible worlds exist. In case $x$ has no accessible worlds then $R(x,y)$ is false but the whole formula is vacuously true: an implication is also true when the antecedent is false.
Standard translation and modal depth
The modal depth of a formula also becomes apparent in the translation to first-order logic. When the modal depth of a formula is k, then the first-order logic formula contains a 'chain' of k transitions from the starting world $x$. The worlds are 'chained' in the sense that these worlds are visited by going from accessible to accessible world. Informally, the number of transitions in the 'longest chain' of transitions in the first-order formula is the modal depth of the formula.
The modal depth of the formula used in the example above is two. The first-order formula indicates that the transitions from $x$ to $y$ and from $y$ to $z$ are needed to verify the validity of the formula. This is also the modal depth of the formula as each modal operator increases the modal depth by one.
References
• Patrick Blackburn and Johan van Benthem (1988), Modal Logic: A Semantic Perspective.
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Narutaka Ozawa
Narutaka Ozawa (小沢登高, Ozawa Narutaka) (born 1974) is a Japanese mathematician, known for his work in operator algebras and discrete groups. He has been a professor at Kyoto University since 2013. He earned a bachelor's degree in mathematics in 1997 from the University of Tokyo and a Ph.D. in mathematics in 2000 from the same institution. One year later he received a Ph.D. in mathematics from Texas A&M University. He was selected for one of the prestigious Sloan Research Fellowships in 2005[1] and was an invited speaker at the 2006 ICM in Madrid where he gave a talk on "Amenable actions and Applications".[2] He has won numerous prizes including the Mathematical Society of Japan (MSJ) Spring Prize and the Japan Society for the Promotion of Science (JSPS) Prize. Before becoming a full professor at Kyoto University in 2013, he was an associate professor at the University of Tokyo and at University of California, Los Angeles.[3]
Narutaka Ozawa
Born1974 (age 48–49)
NationalityJapanese
Scientific career
FieldsMathematics
Notes
1. "Sloan Fellows". Physicalsciences.ucla.edu. Archived from the original on 11 August 2014. Retrieved 8 December 2014.
2. Ozawa, Narutaka (2006), "Amenable actions and applications", Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures, Z\"urich: European Mathematical Society (EMS), doi:10.4171/022-2/74, Zbl 1104.46032
3. "UCLA General Catalog 2009-2010 Mathematics Faculty Roster". Registrar.ucla.edu. Archived from the original on 25 October 2014. Retrieved 8 December 2014.
References
• Brown, Nathanial; Ozawa, Narutaka (2008), C*-Algebras and Finite-Dimensional Approximations, Providence: American Mathematical Society, ISBN 978-0-8218-4381-9, MR 2391387
External links
• Narutaka OZAWA
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Composition algebra
In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies
$N(xy)=N(x)N(y)$
Algebraic structures
Group-like
• Group
• Semigroup / Monoid
• Rack and quandle
• Quasigroup and loop
• Abelian group
• Magma
• Lie group
Group theory
Ring-like
• Ring
• Rng
• Semiring
• Near-ring
• Commutative ring
• Domain
• Integral domain
• Field
• Division ring
• Lie ring
Ring theory
Lattice-like
• Lattice
• Semilattice
• Complemented lattice
• Total order
• Heyting algebra
• Boolean algebra
• Map of lattices
• Lattice theory
Module-like
• Module
• Group with operators
• Vector space
• Linear algebra
Algebra-like
• Algebra
• Associative
• Non-associative
• Composition algebra
• Lie algebra
• Graded
• Bialgebra
• Hopf algebra
for all x and y in A.
A composition algebra includes an involution called a conjugation: $x\mapsto x^{*}.$ The quadratic form $N(x)=xx^{*}$ is called the norm of the algebra.
A composition algebra (A, ∗, N) is either a division algebra or a split algebra, depending on the existence of a non-zero v in A such that N(v) = 0, called a null vector.[1] When x is not a null vector, the multiplicative inverse of x is $ {\frac {x^{*}}{N(x)}}$. When there is a non-zero null vector, N is an isotropic quadratic form, and "the algebra splits".
Structure theorem
Every unital composition algebra over a field K can be obtained by repeated application of the Cayley–Dickson construction starting from K (if the characteristic of K is different from 2) or a 2-dimensional composition subalgebra (if char(K) = 2). The possible dimensions of a composition algebra are 1, 2, 4, and 8.[2][3][4]
• 1-dimensional composition algebras only exist when char(K) ≠ 2.
• Composition algebras of dimension 1 and 2 are commutative and associative.
• Composition algebras of dimension 2 are either quadratic field extensions of K or isomorphic to K ⊕ K.
• Composition algebras of dimension 4 are called quaternion algebras. They are associative but not commutative.
• Composition algebras of dimension 8 are called octonion algebras. They are neither associative nor commutative.
For consistent terminology, algebras of dimension 1 have been called unarion, and those of dimension 2 binarion.[5]
Every composition algebra is an alternative algebra.[3]
Using the doubled form ( _ : _ ): A × A → K by $(a:b)=n(a+b)-n(a)-n(b),$ then the trace of a is given by (a:1) and the conjugate by a* = (a:1)e – a where e is the basis element for 1. A series of exercises prove that a composition algebra is always an alternative algebra.[6]
Instances and usage
When the field K is taken to be complex numbers C and the quadratic form z2, then four composition algebras over C are C itself, the bicomplex numbers, the biquaternions (isomorphic to the 2×2 complex matrix ring M(2, C)), and the bioctonions C ⊗ O, which are also called complex octonions.
The matrix ring M(2, C) has long been an object of interest, first as biquaternions by Hamilton (1853), later in the isomorphic matrix form, and especially as Pauli algebra.
The squaring function N(x) = x2 on the real number field forms the primordial composition algebra. When the field K is taken to be real numbers R, then there are just six other real composition algebras.[3]: 166 In two, four, and eight dimensions there are both a division algebra and a "split algebra":
binarions: complex numbers with quadratic form x2 + y2 and split-complex numbers with quadratic form x2 − y2,
quaternions and split-quaternions,
octonions and split-octonions.
Every composition algebra has an associated bilinear form B(x,y) constructed with the norm N and a polarization identity:
$B(x,y)\ =\ [N(x+y)-N(x)-N(y)]/2.$[7]
History
The composition of sums of squares was noted by several early authors. Diophantus was aware of the identity involving the sum of two squares, now called the Brahmagupta–Fibonacci identity, which is also articulated as a property of Euclidean norms of complex numbers when multiplied. Leonhard Euler discussed the four-square identity in 1748, and it led W. R. Hamilton to construct his four-dimensional algebra of quaternions.[5]: 62 In 1848 tessarines were described giving first light to bicomplex numbers.
About 1818 Danish scholar Ferdinand Degen displayed the Degen's eight-square identity, which was later connected with norms of elements of the octonion algebra:
Historically, the first non-associative algebra, the Cayley numbers ... arose in the context of the number-theoretic problem of quadratic forms permitting composition…this number-theoretic question can be transformed into one concerning certain algebraic systems, the composition algebras...[5]: 61
In 1919 Leonard Dickson advanced the study of the Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain Cayley numbers. He introduced a new imaginary unit e, and for quaternions q and Q writes a Cayley number q + Qe. Denoting the quaternion conjugate by q′, the product of two Cayley numbers is[8]
$(q+Qe)(r+Re)=(qr-R'Q)+(Rq+Qr')e.$
The conjugate of a Cayley number is q' – Qe, and the quadratic form is qq′ + QQ′, obtained by multiplying the number by its conjugate. The doubling method has come to be called the Cayley–Dickson construction.
In 1923 the case of real algebras with positive definite forms was delimited by the Hurwitz's theorem (composition algebras).
In 1931 Max Zorn introduced a gamma (γ) into the multiplication rule in the Dickson construction to generate split-octonions.[9] Adrian Albert also used the gamma in 1942 when he showed that Dickson doubling could be applied to any field with the squaring function to construct binarion, quaternion, and octonion algebras with their quadratic forms.[10] Nathan Jacobson described the automorphisms of composition algebras in 1958.[2]
The classical composition algebras over R and C are unital algebras. Composition algebras without a multiplicative identity were found by H.P. Petersson (Petersson algebras) and Susumu Okubo (Okubo algebras) and others.[11]: 463–81
See also
• Freudenthal magic square
• Pfister form
• Triality
References
Wikibooks has a book on the topic of: Associative Composition Algebra
1. Springer, T. A.; F. D. Veldkamp (2000). Octonions, Jordan Algebras and Exceptional Groups. Springer-Verlag. p. 18. ISBN 3-540-66337-1.
2. Jacobson, Nathan (1958). "Composition algebras and their automorphisms". Rendiconti del Circolo Matematico di Palermo. 7: 55–80. doi:10.1007/bf02854388. Zbl 0083.02702.
3. Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in Symmetries in Complex Analysis by Bruce Gilligan & Guy Roos, volume 468 of Contemporary Mathematics, American Mathematical Society, ISBN 978-0-8218-4459-5
4. Schafer, Richard D. (1995) [1966]. An introduction to nonassociative algebras. Dover Publications. pp. 72–75. ISBN 0-486-68813-5. Zbl 0145.25601.
5. Kevin McCrimmon (2004) A Taste of Jordan Algebras, Universitext, Springer ISBN 0-387-95447-3 MR2014924
6. Associative Composition Algebra/Transcendental paradigm#Categorical treatment at Wikibooks
7. Arthur A. Sagle & Ralph E. Walde (1973) Introduction to Lie Groups and Lie Algebras, pages 194−200, Academic Press
8. Dickson, L. E. (1919), "On Quaternions and Their Generalization and the History of the Eight Square Theorem", Annals of Mathematics, Second Series, Annals of Mathematics, 20 (3): 155–171, doi:10.2307/1967865, ISSN 0003-486X, JSTOR 1967865
9. Max Zorn (1931) "Alternativekörper und quadratische Systeme", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 9(3/4): 395–402
10. Albert, Adrian (1942). "Quadratic forms permitting composition". Annals of Mathematics. 43 (1): 161–177. doi:10.2307/1968887. JSTOR 1968887. Zbl 0060.04003.
11. Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol (1998) "Composition and Triality", chapter 8 in The Book of Involutions, pp. 451–511, Colloquium Publications v 44, American Mathematical Society ISBN 0-8218-0904-0
Further reading
• Faraut, Jacques; Korányi, Adam (1994). Analysis on symmetric cones. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York. pp. 81–86. ISBN 0-19-853477-9. MR 1446489.
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023.
• Harvey, F. Reese (1990). Spinors and Calibrations. Perspectives in Mathematics. Vol. 9. San Diego: Academic Press. ISBN 0-12-329650-1. Zbl 0694.53002.
| Wikipedia |
\begin{document}
\title{{\bf Qualitative Analysis of Partially-observable Markov Decision Processes}}
\author{Krishnendu Chatterjee\inst{1} \and Laurent Doyen\inst{2} \and Thomas A. Henzinger\inst{1} }
\institute{ IST Austria (Institute of Science and Technology Austria) \\ \and LSV, ENS Cachan \& CNRS, France }
\maketitle \thispagestyle{empty}
\begin{abstract} We study observation-based strategies for \emph{partially-observable Markov decision processes} ({\sf POMDP} s) with parity objectives. An observation-based strategy relies on partial information about the history of a play, namely, on the past sequence of observations. We consider qualitative analysis problems: given a {\sf POMDP}\/ with a parity objective, decide whether there exists an observation-based strategy to achieve the objective with probability~1 (almost-sure winning), or with positive probability (positive winning). Our main results are twofold. First, we present a complete picture of the computational complexity of the qualitative analysis problem for {\sf POMDP} s with parity objectives and its subclasses: safety, reachability, B\"uchi, and coB\"uchi objectives. We establish several upper and lower bounds that were not known in the literature, and present efficient and symbolic algorithms for the decidable subclasses. Second, we give, for the first time, optimal bounds (matching upper and lower bounds) for the memory required by pure and randomized observation-based strategies for all classes of objectives. \end{abstract}
\section{Introduction}
\noindent{\bf Markov decision processes.} A \emph{Markov decision process (MDP)} is a model for systems that exhibit both probabilistic and nondeterministic behavior. MDPs have been used to model and solve control problems for stochastic systems: there, nondeterminism represents the freedom of the controller to choose a control action, while the probabilistic component of the behavior describes the system response to control actions. MDPs have also been adopted as models for concurrent probabilistic systems, probabilistic systems operating in open environments~\cite{SegalaT}, and under-specified probabilistic systems \cite{BdA95}.
\noindent{\bf System specifications.} The \emph{specification} describes the set of desired behaviors of the system, and is typically an $\omega$-regular set of paths. Parity objectives are a canonical way to define such specifications in MDPs. They include reachability, safety, B\"uchi and coB\"uchi objectives as special cases. Thus MDPs with parity objectives provide the theoretical framework to study problems such as the verification and the control of stochastic systems.
\noindent{\bf Perfect vs. partial observations.}
Most results about MDPs make the hypothesis of \emph{perfect observation}. In this setting, the controller always knows, while interacting with the system (or MDP), the exact state of the MDP. In practice, this hypothesis is often unrealistic. For example, in the control of multiple processes, each process has only access to the public variables of the other processes, but not to their private variables. In the control of hybrid systems~\cite{BouyerDMP03,DDR06}, or in automated planning~\cite{MadaniHC03}, the controller usually has noisy information about the state of the systems due to finite-precision sensors. In such applications, MDPs with \emph{partial observation} (POMDPs) provide a more appropriate model.
\noindent{\bf Qualitative and quantitative analysis.}
Given an MDP with parity objective, the \emph{qualitative analysis} asks for the computation of the set of \emph{almost-sure winning} states (resp., \emph{positive winning} states) in which the controller can achieve the parity objective with probability~1 (resp., positive probability); the more general \emph{quantitative analysis} asks for the computation at each state of the maximal probability with which the controller can satisfy the parity objective.
The analysis of POMDPs is considerably more complicated than the analysis of MDPs. First, the decision problems for POMDPs usually lie in higher complexity classes than their perfect-observation counterparts: for example, the quantitative analysis of POMDPs with reachability and safety objectives is undecidable~\cite{Paz-Book}, whereas for MDPs with perfect observation, this question can be solved in polynomial time~\cite{luca-thesis,CJH04}. Second, in the context of POMDPs, witness winning strategies for the controller need memory even for the simple objectives of safety and reachability. This is again in contrast to the perfect-observation case, where memoryless strategies suffice for all parity objectives. Since the quantitative analysis of POMDPs is undecidable (even for computing approximations of the maximal probabilities~\cite{MadaniHC03}), we study the qualitative analysis of POMDPs with parity objective and its subclasses.
\noindent{\bf Contribution.}
For the qualitative analysis of POMDPs, the following results are known: (a)~the problems of deciding if a state is almost-sure winning for reachability and B\"uchi objectives can be solved in EXPTIME~\cite{BBG08}; (b)~the problems for almost-sure winning for coB\"uchi objectives and positive winning for B\"uchi objectives are undecidable~\cite{BBG08,GIMBERT:2009:HAL-00403463:3}; and (c)~the EXPTIME-completeness of almost-sure winning for safety objectives follows from the results on games with partial observation~\cite{CDHR07,BD08}. Our new contributions are as follows: \begin{enumerate} \item First, we show that (a)~positive winning for reachability objectives is NLOGSPACE-complete; and (b)~almost-sure winning for reachability and B\"uchi objectives, and positive winning for safety and coB\"uchi objectives are EXPTIME-hard\footnote{A very brief (two line) proof of EXPTIME-hardness is sketched in~\cite{dA99} (see the discussion before Theorem~\ref{thrm_complexity} for more details).\label{pos-safety-hard}}. We also present a new proof that positive winning for safety and coB\"uchi objectives can be solved in EXPTIME\footnote{A different proof that positive safety can be solved in EXPTIME is given in~\cite{GriponS09} (see the discussion after Theorem~\ref{thrm_pos_safe} for a comparison).}. It follows that almost-sure winning for reachability and B\"uchi, and positive winning for safety and coB\"uchi, are EXPTIME-complete. This completes the picture for the complexity of the qualitative analysis for POMDPs with parity objectives. Moreover our new proofs of EXPTIME upper-bound proofs yield efficient and symbolic algorithms to solve positive winning for safety and coB\"uchi objectives in POMDPs.
\item Second, we present a complete characterization of the amount of memory required by pure (deterministic) and randomized strategies for the qualitative analysis of POMDPs. For the first time, we present optimal memory bounds (matching upper and lower bounds) for pure and randomized strategies: we show that (a)~for positive winning of reachability objectives, randomized memoryless strategies suffice, while for pure strategies linear memory is necessary and sufficient; (b)~for almost-sure winning of safety, reachability, and B\"uchi objectives, and for positive winning of safety and coB\"uchi objectives, exponential memory is necessary and sufficient for both pure and randomized strategies.
\end{enumerate}
\noindent{\bf Related work.} Though MDPs have been widely studied under the hypothesis of perfect observations, there are a few works that consider POMDPs, e.g.,~\cite{PapaTsi,Littman-thesis} for several finite-horizon quantitative objectives.
The results of~\cite{BBG08} shows the upper bounds for almost-sure winning for reachability and B\"uchi objectives, and the work of~\cite{CSV09} considers a subclass of {\sf POMDP} s with B\"uchi objectives and presents a PSPACE upper bound for the subclass. The undecidability of almost-sure winning for coB\"uchi and positive winning for B\"uchi objectives is established by~\cite{BBG08,GIMBERT:2009:HAL-00403463:3}. We present a solution to the remaining problems related to the qualitative analysis of POMDPs with parity objectives, and complete the picture. Partial information has been studied in the context of two-player games~\cite{Reif84,CDHR07}, a model that is incomparable to MDPs, though some techniques (like the subset construction) can be adapted to the context of POMDPs. More general models of stochastic games with partial information have been studied in~\cite{BGG09,GriponS09}, and lie in higher complexity classes. For example, a result of~\cite{BGG09} shows that the decision problem for positive winning of safety objectives is 2EXPTIME-complete in the general model, while for POMDPs, we show that the same problem is EXPTIME-complete.
\section{Definitions}\label{sec:definitions}
A \emph{probability distribution} on a finite set $A$ is a function $\kappa: A \to [0,1]$ such that $\sum_{a \in A} \kappa(a) = 1$. The \emph{support} of $\kappa$ is the set ${\sf Supp}(\kappa) = \{a \in A \mid \kappa(a) > 0\}$. We denote by ${\cal D}(A)$ the set of probability distributions on $A$.
\noindent{\em Games and MDPs.} A \emph{two-player game structure} or a \emph{Markov decision process (MDP)} (\emph{of partial observation}) is a tuple $G=\tuple{L,\Sigma,{\delta},{\cal{O}}}$, where $L$ is a finite set of states, $\Sigma$ is a finite set of actions, ${\cal{O}} \subseteq 2^L$ is a set of observations that partition\footnote{A slightly more general model with overlapping observations can be reduced in polynomial time to partitioning observations~\cite{CDHR07}.} the state space $L$. We denote by $\mathsf{obs}({\ell})$ the unique observation $o \in {\cal{O}}$ such that ${\ell} \in o$.
In the case of games, ${\delta} \subseteq L \times \Sigma \times L$ is a set of labeled transitions; in the case of MDPs, ${\delta}: L \times \Sigma \to {\cal D}(L)$ is a probabilistic transition function. For games, we require that for all ${\ell} \in L$ and all $\sigma \in \Sigma$, there exists ${\ell}' \in L$ such that $({\ell}, \sigma, {\ell}') \in {\delta}$.
We refer to a game of partial observation as a {\sf POG}\/ and to an MDP of partial observation as a {\sf POMDP}. We say that $G$ is a game or MDP of \emph{perfect observation} if ${\cal{O}} = \{ \{{\ell}\} \mid {\ell} \in L \}$.
For $\sigma \in \Sigma$ and $s \subseteq L$, define $\mathsf{Post}^G_\sigma(s) = \{{\ell}' \in L \mid \exists {\ell} \in s: ({\ell},\sigma,{\ell}') \in {\delta} \}$ when $G$ is a game, and $\mathsf{Post}^G_\sigma(s) = \{{\ell}' \in L \mid \exists {\ell} \in s: {\delta}({\ell},\sigma)({\ell}') >0 \}$ when $G$ is an MDP.
\noindent{\em Plays.} Games are played in rounds in which Player~$1$ chooses an action in $\Sigma$, and Player~$2$ resolves nondeterminism by choosing the successor state; in MDPs the successor state is chosen according to the probabilistic transition function. A \emph{play} in $G$ is an infinite sequence $\pi={\ell}_0 \sigma_0 {\ell}_1 \ldots \sigma_{n-1} {\ell}_n \sigma_n \ldots$ such that ${\ell}_{i+1} \in \mathsf{Post}^G_{\sigma_i}(\{{\ell}_i\})$ for all $i \geq 0$.
The infinite sequence $\mathsf{obs}(\pi)=\mathsf{obs}({\ell}_0) \sigma_0 \mathsf{obs}({\ell}_1) \ldots \sigma_{n-1} \mathsf{obs}({\ell}_n) \sigma_{n} \ldots$ is the \emph{observation} of $\pi$.
The set of infinite plays in $G$ is denoted ${\sf Plays}(G)$, and the set of finite prefixes ${\ell}_0 \sigma_0 \ldots \sigma_{n-1} {\ell}_n$ of plays is denoted ${\sf Prefs}(G)$. A state ${\ell} \in L$ is \emph{reachable} in $G$ if there exists a prefix $\rho \in {\sf Prefs}(G)$ such that ${\sf Last}(\rho) = {\ell}$ where ${\sf Last}(\rho)$ is the last state of $\rho$.
\begin{comment} \begin{lemma}\label{lem:knowledge} Let $G=\tuple{L,\Sigma,{\delta},{\cal{O}}}$ be a {\sf POG}\/ or a {\sf POMDP}. For $\sigma \in \Sigma$, ${\ell} \in L$, and $\rho,\rho' \in {\sf Prefs}(G)$ with $\rho' = \rho \cdot \sigma \cdot {\ell}$, Then ${\sf K}(\mathsf{obs}(\rho')) = \mathsf{Post}^G_\sigma({\sf K}(\mathsf{obs}(\rho))) \cap \mathsf{obs}({\ell})$. \end{lemma} \end{comment}
\noindent{\em Strategies.} A \emph{pure strategy} in $G$ for Player~$1$ is a function $\alpha:{\sf Prefs}(G) \to \Sigma$. A \emph{randomized strategy} in $G$ for Player~$1$ is a function $\alpha:{\sf Prefs}(G) \to {\cal D}(\Sigma)$. A (pure or randomized) strategy $\alpha$ for Player~$1$ is \emph{observation-based} if for all prefixes $\rho,\rho' \in {\sf Prefs}(G)$, if $\mathsf{obs}(\rho) = \mathsf{obs}(\rho')$, then $\alpha(\rho)=\alpha(\rho')$. In the sequel, we are interested in the existence of observation-based strategies for Player~$1$. A \emph{pure strategy} in $G$ for Player~$2$ is a function $\beta:{\sf Prefs}(G) \times \Sigma \to L$ such that for all $\rho \in {\sf Prefs}(G)$ and all $\sigma \in \Sigma$, we have $({\sf Last}(\rho), \sigma, \beta(\rho,\sigma)) \in {\delta}$. A \emph{randomized strategy} in $G$ for Player~$2$ is a function $\beta:{\sf Prefs}(G) \times \Sigma \to {\cal D}(L)$ such that for all $\rho \in {\sf Prefs}(G)$, all $\sigma \in \Sigma$, and all ${\ell} \in {\sf Supp}(\beta(\rho,\sigma))$, we have $({\sf Last}(\rho), \sigma, {\ell}) \in {\delta}$. We denote by ${\cal A}_G$, ${\cal A}_G^O$, and ${\cal B}_G$ the set of all Player-$1$ strategies, the set of all observation-based Player-$1$ strategies, and the set of all Player-$2$ strategies in $G$, respectively.
\newcommand{{\sf Mem}}{{\sf Mem}}
\noindent{\em Memory requirement of strategies.} An equivalent definition of strategies is as follows. Let ${\sf Mem}$ be a set called \emph{memory}. An observation-based strategy with memory can be described by two functions, a \emph{memory-update} function $\alpha_{u}$: ${\sf Mem} \times {\cal{O}} \times \Sigma \to {\sf Mem}$ that given the current memory, observation and the action updates the memory, and a \emph{next-action} function $\alpha_{n}$: ${\sf Mem} \times {\cal{O}} \to {\cal D}(\Sigma)$ that given the current memory and current observation specifies the probability distribution\footnote{For a pure strategy, the next-action function specifies a single action rather than a probability distribution.} of the next action, respectively.
A strategy is \emph{finite-memory} if the memory ${\sf Mem}$ is finite and the size of a finite-memory strategy $\alpha$ is the size $\abs{{\sf Mem}}$ of its memory. A strategy is \emph{memoryless} if $\abs{{\sf Mem}} = 1$. The memoryless strategies do not depend on the history of a play, but only on the current state. Memoryless strategies for player~1 can be viewed as functions $\alpha$: ${\cal{O}} \to {\cal D}(\Sigma)$.
\begin{comment} The \emph{outcome} of two deterministic strategies $\alpha$ (for Player~$1$) and $\beta$ (for Player~$2$) in $G$ is the play $\pi={\ell}_0 \sigma_0 {\ell}_1 \ldots \sigma_{n-1} {\ell}_n \sigma_n \ldots \in {\sf Plays}(G)$ such that for all $i \geq 0$, we have $\sigma_i = \alpha(\pi(i))$ and ${\ell}_{i+1} = \beta(\pi(i),\sigma_i)$. This play is denoted $\mathsf{outcome}(G,\alpha,\beta)$. The \emph{outcome} of two randomized strategies $\alpha$ (for Player~$1$) and $\beta$ (for Player~$2$) in $G$ is the set of plays $\pi={\ell}_0 \sigma_0 {\ell}_1 \ldots \sigma_{n-1} {\ell}_n \sigma_n \ldots \in {\sf Plays}(G)$ such that for all $i \geq 0$, we have $\alpha(\pi(i))(\sigma_i)>0$ and $\beta(\pi(i),\sigma_i)({\ell}_{i+1})>0$. This set is denoted $\mathsf{outcome}(G,\alpha,\beta)$. The \emph{outcome set} of the deterministic (resp.\ randomized) strategy $\alpha$ for Player~$1$ in $G$ is the set ${\mathsf{Outcome}}_i(G,\alpha)$ of plays $\pi$ such that there exists a deterministic (resp.\ randomized) strategy $\beta$ for Player~$2$ with $\pi=\mathsf{outcome}(G,\alpha,\beta)$ (resp.\ $\pi\in\mathsf{outcome}(G,\alpha,\beta)$). The outcome sets for Player~2 are defined symmetrically.
\noindent{\em Outcome in imperfect-information games.} In an imperfect-information game $G$, the \emph{outcome} of two deterministic strategies $\alpha$ (for Player~$1$) and $\beta$ (for Player~$2$) is the play $\pi=s_0 s_1 s_2 \ldots \in {\sf Plays}(G)$ such that for all $i \geq 0$, we have $s_{i+1} = \alpha(\pi(i))$ if $s_i \in S_1$ and $s_{i+1} = \beta(\pi(i))$ if $s_i \in S_2$. This play is denoted $\mathsf{outcome}(G,\alpha,\beta)$. The \emph{outcome} of two randomized strategies $\alpha$ (for Player~$1$) and $\beta$ (for Player~$2$) in $G$ is the set of plays $\pi=s_0 s_1 s_2 \ldots \in {\sf Plays}(G)$ such that for all $i \geq 0$, we have $\alpha(\pi(i))(s_{i+1})>0$ if $s_i \in S_1$ and $\beta(\pi(i))(s_{i+1})>0$ if $s_i \in S_2$. This set is denoted $\mathsf{outcome}(G,\alpha,\beta)$. In an imperfect-information game, the \emph{outcome set} of the deterministic (resp.\ randomized) strategy $\alpha$ for Player~$1$ in $G$ is the set ${\mathsf{Outcome}}_1(G,\alpha)$ of plays $\pi$ such that there exists a deterministic (resp.\ randomized) strategy $\beta$ for Player~$2$ with $\pi=\mathsf{outcome}(G,\alpha,\beta)$ (resp.\ $\pi\in\mathsf{outcome}(G,\alpha,\beta)$). The outcome sets for Player~2 are defined symmetrically.
\noindent{\em Outcomes in POMDPs.} The outcome of a deterministic strategy $\alpha$ for Player~1 in a POMDP $G$ is the set of plays $\pi=s_0 s_1 s_2 \ldots \in {\sf Plays}(G)$ such that for all $i \geq 0$, we have $s_{i+1} = \alpha(\pi(i))$ if $s_i \in S_1$ and $s_{i+1} \in E(s_i)$ if $s_i \in S_R$. The outcome of a randomized strategy $\alpha$ for Player~1 in a POMDP is the set of plays $\pi=s_0 s_1 s_2 \ldots \in {\sf Plays}(G)$ such that for all $i \geq 0$, we have $ \alpha(\pi(i))(s_{i+1})>0$ if $s_i \in S_1$ and $s_{i+1} \in E(s_i)$ if $s_i \in S_R$. The set of outcomes in a POMDP for a deterministic or a randomized strategy $\alpha$ is denoted as ${\mathsf{Outcome}}_1(G,\alpha)$. \end{comment}
\begin{comment}
\noindent{\em Objectives.} An \emph{objective} for $G$ is a set $\phi$ of infinite sequences of observations and actions, that is, $\phi \subseteq ({\cal{O}}\times \Sigma )^\omega$. A play $\pi = {\ell}_0 \sigma_0 {\ell}_1 \ldots \sigma_{n-1} {\ell}_n \sigma_n \ldots \in {\sf Plays}(G)$ \emph{satisfies} the objective $\phi$, denoted $\pi \models \phi$, if $\gamma^{-1}(\pi) \in \phi$. Objectives are generally Borel measurable: a Borel objective is a Borel set in the Cantor topology on $({\cal{O}}\times\Sigma)^\omega$~\cite{Kechris}. We specifically consider reachability, safety, B\"uchi, coB\"uchi, and parity objectives, all of them being Borel measurable. The parity objectives are a canonical form to express all $\omega$-regular objectives~\cite{Thomas97}. For a play $\pi={\ell}_0\sigma_0{\ell}_1\dots$, we write $\mathsf{Inf}(\pi)$ for the set of observations that appear infinitely often in $\gamma^{-1}(\pi)$, that is, $\mathsf{Inf}(\pi)=\{ o \in {\cal{O}} \mid {\ell}_i \in \gamma(o) \mbox{ for infinitely many } i\mbox{'s} \}$.
\begin{itemize} \item \emph{Reachability and safety objectives.} Given a set ${\cal T} \subseteq {\cal{O}}$ of target observations, the \emph{reachability} objective $\mathsf{Reach}({\cal T})$ requires that an observation in ${\cal T}$ be visited at least once, that is, $\mathsf{Reach}({\cal T})=\set{ {\ell}_0 \sigma_0 {\ell}_1 \sigma_1 \ldots \in {\sf Plays}(G) \mid \exists k \geq 0 \cdot \exists o \in {\cal T}: {\ell}_k \in o}$. Dually, the \emph{safety} objective $\mathsf{Safe}({\cal T})$ requires that only observations in ${\cal T}$ be visited. Formally, $\mathsf{Safe}({\cal T})=\set{ {\ell}_0 \sigma_0 {\ell}_1 \sigma_1 \ldots \in {\sf Plays}(G) \mid \forall k \geq 0 \cdot \exists o \in {\cal T}: {\ell}_k \in \gamma(o)}$. \item \emph{B\"uchi and coB\"uchi objectives.}
The \emph{B\"uchi} objective ${\sf B\ddot{u}chi}({\cal T})$ requires that an observation in ${\cal T}$ be visited infinitely often, that is, ${\sf B\ddot{u}chi}({\cal T})=\set{ \pi \mid \mathsf{Inf}(\pi) \cap {\cal T} \neq \emptyset}$. Dually, the \emph{coB\"uchi} objective $\mathsf{coB\ddot{u}chi}({\cal T})$ requires that only observations in ${\cal T}$ be visited infinitely often.
Formally, $\mathsf{coB\ddot{u}chi}({\cal T}) =\set{\pi \mid \mathsf{Inf}(\pi) \subseteq {\cal T}}$. \item\emph{Parity objectives.} For $d \in \mathbb{N}$, let $p:{\cal{O}} \to \set{0,1,\ldots,d}$ be a \emph{priority function}, which maps each observation to a nonnegative integer priority. The \emph{parity} objective $\mathsf{Parity}(p)$ requires that the minimum priority that appears infinitely often be even. Formally, $\mathsf{Parity}(p)=\set{\pi \mid \min\set{ p(o) \mid o \in \mathsf{Inf}(\pi)} \mbox{ is even} }$. \end{itemize} Note that by definition, for all objectives $\phi$, if $\pi \models \phi$ and $\gamma^{-1}(\pi)=\gamma^{-1}(\pi')$, then $\pi'\models \phi$. Given a B\"uchi objective ${\sf B\ddot{u}chi}({\cal T})$ consider the priority function $p:{\cal{O}} \to \set{0,1}$ such that $p(o)=0$ if $o \in {\cal T}$, and~1 otherwise; then we have $\mathsf{Parity}(p)={\sf B\ddot{u}chi}({\cal T})$. Similarly, given a coB\"uchi objective $\mathsf{coB\ddot{u}chi}({\cal T})$ consider the priority function $p:{\cal{O}} \to \set{1,2}$ such that $p(o)=2$ if $o \in {\cal T}$, and~1 otherwise; then we have $\mathsf{Parity}(p)=\mathsf{coB\ddot{u}chi}({\cal T})$. Hence B\"uchi and coB\"uchi objectives are special cases of parity objectives with two priorities. \end{comment}
\noindent{\em Objectives.} An \emph{objective} for $G$ is a set $\phi$ of infinite sequences of states and actions, that is, $\phi \subseteq (L \times \Sigma)^\omega$.
We consider objectives that are Borel measurable, i.e., sets in the Cantor topology on $(L \times \Sigma)^\omega$~\cite{Kechris}. We specifically consider reachability, safety, B\"uchi, coB\"uchi, and parity objectives, all of them being Borel measurable. The parity objectives are a canonical form to express all $\omega$-regular objectives~\cite{Thomas97}. For a play $\pi={\ell}_0\sigma_0{\ell}_1\dots$, we denote by $\mathsf{Inf}(\pi) = \{ {\ell} \in L \mid {\ell} = {\ell}_i \mbox{ for infinitely many } i\mbox{'s} \}$ the set of states that appear infinitely often in $\pi$.
\begin{itemize}
\item \emph{Reachability and safety objectives.}
Given a set ${\cal T} \subseteq L$ of target states, the \emph{reachability} objective
$\mathsf{Reach}({\cal T}) = \set{ {\ell}_0 \sigma_0 {\ell}_1 \sigma_1 \ldots \in {\sf Plays}(G) \mid \exists k \geq 0: {\ell}_k \in {\cal T}}$
requires that a target state in ${\cal T}$ be visited at least once.
Dually, the \emph{safety} objective $\mathsf{Safe}({\cal T}) = \set{ {\ell}_0 \sigma_0 {\ell}_1 \sigma_1 \ldots \in {\sf Plays}(G) \mid
\forall k \geq 0: {\ell}_k \in {\cal T}}$
requires that only states in ${\cal T}$ be visited; the objective $\mathsf{Until}({\cal T}_1,{\cal T}_2) =
\{ {\ell}_0 \sigma_0 {\ell}_1 \sigma_1 \ldots \in {\sf Plays}(G) \mid \exists k \geq 0: {\ell}_k \in {\cal T}_2 \land \forall j \leq k: {\ell}_j \in {\cal T}_1\}$
requires that only states in ${\cal T}_1$ be visited before a state in ${\cal T}_2$ is visited;
\item \emph{B\"uchi and coB\"uchi objectives.}
The \emph{B\"uchi} objective ${\sf B\ddot{u}chi}({\cal T}) = \set{ \pi \mid \mathsf{Inf}(\pi) \cap {\cal T} \neq \emptyset}$
requires that a state
in ${\cal T}$ be visited infinitely often.
Dually, the \emph{coB\"uchi} objective $\mathsf{coB\ddot{u}chi}({\cal T}) =\set{\pi \mid \mathsf{Inf}(\pi) \subseteq {\cal T}}$
requires that only states
in ${\cal T}$ be visited infinitely often; and
\item\emph{Parity objectives.}
For $d \in \mathbb{N}$, let $p:L \to \set{0,1,\ldots,d}$ be a
\emph{priority function} that maps each state
to a nonnegative integer priority.
The \emph{parity} objective $\mathsf{Parity}(p) = \set{\pi \mid \min\set{ p({\ell}) \mid {\ell} \in \mathsf{Inf}(\pi)}
\text{ is even} }$ requires that the smallest priority that appears infinitely often be even. \end{itemize}
Note that the objectives ${\sf B\ddot{u}chi}({\cal T})$ and $\mathsf{coB\ddot{u}chi}({\cal T})$ are special cases of parity objectives defined by respective priority functions $p_1,p_2$ such that $p_1({\ell}) = 0$ and $p_2({\ell}) = 2$ if ${\ell} \in {\cal T}$, and $p_1({\ell}) = p_2({\ell}) = 1$ otherwise. An objective $\phi$ is \emph{visible} if it depends only on the observations; formally, $\phi$ is visible if, whenever $\pi \in \phi$ and $\mathsf{obs}(\pi)=\mathsf{obs}(\pi')$, then $\pi'\in \phi$. In this work, all our upper bound results are for the general parity objectives (not necessarily visible), and all the lower bound results for {\sf POMDP} s are for the special case of visible objectives (and hence the lower bounds also hold for general objectives).
\begin{comment}
\noindent{\em Outcomes and sure winning.} In an imperfect-information game $G$, the \emph{outcome} of two randomized strategies $\alpha$ (for Player~$1$) and $\beta$ (for Player~$2$) in $G$ from a state ${\ell}$ is the set of plays $\pi={\ell}_0 \sigma_0 {\ell}_1 \ldots \sigma_{n-1} {\ell}_n \sigma_n \ldots \in {\sf Plays}(G)$ such that (a)~${\ell}_0={\ell}$ and (b)~for all $i \geq 0$, we have $\alpha(\pi(i))(\sigma_i)>0$ and $\beta(\pi(i),\sigma_i)({\ell}_{i+1})>0$, where $\pi(i)$ is the prefix of length $i$ of $\pi$. This set is denoted $\mathsf{outcome}(G,\alpha,\beta,{\ell})$. The \emph{outcome set} of the randomized strategy $\alpha$ for Player~$1$ in $G$ is the set ${\mathsf{Outcome}}_1(G,\alpha)$ of plays $\pi$ such that there exists a randomized strategy $\beta$ for Player~$2$ with $\pi\in\mathsf{outcome}(G,\alpha,\beta)$. The outcome sets for Player~2 are defined symmetrically. The outcome of a randomized strategy $\alpha$ for Player~1 in a POMDP from a state ${\ell}$ is the set of plays is the set of plays $\pi={\ell}_0 \sigma_0 {\ell}_1 \ldots \sigma_{n-1} {\ell}_n \sigma_n \ldots \in {\sf Plays}(G)$ such that (a)~${\ell}_0={\ell}$ and (b)~for all $i \geq 0$, we have $\alpha(\pi(i))(\sigma_i)>0$ and ${\delta}({\ell}_i,\sigma_i)({\ell}_{i+1})>0$. The set of outcomes in a POMDP for a randomized strategy $\alpha$ from ${\ell}$ is denoted as ${\mathsf{Outcome}}_1(G,\alpha,{\ell})$. Given an imperfect-information game or an POMDP $G$ and a state ${\ell}$, a strategy $\alpha$ for Player~$1$ is \emph{sure winning} for the objective $\phi$ from ${\ell}$ if ${\mathsf{Outcome}}_1(G,\alpha,{\ell}) \subseteq \phi$ We also say that state ${\ell}$ is sure winning for $\phi$. \end{comment}
\noindent{\em Almost-sure and positive winning.} An \emph{event} is a measurable set of plays, and given strategies $\alpha$ and $\beta$ for the two players (resp., a strategy $\alpha$ for Player~1 in MDPs), the probabilities of events are uniquely defined~\cite{Var85}.
For a Borel objective~$\phi$, we denote by $\mathrm{Pr}_{{\ell}}^{\alpha,\beta}(\phi)$ (resp., $\mathrm{Pr}_{{\ell}}^{\alpha}(\phi)$ for MDPs) the probability that $\phi$ is satisfied from the starting state~${\ell}$ given the strategies~$\alpha$ and $\beta$ (resp., given the strategy~$\alpha$). Given a game $G$ and a state ${\ell}$, a strategy $\alpha$ for Player~$1$ is \emph{almost-sure winning} (resp., \emph{positive winning}) for the objective $\phi$ from ${\ell}$ if for all randomized strategies $\beta$ for Player~$2$, we have $\mathrm{Pr}_{{\ell}}^{\alpha,\beta}(\phi)=1$ (resp., $\mathrm{Pr}_{{\ell}}^{\alpha,\beta}(\phi)>0$). Given an MDP $G$ and a state ${\ell}$, a strategy $\alpha$ for Player~$1$ is almost-sure winning (resp. positive winning) for the objective $\phi$ from ${\ell}$ if we have $\mathrm{Pr}_{{\ell}}^{\alpha}(\phi)=1$ (resp., $\mathrm{Pr}_{{\ell}}^{\alpha}(\phi)>0$). We also say that state ${\ell}$ is almost-sure winning, or positive winning for $\phi$ respectively. We are interested in the problems of deciding the existence of an observation-based strategy for Player~1 that is almost-sure winning (resp., positive winning) from a given state ${\ell}$.
\begin{comment}
\begin{theorem}[Determinacy] {\rm\cite{Mar75}} \label{thrm:boreldeterminacy} For all perfect-information game structures $G$ and all Borel objectives $\phi$, either there exists a deterministic sure-winning strategy for Player~$1$ for the objective $\phi$, or there exists a deterministic sure-winning strategy for Player~$2$ for the complementary objective ${\sf Plays}(G) \setminus \phi$. \end{theorem}
Notice that deterministic strategies suffice for sure winning a game: given a randomized strategy $\alpha$ for Player~$1$, let $\alpha^D$ be the deterministic strategy such that for all $\rho \in {\sf Prefs}(G)$, the strategy $\alpha^D(\rho)$ chooses an action from $\Supp(\alpha(\rho))$. Then ${\mathsf{Outcome}}_1(G,\alpha^D) \subseteq {\mathsf{Outcome}}_1(G,\alpha)$, and thus, if $\alpha$ is sure winning, then so is~$\alpha^D$. The result also holds for observation-based strategies and for perfect-information games. However, for almost winning, randomized strategies are more powerful than deterministic strategies as shown by Example~\ref{ex:example-one}.
\end{comment}
\section{Upper Bounds for the Qualitative Analysis of {\sf POMDP} s} In this section, we present upper bounds for the qualitative analysis of {\sf POMDP} s. We first describe the known results. For qualitative analysis of MDPs, polynomial time upper bounds are known for all parity objectives~\cite{luca-thesis,CJH04}. It follows from the results of~\cite{CDHR07,BBG08} that the decision problems for almost-sure winning for {\sf POMDP} s with reachability, safety, and B\"uchi objectives can be solved in EXPTIME. It also follows from the results of~\cite{BBG08} that the decision problem for almost-sure winning with coB\"uchi objectives and for positive winning with B\"uchi objectives is undecidable if the strategies are restricted to be pure, and the results of~\cite{GIMBERT:2009:HAL-00403463:3} shows that the problem remains undecidable even if randomized strategies are considered. In this section, we complete the results on upper bounds for the qualitative analysis of {\sf POMDP} s: we present complexity upper bounds for the decision problems of positive winning with reachability, safety and coB\"uchi objectives. The following result for reachability objectives is simple, and for a complete and systematic analysis we present the proof.
\begin{theorem}\label{thrm_pos_reach} Given a {\sf POMDP}\/ $G$ with a reachability objective and a starting state ${\ell}$, the problem of deciding whether there is a positive winning strategy from ${\ell}$ in $G$ is NLOGSPACE-complete.
\end{theorem} \begin{myProof} The NLOGSPACE-completeness result for positive reachability for MDPs follows from reductions to and from graph reachability.
\noindent{\em Reduction to graph reachability.} Given a {\sf POMDP}\/ $G= \tuple{L,\Sigma,{\delta},{\cal{O}}}$ and a set of target states ${\cal T} \subseteq L$, consider the graph $\overline{G}=\tuple{L,E}$ where $({\ell},{\ell}') \in E$ if there exists an action $\sigma \in \Sigma$ such that ${\delta}({\ell},\sigma)({\ell}')>0$. Let ${\ell}$ be a starting state, then the following assertions hold: (a)~if there is a path $\pi$ in $\overline{G}$ from ${\ell}$ to a state $t \in {\cal T}$, then the randomized memoryless strategy for Player~1 in $G$ that plays all actions uniformly at random ensures that the path $\pi$ is executed in $G$ with positive probability (i.e., ensures positive winning for $\mathsf{Reach}({\cal T})$ in $G$ from ${\ell}$); and (b)~if there is no path in $\overline{G}$ to reach $T$ from ${\ell}$, then there is no strategy (and hence no observation-based strategy) for Player~1 in $G$ to achieve $\mathsf{Reach}({\cal T})$. This shows that positive winning in {\sf POMDP} s can be decided in NLOGSPACE. Graphs are a special case of {\sf POMDP} s and hence graph reachability can be reduced to reachability with positive probability in {\sf POMDP} s, therefore the problem is NLOGSPACE-complete.
\end{myProof}
\noindent{\bf Positive winning for safety and coB\"uchi objectives.} We now show that the decision problem for positive winning with safety and coB\"uchi objectives for {\sf POMDP} s can be solved in EXPTIME. We first show with an example that the simple approach of reduction to a perfect-information MDP by subset construction and solving the perfect information MDP with safety objective for positive winning does not yield the desired result.
\begin{example} Consider the {\sf POMDP}\/ shown in \figurename~\ref{fig:subset-pomdp}: in every state there exists only one action (which we omit for simplicity). In other words, we have a partially observable Markov chain. States $0$, $1$, and $2$ are safe states and form observation $o_1$, while state $3$ forms observation $o_2$ (which is not in the safe set). The state $0$ in $G$ is positive winning for the safety objective as with positive probability the state $2$ is reached and then the state $2$ is visited forever. In contrast, consider the perfect information MDP $G^{{\sf K}}$ obtained from $G$ by subset construction (in this case $G^{{\sf K}}$ is a Markov chain). In $G^{{\sf K}}$ from the state $\{1,2\}$, the possible successors are $1,2,$ and $3$, and since the observations are different at $1$ and $2$, as compared to $3$, the successors of $\{1,2\}$ are $\{1,2\}$ and $\{3\}$. The reachable set of states in $G^{{\sf K}}$ from the state $\{0\}$ is shown in \figurename~\ref{fig:subset-pomdp}. In $G^{{\sf K}}$, the state $\{0\}$ is not positive winning: the state $\{3\}$ is the only recurrent state reachable from $\{0\}$ and hence from the state $\{0\}$, with probability~1, the state $\{3\}$ is reached and $\{3\}$ is not a safe state. Note that all this holds regardless of the precise value of nonzero probabilities. \begin{figure}\label{fig:subset-pomdp}
\end{figure} \origQED\global\def\qed{} \end{example}
Our result for positive safety and coB\"uchi objectives is based on the computation of almost-sure winning states for safety objectives, and on the following lemma.
\begin{lemma}\label{theo:winningK-then-winning} Let $G=\tuple{L,\Sigma,{\delta},{\cal{O}}}$ be a {\sf POMDP}\/ and let ${\cal T} \subseteq L$ be the set of target states.
If Player~$1$ has an observation-based strategy in $G$ to satisfy $\mathsf{Safe}({\cal T})$ with positive probability from some state ${\ell}$, then there exists a state ${\ell}'$ such that (a) Player~$1$ has an observation-based strategy in $G$ to satisfy $\mathsf{Until}({\cal T},\{{\ell}'\})$ with positive probability from ${\ell}$, and (b) Player~$1$ has an observation-based almost-sure winning strategy in $G$ for $\mathsf{Safe}({\cal T})$ from ${\ell}'$. \end{lemma}
\begin{myProof} We assume without loss of generality that the non-safe states in $G$ are absorbing. Assume that Player~$1$ has an observation-based positive winning strategy $\alpha$ in $G$ for the objective $\mathsf{Safe}({\cal T})$ from~${\ell}$, and towards a contradiction assume that for all states ${\ell}'$ reachable from ${\ell}$ with positive probability using $\alpha$ in $G$, Player~$1$ has no observation-based almost-sure winning strategy for $\mathsf{Safe}({\cal T})$ from~${\ell}'$. A standard argument shows that from every such state ${\ell}'$, regardless of the observation-based strategy of Player~$1$, the probability to stay safe within the next $n$ steps is at most $1 - \eta^n$ where $\eta$ is the least non-zero probability in $G$ and $n$ is the number of states in $G$. Since under strategy $\alpha$, every reachable state has this property, the probability to stay safe within $k\cdot n$ steps is at most $(1 - \eta^n)^k$. This value tends to $0$ when $k \to \infty$, therefore the probability to stay safe using $\alpha$ from ${\ell}$ is $0$, a contradiction. Hence, there exists a state ${\ell}'$ which is almost-sure winning for Player~$1$ (using observation-based strategy $\alpha$) and such that ${\ell}'$ is reached with positive probability from ${\ell}$ while staying in ${\cal T}$ (again using $\alpha$). \end{myProof}
By Lemma~\ref{theo:winningK-then-winning}, positive winning states can be computed as the set of states from which Player~$1$ can force with positive probability to reach an almost-sure winning state while visiting only safe states. Almost-sure winning states can be computed using the following subset construction.
Given a {\sf POMDP}\/ $G=\tuple{L,\Sigma,{\delta},{\cal{O}}}$ and a set ${\cal T} \subseteq L$ of states, the \emph{knowledge-based subset construction} of $G$ is the game of perfect observation \begin{center} $G^{{\sf K}} = \tuple{{\mathcal{L}}, \Sigma, {\delta}^{{\sf K}}}$, \end{center} where ${\mathcal{L}} = 2^L \backslash \{\emptyset\}$, and for all $s_1, s_2 \in {\mathcal{L}}$ (in particular $s_2 \neq \emptyset$) and $\sigma \in \Sigma$, we have $(s_1, \sigma,s_2) \in {\delta}^{{\sf K}}$ iff there exists an observation $o \in {\cal{O}}$ such that either $s_2 = \mathsf{Post}^G_\sigma(s_1) \cap o \cap {\cal T}$, or $s_2 = (\mathsf{Post}^G_\sigma(s_1) \cap o) \setminus {\cal T}$.
We refer to states in $G^{{\sf K}}$ as \emph{cells}. The following result is established using standard techniques (see e.g., Lemma 3.2 and Lemma 3.3 in~\cite{CDHR07}). and the fact that almost-sure winning and sure winning (sure winning is winning with certainty as compared to winning with probability~1 for almost-sure winning, see~\cite{CDHR07} for details of sure winning) coincide for safety objectives.
\begin{lemma}\label{lem:sure-almost-winning} Let $G=\tuple{L,\Sigma,{\delta},{\cal{O}}}$ be a {\sf POMDP}\/ and ${\cal T} \subseteq L$ a set of target states. Let $G^{{\sf K}}$ be the subset construction and $F_{{\cal T}} = \{s \subseteq {\cal T}\}$ the set of safe cells.
Player~$1$ has an almost-sure winning observation-based strategy in $G$ for $\mathsf{Safe}({\cal T})$ from ${\ell}$ if and only if Player~$1$ has an almost-sure winning strategy in $G^{{\sf K}}$ for $\mathsf{Safe}(F)$ from cell $\{{\ell}\}$. \end{lemma}
\begin{remark} Lemma~\ref{lem:sure-almost-winning} also holds if we replace almost-sure winning by sure winning, since for safety objectives almost-sure and sure winning coincide. \end{remark}
\begin{comment} \begin{lemma}\label{theo:winningK-then-winning} Let $G=\tuple{L,\Sigma,{\delta},{\cal{O}}}$ be a {\sf POMDP} and $G^{{\sf K}}$ the MDP constructed by subset construction. Let ${\cal T}$ be the set of target observations, and let $F =\set{s \subseteq L \mid s \subseteq \gamma(o), o \in {\cal T}}$. Let $W$ be the set of cells $s$ in $G^{{\sf K}}$ such that Player~1 has a positive winning strategy from $s$ for the objective $\mathsf{Safe}(F)$. The following assertions hold: \begin{enumerate} \item If the initial knowledge for Player~$1$ is a cell in $W$ in $G$, then there is an observation-based strategy in $G$ to satisfy $\mathsf{Safe}({\cal T})$ with positive probability. \item If the initial knowledge for Player~1 is a cell in $2^L \setminus W$, then there is no observation-based strategy for Player~1 to satisfy $\mathsf{Safe}({\cal T})$ with positive probability. \end{enumerate} \end{lemma}
\begin{myProof} We present the proof in two parts. Without loss of generality, we assume that every state with observations not in ${\cal T}$ is absorbing, and hence cells in $2^L \setminus F$ are also absorbing.
\begin{enumerate} \item Since $G^{{\sf K}}$ is an MDP of perfect observation, it follows that if there is a positive winning strategy for the safety objective $\mathsf{Safe}(F)$, then there is a pure memoryless strategy in $G^{{\sf K}}$ that is also positive winning for $\mathsf{Safe}(F)$~\cite{luca-thesis,CJH04}. Let $\alpha^{{\sf K}}$ be a pure memoryless positive winning strategy for Player~$1$ in $G^{{\sf K}}$ for the objective $\mathsf{Safe}(F)$ from the cells in $W$. Define $\alpha^o$ a strategy for Player~$1$ in $G$ as follows: for every $\rho \in {\sf Prefs}(G)$, let $\alpha^o(\rho) = \alpha^{{\sf K}}(\rho^{{\sf K}})$ where $\rho^{{\sf K}}$ is defined from $\rho= {\ell}_0 \sigma_0 {\ell}_1 \dots \sigma_{n-1} {\ell}_n$ by $\rho^{{\sf K}} = s_0 \sigma_0 s_1 \dots \sigma_{n-1} s_n$ where $s_i = {\sf K}(\gamma^{-1}({\ell}_0 \sigma_0 {\ell}_1 \dots \sigma_{i-1} {\ell}_i))$ for each $0 \leq i \leq n$. Clearly, $\alpha^o$ is a pure observation-based strategy as $\gamma^{-1}(\rho) = \gamma^{-1}(\rho')$ implies $\rho^{{\sf K}} = \rho'^{{\sf K}}$. Since the strategy $\alpha^{{\sf K}}$ is pure memoryless in $G^{{\sf K}}$, the strategy $\alpha^o$ is a finite-memory pure strategy with at most exponential ($2^{O(\abs{L})}$) memory. Once the strategy $\alpha^{{\sf K}}$ is fixed in $G^{{\sf K}}$ we obtain a Markov chain. Since $\alpha^{{\sf K}}$ is positive winning, it follows that in the Markov chain obtained, from every cell in $W$ a closed recurrent set $C \subseteq W \subseteq F$ is reached with positive probability. Hence if the strategy $\alpha^o$ constructed from $\alpha^{{\sf K}}$ is fixed in $G$ and the initial knowledge is a cell in $W$, then it ensures the following: (a)~there exists a subset $O$ of states in $L$ that are labeled by observations in ${\cal T}$, and once the set $O$ is reached, then the set $O$ is never left (this corresponds to the closed recurrent set $C$ in $G^{{\sf K}}$); and (b)~with positive probability a path is executed that goes through only states labeled by observations in ${\cal T}$, and reaches the set $O$. Hence the strategy $\alpha^o$ is positive winning in $G$, given the initial knowledge is a cell in~$W$.
\item Let $\overline{W}=2^L \setminus W$ be the set of cells in $G^{{\sf K}}$ such that there is no positive winning strategy for the objective $\mathsf{Safe}(F)$. Hence from any cell in $\overline{W}$, for any knowledge-based strategy for Player~1 the following property hold: the play remains in $\overline{W}$ and reaches cells in $\overline{W} \setminus F$ with some positive probability $\eta>0$ in $2^{\abs{L}}$ steps. It follows if the initial knowledge for Player~1 in $G$ is a cell in $\overline{W}$, then for any observation-based strategy in $G$, the probability to reach an observation in ${\cal{O}} \setminus{\cal T}$ in $k \cdot 2^{\abs{L}}$ steps is at least $1-(1-\eta')^k$, for some $\eta'>0$. As $k$ goes to $\infty$, the value of $1-(1-\eta')^k$ goes to~1. Hence the probability to stay safe in ${\cal T}$ is~0 for any observation-based Player~1 strategy, with the initial knowledge in $\overline{W}$. \end{enumerate} The result follows. \end{myProof}
In the case of {\sf POMDP} s, given the starting state is a state ${\ell} \in L$, the initial knowledge is the cell $\gamma^{-1}({\ell})$. Hence the decision problem for positive winning in {\sf POMDP} s with safety objectives can be solved by solving the same problem for MDPs with perfect observation of exponential size. Since positive winning in MDPs of perfect observation with safety objectives can be solved in polynomial time~\cite{luca-thesis,CJH04}, we obtain an EXPTIME upper bound for {\sf POMDP} s. \end{comment}
\begin{theorem}\label{thrm_pos_safe} Given a {\sf POMDP}\/ $G$ with a safety objective and a starting state ${\ell}$, the problem of deciding whether there exists a positive winning observation-based strategy from ${\ell}$ can be solved in EXPTIME.
\end{theorem}
\begin{myProof} The almost-sure winning states in $G$ for a safety objective (with observation-based strategy) can be computed in exponential time using the subset construction (by Lemma~\ref{lem:sure-almost-winning} and~\cite{CDHR07}). Then, given the set $W$ of cells that are almost-sure winning in $G^{{\sf K}}$, let ${\cal T}_W = \{{\ell} \in s \mid s \in W \}$ be the almost-sure winning states in $G$. We can compute the states from which Player~$1$ can force ${\cal T}_W$ to be reached with positive probability while staying within the safe states using standard graph analysis algorithms, as in Lemma~\ref{thrm_pos_reach}. Clearly such states are positive winning in $G$, and by Lemma~\ref{theo:winningK-then-winning} all positive winning states in $G$ are obtained in this way. This gives an EXPTIME algorithm to decide from which states there exists a positive winning observation-based strategy for safety objectives. \end{myProof}
\noindent{\bf Algorithms.} The complexity bound of Theorem~\ref{thrm_pos_safe} has been established previously in~\cite{GriponS09}, using an extension of the knowledge-based subset construction which is not necessary (where the state space is $L \times 2^L$). Our proof is simpler and also yield efficient and symbolic algorithms: efficient anti-chain based symbolic algorithm for almost-sure winning for safety objectives can be obtained from~\cite{CDHR07}, and positive reachability is simple graph reachability.
\begin{comment} \noindent{\bf Positive winning for coB\"uchi objectives.}
We now show that we can solve positive winning for {\sf POMDP} s with coB\"uchi objectives by iteratively solving for positive winning with reachability and safety objectives in {\sf POMDP} s. Let $G$ be a {\sf POMDP}\/ with a coB\"uchi objective $\mathsf{coB\ddot{u}chi}({\cal T})$, where ${\cal T} \subseteq {\cal{O}}$. We construct the MDP $G^{{\sf K}}$ of perfect observation, and let $C=\set{s \subseteq L \mid s \subseteq \gamma(o), o \in {\cal T}}$. We consider the positive winning for the coB\"uchi objective $\mathsf{coB\ddot{u}chi}(C)$ in $G^{{\sf K}}$. The set $W$ of positive winning cells in $G^{{\sf K}}$ is obtained as follows: \begin{enumerate} \item let $W_0=\emptyset$; \item we obtain $W_{i+1}$ from $W_i$ as follows: let $Z_i$ be the set of cells in $G^{{\sf K}}$ such that Player~1 can ensure staying safe in $C \cup W_i$ with positive probability, and $W_{i+1}$ is obtained as the set of states that can reach $Z_i$ with positive probability. \end{enumerate} It follows from above that if the current knowledge is a cell in $W_{i+1}$, then either $W_i$ is reached with positive probability or the play eventually only visits states in $C$. It follows from the proof of correctness for positive winning in {\sf POMDP} s with safety objective, that if the current knowledge is a cell in $W_{i+1}$, then there is an observation-based strategy for Player~1 to ensure $\mathsf{coB\ddot{u}chi}({\cal T})$. Let $W$ be the fixpoint of the iteration, i.e., for some $k$ we have $W_{k}= W_{k+1}= W$. Let $\overline{W}=2^L \setminus W$. Then the following assertions hold. \begin{enumerate} \item From any cell $\overline{W}$, Player~1 cannot ensure positive probability to stay safe in $C \cup W$. Otherwise, such a cell would have been included in $Z_{k+1}$ and hence it would contradict that $W_k=W_{k+1}$. Hence for every Player~1 knowledge-based strategy, if the initial knowledge is a cell in $\overline{W}$, then the set $(2^L\setminus C) \cap \overline{W}$ is reached with probability~1. It follows that if the current knowledge is a cell in $\overline{W}$, then for any observation-based strategy in $G$, observations in the set ${\cal{O}} \setminus {\cal T}$ is reached with probability~1.
\item From every cell $\overline{W}$, Player~1 cannot ensure to reach $W$ with positive probability. Hence for every knowledge-based strategy for Player~1, if the initial knowledge is a cell in $\overline{W}$, then the play stays safe in $\overline{W}$ with probability~1. Hence given a knowledge-based strategy for Player~1, from every cell in $\overline{W}$, the set $(2^L \setminus C) \cap \overline{W}$ is reached with probability~1 and the game stays safe in $\overline{W}$. It follows that the set $(2^L \setminus C) \cap \overline{W}$ is visited infinitely often with probability~1, and this ensures that from $\overline{W}$ the coB\"uchi condition is falsified with probability~1. Hence if the initial knowledge is a cell in $\overline{W}$, then for any observation-based strategy in $G$, the coB\"uchi objective $\mathsf{coB\ddot{u}chi}({\cal T})$ is falsified with probability~1. \end{enumerate} Hence by iteratively solving positive winning in {\sf POMDP} s with reachability and safety objectives, the positive winning in {\sf POMDP} s with coB\"uchi objectives can be achieved.
Hence we have the following result. \end{comment}
The positive winning states for a coB\"uchi objective are computed as the set of almost-sure winning states for safety that can be reached with positive probability.
\begin{theorem}\label{thrm_pos_cobuchi} Given a {\sf POMDP}\/ $G$ with a coB\"uchi objective and a starting state ${\ell}$, the problem of deciding whether there exists a positive winning observation-based strategy from ${\ell}$ can be solved in EXPTIME. \end{theorem}
\begin{myProof} Let $\mathsf{coB\ddot{u}chi}({\cal T})$ be a coB\"uchi objective in $G = \tuple{L,\Sigma,{\delta},{\cal{O}}}$. As in the proof of Theorem~\ref{thrm_pos_safe}, we compute in exponential time the set ${\cal T}_W$ of almost-sure winning states in $G$ for $\mathsf{Safe}({\cal T})$, and using Lemma~\ref{thrm_pos_reach} the set $W$ of states from which Player~$1$ is positive winning for $\mathsf{Reach}({\cal T}_W)$. Clearly, all states in $W$ are positive winning for $\mathsf{coB\ddot{u}chi}({\cal T})$, and $W$ can be computed in EXPTIME. We argue that for all states ${\ell} \not \in W$, Player~$1$ is not positive winning for $\mathsf{coB\ddot{u}chi}({\cal T})$ from ${\ell}$. Note that ${\delta}({\ell},\sigma)({\ell}') = 0$ for all ${\ell} \not \in W$, ${\ell}' \in W$, and $\sigma \in \Sigma$, and thus there are no almost-sure winning states for $\mathsf{Safe}({\cal T})$ in $G$ reachable from $L \setminus W$ with positive probability, regardless of the strategy of Player~$1$. Therefore, by an argument similar to the proof of Lemma~\ref{theo:winningK-then-winning}, for all observation-based strategies for Player~$1$, from every state ${\ell} \not \in W$, the set $L \setminus {\cal T}$ is reached with probability~$1$ and the event ${\sf B\ddot{u}chi}(L \setminus {\cal T})$ has probability~$1$. The result follows. \end{myProof}
\section{Lower Bounds for the Qualitative Analysis of {\sf POMDP} s}
In this section we present lower bounds for the qualitative analysis of {\sf POMDP} s. We first present the lower bounds for MDPs with perfect observation.
\noindent{\bf Lower bounds for MDPs with perfect observations.} In the previous section we argued that for reachability objectives even in {\sf POMDP} s the positive winning problem is NLOGSPACE-complete. For safety objectives and almost-sure winning it is known that an MDP can be equivalently considered as a game where Player~2 makes choices of the successors from the support of the probability distribution of the transition function, and the almost-sure winning set is the same in the MDP and the game. Similarly, there is a reduction of games of perfect observations to MDPs of perfect observation for almost-sure winning with safety objectives. The problem of almost-sure winning in games of perfect observation is alternating reachability and is PTIME-complete~\cite{Beeri,Immerman81},. It follows that almost-sure winning for safety objectives in MDPs is PTIME-complete. We now show that the almost-sure winning problem for reachability and the positive winning problem for safety objectives is PTIME-complete for MDPs with perfect observation.
\noindent{\bf Reduction from the {\sc Circuit-Value-Problem}.} Let $N=\set{1,2,\ldots,n}$ be a set of AND and OR gates, and $I$ be a set of inputs. The set of inputs is partitioned into $I_0$ and $I_1$; $I_0$ is the set of inputs set to~0 (false) and $I_1$ is the set of inputs set to~1 (true). Every gate receives two inputs and produces one output; the inputs of a gate are outputs of another gate or an input from the set $I$. The connection graph of the circuit must be acyclic. Let the gate represented by the node~1 be the output node. The {\sc Circuit-Value-Problem} (CVP) is to decide whether the output is~1 or~0. This problem is PTIME-complete. We present a reduction of CVP to MDPs with perfect observation for almost-sure winning with reachability, and positive winning with safety objectives. \begin{enumerate} \item \emph{Almost-sure reachability.} Given the CVP, we construct the MDP of perfect observation as follows: (a)~the set of states is $N \cup I$; (b)~the action set is $\Sigma=\set{l,r}$; (c)~the transition function is as follows: every node in $I$ is absorbing, and for a state that represents a gate, (i)~if it is an OR gate, then for the action $l$ the left input gate is chosen with probability~1, and for the action $r$ the right input gate is chosen with probability~1; and (ii)~if it is an AND gate, then irrespective of the action, the left and right input gate are chosen with probability~$1/2$. The output of the CVP from node~1 is~1 iff the set $I_1$ is reached from the state~1 in the MDP with probability~1 (i.e., the state~1 is almost-sure winning for the reachability objective $\mathsf{Reach}(I_1)$.)
\item \emph{Positive safety.} For positive winning with safety objectives, we take the CVP, apply the same reduction as for almost-sure reachability with the following modifications: every state in $I_0$ remains absorbing and from every state in $I_1$ the next state is the starting state~1 with probability~1 irrespective of the action. The set of safety target is the set $I_1 \cup N$. If the output of the CVP problem is~1, then from the starting state the set $I_1$ is reached with probability~1, and hence the safety objective with the target $N \cup I_1$ is ensured with probability~1. If the output of the CVP problem is~0, then from the starting state the set $I_0$ is reached with positive probability $\eta>0$ in $n$ steps against all strategies. Since from every state in $I_1$ the successor state is the state $1$, it follows that the probability to reach $I_0$ from the starting state~1 in $k\cdot (n+1)$ steps is at least $1-(1-\eta)^k$, and this goes to~1 as $k$ goes to $\infty$. Hence it follows that from state~1, the answer to the positive winning for the safety objective $\mathsf{Safe}(N \cup I_1)$ is YES iff the output to the CVP is~1. \end{enumerate} From the above results it also follows that almost-sure and positive B\"uchi and coB\"uchi objectives are PTIME-hard (and PTIME-completeness follows from the known polynomial time algorithms for qualitative analysis of MDPs with parity objectives~\cite{CJH04,luca-thesis}).
\begin{theorem} Given an MDP $G$ of perfect observation, the following assertions hold: (a)~the positive winning problem for reachability objectives is NLOGSPACE-complete, and the positive winning problem for safety, B\"uchi, coB\"uchi and parity objectives is PTIME-complete; and (b)~the almost-sure winning problem for reachability, safety, B\"uchi, coB\"uchi and parity objectives is PTIME-complete. \end{theorem}
\noindent{\bf Lower bounds for {\sf POMDP} s.} We have already shown that positive winning with reachability objectives in {\sf POMDP} s is NLOGSPACE-complete. As in the case of MDPs with perfect observation, for safety objectives and almost-sure winning a {\sf POMDP}\/ can be equivalently considered as a game of partial observation where Player~2 makes choices of the successors from the support of the probability distribution of the transition function, and the almost-sure winning set is the same in the {\sf POMDP}\/ and the game. Since the problem of almost-sure winning in games of partial observation with safety objective is EXPTIME-complete~\cite{BD08}, the EXPTIME-completeness result follows. We now show that almost-sure winning with reachability objectives and positive winning with safety objectives is EXPTIME-complete. Before the result we first present a discussion on polynomial-space alternating Turing machines (ATM).
\noindent{\em Discussion.} Let $M$ be a polynomial-space ATM and let $w$ be an input word. Then, there is an exponential bound on the number of configurations of the machine. Hence if $M$ can accept the word $w$, then it can do so within some $k_{\abs{w}}$ steps, where $\abs{w}$ is the length of the word $w$, and $k_{\abs{w}}$ is bounded by an exponential in $\abs{w}$. We construct an equivalent polynomial-space ATM $M'$ that behaves as $M$ but keeps track (in polynomial space) of the number of steps executed by $M$, and given a word $\abs{w}$, if the number of steps reaches $k_{\abs{w}}$ without accepting, then the word is rejected. The machine $M'$ is equivalent to $M$ and reaches the accepting or rejecting states in a number of steps bounded by an exponential in the length of the input word. The problem of deciding, given a polynomial-space ATM $M$ and a word $w$, whether $M$ accepts $w$ is EXPTIME-complete.
\noindent{\bf Reduction from Alternating PSPACE Turing machine.} Let $M$ be a polynomial-space ATM such that for every input word $w$, the accepting or the rejecting state is reached within exponential steps in $\abs{w}$. A polynomial-time reduction $R_G$ of a polynomial-space ATM $M$ and an input word $w$ to a game $G = R_G(M,w)$ of partial observation is given in~\cite{CDHR07} such that (a)~there is a special accepting state in $G$, and (b)~$M$ accepts $w$ iff there is an observation-based strategy for Player~1 in $G$ to reach the accepting state with probability~1. If the above reduction is applied to $M$, then the game structure satisfies the following additional properties: there is a special rejecting state that is absorbing, and for every observation-based strategy for Player~1, either (a)~against all Player~2 strategies the accepting state is reached with probability~1; or (b)~there is a pure Player~2 strategy that reaches the rejecting state with positive probability $\eta>0$ in $2^{\abs{L}}$ steps and the accepting or the rejecting state is reached with probability~1 in $2^{\abs{L}}$ steps. We now present the reduction to {\sf POMDP} s: \begin{enumerate}
\item \emph{Almost-sure winning for reachability.} Given a polynomial-space ATM $M$ and $w$ an input word, let $G=R_G(M,w)$. We construct a {\sf POMDP}\/ $G'$ from $G$ as follows: we only modify the transition function in $G'$ by uniformly choosing over the successor choices. Formally, for a state ${\ell} \in L$ and an action $\sigma \in \Sigma$ the probabilistic transition function ${\delta}'$ in $G'$ is as follows: \[ {\delta}'({\ell},\sigma)({\ell}') = \begin{cases} 0 & ({\ell},\sigma,{\ell}') \not \in {\delta}; \\ 1/ \abs{\set{{\ell}_1 \mid ({\ell},\sigma,{\ell}_1) \in {\delta}}} & ({\ell},\sigma,{\ell}') \in {\delta}. \end{cases} \] Given an observation-based strategy for Player~1 in $G$, we consider the same strategy in $G'$: (1)~if the strategy reaches the accepting state with probability~1 against all Player~2 strategies in $G$, then the strategy ensures that in $G'$ the accepting state is reached with probability~1; and (2)~otherwise there is a pure Player~2 strategy $\beta$ in $G$ that ensures the rejecting state is reached in $2^{\abs{L}}$ steps with probability $\eta>0$, and with probability at least $(1/\abs{L})^{2^{\abs{L}}}$ the choices of the successors of strategy $\beta$ is chosen in $G'$, and hence the rejecting state is reached with probability at least $(1/\abs{L})^{2^{\abs{L}}} \cdot \eta>0$. It follows that in $G'$ there is an observation-based strategy for almost-sure winning the reachability objective with target of the accepting state iff there is such a strategy in $G$. The result follows.
\item \emph{Positive winning for safety.} The reduction is same as above. We obtain the {\sf POMDP}\/ $G''$ from the {\sf POMDP}\/ $G'$ above by making the following modification: from the state accepting, the {\sf POMDP}\/ goes back to the initial state with probability~1. If there is an observation-based strategy $\alpha$ for Player~1 in $G'$ to reach the accepting state, then repeating the strategy $\alpha$ each time the accepting state is visited, it can be ensured that the rejecting state is reached with probability~0. Otherwise, against every observation-based strategy for Player~1, the probability to reach the rejecting state in $k \cdot(2^{\abs{L}}+1)$ steps is at least $1-(1-\eta')^k$, where $\eta'=\eta \cdot (1/\abs{L})^{2^{\abs{L}}}>0$ (this is because there is a probability to reach the rejecting state with probability at least $\eta'$ in $2^{\abs{L}}$ steps, and unless the rejecting state is reached the starting state is again reached within $2^{\abs{L}}+1$ steps). Hence the probability to reach the rejecting state is~1. It follows that $G'$ is almost-sure winning for the reachability objective with the target of the accepting state iff in $G''$ there is an observation-based strategy for Player~1 to ensure that the rejecting state is avoided with positive probability. This completes the proof of correctness of the reduction. \end{enumerate}
A very brief (two line proof) sketch was presented as the proof of Theorem~1 of~\cite{dA99} to show that positive winning in {\sf POMDP} s with safety objectives is EXPTIME-hard. We were unable to reconstruct the proof: the proof suggested to simulate a nondeterministic Turing machine. The simulation of a polynomial-space nondeterministic Turing machine only shows PSPACE-hardness, and the simulation of a nondeterministic EXPTIME Turing machine would have shown NEXPTIME-hardness, and an EXPTIME upper bound is known for the problem. Our proof presents a different and detailed proof of the result of Theorem~1 of~\cite{dA99}. Hence we have the following theorem, and the results are summarized in Table~\ref{tab1}.
\begin{theorem}\label{thrm_complexity} Given a {\sf POMDP}\ $G$, the following assertions hold: (a)~the positive winning problem for reachability objectives is NLOGSPACE-complete, the positive winning problem for safety and coB\"uchi objectives is EXPTIME-complete, and the positive winning problem for B\"uchi and parity objectives is undecidable; and (b)~the almost-sure winning problem for reachability, safety and B\"uchi objectives is EXPTIME-complete, and the almost-sure winning problem for coB\"uchi and parity objectives is undecidable. \end{theorem} \begin{myProof} The results are obtained as follows. \begin{enumerate} \item {\em Positive winning.} The NLOGSPACE-completeness for positive winning with reachability objectives is Theorem~\ref{thrm_pos_reach}. Our reduction from Alternating PSPACE Turing machine shows EXPTIME-hardness for positive winning with safety (and hence the lower bound also follows for coB\"uchi objectives), and the upper bounds follow from Theorem~\ref{thrm_pos_safe} and Theorem~\ref{thrm_pos_cobuchi}. The undecidability follows for positive winning for B\"uchi and parity objectives follows from the result of~\cite{BBG08,GIMBERT:2009:HAL-00403463:3}.
\item {\em Almost-sure winning.} It follows from the results of~\cite{CDHR07,BBG08} that the decision problems for almost-sure winning for {\sf POMDP} s with reachability, safety, and B\"uchi objectives can be solved in EXPTIME. Our reduction from Alternating PSPACE Turing machine shows EXPTIME-hardness for almost-sure winning with reachability (and hence the lower bound also follows for B\"uchi objectives). The lower bound for safety objectives follows from the lower bound for partial information games~\cite{CDHR07} and the fact the almost-sure winning for safety coincides with almost-sure winning in games. The undecidability follows for almost-sure winning for coB\"uchi and parity objectives follows from the result of~\cite{BBG08,GIMBERT:2009:HAL-00403463:3}. \end{enumerate}
\end{myProof}
\begin{table} \begin{center}
\begin{tabular}{|c|c|c|} \hline
& Positive & Almost-sure \\ \hline \,Reachability\, & \,NLOGSPACE-complete (up+lo)\, & EXPTIME-complete (lo)\\ \hline Safety & EXPTIME-complete (up+lo) & \,EXPTIME-complete\,~\cite{BD08}\, \\ \hline B\"uchi & Undecidable~\cite{BBG08} & EXPTIME-complete (lo) \\ \hline coB\"uchi & EXPTIME-complete (up+lo) & Undecidable~\cite{BBG08} \\ \hline Parity & Undecidable~\cite{BBG08} & Undecidable~\cite{BBG08} \\ \hline \hline \end{tabular} \end{center} \caption{Computational complexity of {\sf POMDP} s with different classes of parity objectives for positive and almost-sure winning. Our contribution of upper and lower bounds are indicated as ``up'' and ``lo'' respectively in parenthesis.}\label{tab1} \end{table}
\section{Optimal Memory Bounds for Strategies} In this section we present optimal bounds on the memory required by pure and randomized strategies for positive and almost-sure winning for reachability, safety, B\"uchi and coB\"uchi objectives.
\noindent{\bf Bounds for safety objectives.} First, we consider positive and almost-sure winning with safety objectives in {\sf POMDP} s. It follows from the correctness argument of Theorem~\ref{thrm_pos_safe}
that pure strategies with exponential memory are sufficient for positive winning with safety objectives in {\sf POMDP} s, and the exponential upper bound on memory of pure strategies for almost-sure winning with safety objectives in {\sf POMDP} s follows from the reduction to games. We now present a matching exponential lower bound for randomized strategies.
\begin{lemma}\label{lemm_lower_bound_safety} There exists a family $(P_n)_{n \in \mathbb N}$ of {\sf POMDP} s of size $O(p(n))$ for a polynomial $p$ with a safety objective such that the following assertions hold: (a)~Player~$1$ has a (pure) almost-sure (and therefore also positive) winning strategy in each of these {\sf POMDP} s; and (b)~there exists a polynomial $q$ such that every finite-memory randomized strategy for Player~1 that is positive (or almost-sure) winning in $P_n$ has at least $2^{q(n)}$ states. \end{lemma}
\paragraph{{\bf Preliminary.}}
The set of actions of the {\sf POMDP}\/ $P_n$ is $\Sigma_n \cup \{\#\}$ where $\Sigma_n = \{1, \dots,n\}$.
The {\sf POMDP}\/ is composed of an initial state $q_0$ and $n$ sub-MDPs $A_i$ with state space $Q_i$, each consisting of a loop over $p_i$ states $q_1^i,\dots,q_{p_i}^i$ where $p_i$ is the $i$-th prime number. From each state $q_j^i$ ($1 \leq j < p_i$), every action in $\Sigma_n$ leads to the next state $q_{j+1}^i$ with probability $\frac{1}{2}$, and to the initial state $q_0$ with probability $\frac{1}{2}$. The action $\#$ is not allowed. From $q_{p_i}^i$, the action $i$ is not allowed while the other actions in $\Sigma_n$ lead back the first state $q^i_1$ and to the initial state $q_0$ both with probability $\frac{1}{2}$. Moreover, the action $\#$ leads back to the initial state (with probability~$1$). The disallowed actions lead to a bad state. The states of the $A_i$'s are indistinguishable (they have the same observation), while the initial state $q_0$ is visible. We assume that the state spaces $Q_i$ of the $A_i$'s are disjoint.
\begin{figure}
\caption{The {\sf POMDP}\/ $P_2$.}
\label{fig:exp-game}
\caption{The {\sf POMDP}\/ $P'_2$.}
\label{fig:exp-POMDP-reach}
\end{figure}
\paragraph{{\bf {\sf POMDP}\/ family $(P_n)_{n \in\mathbb N}$.}} The state space of $P_n$ is the disjoint union of $Q_1, \dots, Q_n$ and $\{q_0,\mathsf{Bad}\}$. The initial state is $q_0$, the final state is $\mathsf{Bad}$. The probabilistic transition function is as follows: \begin{itemize} \item for all $1 \leq i \leq n$ and $\sigma \in \Sigma_n$, we have ${\delta}(q_0,\sigma)(q^i_1)=\frac{1}{n}$;
\item for all $1 \leq i \leq n$, $1 \leq j < p_i$, and $\sigma \in \Sigma_n$, $\sigma' \in \Sigma_n \setminus \{i\}$, we have ${\delta}(q^i_j,\sigma)(q^i_{j+1})={\delta}(q^i_{j},\sigma)(q_{0})= {\delta}(q^i_{p_i},\sigma')(q^i_{1})={\delta}(q^i_{p_i},\sigma')(q_{0})=\frac{1}{2}$; and
\item for all $1 \leq i \leq n$ and $1 \leq j < p_i$, we have ${\delta}(q_0,\#)(\mathsf{Bad})={\delta}(q^i_{j},\#)(\mathsf{Bad})={\delta}(q^i_{p_i},\#)(q_{0})=1$. \end{itemize}
The initial state is $q_0$. There are two observations, the state $\{q_0\}$ is labelled by observation $o_1$, and the other states in $Q_1 \cup \dots \cup Q_n$ (that we call the loops) by observation $o_2$. \figurename~\ref{fig:exp-game} shows the game $P_2$: the witness family of POMDPs have similarities with analogous constructions for games~\cite{BCDHR08}. However the construction of~\cite{BCDHR08} shows lower bounds only for pure strategies and in games, whereas we present lower bound for randomized strategies and for POMDPs, and hence our proofs are very different.
\paragraph{{\bf Proof of Lemma~\ref{lemm_lower_bound_safety}.}} After the first transition from the initial state, player~$1$ has the following positive winning strategy. Let $p^*_n = \prod_{i=1}^{n} p_i$. While the {\sf POMDP}\/ is in the loops (assume that we have seen $j$ times observation $o_2$ consecutively), if $1 \leq j < p^*_n$, then play any action $i$ such that $j \mod p_i \neq 0$ (this is well defined since $p^*_n$ is the lcm of $p_1, \dots, p_n$), and otherwise play $\#$. It is easy to show that this strategy is winning for the safety condition, with probability $1$.
For the second part of the result, assume towards a contradiction that there exists a finite-memory randomized strategy $\hat{\alpha}$ that is positive winning for Player~$1$ and has less than $p^*_n$ states (since $p^*_n$ is exponential in $s^*_n = \sum_{i=1}^{n} p_i$, the result will follow). Let $\eta$ be the least positive transition probability described by the finite-state strategy $\hat{\alpha}$. Consider any history of a play $\rho$ that ends with $o_1$. We claim that the following properties hold: (a)~with probability~$1$ either observation $o_1$ is visited again from $\rho$ or the state $\mathsf{Bad}$ is reached; and (b)~the state $\mathsf{Bad}$ is reached with a positive probability. The first property (property~(a)) follows from the fact that for all actions the loops are left (the state $q_0$ or $\mathsf{Bad}$ is reached) with probability at least $\frac{1}{2}$. We now prove the second property by showing that the state~$\mathsf{Bad}$ is reached with probability at least $\Delta_n = \frac{1}{n}\cdot\frac{1}{(2\cdot \eta)^{p^*_n}}$. To see this, consider the sequence of actions played by strategy $\hat{\alpha}$ after~$\rho$ when only~$o_2$ is observed. Either $\#$ is never played, and then the action played by $\hat{\alpha}$ after a sequence of $p^*_n$ states leads to $\mathsf{Bad}$ (the current state being then $q^i_{p_i}$ for some $1 \leq i \leq n$). This occurs with probability at least~$\Delta_n$; or~$\#$ is eventually played, but since $\hat{\alpha}$ has less than $p^*_n$ states, it has to be played after less than $p^*_n$ steps, which also leads to $\mathsf{Bad}$ with probability at least~$\Delta_n$. The above two properties that (a)~$o_1 \cup \{\mathsf{Bad}\}$ is reached with probability $1$ from $o_1$, and (b)~within $p^*_n$ steps after a visit to $o_1$, the state $\mathsf{Bad}$ is reached with fixed positive probability, ensures that $\mathsf{Bad}$ is reached with probability~$1$. Hence $\hat{\alpha}$ is not positive winning. It follows that randomized strategies that are almost-sure or positive winning in {\sf POMDP} s with safety objectives may require exponential memory.
\noindent{\bf Bounds for reachability objectives.}
We now argue the memory bounds for pure and randomized strategies for positive winning with reachability objectives. \begin{enumerate}
\item It follows from the correctness argument of Theorem~\ref{thrm_pos_reach} that randomized memoryless strategies suffice for positive winning with reachability objectives in {\sf POMDP} s.
\item We now argue that for pure strategies, memory of size linear in the number of states is sufficient and may be necessary. The upper bound follows from the reduction to graph reachability. Given a {\sf POMDP}\/ $G$, consider the graph $\overline{G}$ constructed from $G$ as in the correctness argument for Theorem~\ref{thrm_pos_reach}. Given the starting state ${\ell}$, if there is path in $\overline{G}$ to the target set $T$ obtained from ${\cal T}$, then there is a path $\pi$ of length at most $\abs{L}$. The pure strategy for Player~1 in $G$ can play the sequence of actions of the path $\pi$ to ensure that the target observations ${\cal T}$ are reached with positive probability in $G$. The family of examples to show that pure strategies require linear memory can be constructed as follows: we construct a {\sf POMDP}\/ with deterministic transition function such that there is a unique path (sequence of actions) of length $O(\abs{L})$ to the target, and any deviation leads to an absorbing state, and other than the target state every other state has the same observation. In this {\sf POMDP}\/ any pure strategy must remember the exact sequence of actions to be played and hence requires $O(\abs{L})$ memory. \end{enumerate}
It follows from the results of~\cite{BBG08} that for almost-sure winning with reachability objectives in {\sf POMDP} s pure strategies with exponential memory suffice, and we now prove an exponential lower bound for randomized strategies.
\begin{lemma}\label{lemm_lower_bound_reachability} There exists a family $(P_n)_{n \in \mathbb N}$ of {\sf POMDP} s of size $O(p(n))$ for a polynomial $p$ with a reachability objective such that the following assertions hold: (a)~Player~$1$ has an almost-sure winning strategy in each of these {\sf POMDP} s; and (b)~there exists a polynomial $q$ such that every finite-memory randomized strategy for Player~1 that is almost-sure winning in $P_n$ has at least $2^{q(n)}$ states. \end{lemma}
Fix the action set as $\Sigma = \{\#,{\sf tick}\}$. The {\sf POMDP}\/ $P'_n$ is composed of an initial state $q_0$ and $n$ sub-MDPs $H_i$, each consisting of a loop over $p_i$ states $q_1^i,\dots,q_{p_i}^i$ where $p_i$ is the $i$-th prime number. From each state in the loops, the action ${\sf tick}$ can be played and leads to the next state in the loop (with probability $1$). The action $\#$ can be played in the last state of each loop and leads to the $\mathsf{Goal}$ state. The objective is to reach $\mathsf{Goal}$ with probability~1. Actions that are not allowed lead to a sink state from which it is impossible to reach $\mathsf{Goal}$. There is a unique observation that consists of the whole state space. \figurename~\ref{fig:exp-POMDP-reach} shows $P'_2$.
\paragraph{{\bf Proof of Lemma~\ref{lemm_lower_bound_reachability}.}} First we show that Player $1$ has an almost-sure winning strategy in $P'_k$ (from $q_0$). As there is only one observation, a strategy for Player $1$ corresponds to a function $\alpha: \mathbb N \to \Sigma$. Consider the strategy $\alpha^*$ as follows: $\alpha^*(j) = {\sf tick}$ for all $0 \leq j < p^*_k$ and $\alpha^*(j) = \#$ for all $j \geq p^*_k$. It is easy to check that $\alpha^*$ ensures winning with certainty and hence almost-sure winning.
For the second part of the result assume, towards a contradiction, that there exists a finite-memory randomized strategy $\hat{\alpha}$ that is almost-sure winning and has less than $p^*_k$ states. Clearly,
$\hat{\alpha}$ cannot play $\#$ before the $(p^*_k+1)$-th round since one of the subMDPs $H_i$ would not be in $q^i_{p_i}$ and therefore Player $1$ would lose with probability at least $\frac{1}{n}$. Note that the state reached by the strategy automaton defining $\hat{\alpha}$ after $p^*_k$ rounds has necessarily been visited in a previous round. Since $\hat{\alpha}$ has to play $\#$ eventually to reach $\mathsf{Goal}$, this means that $\#$ must have been played in some round $j < p^*_k$, when at least one of the subgames~$H_i$ was not in location $q^i_{p_i}$, so that Player~$1$ would have already lost with probability at least $\frac{1}{n}\cdot \eta$, where $\eta$ is the least positive probability specified by $\hat{\alpha}$. This is in contradiction with our assumption that $\hat{\alpha}$ is an almost-sure winning strategy.
\noindent{\bf Bounds for B\"uchi and coB\"uchi objectives.} An exponential upper bound for memory of pure strategies for almost-sure winning of B\"uchi objectives follows from the results of~\cite{BBG08}, and the matching lower bound for randomized strategies follows from our result for reachability objectives. Since positive winning is undecidable for B\"uchi objectives there is no bound on memory for pure or randomized strategies for positive winning. An exponential upper bound for memory of pure strategies for positive winning of coB\"uchi objectives follows from the correctness proof of Theorem~\ref{thrm_pos_cobuchi} that iteratively combines the positive winning strategies for safety and reachability to obtain a positive winning strategy for coB\"uchi objective. The matching lower bound for randomized strategies follows from our result for safety objectives. Since almost-sure winning is undecidable for coB\"uchi objectives there is no bound on memory for pure or randomized strategies for positive winning. This gives us the following theorem (also summarized in Table~\ref{tab2}), which is in contrast to the results for MDPs with perfect observation where pure memoryless strategies suffice for almost-sure and positive winning for all parity objectives.
\begin{theorem} The optimal memory bounds for strategies in {\sf POMDP} s are as follows. \begin{enumerate} \item Reachability objectives: for positive winning randomized memoryless strategies are sufficient, and linear memory is necessary and sufficient for pure strategies; and for almost-sure winning exponential memory is necessary and sufficient for both pure and randomized strategies.
\item Safety objectives: for positive winning and almost-sure winning exponential memory is necessary and sufficient for both pure and randomized strategies.
\item B\"uchi objectives: for almost-sure winning exponential memory is necessary and sufficient for both pure and randomized strategies; and there is no bound on memory for pure and randomized strategies for positive winning.
\item coB\"uchi objectives: for positive winning exponential memory is necessary and sufficient for both pure and randomized strategies; and there is no bound on memory for pure and randomized strategies for almost-sure winning. \end{enumerate}
\end{theorem}
\begin{table} \begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
& Pure Positive & Randomized Positive & Pure Almost & Randomized Almost \\ \hline Reachability & Linear & Memoryless & Exponential & Exponential \\ \hline Safety & Exponential & Exponential & Exponential & Exponential \\ \hline B\"uchi & No Bound & No Bound & Exponential & Exponential \\ \hline coB\"uchi & Exponential & Exponential & No Bound & No Bound \\ \hline Parity & No Bound & No Bound & No Bound & No Bound \\ \hline \hline \end{tabular} \end{center} \caption{Optimal memory bounds for pure and randomized strategies for positive and almost-sure winning.}\label{tab2} \end{table}
\end{document}
\section{Appendix}
\begin{myProof}{\em (of Theorem~\ref{thrm_pos_reach}).} The NLOGSPACE-completeness result for positive reachability for MDPs follows from reductions to and from graph reachability.
\noindent{\em Reduction to graph reachability.} Given a {\sf POMDP}\/ $G= \tuple{L,\Sigma,{\delta},{\cal{O}}}$ and a set of target states ${\cal T} \subseteq L$, consider the graph $\overline{G}=\tuple{L,E}$ where $({\ell},{\ell}') \in E$ if there exists an action $\sigma \in \Sigma$ such that ${\delta}({\ell},\sigma)({\ell}')>0$. Let ${\ell}$ be a starting state, then the following assertions hold: (a)~if there is a path $\pi$ in $\overline{G}$ from ${\ell}$ to a state $t \in {\cal T}$, then the randomized memoryless strategy for Player~1 in $G$ that plays all actions uniformly at random ensures that the path $\pi$ is executed in $G$ with positive probability (i.e., ensures positive winning for $\mathsf{Reach}({\cal T})$ in $G$ from ${\ell}$); and (b)~if there is no path in $\overline{G}$ to reach $T$ from ${\ell}$, then there is no strategy (and hence no observation-based strategy) for Player~1 in $G$ to achieve $\mathsf{Reach}({\cal T})$. This shows that positive winning in {\sf POMDP} s can be decided in NLOGSPACE. Graphs are a special case of {\sf POMDP} s and hence graph reachability can be reduced to reachability with positive probability in {\sf POMDP} s, therefore the problem is NLOGSPACE-complete.
\end{myProof}
\begin{myProof}{\em (of Theorem~\ref{thrm_complexity}).} The results are obtained as follows. \begin{enumerate} \item {\em Positive winning.} The NLOGSPACE-completeness for positive winning with reachability objectives is Theorem~\ref{thrm_pos_reach}. Our reduction from Alternating PSPACE Turing machine shows EXPTIME-hardness for positive winning with safety (and hence the lower bound also follows for coB\"uchi objectives), and the upper bounds follow from Theorem~\ref{thrm_pos_safe} and Theorem~\ref{thrm_pos_cobuchi}. The undecidability follows for positive winning for B\"uchi and parity objectives follows from the result of~\cite{BBG08,GIMBERT:2009:HAL-00403463:3}.
\item {\em Almost-sure winning.} It follows from the results of~\cite{CDHR07,BBG08} that the decision problems for almost-sure winning for {\sf POMDP} s with reachability, safety, and B\"uchi objectives can be solved in EXPTIME. Our reduction from Alternating PSPACE Turing machine shows EXPTIME-hardness for almost-sure winning with reachability (and hence the lower bound also follows for B\"uchi objectives). The lower bound for safety objectives follows from the lower bound for partial information games~\cite{CDHR07} and the fact the almost-sure winning for safety coincides with almost-sure winning in games. The undecidability follows for almost-sure winning for coB\"uchi and parity objectives follows from the result of~\cite{BBG08,GIMBERT:2009:HAL-00403463:3}. \end{enumerate}
\end{myProof}
\end{document} | arXiv |
Recursive language
In mathematics, logic and computer science, a formal language (a set of finite sequences of symbols taken from a fixed alphabet) is called recursive if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language. Equivalently, a formal language is recursive if there exists a Turing machine that, when given a finite sequence of symbols as input, always halts and accepts it if it belongs to the language and halts and rejects it otherwise. In Theoretical computer science, such always-halting Turing machines are called total Turing machines or algorithms (Sipser 1997). Recursive languages are also called decidable.
This article is about a class of formal languages as they are studied in mathematics and theoretical computer science. For computer languages that allow a function to call itself recursively, see Recursion (computer science).
The concept of decidability may be extended to other models of computation. For example, one may speak of languages decidable on a non-deterministic Turing machine. Therefore, whenever an ambiguity is possible, the synonym used for "recursive language" is Turing-decidable language, rather than simply decidable.
The class of all recursive languages is often called R, although this name is also used for the class RP.
This type of language was not defined in the Chomsky hierarchy of (Chomsky 1959). All recursive languages are also recursively enumerable. All regular, context-free and context-sensitive languages are recursive.
Definitions
There are two equivalent major definitions for the concept of a recursive language:
1. A recursive formal language is a recursive subset in the set of all possible words over the alphabet of the language.
2. A recursive language is a formal language for which there exists a Turing machine that, when presented with any finite input string, halts and accepts if the string is in the language, and halts and rejects otherwise. The Turing machine always halts: it is known as a decider and is said to decide the recursive language.
By the second definition, any decision problem can be shown to be decidable by exhibiting an algorithm for it that terminates on all inputs. An undecidable problem is a problem that is not decidable.
Examples
As noted above, every context-sensitive language is recursive. Thus, a simple example of a recursive language is the set L={abc, aabbcc, aaabbbccc, ...}; more formally, the set
$L=\{\,w\in \{a,b,c\}^{*}\mid w=a^{n}b^{n}c^{n}{\mbox{ for some }}n\geq 1\,\}$
is context-sensitive and therefore recursive.
Examples of decidable languages that are not context-sensitive are more difficult to describe. For one such example, some familiarity with mathematical logic is required: Presburger arithmetic is the first-order theory of the natural numbers with addition (but without multiplication). While the set of well-formed formulas in Presburger arithmetic is context-free, every deterministic Turing machine accepting the set of true statements in Presburger arithmetic has a worst-case runtime of at least $2^{2^{cn}}$, for some constant c>0 (Fischer & Rabin 1974). Here, n denotes the length of the given formula. Since every context-sensitive language can be accepted by a linear bounded automaton, and such an automaton can be simulated by a deterministic Turing machine with worst-case running time at most $c^{n}$ for some constant c , the set of valid formulas in Presburger arithmetic is not context-sensitive. On positive side, it is known that there is a deterministic Turing machine running in time at most triply exponential in n that decides the set of true formulas in Presburger arithmetic (Oppen 1978). Thus, this is an example of a language that is decidable but not context-sensitive.
Closure properties
Recursive languages are closed under the following operations. That is, if L and P are two recursive languages, then the following languages are recursive as well:
• The Kleene star $L^{*}$
• The image φ(L) under an e-free homomorphism φ
• The concatenation $L\circ P$
• The union $L\cup P$
• The intersection $L\cap P$
• The complement of $L$
• The set difference $L-P$
The last property follows from the fact that the set difference can be expressed in terms of intersection and complement.
See also
• Recursively enumerable language
• Computable set
• Recursion
References
• Michael Sipser (1997). "Decidability". Introduction to the Theory of Computation. PWS Publishing. pp. 151–170. ISBN 978-0-534-94728-6.
• Chomsky, Noam (1959). "On certain formal properties of grammars". Information and Control. 2 (2): 137–167. doi:10.1016/S0019-9958(59)90362-6.
• Fischer, Michael J.; Rabin, Michael O. (1974). "Super-Exponential Complexity of Presburger Arithmetic". Proceedings of the SIAM-AMS Symposium in Applied Mathematics. 7: 27–41.
• Oppen, Derek C. (1978). "A 222pn Upper Bound on the Complexity of Presburger Arithmetic". J. Comput. Syst. Sci. 16 (3): 323–332. doi:10.1016/0022-0000(78)90021-1.
Automata theory: formal languages and formal grammars
Chomsky hierarchyGrammarsLanguagesAbstract machines
• Type-0
• —
• Type-1
• —
• —
• —
• —
• —
• Type-2
• —
• —
• Type-3
• —
• —
• Unrestricted
• (no common name)
• Context-sensitive
• Positive range concatenation
• Indexed
• —
• Linear context-free rewriting systems
• Tree-adjoining
• Context-free
• Deterministic context-free
• Visibly pushdown
• Regular
• —
• Non-recursive
• Recursively enumerable
• Decidable
• Context-sensitive
• Positive range concatenation*
• Indexed*
• —
• Linear context-free rewriting language
• Tree-adjoining
• Context-free
• Deterministic context-free
• Visibly pushdown
• Regular
• Star-free
• Finite
• Turing machine
• Decider
• Linear-bounded
• PTIME Turing Machine
• Nested stack
• Thread automaton
• restricted Tree stack automaton
• Embedded pushdown
• Nondeterministic pushdown
• Deterministic pushdown
• Visibly pushdown
• Finite
• Counter-free (with aperiodic finite monoid)
• Acyclic finite
Each category of languages, except those marked by a *, is a proper subset of the category directly above it. Any language in each category is generated by a grammar and by an automaton in the category in the same line.
| Wikipedia |
\begin{document}
\begin{abstract} Cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the conditions for classical binary search trees. Based on the bijective correspondence between signed permutations and leveled Cambrian trees, we define the Cambrian Hopf algebra generalizing J.-L.~Loday and M.~Ronco's algebra on binary trees. We describe combinatorially the products and coproducts of both the Cambrian algebra and its dual in terms of operations on Cambrian trees. We also define multiplicative bases of the Cambrian algebra and study structural and combinatorial properties of their indecomposable elements. Finally, we extend to the Cambrian setting different algebras connected to binary trees, in particular S.~Law and N.~Reading's Baxter Hopf algebra on quadrangulations and S.~Giraudo's equivalent Hopf algebra on twin binary trees, and F.~Chapoton's Hopf algebra on all faces of the associahedron. \end{abstract}
\title{Cambrian Hopf Algebras}
\vspace*{-.2cm} \tableofcontents
\section*{Introduction}
The background of this paper is the fascinating interplay between the combinatorial, geometric and algebraic structures of permutations, binary trees and binary sequences (see Table~\ref{tab:structures}): \begin{enumerate}[$\star$] \item \textbf{Combinatorially}, the descent map from permutations to binary sequences factors via binary trees through the BST insertion and the canopy map. These maps define lattice homomorphisms from the weak order via the Tamari lattice to the boolean~lattice. \item \textbf{Geometrically}, the permutahedron is contained in Loday's associahedron~\cite{Loday} which is in turn contained in the parallelepiped generated by the simple roots. These polytopes are just obtained by deleting inequalities from the facet description of the permutahedron. See~\fref{fig:permutahedronAssociahedronCube}. \item \textbf{Algebraically}, these maps translate to Hopf algebra inclusions from M.~Malvenuto and C.~Reute\-nauer's algebra on permutations~\cite{MalvenutoReutenauer} via J.-L.~Loday and M.~Ronco's algebra on binary trees~\cite{LodayRonco} to L.~Solomon's descent algebra~\cite{Solomon}. \end{enumerate}
\begin{table}[h]
\begin{tabular}{c@{\quad}||@{\quad}c@{\quad}|@{\quad}c@{\quad}|@{\quad}c}
Combinatorics & Permutations & Binary trees & Binary sequences \\[.2cm]
\multirow{2}{*}{Geometry} & Permutahedron & Loday's & Parallelepiped \\
& $\conv(\mathfrak{S}_n)$ & associahedron~\cite{Loday} & gen. by $e_{i+1}-e_i$\\[.2cm]
\multirow{2}{*}{Algebra} & Malvenuto-Reutenauer & Loday-Ronco & Solomon \\
& Hopf algebra~\cite{MalvenutoReutenauer} & Hopf algebra~\cite{LodayRonco} & descent algebra~\cite{Solomon}
\end{tabular}
\caption{Related combinatorial, geometric and algebraic structures.}
\label{tab:structures} \end{table}
\begin{figure}
\caption{The $3$-dimensional permutahedron (blue, left), Loday's associahedron (red, middle), and parallelepiped (green, right). Shaded facets are preserved to get the next polytope.}
\label{fig:permutahedronAssociahedronCube}
\end{figure}
These structures and their connections have been partially extended in several directions in particular to the Cambrian lattices of N.~Reading~\cite{Reading-CambrianLattices, ReadingSpeyer} and their polytopal realizations by C.~Hohlweg, C.~Lange, and H.~Thomas~\cite{HohlwegLange, HohlwegLangeThomas}, to the graph associahedra of M.~Carr and S.~Devadoss~\cite{CarrDevadoss, Devadoss}, the nested complexes and their realizations as generalized associahedra by A.~Postnikov~\cite{Postnikov} (see also~\cite{PostnikovReinerWilliams, FeichtnerSturmfels, Zelevinsky}), or to the $m$-Tamari lattices of F.~Bergeron and L.-F.~Pr\'eville-Ratelle~\cite{BergeronPrevilleRatelle} (see also~\cite{BousquetMelouFusyPrevilleRatelle, BousquetMelouChapuyPrevilleRatelle}) and the Hopf algebras on these $m$-structures recently constructed by J.-C.~Novelli and J.-Y.~Thibon~\cite{NovelliThibon, Novelli}.
This paper explores combinatorial and algebraic aspects of Hopf algebras related to the type~$A$ Cambrian lattices. N.~Reading provides in~\cite{Reading-CambrianLattices} a procedure to map a signed permutation of~$\mathfrak{S}_n$ into a triangulation of a certain convex $(n+3)$-gon. The dual trees of these triangulations naturally extend rooted binary trees and were introduced and studied as ``spines''~\cite{LangePilaud} or ``mixed cobinary trees''~\cite{IgusaOstroff}. We prefer here the term ``Cambrian trees'' in reference to N.~Reading's work. The map~$\kappa$ from signed permutations to Cambrian trees is known to encode combinatorial and geometric properties of the Cambrian structures: the Cambrian lattice is the quotient of the weak order under the fibers of~$\kappa$, each maximal cone of the Cambrian fan is the incidence cone of a Cambrian tree~$\tree$ and is refined by the braid cones of the permutations in the fiber~$\kappa^{-1}(\tree)$, etc.
In this paper, we use this map~$\kappa$ for algebraic purposes. In the first part, we introduce the Cambrian Hopf algebra~$\mathsf{Camb}$ as a subalgebra of the Hopf algebra~$\mathsf{FQSym}_\pm$ on signed permutations, and the dual Cambrian algebra~$\mathsf{Camb}^*$ as a quotient algebra of the dual Hopf algebra~$\mathsf{FQSym}_\pm^*$. Their bases are indexed by all Cambrian trees. Our approach extends that of F.~Hivert, \mbox{J.-C.~Novelli} and \mbox{J.-Y.~Thibon}~\cite{HivertNovelliThibon-algebraBinarySearchTrees} to construct J.-L.~Loday and M.~Ronco's Hopf algebra on binary trees~\cite{LodayRonco} as a subalgebra of C.~Malvenuto and C.~Reutenauer's Hopf algebra on permutations~\cite{MalvenutoReutenauer}. We also use this map~$\kappa$ to describe both the product and coproduct in the algebras~$\mathsf{Camb}$ and~$\mathsf{Camb}^*$ in terms of simple combinatorial operations on Cambrian trees. From the combinatorial description of the product in~$\mathsf{Camb}$, we derive multiplicative bases of the Cambrian algebra~$\mathsf{Camb}$ and study the structural and enumerative properties of their indecomposable elements.
In the second part of this paper, we study Baxter-Cambrian structures, extending in the Cambrian setting the constructions of S.~Law and N.~Reading on quadrangulations~\cite{LawReading} and that of S.~Giraudo on twin binary trees~\cite{Giraudo}. We define Baxter-Cambrian lattices as quotients of the weak order under the intersections of two opposite Cambrian congruences. Their elements can be labeled by pairs of twin Cambrian trees, \textit{i.e.}~ Cambrian trees with opposite signatures whose union forms an acyclic graph. We study in detail the number of such pairs of Cambrian trees for arbitrary signatures. Following~\cite{LawReading}, we also observe that the Minkowski sums of opposite associahedra of C.~Hohlweg and C.~Lange~\cite{HohlwegLange} provide polytopal realizations of the Baxter-Cambrian lattices. Finally, we introduce the Baxter-Cambrian Hopf algebra~$\mathsf{BaxCamb}$ as a subalgebra of the Hopf algebra~$\mathsf{FQSym}_\pm$ on signed permutations and its dual~$\mathsf{BaxCamb}^*$ as a quotient algebra of the dual Hopf algebra~$\mathsf{FQSym}_\pm^*$. Their bases are indexed by pairs of twin Cambrian trees, and it is also possible to describe both the product and coproduct in the algebras~$\mathsf{BaxCamb}$ and~$\mathsf{BaxCamb}^*$ in terms of simple combinatorial operations on Cambrian trees. We also extend our results to arbitrary tuples of Cambrian trees, resulting to the Cambrian tuple algebra.
\enlargethispage{.6cm} The last part of the paper is devoted to Schr\"oder-Cambrian structures. We consider Schr\"oder-Cambrian trees which correspond to all faces of all C.~Hohlweg and C.~Lange's associahedra~\cite{HohlwegLange}. We define the Schr\"oder-Cambrian lattice as a quotient of the weak order on ordered partitions defined in~\cite{KrobLatapyNovelliPhanSchwer}, thus extending N.~Reading's type~$A$ Cambrian lattices~\cite{Reading-CambrianLattices} to all faces of the associahedron. Finally, we consider the Schr\"oder-Cambrian Hopf algebra~$\mathsf{SchrCamb}$, generalizing the algebra defined by F.~Chapoton in~\cite{Chapoton}.
\part{The Cambrian Hopf Algebra} \label{part:CambrianAlgebra}
\section{Cambrian trees} \label{sec:CambrianTrees}
In this section, we recall the definition and properties of ``Cambrian trees'', generalizing standard binary search trees. They were introduced independently by K.~Igusa and J.~Ostroff in~\cite{IgusaOstroff} as ``mixed cobinary trees'' in the context of cluster algebras and quiver representation theory and by C.~Lange and V.~Pilaud in~\cite{LangePilaud} as ``spines'' (\textit{i.e.}~ oriented and labeled dual trees) of triangulations of polygons to revisit the multiple realizations of the associahedron of C.~Hohlweg and C.~Lange~\cite{HohlwegLange}. Here, we use the term ``Cambrian trees'' to underline their connection with the type~$A$ Cambrian lattices of N.~Reading~\cite{Reading-CambrianLattices}. Although motivating and underlying this paper, these interpretations are not needed for the combinatorial and algebraic constructions presented here, and we only refer to them when they help to get geometric intuition on our~statements.
\subsection{Cambrian trees and increasing trees}
Consider a directed tree~$\tree$ on a vertex set~$\mathrm{V}$ and a vertex~$v \in \mathrm{V}$. We call \defn{children} (resp.~\defn{parents}) of~$v$ the sources of the incoming arcs (resp.~the targets of the outgoing arcs) at~$v$ and \defn{descendants} (resp.~\defn{ancestor}) \defn{subtrees} of~$v$ the subtrees attached to them. The main characters of our paper are the following trees, which generalize standard binary search trees. Our definition is adapted from~\cite{IgusaOstroff, LangePilaud}.
\begin{definition} A \defn{Cambrian tree} is a directed tree~$\tree$ with vertex set~$\mathrm{V}$ endowed with a bijective vertex labeling $p : \mathrm{V} \to [n]$ such that for each vertex~$v \in \mathrm{V}$, \begin{enumerate}[(i)] \item $v$ has either one parent and two children (its descendant subtrees are called \defn{left and right~subtrees}) or one child and two parents (its ancestor subtrees are called \defn{left and right subtrees}); \item all labels are smaller (resp.~larger) than~$p(v)$ in the left (resp.~right) subtree~of~$v$. \end{enumerate} The \defn{signature} of~$\tree$ is the $n$-tuple~$\varepsilon(\tree) \in \pm^n$ defined by~$\varepsilon(\tree)_{p(v)} = -$ if~$v$ has two children and~$\varepsilon(\tree)_{p(v)} = +$ if~$v$ has two parents. Denote by~$\mathrm{Camb}(\varepsilon)$ the set of Cambrian trees with signature~$\varepsilon$, by~$\mathrm{Camb}(n) = \bigsqcup_{\varepsilon \in \pm^n} \mathrm{Camb}(\varepsilon)$ the set of all Cambrian trees on~$n$ vertices, and by~$\mathrm{Camb} \eqdef \bigsqcup_{n \in \mathbb{N}} \mathrm{Camb}(n)$ the set of all Cambrian trees. \end{definition}
\begin{definition} An \defn{increasing tree} is a directed tree~$\tree$ with vertex set~$\mathrm{V}$ endowed with a bijective vertex labeling~$q : \mathrm{V} \to [n]$ such that~$v \to w$ in~$\tree$ implies~$q(v) < q(w)$. \end{definition}
\begin{definition} A \defn{leveled Cambrian tree} is a directed tree~$\tree$ with vertex set~$\mathrm{V}$ endowed with two bijective vertex labelings~${p,q : V \to [n]}$ which respectively define a Cambrian and an increasing tree. \end{definition}
\begin{figure}
\caption{A Cambrian tree (left), an increasing tree (middle), and a leveled Cambrian tree (right).}
\label{fig:leveledCambrianTree}
\end{figure}
\enlargethispage{-.6cm} In other words, a leveled Cambrian tree is a Cambrian tree endowed with a linear extension of its transitive closure. \fref{fig:leveledCambrianTree} provides examples of a Cambrian tree (left), an increasing tree (middle), and a leveled Cambrian tree (right). All edges are oriented bottom-up. Throughout the paper, we represent leveled Cambrian trees on an $(n \times n)$-grid as follows (see \fref{fig:leveledCambrianTree}): \begin{enumerate}[(i)] \item each vertex~$v$ appears at position $(p(v), q(v))$; \item negative vertices (with one parent and two children) are represented by~$\ominus$, while positive vertices (with one child and two parents) are represented by~$\oplus$; \item we sometimes draw a vertical red wall below the negative vertices and above the positive vertices to mark the separation between the left and right subtrees of each vertex. \end{enumerate}
\begin{remark}[Spines of triangulations] \label{rem:triangulation} Cambrian trees can be seen as spines (\textit{i.e.}~ oriented and labeled dual trees) of triangulations of labeled polygons. Namely, consider an $(n+2)$-gon~$\polygon$ with vertices labeled by~$0, \dots, n+1$ from left to right, and where vertex~$i$ is located above the diagonal~$[0,n+1]$ if~$\varepsilon_i = +$ and below it if~$\varepsilon_i = -$. We associate to a triangulation~$\sigma$ of~$\polygon$ its dual tree, with a node labeled by~$j$ for each triangle~$ijk$ of~$\sigma$ where~$i<j<k$, and an edge between any two adjacent triangles oriented from the triangle below to the triangle above their common diagonal. See \fref{fig:triangulation} and refer to~\cite{LangePilaud} for details. Throughout the paper, we denote by~$\tree^*$ the triangulation of~$\polygon$ dual to the $\varepsilon$-Cambrian tree~$\tree$, and we use this interpretation to provide the reader with some geometric intuition of definitions and results of this paper.
\begin{figure}
\caption{Cambrian trees (left) and triangulations (right) are dual to each other (middle).}
\label{fig:triangulation}
\end{figure} \end{remark}
\begin{proposition}[\cite{LangePilaud, IgusaOstroff}]
For any signature~$\varepsilon \in \pm^n$, the number of $\varepsilon$-Cambrian trees is the Catalan number~$C_n = \frac{1}{n+1}\binom{2n}{n}$. Therefore,~$|\mathrm{Camb}(n)| = 2^n C_n$. See~\href{https://oeis.org/A151374}{\cite[A151374]{OEIS}}. \end{proposition}
There are several ways to prove this statement (to our knowledge, the last two are original): \begin{enumerate}[(i)] \item From the description of~\cite{LangePilaud} given in the previous remark, the number of $\varepsilon$-Cambrian trees is the number of triangulations of a convex $(n+2)$-gon, counted by the Catalan number. \item There are natural bijections between $\varepsilon$-Cambrian trees and binary trees. One simple way is to reorient all edges of a Cambrian tree towards an arbitrary leaf to get a binary tree, but the inverse map is more difficult to explain, see~\cite{IgusaOstroff}. \item Cambrian trees are in bijection with certain pattern avoiding signed permutations, see Section~\ref{subsec:CambrianClasses}. In Proposition~\ref{prop:GeneratingTree}, we show that the shape of the generating tree for these permutations is independent of~$\varepsilon$. \item In Lemma~\ref{lem:switchSign}, we give an explicit bijection between $\varepsilon$- and $\varepsilon'$-Cambrian trees, where~$\varepsilon$ and~$\varepsilon'$ only differ by swapping two consecutive signs or switching the sign of~$1$ (or that of~$n$). \end{enumerate}
\subsection{Cambrian correspondence} \label{subsec:CambrianCorrespondence}
\enlargethispage{-.4cm} We represent graphically a permutation~$\tau \in \mathfrak{S}_n$ by the $(n \times n)$-table, with rows labeled by positions from bottom to top and columns labeled by values from left to right, and with a dot in row~$i$ and column~$\tau(i)$ for all~$i \in [n]$. (This unusual choice of orientation is necessary to fit later with the existing constructions of~\cite{LodayRonco, HivertNovelliThibon-algebraBinarySearchTrees}.)
A \defn{signed permutation} is a permutation table where each dot receives a~$+$ or~$-$ sign, see the top left corner of \fref{fig:insertionAlgorithm}. We could equivalently think of a permutation where the positions or the values receive a sign, but it will be useful later to switch the signature from positions to values. The \defn{p-signature} (resp.~\defn{v-signature}) of a signed permutation~$\tau$ is the sequence~$\signature_p(\tau)$ (resp.~$\signature_v(\tau)$) of signs of~$\tau$ ordered by positions from bottom to top (resp.~by values from left to right). For a signature~$\varepsilon \in \pm^n$, we denote by~$\mathfrak{S}_\varepsilon$ (resp.~by~$\mathfrak{S}^\varepsilon$) the set of signed permutations~$\tau$ with p-signature~$\signature_p(\tau) = \varepsilon$ (resp.~with v-signature~$\signature_v(\tau) = \varepsilon$). Finally, we denote by \[ \mathfrak{S}_\pm \eqdef \bigsqcup_{\substack{n \in \mathbb{N} \\ \varepsilon \in \pm^n}} \mathfrak{S}_\varepsilon = \bigsqcup_{\substack{n \in \mathbb{N} \\ \varepsilon \in \pm^n}} \mathfrak{S}^\varepsilon \] the set of all signed permutations.
In concrete examples, we underline negative positions/values while we overline positive positions/values: for example, we write~$\down{2}\up{7}\down{51}\up{3}\down{4}\up{6}$ for the signed permutation represented on the top left corner of \fref{fig:insertionAlgorithm}, where~${\tau = [2,7,5,1,3,4,6]}$, ${\signature_p = {-}{+}{-}{-}{+}{-}{+}}$ and~${\signature_v = {-}{-}{+}{-}{-}{+}{+}}$.
Following~\cite{LangePilaud}, we now present an algorithm to construct a leveled $\varepsilon$-Cambrian tree~$\Theta(\tau)$ from a signed permutation~$\tau \in \mathfrak{S}^\varepsilon$. \fref{fig:insertionAlgorithm} illustrates this algorithm on the permutation~$\down{2}\up{7}\down{51}\up{3}\down{4}\up{6}$. As a preprocessing, we represent the table of~$\tau$ (with signed dots in positions~$(\tau(i),i)$ for~$i \in [n]$) and draw a vertical wall below the negative vertices and above the positive vertices. We then sweep the table from bottom to top (thus reading the permutation~$\tau$ from left to right) as follows. The procedure starts with an incoming strand in between any two consecutive negative values. A negative dot~$\ominus$ connects the two strands immediately to its left and immediately to its right to form a unique outgoing strand. A positive dot~$\oplus$ separates the only visible strand (not hidden by a wall) into two outgoing strands. The procedure finishes with an outgoing strand in between any two consecutive positive values. See \fref{fig:insertionAlgorithm}.
\begin{figure}
\caption{The insertion algorithm on the signed permutation~$\down{2}\up{7}\down{51}\up{3}\down{4}\up{6}$.}
\label{fig:insertionAlgorithm}
\end{figure}
\begin{proposition}[\cite{LangePilaud}] The map~$\Theta$ is a bijection from signed permutations to leveled Cambrian trees. \end{proposition}
\begin{remark}[Cambrian correspondence] The \defn{Robinson-Schensted correspondence} is a bijection between permutations and pairs of standard Young tableaux of the same shape. Schensted's algorithm~\cite{Schensted} gives an efficient algorithmic way to create the pair of tableaux~$(\mathbf{P}(\tau), \mathbf{Q}(\tau))$ corresponding to a given permutation~$\tau$ by successive insertions: the first tableau~$\mathbf{P}(\tau)$ (\defn{insertion tableau}) remembers the inserted elements of~$\tau$ while the second tableau~$\mathbf{Q}(\tau)$ (\defn{recording tableau}) remembers the order in which the elements have been inserted. F.~Hivert, J.-C.~Novelli and \mbox{J.-Y.~Thibon} defined in~\cite{HivertNovelliThibon-algebraBinarySearchTrees} a similar correspondence, called \defn{sylvester correspondence}, between permutations and pairs of labeled trees of the same shape. In the sylvester correspondence, the first tree (insertion tree) is a standard binary search tree and the second tree (recording tree) is an increasing binary tree. The \defn{Cambrian correspondence} can as well be thought of as a correspondence between signed permutations and pairs of trees of the same shape, where the first tree (insertion tree) is Cambrian and the second tree (recording tree) is increasing. This analogy motivates the following definition. \end{remark}
\begin{definition} Given a signed permutation~$\tau \in \mathfrak{S}^\varepsilon$, its \defn{$\mathbf{P}$-symbol} is the insertion Cambrian tree~$\mathbf{P}(\tau)$ defined by~$\Theta(\tau)$ and its \defn{$\mathbf{Q}$-symbol} is the recording increasing tree~$\mathbf{Q}(\tau)$ defined by~$\Theta(\tau)$. \end{definition}
The following characterization of the fibers of~$\mathbf{P}$ is immediate from the description of the algorithm. We denote by~$\mathcal{L}(\graphG)$ the set of linear extensions of a directed graph~$\graphG$.
\begin{proposition} The signed permutations~$\tau \in \mathfrak{S}^\varepsilon$ such that~$\mathbf{P}(\tau) = \tree$ are precisely the linear extensions of (the transitive closure of)~$\tree$. \end{proposition}
\begin{example} When~$\varepsilon = (+)^n$, the procedure constructs a binary search tree~$\mathbf{P}(\tau)$ pointing up by successive insertions from the left. Equivalently, $\mathbf{P}(\tau)$ can be constructed as the increasing tree of~$\tau^{-1}$. Here, the \defn{increasing tree}~$\mathrm{IT}(\pi)$ of a permutation~$\pi = \pi' 1 \pi''$ is defined inductively by grafting the increasing tree~$\mathrm{IT}(\pi')$ on the left and the increasing tree~$\mathrm{IT}(\pi'')$ on the right of the bottom root labeled by~$1$. When~$\varepsilon = (-)^n$, this procedure constructs bottom-up a binary search tree~$\mathbf{P}(\tau)$ pointing down. This tree would be obtained by successive binary search tree insertions from the right. Equivalently, $\mathbf{P}(\tau)$ can be constructed as the decreasing tree of~$\tau^{-1}$. Here, the \defn{decreasing tree}~$\mathrm{DT}(\pi)$ of a permutation~$\pi = \pi' n \pi''$ is defined inductively by grafting the decreasing tree~$\mathrm{DT}(\pi')$ on the left and the decreasing tree~$\mathrm{DT}(\pi'')$ on the right of the top root labeled by~$n$. These observations are illustrated on~\fref{fig:constantSigns}.
\begin{figure}
\caption{The insertion procedure produces binary search trees when the signature is constant positive (left) or constant negative (right).}
\label{fig:constantSigns}
\end{figure} \end{example}
\begin{remark}[Cambrian correspondence on triangulations] \label{rem:CambrianCorrespondenceTriangulations} N.~Reading~\cite{Reading-CambrianLattices} first described the map~$\mathbf{P}$ on the triangulations of the polygon~$\polygon$ (remember Remark~\ref{rem:triangulation}). Namely, the triangulation~$\mathbf{P}(\tau)^*$ is the union of the paths~$\pi_0, \dots, \pi_n$ where~$\pi_i$ is the path between vertices~$0$ and~$n+1$ of~$\polygon$ passing through the vertices in the symmetric difference~$\varepsilon^{-1}(-) \,\triangle\, \tau([i])$. \end{remark}
\subsection{Cambrian congruence}
Following the definition of the sylvester congruence in~\cite{HivertNovelliThibon-algebraBinarySearchTrees}, we now characterize by a congruence relation the signed permutations~$\tau \in \mathfrak{S}^\varepsilon$ which have the same $\mathbf{P}$-symbol~$\mathbf{P}(\tau)$. This Cambrian congruence goes back to the original definition of N.~Reading~\cite{Reading-CambrianLattices}.
\begin{definition}[\cite{Reading-CambrianLattices}] \label{def:CambrianCongruence} For a signature~$\varepsilon \in \pm^n$, the \defn{$\varepsilon$-Cambrian congruence} is the equivalence relation on~$\mathfrak{S}^\varepsilon$ defined as the transitive closure of the rewriting rules \begin{gather*} UacVbW \equiv_\varepsilon UcaVbW \quad\text{if } a < b < c \text{ and } \varepsilon_b = -, \\ UbVacW \equiv_\varepsilon UbVcaW \quad\text{if } a < b < c \text{ and } \varepsilon_b = +, \end{gather*} where~$a,b,c$ are elements of~$[n]$ while~$U,V,W$ are words on~$[n]$. The \defn{Cambrian congruence} is the equivalence relation on all signed permutations~$\mathfrak{S}_\pm$ obtained as the union of all $\varepsilon$-Cambrian congruences: \[ \equiv \; \eqdef \bigsqcup_{\substack{n \in \mathbb{N} \\ \varepsilon \in \pm^n}} \!\! \equiv_\varepsilon. \] \end{definition}
\begin{proposition} \label{prop:CambrianClass} Two signed permutations~$\tau, \tau' \in \mathfrak{S}^\varepsilon$ are $\varepsilon$-Cambrian congruent if and only if they have the same $\mathbf{P}$-symbol: \[ \tau \equiv_\varepsilon \tau' \iff \mathbf{P}(\tau) = \mathbf{P}(\tau'). \] \end{proposition}
\begin{proof} It boils down to observe that two consecutive vertices~$a,c$ in a linear extension~$\tau$ of a $\varepsilon$-Cambrian tree~$\tree$ can be switched while preserving a linear extension~$\tau'$ of~$\tree$ precisely when they belong to distinct subtrees of a vertex~$b$ of~$\tree$. It follows that the vertices~$a,c$ lie on either sides of~$b$ so that we have~$a < b < c$. If~$\varepsilon_b = -$, then~$a,c$ appear before~$b$ and~$\tau = UacVbW$ can be switched to~$\tau' = UcaVbW$, while if~$\varepsilon_b = +$, then~$a,c$ appear after~$b$ and~$\tau = UbVacW$ can be switched to~$\tau' = UbVcaW$. \end{proof}
\subsection{Cambrian classes and generating trees} \label{subsec:CambrianClasses}
We now focus on the equivalence classes of the Cambrian congruence. Remember that the \defn{(right) weak order} on~$\mathfrak{S}^\varepsilon$ is defined as the inclusion order of coinversions, where a \defn{coinversion} of~$\tau \in \mathfrak{S}^\varepsilon$ is a pair of values~$i < j$ such that~${\tau^{-1}(i) > \tau^{-1}(j)}$ (no matter the signs on~$\tau$). In this paper, we always work with the right weak order, that we simply call weak order for brevity. The following statement is due to N.~Reading~\cite{Reading-CambrianLattices}.
\begin{proposition}[\cite{Reading-CambrianLattices}] All $\varepsilon$-Cambrian classes are intervals of the weak order on~$\mathfrak{S}^\varepsilon$. \end{proposition}
Therefore, the $\varepsilon$-Cambrian trees are in bijection with the weak order maximal permutations of~$\varepsilon$-Cambrian classes. Using Definition~\ref{def:CambrianCongruence} and Proposition~\ref{prop:CambrianClass}, one can prove that these permutations are precisely the permutations in~$\mathfrak{S}^\varepsilon$ that avoid the signed patterns~$b \text{-} ac$ with~$\varepsilon_b = +$ and~$ac \text{-} b$ with~$\varepsilon_b = -$ (for brevity, we write~$\up{b} \text{-} ac$ and~$ac \text{-} \down{b}$). It enables us to construct a generating tree~$\mathcal{T}_{\varepsilon}$ for these permutations. This tree has~$n$ levels, and the nodes at level~$m$ are labeled by the permutations of~$[m]$ whose values are signed by the restriction of~$\varepsilon$ to~$[m]$ and avoiding the two patterns~$\up{b} \text{-} ac$ and~$ac \text{-} \down{b}$. The parent of a permutation in~$\mathcal{T}_\varepsilon$ is obtained by deleting its maximal value. See ~\fref{fig:GeneratingTree} for examples of such trees. The following statement provides another proof that the number of~$\varepsilon$-Cambrian trees on~$n$ nodes is always the Catalan number~$C_n = \frac{1}{n+1}\binom{2n}{n}$, as well as an explicit bijection between~$\varepsilon$- and $\varepsilon'$-Cambrian trees for distinct signatures~$\varepsilon, \varepsilon' \in \pm^n$.
\begin{figure}\label{fig:GeneratingTree}
\end{figure}
\begin{proposition} \label{prop:GeneratingTree} For any signatures~$\varepsilon, \varepsilon' \in \pm^n$, the generating trees~$\mathcal{T}_\varepsilon$ and~$\mathcal{T}_{\varepsilon'}$ are isomorphic. \end{proposition}
For the proof, we consider the possible positions of~$m+1$ in the children of a permutation~$\tau$ at level~$m$ in~$\mathcal{T}_\varepsilon$. Index by~$\{0, \dots, m\}$ from left to right the gaps before the first letter, between two consecutive letters, and after the last letter of~$\tau$. We call \defn{free gaps} the gaps in~$\{0, \dots, m\}$ where placing~$m+1$ does not create a pattern~$ac \text{-} \down{b}$ or~$\up{b} \text{-} ac$. They are marked with a blue point in \fref{fig:GeneratingTree}.
\begin{lemma} \label{lem:GeneratingTree} A permutation with~$k$ free gaps has $k$ children in~$\mathcal{T}_\varepsilon$, whose numbers of free gaps range from~$2$ to~$k+1$. \end{lemma}
\begin{proof} Let~$\tau$ be a permutation at level~$m$ in~$\mathcal{T}_\varepsilon$ with $k$ free gaps. Let~$\sigma$ be the child of~$\tau$ in~$\mathcal{T}_\varepsilon$ obtained by inserting~$m+1$ at a free gap~$j \in \{0, \dots, m\}$. If~$\varepsilon_{m + 1}$ is negative (resp.~positive), then the free gaps of~$\sigma$ are $0$, $j+1$ and the free gaps of~$\tau$ after~$j$ (resp.~before~$j+1$). The result follows. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:GeneratingTree}] Order the children of a permutation of~$\mathcal{T}_\varepsilon$ from left to right by increasing number of free gaps as in \fref{fig:GeneratingTree}. Lemma~\ref{lem:GeneratingTree} shows that the shape of the resulting tree is independent of~$\varepsilon$. It ensures that the trees~$\mathcal{T}_\varepsilon$ and~$\mathcal{T}_{\varepsilon'}$ are isomorphic and provides an explicit bijection between the~$\varepsilon$-Cambrian trees and~$\varepsilon'$-Cambrian trees. \end{proof}
\subsection{Rotations and Cambrian lattices}
We now present rotations in Cambrian trees, a local operation which transforms a $\varepsilon$-Cambrian tree into another $\varepsilon$-Cambrian tree where a single oriented cut differs (see Proposition~\ref{prop:rotation}).
\begin{definition} Let~$i \to j$ be an edge in a Cambrian tree~$\tree$, with~$i < j$. Let~$L$ denote the left subtree of~$i$ and~$B$ denote the remaining incoming subtree of~$i$, and similarly, let~$R$ denote the right subtree of~$j$ and~$A$ denote the remaining outgoing subtree of~$j$. Let~$\tree'$ be the oriented tree obtained from~$\tree$ just reversing the orientation of~$i \to j$ and attaching the subtrees~$L$ and~$A$ to~$i$ and the subtrees~$B$ and~$R$ to~$j$. The transformation from~$\tree$ to~$\tree'$ is called \defn{rotation} of the edge~$i \to j$. See \fref{fig:rotation}.
\begin{figure}
\caption{Rotations in Cambrian trees: the tree~$\tree$ (top) is transformed into the tree~$\tree'$ (bottom) by rotation of the edge~$i \to j$. The four cases correspond to the possible signs of~$i$~and~$j$.}
\label{fig:rotation}
\end{figure} \end{definition}
The following proposition states that rotations are compatible with Cambrian trees and their edge cuts. An \defn{edge cut} in a Cambrian tree~$\tree$ is the ordered partition~$\edgecut{X}{Y}$ of the vertices of~$\tree$ into the set~$X$ of vertices in the source set and the set~$Y$ of vertices in the target set of an oriented edge of~$\tree$.
\begin{proposition}[\cite{LangePilaud}] \label{prop:rotation} The result~$\tree'$ of the rotation of an edge~$i \to j$ in a $\varepsilon$-Cambrian tree~$\tree$ is a $\varepsilon$-Cambrian tree. Moreover, $\tree'$ is the unique $\varepsilon$-Cambrian tree with the same edge cuts as~$\tree$, except the cut defined by the edge~$i \to j$. \end{proposition}
\begin{remark}[Rotations and flips] Rotating an edge~$e$ in a $\varepsilon$-Cambrian tree~$\tree$ corresponds to flipping the dual diagonal~$e^*$ of the dual triangulation~$\tree^*$ of the polygon~$\polygon$. See~\cite[Lemma~13]{LangePilaud}. \end{remark}
Define the \defn{increasing rotation graph} on~$\mathrm{Camb}(\varepsilon)$ to be the graph whose vertices are the $\varepsilon$-Cambrian trees and whose arcs are increasing rotations~$\tree \to \tree'$, \textit{i.e.}~ where the edge~$i \to j$ in~$\tree$ is reversed to the edge~$i \leftarrow j$ in~$\tree'$ for~$i < j$. See \fref{fig:CambrianLattices} for an illustration. The following statement, adapted from N.~Reading's work~\cite{Reading-CambrianLattices}, asserts that this graph is acyclic, that its transitive closure defines a lattice, and that this lattice is closely related to the weak order. See \fref{fig:lattices}.
\hvFloat[floatPos=p, capWidth=h, capPos=r, capAngle=90, objectAngle=90, capVPos=c, objectPos=c]{figure} {\includegraphics[width=1.65\textwidth]{CambrianLattices}} {The $\varepsilon$-Cambrian lattices on $\varepsilon$-Cambrian trees, for the signatures $\varepsilon = {-}{+}{-}{-}$ (left) and ${\varepsilon = {+}{-}{-}{-}}$~(right).} {fig:CambrianLattices}
\begin{proposition}[\cite{Reading-CambrianLattices}] The transitive closure of the increasing rotation graph on~$\mathrm{Camb}(\varepsilon)$ is a lattice, called \defn{$\varepsilon$-Cambrian lattice}. The map~$\mathbf{P} : \mathfrak{S}^\varepsilon \to \mathrm{Camb}(\varepsilon)$ defines a lattice homomorphism from the weak order on~$\mathfrak{S}^\varepsilon$ to the $\varepsilon$-Cambrian lattice on~$\mathrm{Camb}(\varepsilon)$. \end{proposition}
Note that the minimal (resp.~maximal) $\varepsilon$-Cambrian tree is an oriented path from~$1$ to~$n$ (resp.~from~$n$ to~$1$) with an additional incoming leaf at each negative vertex and an additional outgoing leaf at each positive vertex. See \fref{fig:CambrianLattices}.
\begin{example} When~$\varepsilon = (-)^n$, the Cambrian lattice is the classical Tamari lattice~\cite{TamariFestschrift}. It can be defined equivalently by left-to-right rotations in planar binary trees, by slope increasing flips in triangulations of~$\polygon[(-)^n]$, or as the quotient of the weak order by the sylvester congruence. \end{example}
\subsection{Canopy} \label{subsec:canopy}
The canopy of a binary tree was already used by J.-L.~Loday in~\cite{LodayRonco, Loday} but the name was coined by X.~Viennot~\cite{Viennot}. It was then extended to Cambrian trees (or spines) in~\cite{LangePilaud} to define a surjection from the associahedron~$\Asso$ to the parallelepiped~$\Para$ generated by the simple roots. The main observation is that the vertices~$i$ and~$i+1$ are always comparable in a Cambrian tree (otherwise, they would be in distinct subtrees of a vertex~$j$ which should then lie in between~$i$ and~$i+1$).
\begin{definition} The \defn{canopy} of a Cambrian tree~$\tree$ is the sequence~$\mathbf{can}(\tree) \in \pm^{n-1}$ defined by ${\mathbf{can}(\tree)_i = -}$ if $i$ is above~$i+1$ in~$\tree$ and~${\mathbf{can}(\tree)_i = +}$ if $i$ is below~$i+1$ in~$\tree$. \end{definition}
For example, the canopy of the Cambrian tree of \fref{fig:leveledCambrianTree}\,(left) is~${-}{+}{+}{-}{+}{-}$. The canopy of~$\tree$ behaves nicely with the linear extensions of~$\tree$ and with the Cambrian lattice. To state this, we define for a permutation~$\tau \in \mathfrak{S}^\varepsilon$ the sequence~$\mathbf{rec}(\tau) \in \pm^{n-1}$, where~$\mathbf{rec}(\tree)_i = -$ if~$\tau^{-1}(i) > \tau^{-1}(i+1)$ and~$\mathbf{rec}(\tree)_i = +$ otherwise. In other words, $\mathbf{rec}(\tau)$ records the \defn{recoils} of the permutation~$\tau$, \textit{i.e.}~ the \defn{descents} of the inverse permutation of~$\tau$.
\begin{proposition} \label{prop:commutativeDiagram} The maps~$\mathbf{P}, \mathbf{can}$, and~$\mathbf{rec}$ define the following commutative diagram of lattice homomorphisms: \[ \begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=1.5em,column sep=5em,minimum width=2em]
{
\mathfrak{S}^\varepsilon & & \pm^{n-1} \\
& \mathrm{Camb}(\varepsilon) & \\
};
\path[->>]
(m-1-1) edge node [above] {$\mathbf{rec}$} (m-1-3)
edge node [below] {$\mathbf{P}$} (m-2-2.west)
(m-2-2.east) edge node [below] {$\quad\mathbf{can}$} (m-1-3); \end{tikzpicture} \] \end{proposition}
The fibers of these maps on the weak orders of~$\mathfrak{S}_\varepsilon$ for $\varepsilon = {-}{+}{-}{-}$ and $\varepsilon = {+}{-}{-}{-}$ are represented in \fref{fig:lattices}.
\begin{figure}
\caption{The fibers of the maps~$\mathbf{P}$ (red) and~$\mathbf{rec}$ (green) on the weak orders of~$\mathfrak{S}_\varepsilon$ for $\varepsilon = {-}{+}{-}{-}$ (left) and $\varepsilon = {+}{-}{-}{-}$ (right).}
\label{fig:lattices}
\end{figure}
\subsection{Geometric realizations} \label{subsec:geometricRealizations}
We close this section with geometric interpretations of the Cambrian trees, Cambrian classes, Cambrian correspondence, and Cambrian lattices. We denote by~$e_1, \dots, e_n$ the canonical basis of~$\mathbb{R}^n$ and by~$\mathbb{H}$ the hyperplane of~$\mathbb{R}^n$ orthogonal to~$\sum e_i$. Define the \defn{incidence cone}~$\mathrm{C}(\tree)$ and the \defn{braid cone}~$\mathrm{C}^\diamond(\tree)$ of a directed tree~$\tree$ as \[ \mathrm{C}(\tree) \eqdef \cone\set{e_i-e_j}{\text{for all } i \to j \text{ in } \tree} \quad\text{and}\quad \mathrm{C}^\diamond(\tree) \eqdef \set{\b{x} \in \mathbb{H}}{x_i \le x_j \text{ for all } i \to j \text{ in } \tree}. \] These two cones lie in the space~$\mathbb{H}$ and are polar to each other. For a permutation~${\tau \in \mathfrak{S}_n}$, we denote by~$\mathrm{C}(\tau)$ and~$\mathrm{C}^\diamond(\tau)$ the incidence and braid cone of the chain~$\tau(1) \to \dots \to \tau(n)$. Finally, for a sign vector~$\chi \in \pm^{n-1}$, we denote by~$\mathrm{C}(\tau)$ and~$\mathrm{C}^\diamond(\tau)$ the incidence and braid cone of the oriented path~$1 - \dots - n$, where~$i \to i+1$ if~$\chi_i = +$ and $i \leftarrow i+1$ if~$\chi_i = -$.
These cones (together with all their faces) form complete simplicial fans in~$\mathbb{H}$: \begin{enumerate}[(i)] \item the cones~$\mathrm{C}^\diamond(\tau)$, for all permutations~$\tau \in \mathfrak{S}_n$, form the \defn{braid fan}, which is the normal fan of the \defn{permutahedron}~$\Perm \eqdef \conv\bigset{\sum_{i \in [n]} \tau(i) e_i}{\tau \in \mathfrak{S}_n}$; \item the cones~$\mathrm{C}^\diamond(\tree)$, for all $\varepsilon$-Cambrian trees~$\tree$, form the \defn{$\varepsilon$-Cambrian fan}, which is the normal fan of the \defn{$\varepsilon$-associahedron}~$\Asso$ of C.~Hohlweg and C.~Lange~\cite{HohlwegLange} (see also~\cite{LangePilaud}); \item the cones~$\mathrm{C}^\diamond(\chi)$, for all sign vectors~$\chi \in \pm^{n-1}$, form the \defn{boolean fan}, which is the normal fan of the parallelepiped~$\Para \eqdef \bigset{\b{x} \in \mathbb{H}}{i(2n+1-i) \le 2 \sum_{j \le i} x_j \le i(i+1) \text{ for all } i \in [n]}$. \end{enumerate} In fact, $\Asso$ is obtained by deleting certain inequalities in the facet description of~$\Perm$, and similarly, $\Para$ is obtained by deleting facets of~$\Asso$. In particular, we have the geometric inclusions~$\Perm \subset \Asso \subset \Para$. See \fref{fig:permutahedraAssociahedraCubes} for $3$-dimensional examples.
\begin{figure}
\caption{The polytope inclusion~$\Perm[4] \subset \Asso \subset \Para[4]$ for different signatures~$\varepsilon \in \pm^4$. The permutahedron~$\Perm[4]$ is represented in red, the associahedron~$\Asso$ in blue, and the parallelepiped~$\Para[4]$ in green.}
\label{fig:permutahedraAssociahedraCubes}
\end{figure}
\enlargethispage{-.4cm} The incidence and braid cones also characterize the maps~$\mathbf{P}$, $\mathbf{can}$, and~$\mathbf{rec}$ as follows \begin{gather*} \tree = \mathbf{P}(\tau) \iff \mathrm{C}(\tree) \subseteq \mathrm{C}(\tau) \iff \mathrm{C}^\diamond(\tree) \supseteq \mathrm{C}^\diamond(\tau), \\ \chi = \mathbf{can}(\tree) \iff \mathrm{C}(\chi) \subseteq \mathrm{C}(\tree) \iff \mathrm{C}^\diamond(\chi) \supseteq \mathrm{C}^\diamond(\tree), \\ \chi = \mathbf{rec}(\tau) \iff \mathrm{C}(\chi) \subseteq \mathrm{C}(\tau) \iff \mathrm{C}^\diamond(\chi) \supseteq \mathrm{C}^\diamond(\tau). \end{gather*} In particular, Cambrian classes are formed by all permutations whose braid cone belong to the same Cambrian cone. Finally, the $1$-skeleta of the permutahedron~$\Perm$, associahedron~$\Asso$ and parallelepiped~$\Para$, oriented in the direction~$(n, \dots, 1) - (1, \dots, n) = \sum_{i \in [n]} (n+1-2i) \, e_i$ are the Hasse diagrams of the weak order, the Cambrian lattice and the boolean lattice respectively. These geometric properties originally motivated the definition of Cambrian trees in~\cite{LangePilaud}.
\section{Cambrian Hopf Algebra} \label{sec:CambrianAlgebra}
In this section, we introduce the Cambrian Hopf algebra~$\mathsf{Camb}$ as a subalgebra of the Hopf algebra~$\mathsf{FQSym}_\pm$ on signed permutations, and the dual Cambrian algebra~$\mathsf{Camb}^*$ as a quotient algebra of the dual Hopf algebra~$\mathsf{FQSym}_\pm^*$. We describe both the product and coproduct in these algebras in terms of combinatorial operations on Cambrian trees. These results extend the approach of F.~Hivert, J.-C.~Novelli and~J.-Y.~Thibon~\cite{HivertNovelliThibon-algebraBinarySearchTrees} to construct the algebra of \mbox{J.-L.~Loday} and M.~Ronco on binary trees~\cite{LodayRonco} as a subalgebra of the algebra of C.~Malvenuto and C.~Reutenauer on permutations~\cite{MalvenutoReutenauer}.
We immediately mention that a different generalization was studied by N.~Reading in~\cite{Reading-HopfAlgebras}. His idea was to construct a subalgebra of C.~Malvenuto and C.~Reutenauer's algebra~$\mathsf{FQSym}$ using equivalent classes of a congruence relation defined as the union~$\bigcup_{n \in \mathbb{N}} \equiv_{\varepsilon_n}$ of $\varepsilon_n$-Cambrian relation for one fixed signature~$\varepsilon_n \in \pm^n$ for each~$n \in \mathbb{N}$. In order to obtain a valid Hopf algebra, the choice of~$(\varepsilon_n)_{n \in \mathbb{N}}$ has to satisfy certain compatibility relations: N.~Reading characterizes the ``translational'' (resp.~``insertional'') families~$\equiv_n$ of lattice congruences on~$\mathfrak{S}_n$ for which the sums over the elements of the congruence classes of~$(\equiv_n)_{n \in \mathbb{N}}$ form the basis of a subalgebra (resp.~subcoalgebra) of~$\mathsf{FQSym}$. These conditions make the choice of~$(\varepsilon_n)_{n \in \mathbb{N}}$ rather constrained. In contrast, by constructing a subalgebra of~$\mathsf{FQSym}_\pm$ rather than~$\mathsf{FQSym}$, we consider simultaneously all Cambrian relations for all signatures. In particular, our Cambrian algebra contains all Hopf algebras of~\cite{Reading-HopfAlgebras} as subalgebras.
\subsection{Signed shuffle and convolution products} \label{subsec:products}
\enlargethispage{-.2cm} For~$n,n' \in \mathbb{N}$, let \[ \mathfrak{S}^{(n,n')} \eqdef \set{\tau \in \mathfrak{S}_{n+n'}}{\tau(1) < \dots < \tau(n) \text{ and } \tau(n+1) < \dots < \tau(n+n')} \] denote the set of permutations of~$\mathfrak{S}_{n+n'}$ with at most one descent, at position~$n$. The \defn{shifted concatenation}~$\tau\bar\tau'$, the \defn{shifted shuffle product}~$\tau \,\bar\shuffle\, \tau'$, and the \defn{convolution product}~$\tau \star \tau'$ of two (unsigned) permutations~$\tau \in \mathfrak{S}_n$ and~$\tau' \in \mathfrak{S}_{n'}$ are classically defined by \begin{gather*} \tau\bar\tau' \eqdef [\tau(1), \dots, \tau(n), \tau'(1) + n, \dots, \tau'(n') + n] \in \mathfrak{S}_{n+n'}, \\ \tau \,\bar\shuffle\, \tau' \eqdef \bigset{(\tau\bar\tau') \circ \pi^{-1}}{\pi \in \mathfrak{S}^{(n,n')}} \qquad\text{and}\qquad \tau \star \tau' \eqdef \bigset{\pi \circ (\tau\bar\tau')}{\pi \in \mathfrak{S}^{(n,n')}}. \end{gather*} For example, \begin{align*} {\color{red} 12} \,\bar\shuffle\, {\color{darkblue} 231} & = \{ {\color{red} 12}{\color{darkblue} 453}, {\color{red} 1}{\color{darkblue} 4}{\color{red} 2}{\color{darkblue} 53}, {\color{red} 1}{\color{darkblue} 45}{\color{red} 2}{\color{darkblue} 3}, {\color{red} 1}{\color{darkblue} 453}{\color{red} 2}, {\color{darkblue} 4}{\color{red} 12}{\color{darkblue} 53}, {\color{darkblue} 4}{\color{red} 1}{\color{darkblue} 5}{\color{red} 2}{\color{darkblue} 3}, {\color{darkblue} 4}{\color{red} 1}{\color{darkblue} 53}{\color{red} 2}, {\color{darkblue} 45}{\color{red} 12}{\color{darkblue} 3}, {\color{darkblue} 45}{\color{red} 1}{\color{darkblue} 3}{\color{red} 2}, {\color{darkblue} 453}{\color{red} 12} \}, \\ {\color{red} 12} \star {\color{darkblue} 231} & = \{ {\color{red} 12}{\color{darkblue} 453}, {\color{red} 13}{\color{darkblue} 452}, {\color{red} 14}{\color{darkblue} 352}, {\color{red} 15}{\color{darkblue} 342}, {\color{red} 23}{\color{darkblue} 451}, {\color{red} 24}{\color{darkblue} 351}, {\color{red} 25}{\color{darkblue} 341}, {\color{red} 34}{\color{darkblue} 251}, {\color{red} 35}{\color{darkblue} 241}, {\color{red} 45}{\color{darkblue} 231} \}. \end{align*} These operations can be visualized graphically on the tables of the permutations~$\tau, \tau'$. Remember that the table of~$\tau$ contains a dot at coordinates~$(\tau(i),i)$ for each~$i \in [n]$. The table of the shifted concatenation~$\tau\bar\tau'$ contains the table of~$\tau$ as the bottom left block and the table of~$\tau'$ as the top right block. The tables in the shifted shuffle product~$\tau \,\bar\shuffle\, \tau'$ (resp.~in the convolution product~$\tau \star \tau'$) are then obtained by shuffling the rows (resp.~columns) of the table of~$\tau\bar\tau'$. In particular, we obtain the table of~$\tau$ if we erase all dots in the~$n'$ rightmost columns (resp. topmost rows) of a table in the shifted shuffle product~$\tau \,\bar\shuffle\, \tau'$ (resp.~in the convolution product~$\tau \star \tau'$). See~\fref{fig:shuffleConvolution}.
\begin{figure}
\caption{The table of the shifted concatenation~$\tau\bar\tau'$ (left) has two blocks containing the tables of the permutations~$\tau = 12$ and~$\tau' = 231$. Elements of the shifted shuffle product~$\tau \,\bar\shuffle\, \tau'$ (middle) and of the convolution product~$\tau \star \tau'$~(right) are obtained by shuffling respectively the rows and columns of the table of~$\tau\bar\tau'$.}
\label{fig:shuffleConvolution}
\end{figure}
These definitions extend to signed permutations. The \defn{signed shifted shuffle product}~$\tau \,\bar\shuffle\, \tau'$ is defined as the shifted product of the permutations where signs travel with their values, while the \defn{signed convolution product}~$\tau \star \tau'$ is defined as the convolution product of the permutations where signs stay at their positions. For example, \begin{align*} \upr{1}\downr{2} \,\bar\shuffle\, \downb{23}\upb{1} & = \{ \upr{1}\downr{2}\downb{45}\upb{3}, \upr{1}\downb{4}\downr{2}\downb{5}\upb{3}, \upr{1}\downb{45}\downr{2}\upb{3}, \upr{1}\downb{45}\upb{3}\downr{2}, \downb{4}\upr{1}\downr{2}\downb{5}\upb{3}, \downb{4}\upr{1}\downb{5}\downr{2}\upb{3}, \downb{4}\upr{1}\downb{5}\upb{3}\downr{2}, \downb{45}\upr{1}\downr{2}\upb{3}, \downb{45}\upr{1}\upb{3}\downr{2}, \downb{45}\upb{3}\upr{1}\downr{2} \}, \\ \upr{1}\downr{2} \star \downb{23}\upb{1} & = \{ \upr{1}\downr{2}\downb{45}\upb{3}, \upr{1}\downr{3}\downb{45}\upb{2}, \upr{1}\downr{4}\downb{35}\upb{2}, \upr{1}\downr{5}\downb{34}\upb{2}, \upr{2}\downr{3}\downb{45}\upb{1}, \upr{2}\downr{4}\downb{35}\upb{1}, \upr{2}\downr{5}\downb{34}\upb{1}, \upr{3}\downr{4}\downb{25}\upb{1}, \upr{3}\downr{5}\downb{24}\upb{1}, \upr{4}\downr{5}\downb{23}\upb{1} \}. \end{align*} Note that the shifted shuffle is compatible with signed values, while the convolution is compatible with signed positions in the sense that \[ \mathfrak{S}^\varepsilon \,\bar\shuffle\, \mathfrak{S}^{\varepsilon'} = \mathfrak{S}^{\varepsilon\varepsilon'} \qquad\text{while}\qquad \mathfrak{S}_\varepsilon \star \mathfrak{S}_{\varepsilon'} = \mathfrak{S}_{\varepsilon\varepsilon'}. \] In any case, both~$\,\bar\shuffle\,$ and~$\star$ are compatible with the distribution of positive and negative signs,~\textit{i.e.}~ \[
|\tau \,\bar\shuffle\, \tau'|_+ = |\tau|_+ + |\tau'|_+ = |\tau \star \tau'|_+ \qquad\text{and}\qquad
|\tau \,\bar\shuffle\, \tau'|_- = |\tau|_- + |\tau'|_- = |\tau \star \tau'|_-. \]
\subsection{Subalgebra of $\mathsf{FQSym}_\pm$} \label{subsec:subalgebra}
We denote by~$\mathsf{FQSym}_\pm$ the Hopf algebra with basis~$(\mathbb{F}_\tau)_{\tau \in \mathfrak{S}_\pm}$ and whose product and coproduct are defined by \[ \mathbb{F}_\tau \cdot \mathbb{F}_{\tau'} = \sum_{\sigma \in \tau \,\bar\shuffle\, \tau'} \mathbb{F}_\sigma \qquad\text{and}\qquad \triangle \mathbb{F}_\sigma = \sum_{\sigma \in \tau \star \tau'} \mathbb{F}_\tau \otimes \mathbb{F}_{\tau'}. \] This Hopf algebra is bigraded by the size and the number of positive signs of the signed permutations. It naturally extends to signed permutations the Hopf algebra~$\mathsf{FQSym}$ on permutations defined by C.~Malvenuto and C.~Reute\-nauer~\cite{MalvenutoReutenauer}.
We denote by~$\mathsf{Camb}$ the vector subspace of~$\mathsf{FQSym}_\pm$ generated by the elements \[ \mathbb{P}_{\tree} \eqdef \sum_{\substack{\tau \in \mathfrak{S}_\pm \\ \mathbf{P}(\tau) = \tree}} \mathbb{F}_\tau = \sum_{\tau \in \mathcal{L}(\tree)} \mathbb{F}_\tau, \] for all Cambrian trees~$\tree$. For example, for the Cambrian tree of \fref{fig:leveledCambrianTree}\,(left), we have \[ \mathbb{P}_{\!\!\includegraphics{Tex}} = \begin{array}[t]{c} \phantom{+}\mathbb{F}_{\down{21}\up{37}\down{54}\up{6}} + \mathbb{F}_{\down{21}\up{73}\down{54}\up{6}} + \mathbb{F}_{\down{21}\up{7}\down{5}\up{3}\down{4}\up{6}} + \mathbb{F}_{\down{2}\up{7}\down{1}\up{3}\down{54}\up{6}} + \mathbb{F}_{\down{2}\up{7}\down{15}\up{3}\down{4}\up{6}} \\ + \; \mathbb{F}_{\down{2}\up{7}\down{51}\up{3}\down{4}\up{6}} + \mathbb{F}_{\up{7}\down{21}\up{3}\down{54}\up{6}} + \mathbb{F}_{\up{7}\down{215}\up{3}\down{4}\up{6}} + \mathbb{F}_{\up{7}\down{251}\up{3}\down{4}\up{6}} + \mathbb{F}_{\up{7}\down{521}\up{3}\down{4}\up{6}}. \end{array} \]
\begin{theorem} \label{thm:cambSubalgebra} $\mathsf{Camb}$ is a Hopf subalgebra of~$\mathsf{FQSym}_\pm$. \end{theorem}
\begin{proof} We first prove that $\mathsf{Camb}$ is a subalgebra of~$\mathsf{FQSym}_\pm$. To do this, we just need to show that the Cambrian congruence is compatible with the shuffle product, \textit{i.e.}~ that the product of two Cambrian classes can be decomposed into a sum of Cambrian classes. Consider two signatures~${\varepsilon \in \pm^n}$ and~$\varepsilon' \in \pm^{n'}$, two Cambrian trees~$\tree \in \mathrm{Camb}(\varepsilon)$ and~$\tree' \in \mathrm{Camb}(\varepsilon')$, and two congruent permutations~$\sigma \equiv_{\varepsilon\varepsilon'} \tilde\sigma \in \mathfrak{S}^{\varepsilon\varepsilon'}$. We want to show that~$\mathbb{F}_\sigma$ appears in the product~$\mathbb{P}_{\tree} \cdot \mathbb{P}_{\tree'}$ if and only if~$\mathbb{F}_{\tilde\sigma}$ does. We can assume that~$\sigma = UacVbW$ and~$\tilde\sigma = UcaVbW$ for some letters~$a < b < c$ and words~$U,V,W$ with~$(\varepsilon\varepsilon')_b = -$. Suppose moreover that~$\mathbb{F}_\sigma$ appears in the product~$\mathbb{P}_{\tree} \cdot \mathbb{P}_{\tree'}$, and let~$\tau \in \mathcal{L}(\tree)$ and~$\tau' \in \mathcal{L}(\tree')$ such that~$\sigma \in \tau \,\bar\shuffle\, \tau'$. We distinguish three cases: \begin{enumerate}[(i)] \item If~$a \le n$ and~$n < c$, then~$\tilde\sigma$ also belongs~$\tau \,\bar\shuffle\, \tau'$, and thus~$\mathbb{F}_{\tilde\sigma}$ appears in the product~$\mathbb{P}_{\tree} \cdot \mathbb{P}_{\tree'}$. \item If~$a < b < c \le n$, then~$\tau = \hat U ac \hat V b \hat W$ is $\varepsilon$-congruent to~$\tilde\tau = \hat U ca \hat V b \hat W$, and thus~$\tilde\tau \in \mathcal{L}(\tree)$. Since~$\tilde\sigma \in \tilde\tau \,\bar\shuffle\, \tau'$, we obtain that~$\mathbb{F}_{\tilde\sigma}$ appears in the product~$\mathbb{P}_{\tree} \cdot \mathbb{P}_{\tree'}$. \item If~$n < a < b < c$, the argument is similar, exchanging~$ac$ to~$ca$ in~$\tau'$. \end{enumerate} The proof for the other rewriting rule of Definition~\ref{def:CambrianCongruence} is symmetric, and the general case for~${\sigma \equiv_{\varepsilon\varepsilon'} \tilde\sigma}$ follows by transitivity.
We now prove that~$\mathsf{Camb}$ is a subcoalgebra of~$\mathsf{FQSym}^{\pm}$. We just need to show that the Cambrian congruence is compatible with the deconcatenation coproduct, \textit{i.e.}~ that the coproduct of a Cambrian class is a sum of tensor products of Cambrian classes. Consider a Cambrian tree~$\tree \in \mathrm{Camb}(\eta)$, and Cambrian congruent permutations~$\tau \equiv_\varepsilon \tilde\tau \in \mathfrak{S}^\varepsilon$ and~$\tau' \equiv_{\varepsilon'} \tilde\tau' \in \mathfrak{S}^{\varepsilon'}$. We want to show that~$\mathbb{F}_\tau \otimes \mathbb{F}_{\tau'}$ appears in the coproduct~$\triangle(\mathbb{P}_{\tree})$ if and only if~$\mathbb{F}_{\tilde\tau} \otimes \mathbb{F}_{\tilde\tau'}$ does. We can assume that~$\tau = UacVbW$ and~$\tilde\tau = UcaVbW$ for some letters~$a < b < c$ and words~$U,V,W$ with~$\varepsilon_b = -$, while~$\tau' = \tilde\tau'$. Suppose moreover that~$\mathbb{F}_\tau \otimes \mathbb{F}_{\tau'}$ appears in the coproduct~$\triangle(\mathbb{P}_{\tree})$, \textit{i.e.}~ that there exists~$\sigma \in (\tau \star \tau') \cap \mathcal{L}(\tree)$. Since~$\sigma \in \tau \star \tau'$, it can be written as~$\sigma = \hat U \hat a \hat c \hat V \hat b \hat W \hat \tau'$ for some letters~$\hat a < \hat b < \hat c$ and words~$\hat U,\hat V,\hat W,\hat \tau'$ with~$\eta_{\hat b} = -$. Therefore~$\tilde\sigma = \hat U \hat c \hat a \hat V \hat b \hat W \hat \tau'$ is $\eta$-congruent to~$\sigma$ and in the convolution product~$\tilde\tau \star \tilde\tau'$. It follows that~$\mathbb{F}_{\tilde\tau} \otimes \mathbb{F}_{\tilde\tau'}$ also appears in the coproduct~$\triangle(\mathbb{P}_{\tree})$. The proofs for the other rewriting rule on~$\tau$, as well as for both rewriting rules on~$\tau'$, are symmetric, and the general case for~$\tau \equiv_\varepsilon \tilde\tau$ and~$\tau' \equiv_{\varepsilon'} \tilde\tau'$ follows by transitivity. \end{proof}
Another proof of this statement would be to show that the Cambrian congruence yields a $\varphi$-good mono\"id~\cite{Priez}. In the remaining of this section, we provide direct descriptions of the product and coproduct of $\mathbb{P}$-basis elements of~$\mathsf{Camb}$ in terms of combinatorial operations on Cambrian trees.
\para{Product} The product in the Cambrian algebra can be described in terms of intervals in Cambrian lattices. Given two Cambrian trees~$\tree, \tree'$, we denote by~$\raisebox{-6pt}{$\tree$}\nearrow \raisebox{4pt}{$\bar \tree'$}$ the tree obtained by grafting the rightmost outgoing leaf of~$\tree$ on the leftmost incoming leaf of~$\tree$ and shifting all labels of~$\tree'$. Note that the resulting tree is~$\varepsilon\varepsilon'$-Cambrian, where~$\varepsilon\varepsilon'$ is the concatenation of the signatures~$\varepsilon = \varepsilon(\tree)$ and~$\varepsilon' = \varepsilon(\tree')$. We define similarly~$\raisebox{4pt}{$\tree$} \nwarrow \raisebox{-6pt}{$\bar \tree'$}$. Examples are given in \fref{fig:exampleProduct}.
\begin{figure}
\caption{Grafting Cambrian trees.}
\label{fig:exampleProduct}
\end{figure}
\begin{proposition} \label{prop:product} For any Cambrian trees~$\tree, \tree'$, the product~$\mathbb{P}_{\tree} \cdot \mathbb{P}_{\tree'}$ is given by \[ \mathbb{P}_{\tree} \cdot \mathbb{P}_{\tree'} = \sum_{\tree[S]} \mathbb{P}_{\tree[S]}, \] where~$\tree[S]$ runs over the interval between~$\raisebox{-6pt}{$\tree$}\nearrow \raisebox{4pt}{$\bar \tree'$}$ and~$\raisebox{4pt}{$\tree$} \nwarrow \raisebox{-6pt}{$\bar \tree'$}$ in the $\varepsilon(\tree)\varepsilon(\tree')$-Cambrian lattice. \end{proposition}
\begin{proof} For any Cambrian tree~$\tree$, the linear extensions~$\mathcal{L}(\tree)$ form an interval of the weak order~\cite{Reading-CambrianLattices}. Moreover, the shuffle product of two intervals of the weak order is an interval of the weak order. Therefore, the product~$\mathbb{P}_{\tree} \cdot \mathbb{P}_{\tree'}$ is a sum of~$\mathbb{P}_{\tree[S]}$ where~$\tree[S]$ runs over an interval of the Cambrian lattice. It remains to characterize the minimal and maximal elements of this interval.
Let~$\mu_{\tree}$ and~$\omega_{\tree}$ denote respectively the smallest and the greatest linear extension of~$\tree$ in weak order. The product~$\mathbb{P}_{\tree} \cdot \mathbb{P}_{\tree'}$ is the sum of~$\mathbb{P}_{\tree[S]}$ over the interval \[ [\mu_{\tree}, \omega_{\tree}] \,\bar\shuffle\, [\mu_{\tree'}, \omega_{\tree'}] = [\mu_{\tree} \bar \mu_{\tree'}, \bar \omega_{\tree'} \omega_{\tree}], \] where~$\bar~$ denotes as usual the shifting operator on permutations. The result thus follows from the fact that \[ \mathbf{P}(\mu_{\tree} \bar \mu_{\tree'}) = \raisebox{-6pt}{$\tree$}\nearrow \raisebox{4pt}{$\bar \tree'$} \qquad\text{and}\qquad \mathbf{P}(\bar \omega_{\tree'} \omega_{\tree}) = \raisebox{4pt}{$\tree$} \nwarrow \raisebox{-6pt}{$\bar \tree'$}. \qedhere \] \end{proof}
For example, we can compute the product \[ \begin{array}{@{}c@{${} = {}$}c@{+}c@{+}c@{}@{}c} \mathbb{P}_{\!\!\includegraphics{exmProductA}} \cdot \mathbb{P}_{\includegraphics{exmProductB}} &
\multicolumn{3}{l}{\mathbb{F}_{\down{1}\up{2}} \cdot \big( \mathbb{F}_{\up{2}\down{1}\up{3}} + \mathbb{F}_{\up{2}\up{3}\down{1}} \big)}
\\[-.4cm] & \begin{pmatrix} \quad \mathbb{F}_{\down{1}\up{2}\up{4}\down{3}\up{5}} + \mathbb{F}_{\down{1}\up{2}\up{4}\up{5}\down{3}} + \mathbb{F}_{\down{1}\up{4}\up{2}\down{3}\up{5}} \\ + \; \mathbb{F}_{\down{1}\up{4}\up{2}\up{5}\down{3}} + \mathbb{F}_{\down{1}\up{4}\up{5}\up{2}\down{3}} + \mathbb{F}_{\up{4}\down{1}\up{2}\down{3}\up{5}} \\ + \; \mathbb{F}_{\up{4}\down{1}\up{2}\up{5}\down{3}} + \mathbb{F}_{\up{4}\down{1}\up{5}\up{2}\down{3}} + \mathbb{F}_{\up{4}\up{5}\down{1}\up{2}\down{3}} \end{pmatrix} & \begin{pmatrix} \quad \mathbb{F}_{\down{1}\up{4}\down{3}\up{2}\up{5}} + \mathbb{F}_{\down{1}\up{4}\down{3}\up{5}\up{2}} \\ + \; \mathbb{F}_{\down{1}\up{4}\up{5}\down{3}\up{2}} + \mathbb{F}_{\up{4}\down{1}\down{3}\up{2}\up{5}} \\ + \; \mathbb{F}_{\up{4}\down{1}\down{3}\up{5}\up{2}} + \mathbb{F}_{\up{4}\down{1}\up{5}\down{3}\up{2}} \\ + \; \mathbb{F}_{\up{4}\up{5}\down{1}\down{3}\up{2}} \end{pmatrix} & \begin{pmatrix} \quad \mathbb{F}_{\up{4}\down{3}\down{1}\up{2}\up{5}} + \mathbb{F}_{\up{4}\down{3}\down{1}\up{5}\up{2}} \\ + \; \mathbb{F}_{\up{4}\down{3}\up{5}\down{1}\up{2}} + \mathbb{F}_{\up{4}\up{5}\down{3}\down{1}\up{2}} \end{pmatrix} \\[.8cm] & \mathbb{P}_{\!\!\includegraphics{exmProduct1}} & \mathbb{P}_{\!\!\includegraphics{exmProduct2}} & \mathbb{P}_{\!\!\includegraphics{exmProduct3}} & . \end{array} \] The first equality is obtained by computing the linear extensions of the two factors, the second by computing the shuffle product and grouping terms according to their $\mathbf{P}$-symbol, displayed in the last line. Proposition~\ref{prop:product} enables us to shortcut the computation by avoiding to resort to the~$\mathbb{F}$-basis.
\para{Coproduct} The coproduct in the Cambrian algebra can also be described in combinatorial terms. Define a \defn{cut} of a Cambrian tree~$\tree[S]$ to be a set~$\gamma$ of edges such that any geodesic vertical path in~$\tree[S]$ from a down leaf to an up leaf contains precisely one edge of~$\gamma$. Such a cut separates the tree~$\tree$ into two forests, one above~$\gamma$ and one below~$\gamma$, denoted~$A(\tree[S], \gamma)$ and~$B(\tree[S],\gamma)$, respectively. An example is given in \fref{fig:exampleCoproduct}.
\begin{figure}
\caption{A cut~$\gamma$ of a Cambrian tree~$\tree$ defines two forests~$A(\tree, \gamma)$ and~$B(\tree,\gamma)$.}
\label{fig:exampleCoproduct}
\end{figure}
\begin{proposition} \label{prop:coproduct} For any Cambrian tree~$\tree[S]$, the coproduct~$\triangle \mathbb{P}_{\tree[S]}$ is given by \[ \triangle \mathbb{P}_{\tree[S]} = \sum_{\gamma} \bigg( \prod_{\tree \in B(\tree[S],\gamma)} \mathbb{P}_{\tree} \bigg) \otimes \bigg( \prod_{\tree' \in A(\tree[S], \gamma)} \mathbb{P}_{\tree'} \bigg), \] where~$\gamma$ runs over all cuts of~$\tree[S]$. \end{proposition}
\begin{proof} Let~$\sigma$ be a linear extension of~$\tree[S]$ and~$\tau, \tau' \in \mathfrak{S}_\pm$ such that~$\sigma \in \tau \star \tau'$. As discussed in Section~\ref{subsec:products}, the tables of~$\tau$ and~$\tau'$ respectively appear in the bottom and top rows of the table of~$\sigma$. We can therefore associate a cut of~$\tree[S]$ to each element which appears in the coproduct~$\triangle \mathbb{P}_{\tree[S]}$.
Reciprocally, given a cut~$\gamma$ of~$\tree[S]$, we are interested in the linear extensions of~$\tree[S]$ where all indices below~$\gamma$ appear before all indices above~$\gamma$. These linear extensions are precisely the permutations formed by a linear extension of~$B(\tree, \gamma)$ followed by a linear extension of~$A(\tree, \gamma)$. But the linear extensions of a forest are obtained by shuffling the linear extensions of its connected components. The result immediately follows since the product~$\mathbb{P}_{\tree} \cdot \mathbb{P}_{\tree'}$ precisely involves the shuffle of the linear extensions of~$\tree$ with the linear extensions of~$\tree'$. \end{proof}
For example, we can compute the coproduct \[ \begin{array}{@{}c@{${} = {}$}c@{$\,+\,$}c@{$\,+\,$}c@{$\,+\,$}c@{$\,+\,$}c@{$\,+\,$}c@{}} \triangle \mathbb{P}_{\includegraphics{exmProductB}} & \multicolumn{6}{l}{\triangle \big( \mathbb{F}_{\up{2}\down{1}\up{3}} + \mathbb{F}_{\up{2}\up{3}\down{1}} \big)} \\ & 1 \otimes \big( \mathbb{F}_{\up{2}\down{1}\up{3}} + \mathbb{F}_{\up{2}\up{3}\down{1}} \big) & \mathbb{F}_{\up{1}} \otimes \mathbb{F}_{\down{1}\up{2}} & \mathbb{F}_{\up{1}} \otimes \mathbb{F}_{\up{2}\down{1}} & \mathbb{F}_{\up{2}\down{1}} \otimes \mathbb{F}_{\up{1}} & \mathbb{F}_{\up{1}\up{2}} \otimes \mathbb{F}_{\down{1}} & \big( \mathbb{F}_{\up{2}\down{1}\up{3}} + \mathbb{F}_{\up{2}\up{3}\down{1}} \big) \otimes 1 \\[.2cm] & 1 \otimes \mathbb{P}_{\!\includegraphics{exmProductB}} & \mathbb{P}_{\includegraphics{exmTreeY}} \otimes \mathbb{P}_{\!\!\includegraphics{exmTreeAYd}} & \mathbb{P}_{\includegraphics{exmTreeY}} \otimes \mathbb{P}_{\!\includegraphics{exmTreeYAg}} & \mathbb{P}_{\!\includegraphics{exmTreeYAg}} \otimes \mathbb{P}_{\includegraphics{exmTreeY}} & \mathbb{P}_{\includegraphics{exmTreeYYd}} \otimes \mathbb{P}_{\!\includegraphics{exmTreeA}} & \mathbb{P}_{\!\includegraphics{exmProductB}} \otimes 1 \\ & 1 \otimes \mathbb{P}_{\!\includegraphics{exmProductB}} & \multicolumn{2}{c@{$\,+\,$}}{\mathbb{P}_{\includegraphics{exmTreeY}} \otimes \big( \mathbb{P}_{\!\includegraphics{exmTreeA}} \cdot \mathbb{P}_{\includegraphics{exmTreeY}} \big)} & \mathbb{P}_{\!\includegraphics{exmTreeYAg}} \otimes \mathbb{P}_{\includegraphics{exmTreeY}} & \mathbb{P}_{\includegraphics{exmTreeYYd}} \otimes \mathbb{P}_{\!\includegraphics{exmTreeA}} & \mathbb{P}_{\!\includegraphics{exmProductB}} \otimes 1. \end{array} \] \enlargethispage{-.2cm} Proposition~\ref{prop:coproduct} enables us to shortcut the computation by avoiding to resort to the $\mathbb{F}$-basis. We compute directly the last line, which corresponds to the five possible cuts of the Cambrian tree~\raisebox{-.3cm}{\includegraphics{exmProductB}}.
\para{Matriochka algebras} To conclude, we connect the Cambrian algebra to the recoils algebra~$\mathsf{Rec}$, defined as the Hopf subalgebra of~$\mathsf{FQSym}_\pm$ generated by the elements \[ \mathbb{X}_\chi \eqdef \sum_{\substack{\tau \in \mathfrak{S}_\pm \\ \mathbf{rec}(\tau) = \chi}} \mathbb{F}_\tau \] for all sign vectors~$\chi \in \pm^{n-1}$. The commutative diagram of Proposition~\ref{prop:commutativeDiagram} ensures~that \[ \mathbb{X}_\chi = \sum_{\substack{\tree \in \mathrm{Camb} \\ \mathbf{can}(\tree) = \chi}} \mathbb{P}_{\tree}, \] and thus that~$\mathsf{Rec}$ is a subalgebra of~$\mathsf{Camb}$. In other words, the Cambrian algebra is sandwiched between the signed permutation algebra and the recoils algebra~$\mathsf{Rec} \subset \mathsf{Camb} \subset \mathsf{FQSym}_\pm$. This property has to be compared with the polytope inclusions discussed in Section~\ref{subsec:geometricRealizations}.
\subsection{Quotient algebra of~$\mathsf{FQSym}_\pm^*$} \label{subsec:quotientAlgebra}
We switch to the dual Hopf algebra~$\mathsf{FQSym}_\pm^*$ with basis $(\mathbb{G}_\tau)_{\tau \in \mathfrak{S}_\pm}$ and whose product and coproduct are defined by \[ \mathbb{G}_\tau \cdot \mathbb{G}_{\tau'} = \sum_{\sigma \in \tau \star \tau'} \mathbb{G}_\sigma \qquad\text{and}\qquad \triangle \mathbb{G}_\sigma = \sum_{\sigma \in \tau \,\bar\shuffle\, \tau'} \mathbb{G}_\tau \otimes \mathbb{G}_{\tau'}. \] The following statement is automatic from Theorem~\ref{thm:cambSubalgebra}.
\begin{theorem} The graded dual~$\mathsf{Camb}^*$ of the Cambrian algebra is isomorphic to the image of~$\mathsf{FQSym}_\pm^*$ under the canonical projection \[ \pi : \mathbb{C}\langle A \rangle \longrightarrow \mathbb{C}\langle A \rangle / \equiv, \] where~$\equiv$ denotes the Cambrian congruence. The dual basis~$\mathbb{Q}_{\tree}$ of~$\mathbb{P}_{\tree}$ is expressed as~$\mathbb{Q}_{\tree} = \pi(\mathbb{G}_\tau)$, where~$\tau$ is any linear extension of~$\tree$. \end{theorem}
Similarly as in the previous section, we can describe combinatorially the product and coproduct of $\mathbb{Q}$-basis elements of~$\mathsf{Camb}^*$ in terms of operations on Cambrian trees.
\para{Product} Call \defn{gaps} the $n+1$ positions between two consecutive integers of~$[n]$, including the position before~$1$ and the position after~$n$. A gap~$\gamma$ defines a \defn{geodesic vertical path}~$\lambda(\tree,\gamma)$ in a Cambrian tree~$\tree$ from the bottom leaf which lies in the same interval of consecutive negative labels as~$\gamma$ to the top leaf which lies in the same interval of consecutive positive labels as~$\gamma$. See \fref{fig:exampleCoproductDual}. A multiset~$\Gamma$ of gaps therefore defines a \defn{lamination}~$\lambda(\tree,\Gamma)$ of~$\tree$, \textit{i.e.}~ a multiset of pairwise non-crossing geodesic vertical paths in~$\tree$ from down leaves to up leaves. When cut along the paths of a lamination, the Cambrian tree~$\tree$ splits into a forest.
Consider two Cambrian trees~$\tree$ and~$\tree'$ on~$[n]$ and~$[n']$ respectively. For any shuffle~$s$ of their signatures~$\varepsilon$ and~$\varepsilon'$, consider the multiset~$\Gamma$ of gaps of~$[n]$ given by the positions of the negative signs of~$\varepsilon'$ in~$s$ and the multiset~$\Gamma'$ of gaps of~$[n']$ given by the positions of the positive signs of~$\varepsilon$ in~$s$. We denote by~$\tree \,{}_s\!\backslash \tree'$ the Cambrian tree obtained by connecting the up leaves of the forest defined by the lamination~$\lambda(\tree,\Gamma)$ to the down leaves of the forest defined by the lamination~$\lambda(\tree',\Gamma')$.
\begin{example} Consider the Cambrian trees~$\tree^{\text{\color{darkblue}\tiny$\bigcirc$}}$ and~$\tree^\redSquare$ of \fref{fig:exampleProductDual}. To distinguish signs in~$\tree^{\text{\color{darkblue}\tiny$\bigcirc$}}$ and~$\tree^\redSquare$, we circle the signs in~$\varepsilon(\tree^{\text{\color{darkblue}\tiny$\bigcirc$}}) = {\color{darkblue}\ominus}{\color{darkblue}\ominus}{\color{darkblue}\oplus}$ and square the signs in~$\varepsilon(\tree^\redSquare) = \redMinus\redMinus\redPlus\redMinus$. Consider now an arbitrary shuffle~$s = \redMinus{\color{darkblue}\ominus}{\color{darkblue}\ominus}\redMinus\redPlus{\color{darkblue}\oplus}\redMinus$ of these two signatures. The resulting laminations of~$\tree^{\text{\color{darkblue}\tiny$\bigcirc$}}$ and~$\tree^\redSquare$, as well as the Cambrian tree~$\tree^{\text{\color{darkblue}\tiny$\bigcirc$}} {}_s\!\backslash \tree^\redSquare$ are represented in \fref{fig:exampleProductDual}.
\begin{figure}\label{fig:exampleProductDual}
\end{figure} \end{example}
\begin{proposition} \label{prop:productDual} For any Cambrian trees~$\tree, \tree'$, the product~$\mathbb{Q}_{\tree} \cdot \mathbb{Q}_{\tree'}$ is given by \[ \mathbb{Q}_{\tree} \cdot \mathbb{Q}_{\tree'} = \sum_s \mathbb{Q}_{\tree \,{}_s\!\backslash \tree'}, \] where~$s$ runs over all shuffles of the signatures of~$\tree$ and~$\tree'$. \end{proposition}
\begin{proof} Let~$\tau$ and~$\tau'$ be linear extensions of~$\tree$ and~$\tree'$ respectively, let~$\sigma \in \tau \star \tau'$ and let~$\tree[S] = \mathbf{P}(\sigma)$. As discussed in Section~\ref{subsec:products}, the convolution~$\tau \star \tau'$ shuffles the columns of the tables of~$\tau$ and~$\tau'$ while preserving the order of their rows. According to the description of the insertion algorithm~$\Theta$, the tree~$\tree[S]$ thus consists in~$\tree$ below and~$\tree'$ above, except that the vertical walls falling from the negative nodes of~$\tree'$ split~$\tree$ and similarly the vertical walls rising from the positive nodes of~$\tree$ split~$\tree'$. This corresponds to the description of~$\tree \,{}_s\!\backslash \tree'$, where~$s$ is the shuffle of the signatures of~$\tree$ and~$\tree'$ given by~$\sigma$. \end{proof}
For example, we can compute the product \[ \hspace*{-1cm}\begin{array}{@{}c@{${} = {}$}c@{$\,+\,$}c@{$\,+\,$}c@{$\,+\,$}c@{$\,+\,$}c@{$\,+\,$}c@{$\,+\,$}c@{$\,+\,$}c@{$\,+\,$}c@{$\,+\,$}c@{\;}c@{}} \mathbb{Q}_{\includegraphics{exmProductA}} \cdot \mathbb{Q}_{\includegraphics{exmProductB}} &
\multicolumn{10}{l}{\mathbb{G}_{\down{1}\up{2}} \cdot \mathbb{G}_{\up{2}\down{1}\up{3}}}
\\[-.2cm] & \mathbb{G}_{\down{1}\up{2}\up{4}\down{3}\up{5}} & \mathbb{G}_{\down{1}\up{3}\up{4}\down{2}\up{5}} & \mathbb{G}_{\down{1}\up{4}\up{3}\down{2}\up{5}} & \mathbb{G}_{\down{1}\up{5}\up{3}\down{2}\up{4}} & \mathbb{G}_{\down{2}\up{3}\up{4}\down{1}\up{5}} & \mathbb{G}_{\down{2}\up{4}\up{3}\down{1}\up{5}} & \mathbb{G}_{\down{2}\up{5}\up{3}\down{1}\up{4}} & \mathbb{G}_{\down{3}\up{4}\up{2}\down{1}\up{5}} & \mathbb{G}_{\down{3}\up{5}\up{2}\down{1}\up{4}} & \mathbb{G}_{\down{4}\up{5}\up{2}\down{1}\up{3}} \\[.2cm] & \mathbb{Q}_{\!\includegraphics{exmProduct1}} & \mathbb{Q}_{\!\includegraphics{exmProduct4}} & \mathbb{Q}_{\!\includegraphics{exmProduct5}} & \mathbb{Q}_{\!\includegraphics{exmProduct6}} & \mathbb{Q}_{\!\includegraphics{exmProduct7}} & \mathbb{Q}_{\!\includegraphics{exmProduct8}} & \mathbb{Q}_{\!\includegraphics{exmProduct9}} & \mathbb{Q}_{\!\includegraphics{exmProduct10}} & \mathbb{Q}_{\!\includegraphics{exmProduct11}} & \mathbb{Q}_{\!\includegraphics{exmProduct12}} & . \end{array} \] Note that the~$10$ resulting Cambrian trees correspond to the~$10$ possible shuffles of~${-}{+}$ and~${-}{+}{+}$.
\para{Coproduct} To describe the coproduct of~$\mathbb{Q}$-basis elements of~$\mathsf{Camb}^*$, we also use gaps and vertical paths in Cambrian trees. Namely, for a gap~$\gamma$, we denote by~$L(\tree[S],\gamma)$ and~$R(\tree[S],\gamma)$ the left and right Cambrian subtrees of~$\tree[S]$ when split along the path~$\lambda(\tree[S], \gamma)$. An example is given in \fref{fig:exampleCoproductDual}.
\begin{figure}
\caption{A gap~$\gamma$ between~$3$ and~$4$ (left) defines a vertical cut (middle) which splits the Cambrian tree (right).}
\label{fig:exampleCoproductDual}
\end{figure}
\begin{proposition} \label{prop:coproductDual} For any Cambrian tree~$\tree[S]$, the coproduct~$\triangle\mathbb{Q}_{\tree[S]}$ is given by \[ \triangle\mathbb{Q}_{\tree[S]} = \sum_{\gamma} \mathbb{Q}_{L(\tree[S],\gamma)} \otimes \mathbb{Q}_{R(\tree[S],\gamma)}, \] where~$\gamma$ runs over all gaps between vertices of~$\tree[S]$. \end{proposition}
\begin{proof} Let~$\sigma$ be a linear extension of~$\tree[S]$ and~$\tau, \tau' \in \mathfrak{S}_\pm$ such that~$\sigma \in \tau \,\bar\shuffle\, \tau'$. As discussed in Section~\ref{subsec:products}, $\tau$ and~$\tau'$ respectively appear on the left and right columns of~$\sigma$. Let~$\gamma$ denote the vertical gap separating~$\tau$ from~$\tau'$. Applying the insertion algorithm to~$\tau$ and~$\tau'$ separately then yields the trees~$L(\tree[S],\gamma)$ and~$R(\tree[S],\gamma)$. The description follows. \end{proof}
For example, we can compute the coproduct \[ \hspace*{-1cm}\begin{array}{@{}c@{${} = {}$}c@{${} + {}$}c@{${} + {}$}c@{${} + {}$}c@{}} \triangle \mathbb{Q}_{\includegraphics{exmProductB}} &
\multicolumn{4}{l}{\triangle \mathbb{G}_{\up{2}\down{1}\up{3}}}
\\[-.2cm] & 1 \otimes \mathbb{G}_{\up{2}\down{1}\up{3}} & \mathbb{G}_{\down{1}} \otimes \mathbb{G}_{\up{1}\up{2}} & \mathbb{G}_{\up{2}\down{1}} \otimes \mathbb{G}_{\up{1}} & \mathbb{G}_{\up{2}\down{1}\up{3}} \otimes 1 \\[.2cm] & 1 \otimes \mathbb{Q}_{\includegraphics{exmProductB}} & \mathbb{Q}_{\includegraphics{exmTreeA}} \otimes \mathbb{Q}_{\includegraphics{exmTreeYYd}} & \mathbb{Q}_{\includegraphics{exmTreeYAg}} \otimes \mathbb{Q}_{\includegraphics{exmTreeY}} & \mathbb{Q}_{\includegraphics{exmProductB}} \otimes 1. \end{array} \] Note that the last line can indeed be directly computed using the paths defined by the four possible gaps of the Cambrian tree~\raisebox{-.3cm}{\includegraphics{exmProductB}}.
\subsection{Duality}
As proven in~\cite{HivertNovelliThibon-algebraBinarySearchTrees}, the duality~$\tau \mapsto \tau^{-1}$ between the Hopf algebras~$\mathsf{FQSym}$ and~$\mathsf{FQSym}^*$ induces a duality between the Hopf algebras~$\mathsf{PBT}$ and~$\mathsf{PBT}^*$. That is to say that the composition~$\Psi$ of the applications \[ \begin{array}{ccccccc} \mathsf{PBT} & \verylonghookrightarrow & \mathsf{FQSym} & \verylongleftrightarrow & \mathsf{FQSym}^* & \verylongtwoheadrightarrow & \mathsf{PBT}^* \\ & \mathbb{P}_{\tree} \mapsto \!\!\sum\limits_{\tau \in \mathcal{L}(\tree)} \mathbb{F}_\tau & & \tau \mapsto \tau^{-1} & & \mathbb{G}_\tau \mapsto \mathbb{Q}_{\mathbf{P}(\tau)} \end{array} \] is an isomorphism between~$\mathsf{PBT}$ and~$\mathsf{PBT}^*$. This property is no longer true for the Cambrian algebra~$\mathsf{Camb}$ and its dual~$\mathsf{Camb}^*$. Namely, the composition~$\Psi$ of the applications \[ \begin{array}{ccccccc} \mathsf{Camb} & \verylonghookrightarrow & \mathsf{FQSym}_\pm & \verylongleftrightarrow & \mathsf{FQSym}_\pm^* & \verylongtwoheadrightarrow & \mathsf{Camb}^* \\ & \mathbb{P}_{\tree} \mapsto \!\!\sum\limits_{\tau \in \mathcal{L}(\tree)} \mathbb{F}_\tau & & \mathbb{F}_\tau \mapsto \mathbb{G}_\tau^{-1} & & \mathbb{G}_\tau \mapsto \mathbb{Q}_{\mathbf{P}(\tau)} \end{array} \] is not an isomorphism. It is indeed not injective as \[ \Psi \big( \mathbb{P}_{\raisebox{-.25cm}{\includegraphics{ctrexmDualityA}}} \big) = \mathbb{Q}_{\raisebox{-.25cm}{\includegraphics{exmProductB}}} = \Psi \big( \mathbb{P}_{\raisebox{-.25cm}{\includegraphics{ctrexmDualityB}}} \big). \] Indeed, their images along the three maps are given by \[ \begin{array}{ccccccc@{\,}l} \mathbb{P}_{\raisebox{-.25cm}{\includegraphics{ctrexmDualityA}}} & \longmapsto & \mathbb{F}_{\down{2}\up{13}} & \longmapsto & \mathbb{G}_{\up{2}\down{1}\up{3}} & \longmapsto & \mathbb{Q}_{\raisebox{-.25cm}{\includegraphics{exmProductB}}} & , \text{ and}\\ \mathbb{P}_{\raisebox{-.25cm}{\includegraphics{ctrexmDualityB}}} & \longmapsto & \mathbb{F}_{\down{3}\up{12}} & \longmapsto & \mathbb{G}_{\up{23}\down{1}} & \longmapsto & \mathbb{Q}_{\raisebox{-.25cm}{\includegraphics{exmProductB}}} & . \end{array} \]
\section{Multiplicative bases} \label{sec:multiplicativeBases}
In this section, we define multiplicative bases of~$\mathsf{Camb}$ and study the indecomposable elements of~$\mathsf{Camb}$ for these bases. We prove in Sections~\ref{subsec:structuralProp} and~\ref{subsec:enumerativeProp} both structural and enumerative properties of the set of indecomposable elements.
\subsection{Multiplicative bases and indecomposable elements}
For a Cambrian tree~$\tree$, we define \[ \mathbb{E}^{\tree} \eqdef \sum_{\tree \le \tree'} \mathbb{P}_{\tree'} \qquad\text{and}\qquad \mathbb{H}^{\tree} \eqdef \sum_{\tree' \le \tree} \mathbb{P}_{\tree'}. \] To describe the product of two elements of the~$\mathbb{E}$- or $\mathbb{H}$-basis, remember that the Cambrian trees \[ \raisebox{-6pt}{$\tree$}\nearrow \raisebox{4pt}{$\bar \tree'$} \qquad\text{and}\qquad \raisebox{4pt}{$\tree$} \nwarrow \raisebox{-6pt}{$\bar \tree'$} \] are defined to be the trees obtained by shifting all labels of~$\tree'$ and grafting for the first one the rightmost outgoing leaf of~$\tree$ on the leftmost incoming leaf of~$\tree'$, and for the second one the rightmost incoming leaf of~$\tree$ on the leftmost outgoing leaf of~$\tree'$. Examples are given in \fref{fig:exampleProduct}.
\begin{proposition} \label{prop:productMultiplicativeBases} $(\mathbb{E}^{\tree})_{\tree \in \mathrm{Camb}}$ and~$(\mathbb{H}^{\tree})_{\tree \in \mathrm{Camb}}$ are multiplicative bases of~$\mathsf{Camb}$: \[ \mathbb{E}^{\tree} \cdot \mathbb{E}^{\tree'} = \mathbb{E}^{\raisebox{-5pt}{\scriptsize$\tree$}\nearrow \raisebox{4pt}{\scriptsize$\bar \tree'$}} \qquad\text{and}\qquad \mathbb{H}^{\tree} \cdot \mathbb{H}^{\tree'} = \mathbb{H}^{\raisebox{4pt}{\scriptsize$\tree$}\nwarrow \raisebox{-5pt}{\scriptsize$\bar \tree'$}}. \] \end{proposition}
\begin{proof} Let~$\omega_{\tree}$ denote the maximal linear extension of~$\tree$ in weak order. Since~$\bigset{\mathcal{L}(\tilde\tree)}{\tilde\tree \le \tree}$ partitions the weak order interval~$[12 \cdots n, \omega_{\tree}]$, we have \[ \mathbb{H}^{\tree} = \sum_{\tilde\tree \le \tree} \mathbb{P}_{\tilde\tree} = \sum_{\tilde\tree \le \tree} \sum_{\tau \in \mathcal{L}(\tilde\tree)} \mathbb{F}_\tau = \sum_{\tau \le \omega_{\tree}} \mathbb{F}_\tau. \] Since the shuffle product of two intervals of the weak order is an interval of the weak order, the product~$\mathbb{H}^{\tree} \cdot \mathbb{H}^{\tree'}$ is the sum of~$\mathbb{F}_\tau$ over the interval \[ [12 \cdots n, \omega_{\tree}] \,\bar\shuffle\, [12 \cdots n', \omega_{\tree'}] = [12 \cdots (n+n'), \bar \omega_{\tree'}\omega_{\tree}]. \] The result thus follows from the fact that \[ \mathbf{P}(\bar\omega_{\tree'}\omega_{\tree}) = \raisebox{4pt}{$\tree$} \nwarrow \raisebox{-6pt}{$\bar \tree'$}. \] The proof is symmetric for~$\mathbb{E}^{\tree}$, replacing lower interval and~$[12 \cdots n, \omega_{\tree}]$ by the upper interval~$[\mu_{\tree}, n \cdots 21]$. \end{proof}
As the multiplicative bases~$(\mathbb{E}^{\tree})_{\tree \in \mathrm{Camb}}$ and~$(\mathbb{H}^{\tree})_{\tree \in \mathrm{Camb}}$ have symmetric properties, we focus our analysis on the $\mathbb{E}$-basis. The reader is invited to translate the results below to the $\mathbb{H}$-basis. We consider multiplicative decomposability. Remember that an \defn{edge cut} in a Cambrian tree~$\tree[S]$ is the ordered partition~$\edgecut{X}{Y}$ of the vertices of~$\tree[S]$ into the set~$X$ of vertices in the source set and the set~$Y$ of vertices in the target set of an oriented edge~$e$ of~$\tree[S]$.
\begin{proposition} The following properties are equivalent for a Cambrian tree~$\tree[S]$: \begin{enumerate}[(i)] \item $\mathbb{E}^{\tree[S]}$ can be decomposed into a product~$\mathbb{E}^{\tree[S]} = \mathbb{E}^{\tree} \cdot \mathbb{E}^{\tree'}$ for non-empty Cambrian trees~$\tree, \tree'$; \label{enum:decomposable} \item $\edgecut{[k]}{[n] \smallsetminus [k]}$ is an edge cut of~$\tree[S]$ for some~$k \in [n]$; \label{enum:cut} \item at least one linear extension~$\tau$ of~$\tree[S]$ is decomposable, \textit{i.e.}~ $\tau([k]) = [k]$ for some~$k \in [n]$. \label{enum:perm} \end{enumerate} The tree~$\tree[S]$ is then called \defn{$\mathbb{E}$-decomposable} and the edge cut~$\edgecut{[k]}{[n] \smallsetminus [k]}$ is called \defn{splitting}. \end{proposition}
\begin{proof} The equivalence~\eqref{enum:decomposable} $\iff$ \eqref{enum:cut} is an immediate consequence of the description of the product~$\mathbb{E}^{\tree} \cdot \mathbb{E}^{\tree'}$ in Proposition~\ref{prop:productMultiplicativeBases}. The implication \eqref{enum:cut} $\Rightarrow$ \eqref{enum:perm} follows from the fact that for any cut~$\edgecut{X}{Y}$ of a directed acyclic graph~$G$, there exists a linear extension of~$G$ which starts with~$X$ and finishes with~$Y$. Reciprocally, if~$\tau$ is a decomposable linear extension of~$\tree[S]$, then the insertion algorithm creates two blocks and necessarily relates the bottom-left block to the top-right block by a splitting edge. \end{proof}
For example, \fref{fig:exampleProduct} shows that~$\mathbf{P}(\down{2}\up{7}\down{51}\up{3}\down{4}\up{6})$ is both $\mathbb{E}$- and $\mathbb{H}$-decomposable. In the remaining of this section, we study structural and enumerative properties of $\mathbb{E}$-indecompo\-sable elements of~$\mathrm{Camb}$. We denote by~$\mathrm{Ind}_\varepsilon$ the set of $\mathbb{E}$-indecomposable elements of~$\mathrm{Camb}(\varepsilon)$.
\begin{example} \label{exm:rightTilting} For~$\varepsilon = (-)^n$, the $\mathbb{E}$-indecomposable $\varepsilon$-Cambrian trees are \defn{right-tilting} binary trees, \textit{i.e.}~ binary trees whose root has no left child. Similarly, for~$\varepsilon = (+)^n$, the $\mathbb{E}$-indecomposable $\varepsilon$-Cambrian trees are \defn{left-tilting} binary trees oriented upwards. See \fref{fig:minIndecomposable} for illustrations. \end{example}
\subsection{Structural properties} \label{subsec:structuralProp}
The objective of this section is to prove the following property of the $\mathbb{E}$-indecomposable elements of~$\mathrm{Camb}(\varepsilon)$.
\begin{proposition} \label{prop:upperIdeal} For any signature~$\varepsilon \in \pm^n$, the set~$\mathrm{Ind}_\varepsilon$ of~$\mathbb{E}$-indecomposable $\varepsilon$-Cambrian trees forms a principal upper ideal of the $\varepsilon$-Cambrian lattice. \end{proposition}
To prove this statement, we need the following result.
\begin{lemma} \label{lem:rotationIndecomposable} Let~$\tree$ be a $\varepsilon$-Cambrian tree, let~$i \to j$ be an edge of~$\tree$ with~$i < j$, and let~$\tree'$ be the $\varepsilon$-Cambrian tree obtained by rotating~$i \to j$ in~$\tree$. Then \begin{enumerate}[(i)] \item if~$\tree$ is $\mathbb{E}$-indecomposable, then so is~$\tree'$; \item if~$\tree$ is $\mathbb{E}$-decomposable while~$\tree'$ is not, then~$\varepsilon_i = +$ or~$i = 1$, and~$\varepsilon_j = -$ or~$j = n$. \end{enumerate} \end{lemma}
\begin{proof} As observed in Proposition~\ref{prop:rotation}, the Cambrian trees~$\tree$ and~$\tree'$ have the same edge cuts, except the cut defined by edge~$i \to j$. Using notations of \fref{fig:rotation}, the edge cut~$C \eqdef \edgecut{i \cup L \cup B}{j \cup R \cup A}$ of~$\tree$ is replaced by the edge cut~$C' \eqdef \edgecut{j \cup R \cup B}{i \cup L \cup A}$ of~$\tree'$. Since $i < j$, the edge cut~$C'$ cannot be splitting. Therefore, $\tree'$ is always $\mathbb{E}$-indecomposable when~$\tree$ is $\mathbb{E}$-indecomposable.
Assume conversely that~$\tree$ is $\mathbb{E}$-decomposable while $\tree'$ is not. This implies that~$C$ is splitting while~$C'$ is not. Since~$C$ is splitting we have~$i \cup L \cup B < j \cup R \cup A$ (where we write~$X < Y$ if~$x < y$ for all~$x \in X$ and~$y \in Y$). If~$\varepsilon_i = -$, then~$L < i < B$, and thus~$L < \{i,j\} \cup R \cup A \cup B$. If moreover~$1 < i$, then~$1 < \{i,j\} \cup R \cup A \cup B$ and thus~$1 \in L \ne \varnothing$. This would imply that the cut of~$\tree'$ defined by the edge~$L \to i$ would be splitting. Contradiction. We prove similarly that~$\varepsilon_j = -$ or~$j = n$. \end{proof}
\begin{proof}[Proof or Proposition~\ref{prop:upperIdeal}] We already know from Lemma~\ref{lem:rotationIndecomposable}\,(i) that~$\mathrm{Ind}_\varepsilon$ is an upper set of the $\varepsilon$-Cambrian lattice. To see that this upper set is a principal upper ideal, we characterize the unique $\mathbb{E}$-indecomposable $\varepsilon$-Cambrian tree~$\tree_\bullet$ whose decreasing rotations all create a splitting edge cut. We proceed in three steps.
\para{Claim~A} All negative vertices~$i > 1$ of~$\tree_\bullet$ have no right child, while all positive vertices~$j < n$ of~$\tree_\bullet$ have no left child. \\[.1cm] \textit{Proof.} Assume by means of contradiction that a negative vertex~$i > 1$ has a right child~$j$. Let~$\tree$ be the Cambrian tree obtained by rotation of the edge~$i \leftarrow j$ in~$\tree_\bullet$. Since this rotation is decreasing (because~$i < j$), $\tree$ is decomposable while~$\tree_\bullet$ is not. This contradicts Lemma~\ref{lem:rotationIndecomposable}\,(ii).
Claim~A ensures that the Cambrian tree~$\tree_\bullet$ is a path with additional leaves incoming at negative vertices and outgoing at positive vertices. Therefore, $\tree_\bullet$ admits a unique linear extension~$\tau_\bullet$. The next two claims determine~$\tau_\bullet$ and thus~$\tree_\bullet = \mathbf{P}(\tau_\bullet)$.
As vertex~$1$ has no left child and vertex~$n$ has no right child, we consider that~$1$ behaves as a positive vertex and~$n$ behaves as a negative vertex. We thus define~${N \eqdef \{n_1 < \dots < n_{N-1} < n_N = n\}}$ and~$P \eqdef \{1 = p_1 < p_2 < \dots < p_P\}$, where~$n_1 < \dots < n_{N-1}$ are the negative vertices and~$p_2 < \dots < p_P$ are the positive vertices among~$\{2, \dots, n-1\}$.
\para{Claim~B} The sets~$N$ and~$P$ both appear in increasing order in~$\tau_\bullet$. \\[.1cm] \textit{Proof.} If~~$i$ appears in~$\tau_\bullet$ before~$j \in N$, then~$i$ lies in the left child of~$j$ (since~$j$ has no right child), so that~$i < j$. In particular, $N$ is sorted in~$\tau_\bullet$. The proof is symmetric for positive vertices.
\para{Claim~C} In~$\tau_\bullet$, vertex~$p_k$ appears immediately after the first vertex in~$N$ larger than~$p_{k+1}$. \\[.1cm] \textit{Proof.} Let~$n_\ell$ denote the first vertex in~$N$ larger than~$p_{k+1}$. If~$p_k$ appears before~$n_\ell$ in~$\tau_\bullet$, then~$\tau_\bullet$ is a decomposable permutation (since~$\tau([p_{k+1}-1]) = [p_{k+1}-1]$). If~$p_k$ appears after~$n_{\ell+1}$ in~$\tau_\bullet$, then the Cambrian tree obtained by rotation of the incoming edge at~$p_k$ in~$\tree_\bullet$ remains indecomposable. Therefore, $p_k$ appears precisely in between~$n_\ell$ and~$n_{\ell+1}$. \end{proof}
\enlargethispage{.1cm} For example, \fref{fig:minIndecomposable} illustrates the generator of the $\mathbb{E}$-indecomposable $\varepsilon$-Cambrian trees for~$\varepsilon = {-}{-}{+}{-}{-}{+}{+}$, $\varepsilon = (-)^7$, and~$\varepsilon = (+)^7$. The last two are right- and left-tilting trees respectively.
\begin{figure}
\caption{The generators of the principal upper ideals of $\mathbb{E}$-indecomposable $\varepsilon$-Cambrian trees for~$\varepsilon = {-}{-}{+}{-}{-}{+}{+}$ (left), $\varepsilon = (-)^7$ (middle), $\varepsilon = (+)^7$~(right).}
\label{fig:minIndecomposable}
\end{figure}
\subsection{Enumerative properties} \label{subsec:enumerativeProp}
We now consider enumerative properties of $\mathbb{E}$-indecomposable elements. We want to show that the number of $\mathbb{E}$-indecomposable $\varepsilon$-Cambrian trees is independent of the signature~$\varepsilon$.
\begin{proposition} \label{prop:numberIndecomposables} For any signature~$\varepsilon \in \pm^n$, there are~$C_{n-1}$ $\mathbb{E}$-indecomposable $\varepsilon$-Cambrian trees. Therefore, there are~$2^n C_{n-1}$ $\mathbb{E}$-indecomposable Cambrian trees on~$n$ vertices. \end{proposition}
This result is immediate for the signature~$\varepsilon = (-)^n$ as $\mathbb{E}$-indecomposable elements are right-tilting binary trees (see Example~\ref{exm:rightTilting}), which are clearly counted by the Catalan number~$C_{n-1}$. To show Proposition~\ref{prop:numberIndecomposables}, we study the behavior of Cambrian trees and their decompositions under local transformations on signatures of~$[n]$. We believe that these transformations are interesting \textit{per se}. For example, they provide an alternative proof that there are $C_n$ $\varepsilon$-Cambrian trees for any signature~$\varepsilon \in \pm^n$.
Let~$\chi_0 : \pm^n \to \pm^n$ and~$\chi_n : \pm^n \to \pm^n$ denote the transformations which switch the signs of~$1$ and~$n$, respectively. Denote by~$\Psi_0(\tree)$ and~$\Psi_n(\tree)$ the trees obtained from a Cambrian tree~$\tree$ by changing the direction of the leftmost and rightmost leaf of~$\tree$ respectively. For~$i \in [n-1]$, let~$\chi_i : \pm^n \to \pm^n$ denote the transformation which switches the signs at positions~$i$ and~$i+1$. The transformation~$\varepsilon \to \chi_i(\varepsilon)$ is only relevant when~$\varepsilon_i \ne \varepsilon_{i+1}$. In this situation, we denote by~$\Psi_i(\tree)$ the tree obtained from a $\varepsilon$-Cambrian tree~$\tree$ by \begin{itemize} \item reversing the arc from the positive to the negative vertex of~$\{i,i+1\}$ if it exists, \item exchanging the labels of~$i$ and~$i+1$ otherwise. \end{itemize} This transformation is illustrated on \fref{fig:switchSign} when~$\varepsilon_i = +$ and~$\varepsilon_{i+1} = -$.
\begin{figure}
\caption{The transformation~$\Psi_i$ when~$\varepsilon_i = +$ and~$\varepsilon_{i+1} = -$. The tree~$\Psi_i(\tree)$ is obtained by reversing the arc from ~$i$ to~$i+1$ if it exists (left), and just exchanging the labels of~$i$ and~$i+1$ otherwise (right).}
\label{fig:switchSign}
\end{figure}
To show that~$\Psi_i$ transforms $\varepsilon$-Cambrian trees to $\chi_i(\varepsilon)$-Cambrian trees and preserves the number of $\mathbb{E}$-indecomposable elements, we need the following lemma. Note that this lemma also explains why \fref{fig:switchSign} covers all possibilities when~$\varepsilon_i = +$ and~$\varepsilon_{i+1} = -$.
\begin{lemma} If~$\varepsilon_i = +$ and~$\varepsilon_{i+1} = -$, then the following assertions are equivalent for a $\varepsilon$-Cambrian tree~$\tree$: \begin{enumerate}[(i)] \item $\edgecut{[i]}{[n] \smallsetminus [i]}$ is an edge cut of~$\tree$; \label{enum:cut2} \item $i$ is smaller than~$i+1$ in~$\tree$; \label{enum:smaller} \item $i$ is in the left subtree of~$i+1$ and~$i+1$ is in the right subtree of~$i$; \label{enum:subtrees} \item $i$ is the left child of~$i+1$ and~$i+1$ is the right parent of~$i$. \label{enum:edge} \end{enumerate} A similar statement holds in the case when~$\varepsilon_i = -$ and~$\varepsilon_{i+1} = +$. \end{lemma}
\begin{proof} Since~$i$ and~$i+1$ are comparable in~$\tree$ (see Section~\ref{subsec:canopy}), the fact that~$\edgecut{[i]}{[n] \smallsetminus [i]}$ is an edge cut of~$\tree$ implies that~$i$ is smaller than~$i+1$ in~$\tree$. This shows that~\eqref{enum:cut2} $\Rightarrow$ \eqref{enum:smaller}.
If~$i$ is smaller than~$i+1$ in~$\tree$, then~$i$ is in a subtree of~$i+1$, and thus in the left one, and similarly, $i+1$ is in the right subtree of~$i$. This shows that~\eqref{enum:smaller} $\Rightarrow$ \eqref{enum:subtrees}.
Assume now that~$i$ is in the left subtree of~$i+1$ and~$i+1$ is in the right subtree of~$i$, and consider the path from~$i$ to~$i+1$ in~$\tree$. Since it lies in the right subtree of~$i$ and in the left subtree of~$i+1$, any label along this path should be greater than~$i$ and smaller than~$i+1$. This path is thus a single arc. This shows that~\eqref{enum:subtrees} $\Rightarrow$ \eqref{enum:edge}.
Finally, assume that $i$ is the left child of~$i+1$ and~$i+1$ is the right parent of~$i$ in~$\tree$. Then the cut corresponding to the arc~$e$ of~$\tree$ from~$i$ to~$i+1$ is~$\edgecut{[i]}{[n] \smallsetminus [i]}$. Indeed, all elements in the source of~$e$ are in the left subtree and thus smaller than~$i+1$, while all elements in the target of~$e$ are in the right subtree and thus greater than~$i$. This shows that~\mbox{\eqref{enum:edge} $\Rightarrow$ \eqref{enum:cut2}}. \end{proof}
\begin{lemma} \label{lem:switchSign} For~$0 \le i \le n$, the map~$\Psi_i$ defines a bijection from~$\varepsilon$-Cambrian trees to~$\chi_i(\varepsilon)$-Cambrian trees and preserves the number of $\mathbb{E}$-indecomposable elements. \end{lemma}
\begin{proof} The result is immediate for~$i = 0$ and~$i = n$. Assume thus that~$i \in [n-1]$ and that~$\varepsilon_i = +$ while~$\varepsilon_{i+1} = -$. We first prove that~$\Psi_i$ sends~$\varepsilon$-Cambrian trees to~$\chi_i(\varepsilon)$-Cambrian trees. It clearly transforms trees to trees. To see that~$\Psi_i(\tree)$ is $\chi_i(\varepsilon)$-Cambrian, we distinguish two cases: \begin{itemize} \item \fref{fig:switchSign}\,(left) illustrates the case when $\tree$ has an arc in from~$i$ to~$i+1$. All labels in~$B$ are smaller than~$i$ since they are distinct from~$i$ and in the left subtree of~$i+1$, and all labels in the right subtree of~$i$ in~$\Psi_i(\tree)$ are greater than~$i$ since they were in the right subtree of~$i$ in~$\tree$. Therefore, the labels around vertex~$i$ of~$\Psi_i(\tree)$ respect the Cambrian rules. We argue similarly around~$i+1$. All other vertices have the same signs and subtrees. \item \fref{fig:switchSign}\,(right) illustrates the case when $\tree$ has no arc in from~$i$ to~$i+1$. All labels in~$B$ (resp.~$D$) are smaller (resp.~greater) than~$i$ since they are distinct from~$i$ and in the left (resp.~ right) subtree of~$i+1$, so the labels around vertex~$i$ of~$\Psi_i(\tree)$ respect the Cambrian rules. We argue similarly around~$i+1$. All other vertices have the same signs and subtrees. \end{itemize} Alternatively, it is also easy to see~$\Psi_i$ transforms~$\varepsilon$-Cambrian trees to~$\chi_i(\varepsilon)$-Cambrian trees using the interpretation of Cambrian trees as dual trees of triangulations (see Remark~\ref{rem:triangulation}).
Although~$\Psi_i$ does not preserve $\mathbb{E}$-indecomposable elements, we now check that~$\Psi_i$ preserves the number of $\mathbb{E}$-indecomposable elements. Write~$\varepsilon = \underline{\varepsilon}\overline{\varepsilon}$ with~$\underline{\varepsilon} : [i] \to \{\pm\}$ and~$\overline{\varepsilon} : [n] \smallsetminus [i] \to \{\pm\}$, and let~$\underline{I} = |\mathrm{Ind}_{\underline{\varepsilon}}|$ and~$\overline{I} = |\mathrm{Ind}_{\overline{\varepsilon}}|$. We claim that \begin{itemize} \item the map~$\Psi_i$ transforms precisely $\underline{I} \cdot \overline{I}$\, $\mathbb{E}$-decomposable $\varepsilon$-Cambrian trees to $\mathbb{E}$-indecomposable $\chi_i(\varepsilon)$-Cambrian trees. Indeed, $\tree$ is $\mathbb{E}$-decomposable while $\Psi_i(\tree)$ is $\mathbb{E}$-indecomposable if and only of~$\tree$ has an arc from~$i$ to~$i+1$ whose source and target subtrees are $\mathbb{E}$-indecomposable $\underline{\varepsilon}$- and $\overline{\varepsilon}$-Cambrian trees, respectively. \item the map~$\Psi_i$ transforms precisely $\underline{I} \cdot \overline{I}$\, $\mathbb{E}$-indecomposable $\varepsilon$-Cambrian trees to $\mathbb{E}$-decomposable $\chi_i(\varepsilon)$-Cambrian trees. Indeed, assume that~$\tree$ is $\mathbb{E}$-indecomposable while $\Psi_i(\tree)$ is $\mathbb{E}$-decomposable. We claim that~$\edgecut{[i]}{[n] \smallsetminus [i]}$ is the only splitting edge cut of~$\Psi_i(\tree)$. Indeed, for~$j \ne i$, both $i$ and~$i+1$ belong either to~$[j]$ or to~$[n] \smallsetminus [j]$, and~$\edgecut{[j]}{[n] \smallsetminus [j]}$ is an edge cut of~$\Psi_i(\tree)$ if and only if it is an edge cut of~$\tree$. Moreover, the $\underline{\varepsilon}$- and~$\overline{\varepsilon}$-Cambrian trees~$\underline{\tree[S]}$ and~$\overline{\tree[S]}$ induced by~$\Psi_i(\tree)$ on~$[i]$ and~$[n] \smallsetminus [i]$ are both $\mathbb{E}$-indecomposable. Otherwise, a splitting edge cut~$\edgecut{[j]}{[i] \smallsetminus [j]}$ of~$\underline{\tree[S]}$ would define a splitting edge cut~$\edgecut{[j]}{[n] \smallsetminus [j]}$ of~$\Psi_i(\tree)$. Conversely, if~$\underline{\tree[S]}$ and~$\overline{\tree[S]}$ are both $\mathbb{E}$-indecomposable, then so is~$\tree$. \end{itemize} We conclude that~$\Psi_i$ globally preserves the number of $\mathbb{E}$-indecomposable Cambrian trees. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:numberIndecomposables}] Starting from the fully negative signature~$(-)^n$, we can reach any signature~$\varepsilon$ by the transformations~$\chi_0, \dots, \chi_{n-1}$: we can make positive signs appear on vertex~$1$ (using the map~$\chi_0$) and make these positive signs travel towards their final position in~$\varepsilon$ (using the maps~$\chi_i$). More precisely, if~$p_1 < \dots < p_P$ denote the positions of the positive signs of~$\varepsilon$, then~$\varepsilon = \big( \prod_{j \in [P]} \chi_{p_j} \circ \chi_{p_{j-1}} \circ \dots \circ \chi_{p_1} \circ \chi_0 \big) \big( (-)^n \big)$. The result thus follows from Lemma~\ref{lem:switchSign}. \end{proof}
\begin{proposition} The Cambrian algebra~$\mathsf{Camb}$ is free. \end{proposition}
\begin{proof} As the generating function~$B(u)$ of the Catalan numbers satisfies the functional equation~$B(u) = 1 + uB(u)^2$, we obtain by substitution~$u = 2t$ that \[ \frac{1}{1 - \sum_{n \ge 1} 2^n C_{n-1} t^n} = \sum_{n \ge 0} 2^n C_n t^n. \] The result immediately follows from Proposition~\ref{prop:numberIndecomposables}. \end{proof}
\part{The Baxter-Cambrian Hopf Algebra} \label{part:BaxterCambrian}
\section{Twin Cambrian trees} \label{sec:BaxterTrees}
We now consider twin Cambrian trees and the resulting Baxter-Cambrian algebra. It provides a straightforward generalization to the Cambrian setting of the work of S.~Law and N.~Reading on quadrangulations~\cite{LawReading} and S.~Giraudo on twin binary trees~\cite{Giraudo}. The bases of these algebras are counted by the Baxter numbers. In Section~\ref{subsec:BaxterCambrianNumbers} we provide references for the various Baxter families and their bijective correspondences, and we discuss the Cambrian counterpart of these numbers. Definitions and combinatorial properties of twin Cambrian trees are given in this section, while the algebraic aspects are treated in the next section.
\subsection{Twin Cambrian trees} \label{subsec:twinCambrianTrees}
This section deals with the following pairs of Cambrian trees.
\begin{definition} \label{def:twinCambrianTrees} Two $\varepsilon$-Cambrian trees~$\tree_\circ, \tree_\bullet$ are \defn{twin} if the union~$\graphTwin$ of~$\tree_\circ$ with the reverse of~$\tree_\bullet$ (reversing the orientations of all edges) is acyclic. \end{definition}
\begin{definition} Let~$\tree_{\circ}, \tree_{\bullet}$ be two leveled~$\varepsilon$-Cambrian trees with labelings~$p_{\circ}, q_{\circ}$ and $p_{\bullet}, q_{\bullet}$ respectively. We say that they are \defn{twin} if $q_{\circ}(p_{\circ}^{-1}(i)) = n - q_{\bullet}(p_{\bullet}^{-1}(i))$ for all~$i \in [n]$. In other words, when labeled as Cambrian trees, the bottom-up order of the vertices of~$\tree_{\circ}$ and $\tree_{\bullet}$ are opposite. \end{definition}
Examples of twin Cambrian trees and twin leveled Cambrian trees are represented in \fref{fig:twinCambrianTrees}. Note that twin leveled Cambrian trees are twin Cambrian trees~$\tree_\circ, \tree_\bullet$ endowed with a linear extension of the transitive closure of~$\graphTwin$.
\begin{figure}
\caption{A pair of twin Cambrian trees (left), and a pair of twin leveled Cambrian trees (right).}
\label{fig:twinCambrianTrees}
\end{figure}
If~$\tree_\circ, \tree_\bullet$ are two $\varepsilon$-Cambrian trees, they necessarily have opposite canopy (see Section~\ref{subsec:canopy}), meaning that $\mathbf{can}(\tree_\circ)_i = -\mathbf{can}(\tree_\bullet)_i$ for all~$i \in [n-1]$. The reciprocal statement for the constant signature~$(-)^n$ is proved by S.~Giraudo in~\cite{Giraudo}.
\begin{proposition}[\cite{Giraudo}] \label{prop:twinBinaryTrees} Two binary trees are twin if and only if they have opposite canopy. \end{proposition}
We conjecture that this statement holds for general signatures. Consider two $\varepsilon$-Cambrian trees~$\tree_\circ, \tree_\bullet$ with opposite canopies. It is easy to show that~$\graphTwin$ cannot have trivial cycles, meaning that~$\tree_\circ$ and~$\tree_\bullet$ cannot both have a path from~$i$ to~$j$ for~$i \ne j$. To prove that~$\graphTwin$ has no cycles at all, a good method is to design an algorithm to extract a linear extension of~$\graphTwin$. This approach was used in~\cite{Giraudo} for the signature~$(-)^n$. In this situation, it is clear that the root of~$\tree_\bullet$ is minimal in~$\tree_\circ$ (by the canopy assumption), and we therefore pick it as the first value of a linear extension of~$\graphTwin$. The remaining of the linear extension is constructed inductively. In the general situation, it turns out that not all maximums in~$\tree_\bullet$ are minimums in~$\tree_\circ$ (and reciprocally). It is thus not clear how to choose the first value of a linear extension of~$\graphTwin$.
\begin{remark}[Reversing~$\tree_\bullet$] \label{rem:reversing} \enlargethispage{.2cm} It is sometimes useful to reverse the second tree~$\tree_\bullet$ in a pair~$[\tree_\circ,\tree_\bullet]$ of twin Cambrian trees. The resulting Cambrian trees have opposite signature and their union is acyclic. In this section, we have chosen the orientation of Definition~\ref{def:twinCambrianTrees} to fit with the notations and results in~\cite{Giraudo}. We will have to switch to the opposite convention in Section~\ref{sec:tuples} when we will extend our results on twin Cambrian trees to arbitrary tuples of Cambrian trees. \end{remark}
\subsection{Baxter-Cambrian correspondence}
We obtain the \defn{Baxter-Cambrian correspondence} between permutations of~$\mathfrak{S}^\varepsilon$ and pairs of twin leveled $\varepsilon$-Cambrian trees by inserting with the map~$\Theta$ from Section~\ref{subsec:CambrianCorrespondence} a permutation~$\tau = \tau_1 \cdots \tau_n \in \mathfrak{S}^\varepsilon$ and its \defn{mirror}~${\mirror{\tau} = \tau_n \cdots \tau_1 \in \mathfrak{S}^\varepsilon}$.
\begin{proposition} The map~$\CambCorresp\Bax$ defined by~$\CambCorresp\Bax(\tau) = \big[ \Theta(\tau), \Theta(\mirror{\tau}) \big]$ is a bijection from signed permutations to pairs of twin leveled Cambrian trees. \end{proposition}
\begin{proof} If~$p,q : V \to [n]$ denote the Cambrian and increasing labelings of the Cambrian tree~$\Theta(\tau)$, then~$\tau = q \circ p$. This yields that the leveled $\varepsilon$-Cambrian trees~$\Theta(\tau)$ and~$\Theta(\mirror{\tau})$ are twin and the map~$\CambCorresp\Bax$ is bijective. \end{proof}
As for Cambrian trees, we focus on the $\mathbf{P}^{\upharpoonleft\!\!\downharpoonright}$-symbol of this correspondence.
\begin{proposition} The map~$\surjectionPermAsso\Bax$ defined by~$\surjectionPermAsso\Bax(\tau) = \big[ \mathbf{P}(\tau), \mathbf{P}(\mirror{\tau}) \big]$ is a surjection from signed permutations to pairs of twin Cambrian trees. \end{proposition}
\begin{proof} The fiber~$(\surjectionPermAsso\Bax)^{-1}([\tree_\circ, \tree_\bullet])$ of a pair of twin $\varepsilon$-Cambrian trees~$\tree_\circ, \tree_\bullet$ is the set~$\mathcal{L}(\graphTwin)$ of linear extensions of the graph~$\graphTwin$. This set is non-empty since~$\graphTwin$ is acyclic by definition of twin Cambrian trees. \end{proof}
\subsection{Baxter-Cambrian congruence} \label{subsec:BaxterCongruence}
We now characterize by a congruence relation the signed permutations~$\tau \in \mathfrak{S}^{\varepsilon}$ which have the same $\mathbf{P}^{\upharpoonleft\!\!\downharpoonright}$-symbol.
\begin{definition} For a signature~${\varepsilon \in \pm^n}$, the \defn{$\varepsilon$-Baxter-Cambrian congruence} is the equivalence relation on $\mathfrak{S}^{\varepsilon}$ defined as the transitive closure of the rewriting rules \begin{gather*} UbVadWcX \equiv^{\upharpoonleft\!\!\downharpoonright}_\varepsilon UbVdaWcX \quad\text{if } a < b, c < d \text{ and } \varepsilon_b = \varepsilon_c, \\ UbVcWadX \equiv^{\upharpoonleft\!\!\downharpoonright}_\varepsilon UbVcWdaX \quad\text{if } a < b, c < d \text{ and } \varepsilon_b \not= \varepsilon_c, \\ UadVbWcX \equiv^{\upharpoonleft\!\!\downharpoonright}_\varepsilon UdaVbWcX \quad\text{if } a < b, c < d \text{ and } \varepsilon_b \not= \varepsilon_c, \end{gather*} where~$a, b, c, d$ are elements of~$[n]$ while~$U, V, W, X$ are words on~$[n]$. The \defn{Baxter-Cambrian congruence} is the equivalence relation on all signed permutations~$\mathfrak{S}_{\pm}$ obtained as the union of all~$\varepsilon$-Baxter-Cambrian congruences: \[ \equiv^{\upharpoonleft\!\!\downharpoonright} \; \eqdef \bigsqcup_{\substack{n \in \mathbb{N} \\ \varepsilon \in \pm^n}} \!\! \equiv^{\upharpoonleft\!\!\downharpoonright}_\varepsilon. \] \end{definition}
\begin{proposition} Two signed permutations~$\tau, \tau' \in \mathfrak{S}^{\varepsilon}$ are~$\varepsilon$-Baxter-Cambrian congruent if and only if they have the same $\mathbf{P}^{\upharpoonleft\!\!\downharpoonright}$-symbol: \[ \tau \equiv^{\upharpoonleft\!\!\downharpoonright}_\varepsilon \tau' \iff \surjectionPermAsso\Bax(\tau) = \surjectionPermAsso\Bax(\tau'). \] \end{proposition}
\begin{proof} The proof of this proposition consists essentially in seeing that~$\surjectionPermAsso\Bax(\tau) = \surjectionPermAsso\Bax(\tau')$ if and only if~$\tau \equiv \tau'$ and~$\mirror{\tau} \, \equiv \, \mirror{\tau}\!\!'$ (by definition of $\surjectionPermAsso\Bax$). The definition of the~$\varepsilon$-Baxter-Cambrian equivalence $\equiv^{\upharpoonleft\!\!\downharpoonright}_{\varepsilon}$ is exactly the translation of this observation in terms of rewriting rules. \end{proof}
\begin{proposition} \label{prop:intersectionCambrianClasses} The~$\varepsilon$-Baxter-Cambrian class indexed by a pair~$[\tree_\circ, \tree_\bullet]$ of twin $\varepsilon$-Cambrian trees is the intersection of the $\varepsilon$-Cambrian class indexed by~$\tree_\circ$ with the $(-\varepsilon)$-Cambrian class indexed by the reverse of~$\tree_\bullet$. \end{proposition}
\begin{proof} The~$\varepsilon$-Baxter-Cambrian class indexed by~$[\tree_\circ, \tree_\bullet]$ is the set of linear extensions of~$\graphTwin$, \textit{i.e.}~ of permutations which are both linear extensions of~$\tree_\circ$ and linear extensions of the reverse of~$\tree_\bullet$. The former form the $\varepsilon$-Cambrian class indexed by~$\tree_\circ$ while the latter form the $(-\varepsilon)$-Cambrian class indexed by the reverse of~$\tree_\bullet$. This is illustrated in \fref{fig:twoOppositeCambrians}.
\begin{figure}\label{fig:twoOppositeCambrians}
\end{figure} \end{proof}
\subsection{Rotations and Baxter-Cambrian lattices}
We now present the rotation operation on pairs of twin $\varepsilon$-Cambrian trees.
\begin{definition} \label{def:rotationBaxter} Let~$[\tree_\circ, \tree_\bullet]$ be a pair of $\varepsilon$-Cambrian trees and~$i \to j$ be an edge of~$\graphTwin$. We say that the edge~$i \to j$ is \defn{rotatable} if \begin{itemize} \item either~$i \to j$ is an edge in~$\tree_\circ$ and~$j \to i$ is an edge in~$\tree_\bullet$, \item or~$i \to j$ is an edge in~$\tree_\circ$ while~$i$ and~$j$ are incomparable in~$\tree_\bullet$, \item or $i$ and~$j$ are incomparable in~$\tree_\circ$ while $j \to i$ is an edge in~$\tree_\bullet$. \end{itemize} If~$i \to j$ is rotatable in~$[\tree_\circ, \tree_\bullet]$, its \defn{rotation} transforms~$[\tree_\circ, \tree_\bullet]$ to the pair of trees~$[\tree'_\circ, \tree'_\bullet]$,~where \begin{itemize} \item $\tree'_\circ$ is obtained by rotation of~$i \to j$ in~$\tree_\circ$ if possible and~$\tree'_\circ = \tree_\circ$ otherwise, and \item $\tree'_\bullet$ is obtained by rotation of~$j \to i$ in~$\tree_\bullet$ if possible and~$\tree'_\bullet = \tree_\bullet$ otherwise. \end{itemize} \end{definition}
\begin{proposition} Rotating a rotatable edge~$i \to j$ in a pair~$[\tree_\circ, \tree_\bullet]$ of twin $\varepsilon$-Cambrian trees yields a pair~$[\tree'_\circ, \tree'_\bullet]$ of twin $\varepsilon$-Cambrian trees. \end{proposition}
\begin{proof} By Proposition~\ref{prop:rotation}, the trees~$\tree_\circ, \tree_\bullet$ are $\varepsilon$-Cambrian trees. To see that they are twins, observe that switching~$i$ and~$j$ in a linear extension of~$\graphTwin$ yields a linear extension of~$\unionOp{\tree'_\circ}{\tree'_\bullet}$. \end{proof}
\begin{remark}[Number of rotatable edges] Note that a pair~$[\tree_\circ, \tree_\bullet]$ of~$\varepsilon$-Cambrian trees has always at least~$n-1$ rotatable edges. This will be immediate from the considerations of Section~\ref{subsec:geometricRealizationsBaxter}. \end{remark}
Consider the \defn{increasing rotation graph} whose vertices are pairs of twin $\varepsilon$-Cambrian trees and whose arcs are increasing rotations~$[\tree_\circ, \tree_\bullet] \to [\tree'_\circ, \tree'_\bullet]$, \textit{i.e.}~ for which~$i < j$ in Definition~\ref{def:rotationBaxter}. This graph is illustrated on \fref{fig:BaxterCambrianLattices} for the signatures $\varepsilon = {-}{+}{-}{-}$ and ${\varepsilon = {+}{-}{-}{-}}$.
\hvFloat[floatPos=p, capWidth=h, capPos=r, capAngle=90, objectAngle=90, capVPos=c, objectPos=c]{figure} {\includegraphics[width=1.65\textwidth]{BaxterCambrianLattices}} {The $\varepsilon$-Baxter-Cambrian lattices on pairs of twin $\varepsilon$-Cambrian trees, for the signatures $\varepsilon = {-}{+}{-}{-}$ (left) and ${\varepsilon = {+}{-}{-}{-}}$~(right).} {fig:BaxterCambrianLattices}
\begin{proposition} For any cover relation~$\tau < \tau'$ in the weak order on~$\mathfrak{S}^\varepsilon$, either~$\surjectionPermAsso\Bax(\tau) = \surjectionPermAsso\Bax(\tau')$ or~$\surjectionPermAsso\Bax(\tau) \to \surjectionPermAsso\Bax(\tau')$ in the increasing rotation graph. \end{proposition}
\begin{proof} Let~$i,j \in [n]$ be such that~$\tau'$ is obtained from~$\tau$ by switching two consecutive values~$ij$ to~$ji$. If~$i$ and~$j$ are incomparable in~$\mathbf{P}(\tau)$, then~$\mathbf{P}(\tau) = \mathbf{P}(\tau')$. Otherwise, there is an edge~$i \to j$ in~$\mathbf{P}(\tau)$, and~$\mathbf{P}(\tau')$ is obtained by rotating~$i \to j$ in~$\mathbf{P}(\tau)$. The same discussion is valid for the trees~$\mathbf{P}(\mirror{\tau})$ and~$\mathbf{P}(\mirror{\tau}\!\!')$ and edge~$j \to i$. The result immediately follows. \end{proof}
It follows that the increasing rotation graph on pairs of twin $\varepsilon$-Cambrian trees is acyclic and we call \defn{$\varepsilon$-Baxter-Cambrian poset} its transitive closure. In other words, the previous statement says that the map~$\surjectionPermAsso\Bax$ defines a poset homomorphism from the weak order on~$\mathfrak{S}^\varepsilon$ to the $\varepsilon$-Baxter-Cambrian poset. The following statement extends the results of N.~Reading~\cite{Reading-CambrianLattices} on Cambrian lattices and S.~Law and N.~Reading~\cite{LawReading} on the lattice of diagonal rectangulations.
\begin{proposition} The~$\varepsilon$-Baxter-Cambrian poset is a lattice quotient of the weak order on~$\mathfrak{S}^\varepsilon$. \end{proposition}
\begin{proof} By Proposition~\ref{prop:intersectionCambrianClasses}, the $\varepsilon$-Baxter-Cambrian congruence is the intersection of two Cambrian congruences. The statement follows since the Cambrian congruences are lattice congruences of the weak order~\cite{Reading-CambrianLattices} and an intersection of lattice congruences is a lattice congruence. \end{proof}
\begin{remark}[Cambrian \textit{vs.}~ Baxter-Cambrian lattices] \enlargethispage{.1cm} Using the definition of~$\CambCorresp\Bax$, we also notice that the~$\varepsilon$-Cambrian classes are unions of~$\varepsilon$-Baxter-Cambrian classes, therefore the Cambrian lattice is a lattice quotient of the Baxter-Cambrian lattice. \fref{fig:latticesBis} illustrates the Baxter-Cambrian, Cambrian, and boolean congruence classes on the weak orders of~$\mathfrak{S}_\varepsilon$ for the signatures $\varepsilon = {-}{+}{-}{-}$ and ${\varepsilon = {+}{-}{-}{-}}$.
\begin{figure}\label{fig:latticesBis}
\end{figure} \end{remark}
\begin{remark}[Extremal elements and pattern avoidance] \label{rem:patternsBaxterCambrian} Since the Baxter-Cambrian classes are generated by rewriting rules, we immediately obtain that the minimal elements of the Baxter-Cambrian classes are precisely the signed permutations avoiding the patterns: \[ \down{b} \text{-} da \text{-} \down{c}, \; \up{b} \text{-} da \text{-} \up{c}, \; \down{c} \text{-} da \text{-} \down{b}, \; \up{c} \text{-} da \text{-} \up{b}, \; \down{b} \text{-} \up{c} \text{-} da, \; \up{b} \text{-} \down{c} \text{-} da, \; \down{c} \text{-} \up{b} \text{-} da, \; \up{c} \text{-} \down{b} \text{-} da, \; da \text{-} \down{b} \text{-} \up{c}, \; da \text{-} \up{b} \text{-} \down{c}, \; da \text{-} \down{c} \text{-} \up{b}, \; da \text{-} \up{c} \text{-} \down{b}. \] Similarly, the maximal elements of the Baxter-Cambrian classes are precisely the signed permutations avoiding the patterns: \begin{equation} \label{eqn:patternsBaxterCambrian}\tag{$\star$} \down{b} \text{-} ad \text{-} \down{c}, \; \up{b} \text{-} ad \text{-} \up{c}, \; \down{c} \text{-} ad \text{-} \down{b}, \; \up{c} \text{-} ad \text{-} \up{b}, \; \down{b} \text{-} \up{c} \text{-} ad, \; \up{b} \text{-} \down{c} \text{-} ad, \; \down{c} \text{-} \up{b} \text{-} ad, \; \up{c} \text{-} \down{b} \text{-} ad, \; ad \text{-} \down{b} \text{-} \up{c}, \; ad \text{-} \up{b} \text{-} \down{c}, \; ad \text{-} \down{c} \text{-} \up{b}, \; ad \text{-} \up{c} \text{-} \down{b}. \end{equation} \end{remark}
\subsection{Baxter-Cambrian numbers} \label{subsec:BaxterCambrianNumbers}
In contrast to the number of $\varepsilon$-Cambrian trees, the number of pairs of twin $\varepsilon$-Cambrian trees does depend on the signature~$\varepsilon$. For example, there are~$22$ pairs of twin $({-}{-}{-}{-})$-Cambrian trees and only~$20$ pairs of twin $({-}{+}{-}{-})$-Cambrian trees. See Figures~\ref{fig:BaxterCambrianLattices}, \ref{fig:latticesBis} and~\ref{fig:GeneratingTreeBaxterCambrian}.
\enlargethispage{-.2cm} For a signature~$\varepsilon$, we define the \defn{$\varepsilon$-Baxter-Cambrian number}~$B_\varepsilon$ to be the number of pairs of twin $\varepsilon$-Cambrian trees. We immediately observe that~$B_\varepsilon$ is preserved when we change the first and last sign of~$\varepsilon$, inverse simultaneously all signs of~$\varepsilon$, or reverse the signature~$\varepsilon$: \[ B_\varepsilon = B_{\chi_0(\varepsilon)} = B_{\chi_n(\varepsilon)} = B_{-\varepsilon} = B_{\mirror{\varepsilon}}, \] where~$\chi_0$ and~$\chi_n$ change the first and last sign,~$(-\varepsilon)_i = -\varepsilon_i$ and $(\mirror{\varepsilon})_i = \varepsilon_{n+1-i}$. Table~\ref{table:numberBaxterCambrianEachSignature} shows the $\varepsilon$-Baxter-Cambrian number~$B_\varepsilon$ for all small signatures~$\varepsilon$ up to these transformations. Table~\ref{table:numbersBaxterCambrian} records all possible $\varepsilon$-Baxter-Cambrian numbers~$B_\varepsilon$ for signatures~$\varepsilon$ of sizes~$n \le 10$.
\begin{table}[h]
\centerline{
\begin{tabular}{c|l@{\qquad}l@{\qquad}l}
$n = 4$ & $B_{{+}{+}{+}{+}} = 22$ & $B_{{+}{+}{-}{+}} = 20$ & \\[.1cm]
\hline \\[-.3cm]
$n = 5$ & $B_{{+}{+}{+}{+}{+}} = 92$ & $B_{{+}{+}{+}{-}{+}} = 78$ & $B_{{+}{+}{-}{+}{+}} = 70$ \\[.1cm]
\hline \\[-.3cm]
$n = 6$ & $B_{{+}{+}{+}{+}{+}{+}} = 422$ & $B_{{+}{+}{+}{+}{-}{+}} = 342$ & $B_{{+}{+}{+}{-}{-}{+}} = 316$ \\
& $B_{{+}{+}{-}{-}{+}{+}} = 284$ & $B_{{+}{+}{+}{-}{+}{+}} = 282$ & $B_{{+}{+}{-}{+}{-}{+}} = 252$ \\[.1cm]
\hline \\[-.3cm]
$n = 7$ & $B_{{+}{+}{+}{+}{+}{+}{+}} = 2074$ & $B_{{+}{+}{+}{+}{+}{-}{+}} = 1628$ & $B_{{+}{+}{+}{+}{-}{-}{+}} = 1428$ \\
& $B_{{+}{+}{-}{-}{-}{+}{+}} = 1298$ & $B_{{+}{+}{+}{+}{-}{+}{+}} = 1270$ & $B_{{+}{+}{+}{-}{-}{+}{+}} = 1172$ \\
& $B_{{+}{+}{+}{-}{+}{+}{+}} = 1162$ & $B_{{+}{+}{-}{+}{+}{-}{+}} = 1044$ & $B_{{+}{+}{+}{-}{+}{-}{+}} = 1036$ \\
& & $B_{{+}{+}{-}{+}{-}{+}{+}} = 924$ & \\
\end{tabular}
}
\caption{The number~$B_\varepsilon$ of twin $\varepsilon$-Cambrian trees for all small signatures~$\varepsilon$ (up to first/last sign change, simultaneous inversion of all signs, and reverse).}
\label{table:numberBaxterCambrianEachSignature} \end{table}
\begin{table}[h]
\centerline{
\begin{tabular}{c|l}
$n = 4$ & $22$ {\scriptsize $(1)$}, $20$ {\scriptsize $(1)$} \\[.1cm]
\hline \\[-.3cm]
$n = 5$ & $92$ {\scriptsize $(1)$}, $78$ {\scriptsize $(2)$}, $70$ {\scriptsize $(1)$} \\[.1cm]
\hline \\[-.3cm]
$n = 6$ & $422$ {\scriptsize $(1)$}, $342$ {\scriptsize $(2)$}, $316$ {\scriptsize $(1)$}, $284$ {\scriptsize $(1)$}, $282$ {\scriptsize $(2)$}, $252$ {\scriptsize $(1)$} \\[.1cm]
\hline \\[-.3cm]
$n = 7$ & $2074$ {\scriptsize $(1)$}, $1628$ {\scriptsize $(2)$}, $1428$ {\scriptsize $(2)$}, $1298$ {\scriptsize $(1)$}, $1270$ {\scriptsize $(2)$}, $1172$ {\scriptsize $(2)$}, $1162$ {\scriptsize $(1)$}, $1044$ {\scriptsize $(2)$}, $1036$ {\scriptsize $(2)$}, $924$ {\scriptsize $(1)$} \\[.1cm]
\hline \\[-.3cm]
$n = 8$ & $10754$ {\scriptsize $(1)$}, $8244$ {\scriptsize $(2)$}, $6966$ {\scriptsize $(2)$}, $6612$ {\scriptsize $(1)$}, $6388$ {\scriptsize $(1)$}, $6182$ {\scriptsize $(2)$}, $5498$ {\scriptsize $(2)$}, $5380$ {\scriptsize $(2)$}, $5334$ {\scriptsize $(2)$}, $4902$ {\scriptsize $(1)$}, \\
& $4884$ {\scriptsize $(2)$}, $4748$ {\scriptsize $(2)$}, $4392$ {\scriptsize $(1)$}, $4362$ {\scriptsize $(2)$}, $4356$ {\scriptsize $(2)$}, $4324$ {\scriptsize $(1)$}, $3882$ {\scriptsize $(1)$}, $3880$ {\scriptsize $(2)$}, $3852$ {\scriptsize $(2)$}, $3432$ {\scriptsize $(1)$}\\[.1cm]
\hline \\[-.3cm]
$n = 9$ & $58202$ {\scriptsize $(1)$}, $43812$ {\scriptsize $(2)$}, $35998$ {\scriptsize $(2)$}, $33240$ {\scriptsize $(1)$}, $32908$ {\scriptsize $(2)$}, $31902$ {\scriptsize $(2)$}, $27660$ {\scriptsize $(2)$}, $26602$ {\scriptsize $(2)$}, $26392$ {\scriptsize $(2)$}, \\
& $25768$ {\scriptsize $(2)$}, $24888$ {\scriptsize $(1)$}, $24528$ {\scriptsize $(2)$}, $23530$ {\scriptsize $(1)$}, $23466$ {\scriptsize $(2)$}, $22768$ {\scriptsize $(2)$}, $20888$ {\scriptsize $(2)$}, $20886$ {\scriptsize $(2)$}, $20718$ {\scriptsize $(2)$}, \\
& $20244$ {\scriptsize $(2)$}, $20218$ {\scriptsize $(2)$}, $20082$ {\scriptsize $(2)$}, $18544$ {\scriptsize $(1)$}, $18518$ {\scriptsize $(2)$}, $18430$ {\scriptsize $(2)$}, $18376$ {\scriptsize $(2)$}, $17874$ {\scriptsize $(2)$}, $16470$ {\scriptsize $(2)$}, \\
& $16454$ {\scriptsize $(1)$}, $16358$ {\scriptsize $(2)$}, $16344$ {\scriptsize $(2)$}, $16342$ {\scriptsize $(2)$}, $16234$ {\scriptsize $(1)$}, $14550$ {\scriptsize $(4)$}, $14454$ {\scriptsize $(2)$}, $12870$ {\scriptsize $(1)$} \\[.1cm]
\hline \\[-.3cm]
$n = 10$ & $326240$ {\scriptsize $(1)$}, $242058$ {\scriptsize $(2)$}, $194608$ {\scriptsize $(2)$}, $180678$ {\scriptsize $(1)$}, $172950$ {\scriptsize $(2)$}, $172304$ {\scriptsize $(2)$}, $166568$ {\scriptsize $(1)$}, $146622$ {\scriptsize $(2)$}, \\
& $139100$ {\scriptsize $(2)$}, $138130$ {\scriptsize $(2)$}, $131994$ {\scriptsize $(2)$}, $129870$ {\scriptsize $(2)$}, $129600$ {\scriptsize $(2)$}, $124896$ {\scriptsize $(2)$}, $122716$ {\scriptsize $(2)$}, $120800$ {\scriptsize $(1)$}, \\
& $113754$ {\scriptsize $(2)$}, $111274$ {\scriptsize $(2)$}, $107072$ {\scriptsize $(2)$}, $106854$ {\scriptsize $(1)$}, $106382$ {\scriptsize $(2)$}, $105606$ {\scriptsize $(2)$}, $101084$ {\scriptsize $(3)$}, $101028$ {\scriptsize $(2)$}, \\
& $100426$ {\scriptsize $(2)$}, $98730$ {\scriptsize $(2)$}, $97524$ {\scriptsize $(2)$}, $94908$ {\scriptsize $(1)$}, $94372$ {\scriptsize $(1)$}, $93854$ {\scriptsize $(2)$}, $89952$ {\scriptsize $(2)$}, $89324$ {\scriptsize $(2)$}, $89276$ {\scriptsize $(2)$}, \\
& $88966$ {\scriptsize $(2)$}, $86638$ {\scriptsize $(2)$}, $86034$ {\scriptsize $(2)$}, $86026$ {\scriptsize $(2)$}, $79826$ {\scriptsize $(2)$}, $79384$ {\scriptsize $(2)$}, $79226$ {\scriptsize $(2)$}, $79076$ {\scriptsize $(2)$}, $79018$ {\scriptsize $(2)$}, \\
& $78580$ {\scriptsize $(1)$}, $78528$ {\scriptsize $(2)$}, $76542$ {\scriptsize $(2)$}, $76526$ {\scriptsize $(2)$}, $76484$ {\scriptsize $(2)$}, $76072$ {\scriptsize $(2)$}, $70450$ {\scriptsize $(2)$}, $70316$ {\scriptsize $(1)$}, $69866$ {\scriptsize $(4)$}, \\
& $69838$ {\scriptsize $(2)$}, $69810$ {\scriptsize $(2)$}, $69400$ {\scriptsize $(2)$}, $69314$ {\scriptsize $(1)$}, $67694$ {\scriptsize $(2)$}, $62124$ {\scriptsize $(3)$}, $62120$ {\scriptsize $(1)$}, $62096$ {\scriptsize $(2)$}, $61766$ {\scriptsize $(2)$}, \\
& $61746$ {\scriptsize $(2)$}, $61706$ {\scriptsize $(2)$}, $61682$ {\scriptsize $(2)$}, $61376$ {\scriptsize $(1)$}, $54956$ {\scriptsize $(2)$}, $54920$ {\scriptsize $(2)$}, $54892$ {\scriptsize $(1)$}, $54626$ {\scriptsize $(2)$}, $48620$ {\scriptsize $(1)$}
\end{tabular}
}
\caption{All possible $\varepsilon$-Baxter-Cambrian numbers~$B_\varepsilon$ for signatures~$\varepsilon$ of sizes~$n \le 10$. Numbers in parenthesis indicate the multiplicity of each Baxter number: for example, the second line indicates that there are~$8$ (resp.~$16$, resp.~$8$) signatures~$\varepsilon$ of~$\pm^5$ such that~$B_\varepsilon = 92$ (resp.~$78$, resp.~$70$).}
\label{table:numbersBaxterCambrian} \end{table}
In the following statements, we provide an inductive formula to compute all $\varepsilon$-Baxter-Cambrian numbers, using a two-parameters refinement. The proof is based on ideas similar to Proposition~\ref{prop:GeneratingTree}. The pairs of twin $\varepsilon$-Cambrian trees are in bijection with the weak order maximal permutations of $\varepsilon$-Baxter-Cambrian classes. These permutations are precisely the permutations avoiding the patterns~\eqref{eqn:patternsBaxterCambrian} in Remark~\ref{rem:patternsBaxterCambrian}. We consider the generating tree~$\mathcal{T}_\varepsilon^{\upharpoonleft\!\!\downharpoonright}$ for these permutations. This tree has~$n$ levels, and the nodes at level~$m$ are labeled by permutations of~$[m]$ whose values are signed by the restriction of~$\varepsilon$ to~$[m]$ and avoiding the patterns~\eqref{eqn:patternsBaxterCambrian}. The parent of a permutation in~$\mathcal{T}_\varepsilon^{\upharpoonleft\!\!\downharpoonright}$ is obtained by deleting its maximal value. See \fref{fig:GeneratingTreeBaxterCambrian}.
As in the proof of Proposition~\ref{prop:GeneratingTree}, we consider the possible positions of~$m + 1$ in the children of a permutation~$\tau$ at level~$m$ in this generating tree~$\mathcal{T}_\varepsilon^{\upharpoonleft\!\!\downharpoonright}$. Index by~$\{0, \dots, m\}$ from left to right the gaps before the first letter, between two consecutive letters, and after the last letter of~$\tau$. \defn{Free gaps} are those where placing~$m + 1$ does not create a pattern of~\eqref{eqn:patternsBaxterCambrian}. Free gaps are marked with a blue dot in \fref{fig:GeneratingTreeBaxterCambrian}. It is important to observe that gap~$0$ as well as the gaps immediately after~$m-1$ and~$m$ are always free, no matter~$\tau$ or the signature~$\varepsilon$.
Define the \defn{free-gap-type} of~$\tau$ to be the pair~$(\ell,r)$ where~$\ell$ (resp.~$r$) denote the number of free gaps on the left (resp.~right) of~$m$ in~$\tau$. For a signature~$\varepsilon$, let~$B_\varepsilon(\ell,r)$ denote the number of free-gap-type~$(\ell,r)$ weak order maximal permutations of $\varepsilon$-Baxter-Cambrian classes. These refined Baxter-Cambrian numbers enables us to write inductive equations.
\begin{proposition} \label{prop:inductionBaxterCambrian} Consider two signatures~$\varepsilon \in \pm^n$ and~$\varepsilon' \in \pm^{n-1}$, where~$\varepsilon'$ is obtained by deleting the last sign of~$\varepsilon$. Then \[ B_\varepsilon(\ell,r) = \begin{cases} \sum\limits_{\ell' \ge \ell} B_{\varepsilon'}(\ell',r-1) \; + \; \sum\limits_{r' \ge r} B_{\varepsilon'}(\ell-1,r') & \text{if } \varepsilon_{n-1} = \varepsilon_{n}, \vspace*{.2cm}\quad(=) \\ \delta_{\ell = 1} \cdot \delta_{r \ge 2} \cdot\!\!\! \sum\limits_{\substack{\ell' \ge r-1 \\ r' \ge 1}} B_{\varepsilon'}(\ell',r') \; + \; \delta_{\ell \ge 2} \cdot \delta_{r = 1} \cdot\!\!\! \sum\limits_{\substack{\ell' \ge 1 \\ r' \ge \ell-1}} B_{\varepsilon'}(\ell',r') \; & \text{if } \varepsilon_{n-1} \ne \varepsilon_{n}, \quad(\ne) \end{cases} \] where~$\delta$ denote the Kronecker~$\delta$ (defined by~$\delta_X = 1$ if~$X$ is satisfied and~$0$ otherwise). \end{proposition}
\begin{proof}
\hvFloat[floatPos=p, capWidth=h, capPos=r, capAngle=90, objectAngle=90, capVPos=c, objectPos=c]{figure} {\input{generatingTreeBaxterCambrian}} {The generating trees~$\mathcal{T}_\varepsilon^{\upharpoonleft\!\!\downharpoonright}$ for the signatures~$\varepsilon = {-}{+}{-}{-}$ (top) and~$\varepsilon = {+}{-}{-}{-}$ (bottom). Free gaps are marked with a blue dot.} {fig:GeneratingTreeBaxterCambrian}
Assume first that~$\varepsilon_{n-1} = \varepsilon_n$. Consider two permutations~$\tau$ and~$\tau'$ at level~$n$ and~$n-1$ in~$\mathcal{T}_\varepsilon^{\upharpoonleft\!\!\downharpoonright}$ such that~$\tau'$ is obtained by deleting~$n$ in~$\tau$. Denote by~$\alpha$ and~$\beta$ the gaps immediately after~$n-1$ and~$n$ in~$\tau$, by~$\alpha'$ the gap immediately after~$n-1$ in~$\tau'$, and by~$\beta'$ the gap in~$\tau'$ where we insert~$n$ to get~$\tau$. Then, besides gaps~$0$, $\alpha$ and~$\beta$, the free gaps of~$\tau$ are precisely the free gaps of~$\tau'$ not located between gaps~$\alpha'$ and~$\beta'$. Indeed, \begin{itemize} \item inserting~$d \eqdef n+1$ just after a value~$a$ located between~$b \eqdef n-1$ and~$c \eqdef n$ in~$\tau$ would create a pattern~$b \text{-} ad \text{-} c$ or~$c \text{-} ad \text{-} b$ with~$\varepsilon_b = \varepsilon_c$; \item conversely, consider a gap~$\gamma$ of~$\tau$ not located between~$\alpha$ and~$\beta$. If inserting~$n+1$ at~$\gamma$ in~$\tau$ creates a forbidden pattern of~\eqref{eqn:patternsBaxterCambrian} with~$c = n$, then inserting~$n$ at~$\gamma$ in~$\tau'$ would also create the same forbidden pattern of~\eqref{eqn:patternsBaxterCambrian} with~$c = n-1$. Therefore, all free gaps not located between gaps~$\alpha'$ and~$\beta'$ remain free. \end{itemize} Let~$(\ell,r)$ denote the free-gap-type of~$\tau$ and~$(\ell',r')$ denote the free-gap-type of~$\tau'$. We obtain that \begin{itemize} \item $\ell' \ge \ell$ and~$r' = r-1$ if~$n$ is inserted on the left of~$n-1$; \item $\ell' = \ell-1$ and~$r' \ge r$ if~$n$ is inserted on the right of~$n-1$. \end{itemize} The formula follows immediately when~$\varepsilon_{n-1} = \varepsilon_n$.
Assume now that~$\varepsilon_{n-1} = -\varepsilon_n$, and keep the same notations as before. Using similar arguments, we observe that besides gaps~$0$, $\alpha$ and~$\beta$, the free gaps of~$\tau$ are precisely the free gaps of~$\tau'$ located between gaps~$\alpha'$ and~$\beta'$. Therefore, we obtain that \begin{itemize} \item $\ell = 1$, $r \ge 2$, and~$\ell' \ge r-1$ if~$n$ is inserted on the left of~$n-1$; \item $\ell \ge 2$, $r = 1$, and~$r' \ge \ell-1$ if~$n$ is inserted on the right of~$n-1$. \end{itemize} The formula follows for~$\varepsilon_{n-1} = -\varepsilon_n$. \end{proof}
\enlargethispage{.3cm} Before applying these formulas to obtain bounds on~$B_\varepsilon$ for arbitrary signatures~$\varepsilon$, let us consider two special signatures: the constant and the alternating signature.
\para{Alternating signature} Since it is the easiest, we start with the \defn{alternating signature}~$({+}{-})^{\frac{n}{2}}$ (where we define~$({+}{-})^{\frac{n}{2}}$ to be~$({+}{-})^{m}{+}$ when~$n = 2m+1$ is odd).
\begin{proposition} \label{prop:BaxterAlternating} The Baxter-Cambrian numbers for alternating signatures are central binomial coefficients (see \href{https://oeis.org/A000984}{\cite[A000984]{OEIS}}): \[ B_{({+}{-})^{\frac{n}{2}}} = \binom{2n-2}{n-1}. \] \end{proposition}
\begin{proof} We prove by induction on~$n$ that the refined Baxter-Cambrian numbers are \[ B_{({+}{-})^{\frac{n}{2}}}(\ell,r) = \delta_{\ell = 1} \cdot \delta_{r \ge 2} \cdot \binom{2n-2-r}{n-r} + \delta_{\ell \ge 2} \cdot \delta_{r = 1} \cdot \binom{2n-2-\ell}{n-\ell}. \] This is true for~$n = 2$ since~$B_{{+}{-}}(1,2) = 1$ (counting the permutation~$21$) and~$B_{{+}{-}}(2,1) = 1$ (counting the permutation~$12$). Assume now that it is true for some~$n \in \mathbb{N}$. Then Equation~$(\ne)$ of Proposition~\ref{prop:inductionBaxterCambrian} shows that \begin{align*} B_{({+}{-})^{\frac{n+1}{2}}}(\ell,r) & = \delta_{\ell = 1} \cdot \delta_{r \ge 2} \cdot\!\!\! \sum_{\ell' \ge r-1} \binom{2n-2-\ell'}{n-\ell'} + \delta_{\ell \ge 2} \cdot \delta_{r = 1} \cdot\!\!\! \sum_{r' \ge \ell-1} \binom{2n-2-r'}{n-r'} \\ & = \delta_{\ell = 1} \cdot \delta_{r \ge 2} \cdot \binom{2n - r}{n + 1 - r} + \delta_{\ell \ge 2} \cdot \delta_{r = 1} \cdot \binom{2n - \ell}{n + 1 - \ell}, \end{align*} since a sum of binomial coefficients along a diagonal~$\sum_{i = 0}^p \binom{q+i}{i}$ simplifies to the binomial coefficient~$\binom{q+p+1}{p}$ by multiple applications of Pascal's rule. Finally, we conclude observing that \[ B_{({+}{-})^{\frac{n}{2}}} = \sum_{\ell,r \in [n]} B_{({+}{-})^{\frac{n}{2}}}(\ell,r) = 2 \sum_{u \ge 2} \binom{2n-2-u}{n-u} = 2 \binom{2n-3}{n-2} = \binom{2n-2}{n-1}. \] Remark~\ref{rem:BaxterGeneratingFunctions} provides an alternative analytic proof for this result. \end{proof}
\begin{remark}[Properties of the generating tree~$\mathcal{T}_{({+}{-})^{\frac{n}{2}}}^{\upharpoonleft\!\!\downharpoonright}$] Observe that: \begin{enumerate}[(i)] \item A permutation at level~$m$ with $k$ free gaps has~$k$ children, whose numbers of free gaps are~$3, 3, 4, 5, \dots, k+1$ respectively (compare to Lemma~\ref{lem:GeneratingTree}). This can already be observed on the generating tree~$\mathcal{T}_{{+}{-}{+}{-}}^{\upharpoonleft\!\!\downharpoonright}$ of \fref{fig:GeneratingTreeBaxterCambrian}. \item For a permutation~$\tau$ at level~$m$ with $k$ free gaps, there are precisely~$\binom{k + 2p -2}{p}$ permutations at level~$m + p$ that have~$\tau$ as a subword, for any~$p \in \mathbb{N}$. \item The number of permutations at level~$m$ with~$k$ free gaps is~$2 \binom{2m - 1 - k}{m + 1 - k}$. Counting permutations at level~$m$ and~$m+1$ according to their number of free gaps gives \[ \binom{2m-2}{m-1} = \sum_{k \ge 3} 2 \binom{2m - 1 - k}{m + 1 - k} \qquad\text{and}\qquad \binom{2m}{m} = \sum_{k \ge 3} 2k \binom{2m - 1 - k}{m + 1 - k}. \] \item Slight perturbations of the alternating signature yields interesting signatures for which we can give closed formulas for the Baxter-Cambrian number. For example, consider the signature~${+}{+}({+}{-})^{\frac{n}{2}-1}$ obtained from the alternating one by switching the second sign. Its Baxter-Cambrian number is given by a sum of three almost-central binomial coefficients: \[ B_{{+}{+}({+}{-})^{\frac{n}{2}-1}} = 2\binom{2n - 6}{n - 4} + \binom{2n - 2}{n - 1}. \] \end{enumerate} \end{remark}
\para{Constant signature} We now consider the \defn{constant signature}~$(+)^n$. The number~$B_{({+})^n}$ is the classical \defn{Baxter number} (see \href{https://oeis.org/A001181}{\cite[A001181]{OEIS}}) defined by \[ B_{(+)^n} = B_n = \binom{n+1}{1}^{-1} \binom{n+1}{2}^{-1} \sum_{k=1}^n \binom{n+1}{k-1} \binom{n+1}{k} \binom{n+1}{k+1}. \] These numbers have been extensively studied, see in particular~\cite{ChungGrahamHoggattKleiman, Mallows, DulucqGuibert1, DulucqGuibert2, YaoChenChengGraham, FelsnerFusyNoyOrden, BonichonBousquetMelouFusy, LawReading, Giraudo}. The Baxter number~$B_n$ counts several families: \begin{itemize} \item Baxter permutations of~$[n]$, \textit{i.e.}~ permutations avoiding the patterns~$b \text{-} da \text{-} c$ and~$c \text{-} ad \text{-} b$, \item weak order maximal (resp.~minimal) permutations of Baxter congruence classes on~$\mathfrak{S}_n$, \textit{i.e.}~ permutations avoiding the patterns~$b \text{-} ad \text{-} c$ and~$c \text{-} ad \text{-} b$ (resp.~$b \text{-} da \text{-} c$ and~$c \text{-} da \text{-} b$), \item pairs of twin binary trees on~$n$ nodes, \item diagonal rectangulations of an~$n \times n$ grid, \item plane bipolar orientations with~$n$ edges, \item non-crossing triples of path with~$k-1$ north steps and~$n-k$ east steps, for all~$k \in [n]$, \item etc. \end{itemize} Bijections between all these \defn{Baxter families} are discussed in~\cite{DulucqGuibert1, DulucqGuibert2, FelsnerFusyNoyOrden, BonichonBousquetMelouFusy}.
\begin{remark}[Two proofs of the summation formula] \label{rem:BaxterGeneratingFunctions} There are essentially two ways to obtain the above summation formula for Baxter numbers: it was first proved analytically in~\cite{ChungGrahamHoggattKleiman}, and then bijectively in~\cite{Viennot-Baxter, DulucqGuibert2, FelsnerFusyNoyOrden}. Let us shortly comment on these two techniques and discuss the limits of their extension to the Baxter-Cambrian setting. \begin{enumerate}[(i)] \item The \defn{bijective proofs} in~\cite{DulucqGuibert2, FelsnerFusyNoyOrden} transform pairs of binary trees to triples of non-crossing paths, and then use the Gessel-Viennot determinant lemma~\cite{GesselViennot} to get the summation formula. The middle path of these triples is given by the canopy of the twin binary trees, while the other two paths are given by the structure of the trees. We are not yet able to adapt this technique to provide summation formulas for all Baxter-Cambrian numbers. \item The \defn{analytic proof} in~\cite{ChungGrahamHoggattKleiman} is based on Equation~$(=)$ of Proposition~\ref{prop:inductionBaxterCambrian} and can be partially adapted to arbitrary signatures as follows. Define the \defn{extension} of a signature~${\varepsilon \in \pm^n}$ by a signature~$\delta \in \pm^m$ to be the signature~$\varepsilon \vartriangleleft \delta \in \pm^{n+m}$ such that~$(\varepsilon \vartriangleleft \delta)_i = \varepsilon_i$ for~$i \in [n]$ and~$(\varepsilon \vartriangleleft \delta)_{n+j} = \delta_j \cdot (\varepsilon \vartriangleleft \delta)_{n+j-1}$ for~$j \in m$. For example, ${+}{+}{-} \vartriangleleft {+}{-}{-}{+} = {+}{+}{-}{-}{+}{-}{-}$. Then for any~$\varepsilon \in \pm^n$ and~$\delta \in \pm^m$, we have \[ B_{\varepsilon\vartriangleleft\delta} = \sum_{\ell, r \ge 1} X_\delta(\ell, r) \, B_\varepsilon(\ell,r), \] where the coefficients~$X_\delta(\ell,r)$ are obtained inductively from the formulas of Proposition~\ref{prop:inductionBaxterCambrian}. Namely, for any~$\ell, r \ge 1$, we have~${X_\varnothing(\ell,r) = 1}$ and \begin{align*} X_{(+\delta)}(\ell,r) & = \sum\limits_{1 \le \ell' \le \ell} X_{\delta}(\ell',r+1) \; + \; \sum\limits_{1 \le r' \le r} X_{\delta}(\ell+1,r'), \\ X_{(-\delta)}(\ell,r) & = \sum\limits_{2 \le \ell' \le r+1} X_{\delta}(\ell',1) \; + \; \sum\limits_{2 \le r' \le \ell+1} X_{\delta}(1,r'). \end{align*} These equations translate on the generating function~$\mathfrak{X}_\delta(u,v) \eqdef \sum_{\ell, r \ge 1} X_\delta(\ell,r) u^{\ell-1} v^{r-1}$ to the formulas~$\mathfrak{X}_\varnothing(u,v) = \frac{1}{(1-u)(1-v)}$ and \begin{align*} \mathfrak{X}_{(+\delta)}(u,v) & = \frac{\mathfrak{X}_\delta(u,v) - \mathfrak{X}_\delta(u,0)}{(1-u) v} + \frac{\mathfrak{X}_\delta(u,v)-\mathfrak{X}_\delta(0,v)}{u (1-v)}, \\ \mathfrak{X}_{(-\delta)}(u,v) & = \frac{\mathfrak{X}_\delta(v,0) - \mathfrak{X}_\delta(0,0)}{(1-u) (1-v) v} + \frac{\mathfrak{X}_\delta(0,u) - \mathfrak{X}_\delta(0,0)}{u (1-u) (1-v)}. \end{align*} Note that the~$u/v$-symmetry of~$\mathfrak{X}_\delta(u,v)$ is reflected in a symmetry on these inductive equations. We can thus write this generating function~$\mathfrak{X}_\delta(u,v)$ as \[
\mathfrak{X}_\delta(u,v) = \sum_{\substack{i,j \ge 0 \\ k \in [|\delta|+1]}} Y_\delta^{i, j, k} \frac{(-u)^i \, (-v)^j}{(1-u)^{|\delta|+2-k} (1-v)^k}, \] where the non-vanishing coefficients~$Y_\delta^{i,j,k}$ are computed inductively by~$Y_\varnothing^{0,0,1} = 1$~and \begin{align*}
Y_{(+\delta)}^{i,j,k} & = \binom{k}{j+1} Y_\delta^{i,0,k} - Y_\delta^{i,j+1,k} + \binom{|\delta|+3-k}{i+1} Y_\delta^{0,j,k-1} - Y_\delta^{i+1,j,k-1}, \\
Y_{(-\delta)}^{i,j,k} & = \binom{k-1}{j} \! \bigg[ \! \binom{|\delta|+2-k}{i+1} Y_\delta^{0,0,k} - Y_\delta^{i+1,0,k} \bigg] + \binom{|\delta|+2-k}{i} \! \bigg[ \! \binom{k-1}{j+1} Y_\delta^{0,0,k-1} - Y_\delta^{0,j+1,k-1} \bigg]. \end{align*}
We used that~$Y_\delta^{i,j,k} = Y_\delta^{j,i,|\delta|+2-k}$ to simplify the second equation. Note that this decomposition of~$\mathfrak{X}_\delta$ is not unique and the inductive equations on~$Y_\delta^{i,j,k}$ follow from a particular choice of such a decomposition.
At that stage, F.~Chung, R.~Graham, V.~Hoggatt, and M.~Kleiman~\cite{ChungGrahamHoggattKleiman}, guess and check that the first equation is always satisfied by \[ Y_{(+)^{n-1}}^{i,j,k} = \frac{\binom{n+1}{k} \binom{n+1}{k+i+1} \binom{n+1}{k-j-1} \big[ \! \binom{k+i-2}{i} \binom{n+j-k-1}{j} - \binom{k+i-2}{i-1} \binom{n+j-k-1}{j-1} \! \big]}{\binom{n+1}{1} \binom{n+1}{2}} \] from which they derive immediately that \begin{align*} B_{(+)^n} & = B_{+\vartriangleleft(+)^{n-1}} = \sum_{\ell, r \ge 1} X_{(+)^{n-1}}(\ell, r) \, B_+(\ell,r) = X_{(+)^{n-1}}(1,1) = \mathfrak{X}_{(+)^{n-1}}(0,0) \\ & = \sum_{k \in [n]} Y_{(+)^{n-1}}^{0,0,k} = \binom{n+1}{1}^{-1} \binom{n+1}{2}^{-1} \sum_{k=1}^n \binom{n+1}{k-1} \binom{n+1}{k} \binom{n+1}{k+1}. \end{align*}
Unfortunately, we have not been able to guess a closed formula for the coefficients~$Y_{(-)^n}^{i,j,k}$. Note that it would be sufficient to understand the coefficients~$Y_{(-)^n}^{i,0,k}$ for which we observed empirically that \[ Y_{(-)^n}^{0,0,k} = C_n, \quad Y_{(-)^n}^{i,0,0} = Y_{(-)^n}^{i,0,1} = \binom{2n}{n-1-i}\binom{n-1+i}{i}\bigg/n \quad\text{and}\quad Y_{(-)^n}^{i,0,n+1-i} = \sum_{p = i}^{n-1} C_{n-1-p}C_p. \] See~\href{https://oeis.org/A000108}{\cite[A000108]{OEIS}}, \href{https://oeis.org/A234950}{\cite[A234950]{OEIS}}, and \href{https://oeis.org/A028364}{\cite[A028364]{OEIS}}. This would provide an alternative proof of Proposition~\ref{prop:BaxterAlternating} as we would obtain that \[ B_{({+}{-})^{\frac{n}{2}}} = B_{+\vartriangleleft(-)^{n-1}} = \mathfrak{X}_{(-)^{n-1}}(0,0) = \sum_{k \in [n]} Y_{(-)^{n-1}}^{0,0,k} = n C_{n-1} = \binom{2n-2}{n-1}. \] However, even if we were not able to work out the coefficients~$Y_{(-)^n}^{i,0,k}$, we still obtain another proof Proposition~\ref{prop:BaxterAlternating} by checking directly on the inductive equations on~$\mathfrak{X}_\delta(u,v)$ that \[ \mathfrak{X}_{(-)^n}(u,v) = \sum_{k \in [n]} \binom{2n-1-k}{n-1} \bigg[ \frac{1}{(1-u) (1-v)^{k+1}} + \frac{1}{(1-u)^{k+1} (1-v)} \bigg], \] from which we obtain \begin{align*} B_{({+}{-})^{\frac{n}{2}}} & = B_{+\vartriangleleft(-)^{n-1}} = \mathfrak{X}_{(-)^{n-1}}(0,0) = \sum_{k \in [n-1]} 2\binom{2n-3-k}{n-2} \\ & = 2 \sum_{k = 2}^{n} \binom{2n-2-k}{n-2} = 2\binom{2n-2}{n-1} - 2\binom{2n-3}{n-2} = \binom{2n-2}{n-1}. \end{align*} For the prior to last equality, choose $n-1$ positions among~$2n-2$ and group according to the first position~$k$. \end{enumerate} \end{remark}
\para{Arbitrary signatures} We now come back to an arbitrary signature~$\varepsilon$. We were not able to derive summation formulas for arbitrary signatures using the techniques presented in Remark~\ref{rem:BaxterGeneratingFunctions} above. However, we use here the inductive formulas of Proposition~\ref{prop:inductionBaxterCambrian} to bound the Baxter-Cambrian number~$B_\varepsilon$ for an arbitrary signature~$\varepsilon$.
For this, we consider the matrix~$\mathbf{B}_\varepsilon \eqdef \big( B_\varepsilon(\ell, r) \big)_{\ell, r \in [n]}$. The inductive formulas of Proposition~\ref{prop:inductionBaxterCambrian} provide an efficient inductive algorithm to compute this matrix~$\mathbf{B}_\varepsilon$ and thus the~$\varepsilon$-Baxter-Cambrian number~${B_\varepsilon = \sum_{\ell, r \in [n]} B_\varepsilon(\ell,r)}$. Namely, if~$\varepsilon$ is obtained by adding a sign at the end of~$\varepsilon'$, then each entry of~$\mathbf{B}_\varepsilon$ is the sum of entries of~$\mathbf{B}_{\varepsilon'}$ in a region depending on whether~$\varepsilon_n = \varepsilon_{n-1}$. These regions are sketched in \fref{fig:rulesMatrixComputation} and examples of such computations appear in \fref{fig:examplesComputationsMatrices}.
\begin{figure}
\caption{Inductive computation of~$\mathbf{B}_\varepsilon$: the black entry of~$\mathbf{B}_\varepsilon$ is the sum of the entries of~$\mathbf{B}_{\varepsilon'}$ over the shaded region. Entries outside the upper triangular region always vanish. When~$\varepsilon_n = -\varepsilon_{n-1}$, the only non-vanishing entries of~$\mathbf{B}_\varepsilon$ are in the first row or in the first column.}
\label{fig:rulesMatrixComputation}
\end{figure}
\begin{figure}
\caption{Inductive computation of~$\mathbf{B}_\varepsilon$, for~$\varepsilon = ({+})^6$, $({+})^3({-})^3$ and~$({+}{-})^3$.}
\label{fig:examplesComputationsMatrices}
\end{figure}
We observe that the transformations of \fref{fig:rulesMatrixComputation} are symmetric with respect to the diagonal of the matrix. Since~$\mathbf{B}_{\varepsilon_1\varepsilon_2} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ is symmetric, and~$\mathbf{B}_\varepsilon$ is obtained from~$\mathbf{B}_{\varepsilon_1\varepsilon_2}$ by successive applications of these symmetric transformations, we obtain that~$\mathbf{B}_\varepsilon$ is always symmetric. Although this fact may seem natural to the reader, it is not at all immediate as there is an asymmetry on the three forced free gaps: for example gap~$0$ is always free.
\newcommand{^\textsc{se}}{^\textsc{se}} For a matrix~$M \eqdef (m_{i,j})$, we consider the matrix~$M^\textsc{se} \eqdef \big( m^\textsc{se}_{i,j} \big)$ where \[ m^\textsc{se}_{i,j} \eqdef \sum_{p \ge i, \; q \ge j} m_{p,q} \] is the sum of all entries located south-east of~$(i,j)$ (in matrix notation). Observe that~$(\mathbf{B}_\varepsilon)^\textsc{se}_{1,1}$ is the sum of all entries of~$\mathbf{B}_\varepsilon$, and thus equals the $\varepsilon$-Baxter-Cambrian number~$B_\varepsilon$. Using \fref{fig:rulesMatrixComputation}, we obtain a similar rule to compute the entries of~$B_\varepsilon^\textsc{se}$ as sums of entries of~$B_{\varepsilon'}^\textsc{se}$ when~$\varepsilon$ is obtained by adding a sign at the end of~$\varepsilon'$. This rule is presented in \fref{fig:rulesSESMatrixComputation}.
\begin{figure}
\caption{Inductive computation of~$\mathbf{B}_\varepsilon^\textsc{se}$: the black entry of~$\mathbf{B}_\varepsilon^\textsc{se}$ is the sum of the entries of~$\mathbf{B}_{\varepsilon'}^\textsc{se}$ over the shaded region. Entries outside the triangular shape always vanish. When~$\varepsilon_n = -\varepsilon_{n-1}$, the only non-vanishing entries of~$\mathbf{B}_\varepsilon^\textsc{se}$ are in the first row or in the first column.}
\label{fig:rulesSESMatrixComputation}
\end{figure}
This matrix interpretation of the formulas of Proposition~\ref{prop:inductionBaxterCambrian} provides us with tools to bound the Baxter-Cambrian numbers. For a signature~$\varepsilon$, we denote by~$\mathsf{switch}(\varepsilon)$ the set of gaps where~$\varepsilon$ switches sign.
\begin{proposition} \label{prop:BCSE} For any two signatures~$\varepsilon, \tilde\varepsilon \in \pm^n$, if~$\mathsf{switch}(\varepsilon) \subset \mathsf{switch}(\tilde\varepsilon)$ then~$B_\varepsilon > B_{\tilde\varepsilon}$. \end{proposition}
\begin{proof} For two matrices~$M \eqdef (m_{i,j})$ and~$\tilde M \eqdef (\tilde m_{i,j})$, we write~$M \succcurlyeq \tilde M$ when~$m_{i,j} \ge \tilde m_{i,j}$ for all indices~$i,j$ (entrywise comparison), and we write~$M \succ \tilde M$ when~$M \succcurlyeq \tilde M$ and~$M \ne \tilde M$. Consider four signatures~$\varepsilon, \tilde\varepsilon \in \pm^n$ and~$\varepsilon', \tilde\varepsilon' \in \pm^{n-1}$ such that~$\varepsilon'$ (resp.~$\tilde\varepsilon'$) is obtained by deleting the last sign of~$\varepsilon$ (resp.~$\tilde\varepsilon$). From \fref{fig:rulesSESMatrixComputation}, and using the fact that~$\mathbf{B}_\varepsilon$ is symmetric, we obtain that: \begin{itemize} \item if~$\varepsilon_n = \varepsilon_{n-1}$ while~$\tilde\varepsilon_n = -\tilde\varepsilon_{n-1}$, then~$\mathbf{B}_{\varepsilon'}^\textsc{se} \succcurlyeq \mathbf{B}_{\tilde\varepsilon'}^\textsc{se}$ implies $\mathbf{B}_{\varepsilon}^\textsc{se} \succ \mathbf{B}_{\tilde\varepsilon}^\textsc{se}$. \item if either both~$\varepsilon_n = \varepsilon_{n-1}$ and~$\tilde\varepsilon_n = \tilde\varepsilon_{n-1}$, or both~$\varepsilon_n = -\varepsilon_{n-1}$ and~$\tilde\varepsilon_n = -\tilde\varepsilon_{n-1}$, then~$\mathbf{B}_{\varepsilon'}^\textsc{se} \succ \mathbf{B}_{\tilde\varepsilon'}^\textsc{se}$ implies $\mathbf{B}_{\varepsilon}^\textsc{se} \succ \mathbf{B}_{\tilde\varepsilon}^\textsc{se}$. \end{itemize} By repeated applications of these observations, we therefore obtain that~$\mathsf{switch}(\varepsilon) \subset \mathsf{switch}(\tilde\varepsilon)$ implies~$\mathbf{B}_{\varepsilon}^\textsc{se} \succ \mathbf{B}_{\tilde\varepsilon}^\textsc{se}$, and thus~$B_\varepsilon > B_{\tilde\varepsilon}$. \end{proof}
\begin{corollary} \label{coro:boundsBaxterCambrianNumbers} Among all signatures of~$\pm^n$, the constant signature maximizes the Baxter-Cambrian number, while the alternating signature minimizes it: for all~$\varepsilon \in \pm^n$, \[ \binom{2n-2}{n-1} = B_{({+}{-})^{\frac{n}{2}}} \le B_\varepsilon \le B_{(+)^n} = \binom{n+1}{1}^{-1} \binom{n+1}{2}^{-1} \sum_{k=1}^n \binom{n+1}{k-1} \binom{n+1}{k} \binom{n+1}{k-1}. \] \end{corollary}
\begin{remark} The proof of Proposition~\ref{prop:BCSE} may seem unnecessarily intricate. Observe however that the situation is rather subtle: \begin{itemize}
\item If~$\mathsf{switch}(\varepsilon) \not\subseteq \mathsf{switch}(\tilde\varepsilon)$, we may have~$B_\varepsilon < B_{\tilde\varepsilon}$ even if~$|\mathsf{switch}(\varepsilon)| < |\mathsf{switch}(\tilde\varepsilon)|$. The smallest example is given by~$B_{{+}{+}{+}{-}{+}{+}{-}{-}{-}} = 18376 < 18544 = B_{{+}{+}{-}{+}{+}{+}{-}{+}{+}}$. \item We may have~$\mathbf{B}_\varepsilon^\textsc{se} \succcurlyeq \mathbf{B}_{\tilde\varepsilon}^\textsc{se}$ but~$\mathbf{B}_\varepsilon \not\succcurlyeq \mathbf{B}_{\tilde\varepsilon}$. See the third column of \fref{fig:examplesComputationsMatrices}. \end{itemize} \end{remark}
\subsection{Geometric realizations} \label{subsec:geometricRealizationsBaxter}
Using similar tools as in Section~\ref{subsec:geometricRealizations} and following~\cite{LawReading}, we present geometric realizations for pairs of twin Cambrian trees, for the Baxter-Cambrian lattice, and for the Baxter-Cambrian $\mathbf{P}^{\upharpoonleft\!\!\downharpoonright}$-symbol. For a partial order~$\prec$ on~$[n]$, we still define its \defn{incidence cone}~$\mathrm{C}(\prec)$ and its \defn{braid cone}~$\mathrm{C}^\diamond(\prec)$ as \[ \mathrm{C}(\prec) \eqdef \cone\set{e_i-e_j}{\text{for all } i \prec j} \quad\text{and}\quad \mathrm{C}^\diamond(\prec) \eqdef \set{\b{x} \in \mathbb{H}}{x_i \le x_j \text{ for all } i \prec j}. \] The cones~$\mathrm{C}(\graphTwin)$ for all pairs~$[\tree_\circ, \tree_\bullet]$ of twin $\varepsilon$-Cambrian trees form (together with all their faces) a complete polyhedral fan that we call the \defn{$\varepsilon$-Baxter-Cambrian fan}. It is the common refinement of the $\varepsilon$- and~$(-\varepsilon)$-Cambrian fans. It is therefore the normal fan of the Minkowski sum of the associahedra~$\Asso$ and~$\Asso[-\varepsilon]$. We call this polytope \defn{Baxter-Cambrian associahedron} and denote it by~$\BaxAsso$. Note that~$\BaxAsso$ is clearly centrally symmetric (since~$\Asso = -\Asso[-\varepsilon]$) but not necessarily simple. Examples are illustrated on \fref{fig:MinkowskiSums}. The graph of~$\BaxAsso$, oriented in the direction ${(n, \dots, 1)-(1, \dots, n) = \sum_{i \in [n]} (n+1-2i) \, e_i}$, is the Hasse diagram of the $\varepsilon$-Baxter-Cambrian lattice. Finally, the Baxter-Cambrian $\mathbf{P}^{\upharpoonleft\!\!\downharpoonright}$-symbol can be read geometrically as \[ [\tree_\circ, \tree_\bullet] = \surjectionPermAsso\Bax(\tau) \iff \mathrm{C}(\graphTwin) \subseteq \mathrm{C}(\tau) \iff \mathrm{C}^\diamond(\graphTwin) \supseteq \mathrm{C}^\diamond(\tau). \]
\begin{figure}\label{fig:MinkowskiSums}
\end{figure}
\section{Baxter-Cambrian Hopf Algebra} \label{sec:BaxterCambrianAlgebra}
In this section, we define the Baxter-Cambrian Hopf algebra~$\mathsf{BaxCamb}$, extending simultaneously the Cambrian Hopf algebra and the Baxter Hopf algebra studied by S.~Law and N.~Reading~\cite{LawReading} and S.~Giraudo~\cite{Giraudo}. We present again the construction of~$\mathsf{BaxCamb}$ as a subalgebra of~$\mathsf{FQSym}_\pm$ and that of its dual~$\mathsf{BaxCamb}^*$ as a quotient of~$\mathsf{FQSym}_\pm^*$.
\subsection{Subalgebra of $\mathsf{FQSym}_\pm$}
We denote by~$\mathsf{BaxCamb}$ the vector subspace of~$\mathsf{FQSym}_\pm$ generated by the elements \[ \mathbb{P}_{[\tree_\circ, \tree_\bullet]} \eqdef \sum_{\substack{\tau \in \mathfrak{S}_\pm \\ \surjectionPermAsso\Bax(\tau) = [\tree_\circ, \tree_\bullet]}} \mathbb{F}_\tau = \sum_{\tau \in \mathcal{L}(\unionOp{\tree_\circ\;}{\,\tree_\bullet})} \mathbb{F}_\tau, \] for all pairs of twin Cambrian trees~$[\tree_\circ, \tree_\bullet]$. For example, for the pair of twin Cambrian trees of \fref{fig:twinCambrianTrees}\,(left), we have \[ \mathbb{P}_{\left[ \raisebox{-.45cm}{\includegraphics{Tex}}, \raisebox{-.45cm}{\includegraphics{TexTwin}} \right]} = \mathbb{F}_{\down{21}\up{7}\down{5}\up{3}\down{4}\up{6}} + \mathbb{F}_{\down{2}\up{7}\down{15}\up{3}\down{4}\up{6}} + \mathbb{F}_{\down{2}\up{7}\down{51}\up{3}\down{4}\up{6}} + \mathbb{F}_{\up{7}\down{215}\up{3}\down{4}\up{6}} + \mathbb{F}_{\up{7}\down{251}\up{3}\down{4}\up{6}} + \mathbb{F}_{\up{7}\down{521}\up{3}\down{4}\up{6}}. \]
\begin{theorem} \label{thm:baxSubalgebra} $\mathsf{BaxCamb}$ is a Hopf subalgebra of~$\mathsf{FQSym}_\pm$. \end{theorem}
\begin{proof} The proof of this theorem is left to the reader as it is very similar to that of Theorem~\ref{thm:cambSubalgebra}. Exchanges in a permutation~$\tau$ of the product~$\mathbb{P}_{[\tree_\circ, \tree_\bullet]} \cdot \mathbb{P}_{[\tree'_\circ, \tree'_\bullet]}$ are due to exchanges either in the linear extensions of~$\unionOp{\tree_\circ}{\tree_\bullet}$ and $\unionOp{\tree'_\circ}{\tree'_\bullet}$ or in the shuffle product of these linear extensions. The coproduct is treated similarly. \end{proof}
\enlargethispage{-.7cm} As for the Cambrian algebra, we can describe combinatorially the product and coproduct of $\mathbb{P}$-basis elements of~$\mathsf{BaxCamb}$ in terms of operations on pairs of twin Cambrian trees.
\para{Product} The product in the Baxter-Cambrian algebra~$\mathsf{BaxCamb}$ can be described in terms of intervals in Baxter-Cambrian lattices.
\begin{proposition} For any two pairs~$[\tree_\circ, \tree_\bullet]$ and~$[\tree_\circ', \tree_\bullet']$ of twin Cambrian trees, the product~$\mathbb{P}_{[\tree_\circ, \tree_\bullet]} \cdot \mathbb{P}_{[\tree_\circ', \tree_\bullet']}$ is given by \[ \mathbb{P}_{[\tree_\circ, \tree_\bullet]} \cdot \mathbb{P}_{[\tree_\circ', \tree_\bullet']} = \sum_{[\tree[S]_\circ, \tree[S]_\bullet]} \mathbb{P}_{[\tree[S]_\circ, \tree[S]_\bullet]}, \] where~$[\tree[S]_\circ, \tree[S]_\bullet]$ runs over the interval between~$\left[ \raisebox{-6pt}{$\tree_\circ$} \nearrow \raisebox{4pt}{$\bar\tree'_\circ$},\raisebox{4pt}{$\tree_\bullet$} \nwarrow \raisebox{-6pt}{$\bar \tree_\bullet'$} \right]$ and~$\left[ \raisebox{4pt}{$\tree_\circ$} \nwarrow \raisebox{-6pt}{$\bar\tree_\circ'$}, \raisebox{-6pt}{$\tree_\bullet$} \nearrow \raisebox{4pt}{$\bar\tree_\bullet'$} \right]$ in the~$\varepsilon(\tree_\circ)\varepsilon(\tree_\circ')$-Baxter-Cambrian lattice. \end{proposition}
\begin{proof} The result relies on the fact that the $\varepsilon$-Baxter-Cambrian classes are intervals of the weak order on~$\mathfrak{S}^\varepsilon$, and that the shuffle product of two intervals of the weak order is again an interval of the weak order. See the similar proof of Proposition~\ref{prop:product}. \end{proof}
For example, we can compute the product \begin{gather*} \hspace*{-1.7cm} \mathbb{P}_{\left[ \raisebox{-.25cm}{\includegraphics{exmTreeYAg}}, \raisebox{-.25cm}{\includegraphics{exmTreeAYd}} \right]} \cdot \mathbb{P}_{\left[ \raisebox{-.3cm}{\includegraphics{exmProductTwinA}}, \raisebox{-.3cm}{\includegraphics{exmProductTwinB}} \right]} = \mathbb{F}_{\up{2}\down{1}} \cdot \big( \mathbb{F}_{\up{2}\down{34}\up{1}} + \mathbb{F}_{\up{2}\down{3}\up{1}\down{4}} \big) \hspace*{\textwidth} \\[-.2cm] \hspace*{-2cm} \begin{array}{@{$\quad{} = {}$}c@{+}c@{+}c@{+}c@{+}c@{}c@{}c} \begin{pmatrix} \quad \mathbb{F}_{\up{2}\down{1}\up{4}\down{5}\up{3}\down{6}} + \mathbb{F}_{\up{24}\down{15}\up{3}\down{6}} \\ + \mathbb{F}_{\up{24}\down{51}\up{3}\down{6}} + \mathbb{F}_{\up{2}\down{1}\up{4}\down{56}\up{3}} \\ + \mathbb{F}_{\up{24}\down{156}\up{3}} + \mathbb{F}_{\up{24}\down{516}\up{3}} \\ + \mathbb{F}_{\up{24}\down{561}\up{3}} \end{pmatrix} & \begin{pmatrix} \mathbb{F}_{\up{24}\down{5}\up{3}\down{16}} + \mathbb{F}_{\up{24}\down{5}\up{3}\down{61}} \\ + \mathbb{F}_{\up{24}\down{56}\up{3}\down{1}} \end{pmatrix} & \begin{pmatrix} \quad \mathbb{F}_{\up{42}\down{15}\up{3}\down{6}} + \mathbb{F}_{\up{42}\down{51}\up{3}\down{6}} \\ + \mathbb{F}_{\up{42}\down{156}\up{3}} + \mathbb{F}_{\up{42}\down{516}\up{3}} \\ + \mathbb{F}_{\up{42}\down{561}\up{3}} + \mathbb{F}_{\up{4}\down{5}\up{2}\down{1}\up{3}\down{6}} \\ + \mathbb{F}_{\up{4}\down{5}\up{2}\down{16}\up{3}} + \mathbb{F}_{\up{4}\down{5}\up{2}\down{61}\up{3}} \\ + \mathbb{F}_{\up{4}\down{56}\up{2}\down{1}\up{3}} \end{pmatrix} & \begin{pmatrix} \quad \mathbb{F}_{\up{42}\down{5}\up{3}\down{16}} + \mathbb{F}_{\up{42}\down{5}\up{3}\down{61}} \\ + \mathbb{F}_{\up{42}\down{56}\up{3}\down{1}} + \mathbb{F}_{\up{4}\down{5}\up{23}\down{16}} \\ + \mathbb{F}_{\up{4}\down{5}\up{23}\down{61}} + \mathbb{F}_{\up{4}\down{5}\up{2}\down{6}\up{3}\down{1}} \\ + \mathbb{F}_{\up{4}\down{56}\up{23}\down{1}} \end{pmatrix} & \begin{pmatrix} \quad \mathbb{F}_{\up{4}\down{5}\up{32}\down{16}} + \mathbb{F}_{\up{4}\down{5}\up{32}\down{61}} \\ + \mathbb{F}_{\up{4}\down{5}\up{3}\down{6}\up{2}\down{1}} + \mathbb{F}_{\up{4}\down{56}\up{32}\down{1}} \end{pmatrix} \\[.8cm] \mathbb{P}_{\left[ \raisebox{-.35cm}{\includegraphics{exmProductTwin1}}, \raisebox{-.35cm}{\includegraphics{exmProductTwin2}} \right]} & \mathbb{P}_{\left[ \raisebox{-.35cm}{\includegraphics{exmProductTwin1}}, \raisebox{-.35cm}{\includegraphics{exmProductTwin3}} \right]} & \mathbb{P}_{\left[ \raisebox{-.35cm}{\includegraphics{exmProductTwin4}}, \raisebox{-.35cm}{\includegraphics{exmProductTwin2}} \right]} & \mathbb{P}_{\left[ \raisebox{-.35cm}{\includegraphics{exmProductTwin4}}, \raisebox{-.35cm}{\includegraphics{exmProductTwin3}} \right]} & \mathbb{P}_{\left[ \raisebox{-.35cm}{\includegraphics{exmProductTwin5}}, \raisebox{-.35cm}{\includegraphics{exmProductTwin6}} \right]} & . \end{array} \end{gather*}
\begin{remark}[Multiplicative bases] \label{rem:multiplicativeBasesBaxter} Similar to the multiplicative bases defined in Section~\ref{sec:multiplicativeBases}, the bases~$\mathbb{E}^{[\tree_\circ, \tree_\bullet]}$ and~$\mathbb{H}^{[\tree_\circ, \tree_\bullet]}$ defined by \[ \mathbb{E}^{[\tree_\circ, \tree_\bullet]} \eqdef \sum_{[\tree_\circ, \tree_\bullet] \le [\tree'_\circ, \tree'_\bullet]} \mathbb{P}_{[\tree'_\circ, \tree'_\bullet]} \qquad\text{and}\qquad \mathbb{H}^{[\tree_\circ, \tree_\bullet]} \eqdef \sum_{[\tree'_\circ, \tree'_\bullet] \le [\tree_\circ, \tree_\bullet]} \mathbb{P}_{[\tree'_\circ, \tree'_\bullet]} \] are multiplicative since \[ \mathbb{E}^{[\tree_\circ, \tree_\bullet]} \cdot \mathbb{E}^{[\tree'_\circ, \tree'_\bullet]} = \mathbb{E}^{\left[ \raisebox{-5pt}{\scriptsize$\tree_\circ$}\nearrow \raisebox{4pt}{\scriptsize$\bar \tree'_\circ$}, \raisebox{4pt}{\scriptsize$\tree_\bullet$}\nwarrow \raisebox{-5pt}{\scriptsize$\bar \tree'_\bullet$} \right]} \qquad\text{and}\qquad \mathbb{H}^{[\tree_\circ, \tree_\bullet]} \cdot \mathbb{H}^{[\tree'_\circ, \tree'_\bullet]} = \mathbb{H}^{\left[ \raisebox{4pt}{\scriptsize$\tree_\circ$}\nwarrow \raisebox{-5pt}{\scriptsize$\bar \tree'_\circ$}, \raisebox{-5pt}{\scriptsize$\tree_\bullet$}\nearrow \raisebox{4pt}{\scriptsize$\bar \tree'_\bullet$} \right]}. \] The $\mathbb{E}$-indecomposable elements are precisely the pairs~$[\tree_\circ, \tree_\bullet]$ for which all linear extensions of~$\graphTwin$ are indecomposable. In particular, $[\tree_\circ, \tree_\bullet]$ is $\mathbb{E}$-indecomposable as soon as~$\tree_\circ$ is \mbox{$\mathbb{E}$-inde}\-composable or~$\tree_\bullet$ is $\mathbb{H}$-indecomposable. This condition is however not necessary. For example $\surjectionPermAsso\Bax(\down{3142})$ is $\mathbb{E}$-indecomposable while~$\mathbf{P}(\down{3142}) = \mathbf{P}(\down{1342})$ is $\mathbb{E}$-decomposable and~${\mathbf{P}(\down{2413}) = \mathbf{P}(\down{4213})}$ is $\mathbb{H}$-decomposable. The enumerative and structural properties studied in Section~\ref{sec:multiplicativeBases} do not hold anymore for the set of $\mathbb{E}$-indecomposable pairs of twin Cambrian trees: they form an ideal of the Baxter-Cambrian lattice, but this ideal is not principal as in Proposition~\ref{prop:upperIdeal}, and they are not counted by simple formulas as in Proposition~\ref{prop:numberIndecomposables}. Let us however mention that \begin{itemize} \item the numbers of $\mathbb{E}$-indecomposable elements with constant signature~$(-)^n$ are given by $1$, $1$, $3$, $11$, $47$, $221$, $\dots$ See~\href{https://oeis.org/A217216}{\cite[A217216]{OEIS}}. \item the numbers of $\mathbb{E}$-indecomposable elements with constant signature~$({+}{-})^{n/2}$ are given by~$1$, $1$, $3$, $9$, $29$, $97$, $333$, $1165$, $4135$, $\dots$ These numbers are the coefficients of the Taylor series of~$\frac{1}{x + \sqrt{1-4x}}$. See~\href{https://oeis.org/A081696}{\cite[A081696]{OEIS}} for references and details. \end{itemize} \end{remark}
\para{Coproduct} A \defn{cut} of a pair of twin Cambrian trees~$[\tree[S]_\circ, \tree[S]_\bullet]$ is a pair~$\gamma = [\gamma_\circ, \gamma_\bullet]$ where~$\gamma_\circ$ is a cut of~$\tree[S]_\circ$ and~$\gamma_\bullet$ is a cut of~$\tree[S]_\bullet$ such that the labels of~$\tree[S]_\circ$ below~$\gamma_\circ$ coincide with the labels of~$\tree[S]_\bullet$ above~$\gamma_\bullet$. Equivalently, it can be seen as a lower set of~$\graphTwin$. An example is illustrated in \fref{fig:exampleCoproductTwin}.
\begin{figure}
\caption{A cut~$\gamma$ of a pair of twin Cambrian trees.}
\label{fig:exampleCoproductTwin}
\end{figure}
We denote by~$AB([\tree[S]_\circ, \tree[S]_\bullet], [\gamma_\circ, \gamma_\bullet])$ the set of pairs~$[A_\circ, B_\bullet]$, where~$A_\circ$ appears in the product~$\prod_{\tree \in A(\tree[S]_\circ)} \mathbb{P}_{\tree}$ while~$B_\bullet$ appears in the product~$\prod_{\tree \in B(\tree[S]_\circ)} \mathbb{P}_{\tree}$, and~$A_\circ$ and~$B_\bullet$ are twin Cambrian trees. We define~$BA([\tree[S]_\circ, \tree[S]_\bullet], [\gamma_\circ, \gamma_\bullet])$ similarly exchanging the role of~$A$ and~$B$. We obtain the following description of the coproduct in the Baxter-Cambrian algebra~$\mathsf{BaxCamb}$.
\begin{proposition} For any pair of twin Cambrian trees~$[\tree[S]_\circ, \tree[S]_\bullet]$, the coproduct~$\triangle \mathbb{P}_{[\tree[S]_\circ, \tree[S]_\bullet]}$ is given~by \[ \triangle \mathbb{P}_{[\tree[S]_\circ, \tree[S]_\bullet]} = \sum_{\gamma} \bigg( \sum_{[B_\circ, A_\bullet]} \mathbb{P}_{[B_\circ, A_\bullet]} \bigg) \otimes \bigg( \sum_{[A_\circ, B_\bullet]} \mathbb{P}_{[A_\circ, B_\bullet]} \bigg), \] where~$\gamma$ runs over all cuts of~$[\tree[S]_\circ, \tree[S]_\bullet]$, $[B_\circ, A_\bullet]$ runs over~$BA([\tree[S]_\circ, \tree[S]_\bullet], [\gamma_\circ, \gamma_\bullet])$ and $[A_\circ, B_\bullet]$ runs over~$AB([\tree[S]_\circ, \tree[S]_\bullet], [\gamma_\circ, \gamma_\bullet])$. \end{proposition}
\begin{proof} The proof is similar to that of Proposition~\ref{prop:coproduct}. The difficulty here is to describe the linear extensions of the union of the forest~$A(\tree[S]_\circ, \gamma_\circ)$ with the opposite of the forest~$B(\tree[S]_\bullet, \gamma_\bullet)$. This difficulty is hidden in the definition of~$AB([\tree[S]_\circ, \tree[S]_\bullet], [\gamma_\circ, \gamma_\bullet])$. \end{proof}
For example, we can compute the coproduct \begin{align*} & \hspace*{-1cm} \triangle \mathbb{P}_{\left[ \raisebox{-.3cm}{\includegraphics{exmProductTwinA}}, \raisebox{-.3cm}{\includegraphics{exmProductTwinB}} \right]} = \triangle \big( \mathbb{F}_{\up{2}\down{3}\up{1}\down{4}} + \mathbb{F}_{\up{2}\down{34}\up{1}} \big) \hspace*{\textwidth} \\ & \hspace*{-.5cm} = 1 \otimes \big( \mathbb{F}_{\up{2}\down{3}\up{1}\down{4}} + \mathbb{F}_{\up{2}\down{34}\up{1}} \big) + \mathbb{F}_{\up{1}} \otimes \big( \mathbb{F}_{\down{2}\up{1}\down{3}} + \mathbb{F}_{\down{23}\up{1}} \big) + \mathbb{F}_{\up{1}\down{2}} \otimes \big( \mathbb{F}_{\up{1}\down{2}} + \mathbb{F}_{\down{2}\up{1}} \big) + \mathbb{F}_{\up{2}\down{3}\up{1}} \otimes \mathbb{F}_{\down{1}} + \mathbb{F}_{\up{1}\down{2}\down{3}} \otimes \mathbb{F}_{\up{1}} + \big( \mathbb{F}_{\up{2}\down{3}\up{1}\down{4}} + \mathbb{F}_{\up{2}\down{34}\up{1}} \big) \otimes 1 \\ & \hspace*{-.5cm} = 1 \otimes \mathbb{P}_{\left[ \raisebox{-.3cm}{\includegraphics{exmProductTwinA}}, \raisebox{-.3cm}{\includegraphics{exmProductTwinB}} \right]} \;\; + \;\; \mathbb{P}_{\left[ \raisebox{-.15cm}{\includegraphics{exmTreeY}}, \raisebox{-.15cm}{\includegraphics{exmTreeY}} \right]} \otimes \big( \mathbb{P}_{\left[ \raisebox{-.25cm}{\includegraphics{exmCoproductTwin1}}, \raisebox{-.25cm}{\includegraphics{exmCoproductTwin2}} \right]} + \mathbb{P}_{\left[ \raisebox{-.25cm}{\includegraphics{exmCoproductTwin3}}, \raisebox{-.25cm}{\includegraphics{exmCoproductTwin2}} \right]} \big) \;\; + \;\; \mathbb{P}_{\left[ \raisebox{-.25cm}{\includegraphics{exmTreeYAd}}, \raisebox{-.25cm}{\includegraphics{exmTreeAYg}} \right]} \otimes \big( \mathbb{P}_{\left[ \raisebox{-.25cm}{\includegraphics{exmTreeYAd}}, \raisebox{-.25cm}{\includegraphics{exmTreeAYg}} \right]} + \mathbb{P}_{\left[ \raisebox{-.25cm}{\includegraphics{exmTreeAYg}}, \raisebox{-.25cm}{\includegraphics{exmTreeYAd}} \right]} \big) \\[.2cm] & \qquad\qquad\qquad + \mathbb{P}_{\left[ \raisebox{-.25cm}{\includegraphics{exmCoproductTwin4}}, \raisebox{-.25cm}{\includegraphics{exmCoproductTwin5}} \right]} \otimes \mathbb{P}_{\left[ \raisebox{-.15cm}{\includegraphics{exmTreeA}}, \raisebox{-.15cm}{\includegraphics{exmTreeA}} \right]} \;\; + \;\; \mathbb{P}_{\left[ \raisebox{-.25cm}{\includegraphics{exmCoproductTwin6}}, \raisebox{-.25cm}{\includegraphics{exmCoproductTwin7}} \right]} \otimes \mathbb{P}_{\left[ \raisebox{-.15cm}{\includegraphics{exmTreeY}}, \raisebox{-.15cm}{\includegraphics{exmTreeY}} \right]} \;\; + \;\; \mathbb{P}_{\left[ \raisebox{-.3cm}{\includegraphics{exmProductTwinA}}, \raisebox{-.3cm}{\includegraphics{exmProductTwinB}} \right]} \otimes 1. \end{align*}
In the result line, we have grouped the summands according to the six possible cuts of the pair of twin Cambrian trees~$\left[ \raisebox{-.3cm}{\includegraphics{exmProductTwinA}}, \raisebox{-.3cm}{\includegraphics{exmProductTwinB}} \right]$.
\para{Matriochka algebras} As the Baxter-Cambrian classes refine the Cambrian classes, the Baxter-Cambrian Hopf algebra is sandwiched between the Hopf algebra on signed permutations and the Cambrian Hopf algebra. It completes our sequence of subalgebras: \[ \mathsf{Rec} \subset \mathsf{Camb} \subset \mathsf{BaxCamb} \subset \mathsf{FQSym}_\pm. \]
\subsection{Quotient algebra of $\mathsf{FQSym}_\pm^*$}
As for the Cambrian algebra, the following result is automatic from Theorem~\ref{thm:baxSubalgebra}.
\begin{theorem} The graded dual~$\mathsf{BaxCamb}^*$ of the Baxter-Cambrian algebra is isomorphic to the image of~$\mathsf{FQSym}_\pm^*$ under the canonical projection \[ \pi : \mathbb{C}\langle A \rangle \longrightarrow \mathbb{C}\langle A \rangle / \equiv^{\upharpoonleft\!\!\downharpoonright}, \] where~$\equiv^{\upharpoonleft\!\!\downharpoonright}$ denotes the Baxter-Cambrian congruence. The dual basis~$\mathbb{Q}_{[\tree_\circ, \tree_\bullet]}$ of~$\mathbb{P}_{[\tree_\circ, \tree_\bullet]}$ is expressed as~$\mathbb{Q}_{[\tree_\circ, \tree_\bullet]} = \pi(\mathbb{G}_\tau)$, where~$\tau$ is any linear extension of~$\graphTwin$. \end{theorem}
We now describe the product and coproduct in~$\mathsf{BaxCamb}^*$ by combinatorial operations on pairs of twin Cambrian trees. We use the definitions and notations introduced in Section~\ref{subsec:quotientAlgebra}.
\para{Product} The product in~$\mathsf{BaxCamb}^*$ can be described using gaps and laminations similarly to Proposition~\ref{prop:productDual}. An example is illustrated on \fref{fig:exampleProductDualTwin}. For two Cambrian trees~$\tree$ and~$\tree'$ and a shuffle~$s$ of the signatures~$\varepsilon(\tree)$ and~$\varepsilon(\tree')$, we still denote by~$\tree \,{}_s\!\backslash \tree'$ the tree described in Section~\ref{subsec:quotientAlgebra}.
\begin{proposition} For any two pairs of twin Cambrian trees~$[\tree_\circ, \tree_\bullet]$ and~$[\tree'_\circ, \tree'_\bullet]$, the product~$\mathbb{Q}_{[\tree_\circ, \tree_\bullet]} \cdot \mathbb{Q}_{[\tree'_\circ, \tree'_\bullet]}$ is given by \[ \mathbb{Q}_{[\tree_\circ, \tree_\bullet]} \cdot \mathbb{Q}_{[\tree'_\circ, \tree'_\bullet]} = \sum_s \mathbb{Q}_{[\tree_\circ \,{}_s\!\backslash \tree'_\circ, \tree'_\bullet \,{}_s\!\backslash \tree_\bullet]}, \] where~$s$ runs over all shuffles of the signatures~$\varepsilon(\tree_\circ) = \varepsilon(\tree_\bullet)$ and~$\varepsilon(\tree'_\circ) = \varepsilon(\tree'_\bullet)$. \end{proposition}
\begin{proof} The proof follows the same lines as that of Proposition~\ref{prop:productDual}. The only difference is that if~$\tau \in \mathcal{L}(\unionOp{\tree_\circ}{\tree_\bullet})$, $\tau' \in \mathcal{L}(\unionOp{\tree'_\circ}{\tree'_\bullet})$, and~$\sigma \in \tau \star \tau'$, then~$\tree_\circ = \mathbf{P}(\tau)$ appears below~$\tree'_\circ = \mathbf{P}(\tau')$ in~$\mathbf{P}(\sigma)$ since~$\sigma$ is inserted from left to right in~$\mathbf{P}(\sigma)$, while $\tree_\bullet = \mathbf{P}(\mirror{\tau})$ appears above~$\tree'_\bullet = \mathbf{P}(\mirror{\tau}\!\!')$ in~$\mathbf{P}(\mirror{\sigma})$ since~$\sigma$ is inserted from right to left in~$\mathbf{P}(\mirror{\sigma})$. \end{proof}
\begin{figure}
\caption{Two pairs of twin Cambrian trees~$[\tree_\circ, \tree_\bullet]$ and~$[\tree'_\circ, \tree'_\bullet]$ (left), and a pair of twin Cambrian tree which appear in the product~$\mathbb{Q}_{[\tree_\circ, \tree_\bullet]} \cdot \mathbb{Q}_{[\tree'_\circ, \tree'_\bullet]}$ (right).}
\label{fig:exampleProductDualTwin}
\end{figure}
For example, we can compute the product \begin{align*} & \hspace*{-.5cm} \mathbb{Q}_{\left[ \raisebox{-.25cm}{\includegraphics{exmTreeYAg}}, \raisebox{-.25cm}{\includegraphics{exmTreeAYd}} \right]} \cdot \mathbb{Q}_{\left[ \raisebox{-.25cm}{\includegraphics{exmTreeYAd}}, \raisebox{-.25cm}{\includegraphics{exmTreeAYg}} \right]} = \mathbb{G}_{\up{2}\down{1}} \cdot \mathbb{G}_{\up{1}\down{2}} \hspace*{\textwidth} \\ & \hspace*{-.5cm} \begin{array}{@{$\quad{} = {}$}c@{+\,}c@{+\,}c@{+\,}c@{+\,}c@{+\,}c} \mathbb{G}_{\up{2}\down{1}\up{3}\down{4}} & \mathbb{G}_{\up{3}\down{1}\up{2}\down{4}} & \mathbb{G}_{\up{4}\down{1}\up{2}\down{3}} & \mathbb{G}_{\up{3}\down{2}\up{1}\down{4}} & \mathbb{G}_{\up{4}\down{2}\up{1}\down{3}} & \mathbb{G}_{\up{4}\down{3}\up{1}\down{2}} \\ \mathbb{Q}_{\left[ \raisebox{-.28cm}{\includegraphics{exmProductTwin7}}, \raisebox{-.28cm}{\includegraphics{exmProductTwin8}} \right]} & \mathbb{Q}_{\left[ \raisebox{-.28cm}{\includegraphics{exmProductTwin9}}, \raisebox{-.28cm}{\includegraphics{exmProductTwin10}} \right]} & \mathbb{Q}_{\left[ \raisebox{-.28cm}{\includegraphics{exmProductTwin11}}, \raisebox{-.28cm}{\includegraphics{exmProductTwin12}} \right]} & \mathbb{Q}_{\left[ \raisebox{-.28cm}{\includegraphics{exmProductTwin13}}, \raisebox{-.28cm}{\includegraphics{exmProductTwin10}} \right]} & \mathbb{Q}_{\left[ \raisebox{-.28cm}{\includegraphics{exmProductTwin14}}, \raisebox{-.28cm}{\includegraphics{exmProductTwin15}} \right]} & \mathbb{Q}_{\left[ \raisebox{-.28cm}{\includegraphics{exmProductTwin16}}, \raisebox{-.28cm}{\includegraphics{exmProductTwin17}} \right]}. \end{array} \end{align*}
\para{Coproduct} The coproduct in~$\mathsf{BaxCamb}^*$ can be described combinatorially as in Proposition~\ref{prop:coproductDual}. For a Cambrian tree~$\tree[S]$ and a gap~$\gamma$ between two consecutive vertices of~$\tree[S]$, we still denote by~$L(\tree[S], \gamma)$ and~$R(\tree[S], \gamma)$ the left and right Cambrian subtrees of~$\tree[S]$ when split along the path~$\lambda(\tree[S], \gamma)$.
\begin{proposition} For any pair of twin Cambrian trees~$[\tree[S]_\circ, \tree[S]_\bullet]$, the coproduct~$\triangle\mathbb{Q}_{[\tree[S]_\circ, \tree[S]_\bullet]}$ is given~by \[ \triangle\mathbb{Q}_{[\tree[S]_\circ, \tree[S]_\bullet]} = \sum_{\gamma} \mathbb{Q}_{[L(\tree[S]_\circ,\gamma), L(\tree[S]_\bullet, \gamma)]} \otimes \mathbb{Q}_{[R(\tree[S]_\circ,\gamma), R(\tree[S]_\bullet, \gamma)]}, \] where~$\gamma$ runs over all gaps between consecutive positions in~$[n]$. \end{proposition}
\begin{proof} The proof is identical to that of Proposition~\ref{prop:coproductDual}. \end{proof}
For example, we can compute the coproduct \begin{align*} \triangle \mathbb{Q}_{\left[ \raisebox{-.3cm}{\includegraphics{exmProductTwinA}}, \raisebox{-.3cm}{\includegraphics{exmProductTwinB}} \right]} & = \triangle \mathbb{G}_{\up{2}\down{34}\up{1}} \\[-.3cm] & = 1 \otimes \mathbb{G}_{\up{2}\down{34}\up{1}} + \mathbb{G}_{\up{1}} \otimes \mathbb{G}_{\up{1}\down{23}} + \mathbb{G}_{\up{21}} \otimes \mathbb{G}_{\down{12}} + \mathbb{G}_{\up{2}\down{3}\up{1}} \otimes \mathbb{G}_{\down{1}} + \mathbb{G}_{\up{2}\down{34}\up{1}} \otimes 1 \\[.2cm] & = 1 \otimes \mathbb{Q}_{\left[ \raisebox{-.3cm}{\includegraphics{exmProductTwinA}}, \raisebox{-.3cm}{\includegraphics{exmProductTwinB}} \right]} \;\; + \;\; \mathbb{Q}_{\left[ \raisebox{-.15cm}{\includegraphics{exmTreeY}}, \raisebox{-.15cm}{\includegraphics{exmTreeY}} \right]} \otimes \mathbb{Q}_{\left[ \raisebox{-.25cm}{\includegraphics{exmCoproductTwin6}}, \raisebox{-.25cm}{\includegraphics{exmCoproductTwin7}} \right]} \;\; + \;\; \mathbb{Q}_{\left[ \raisebox{-.25cm}{\includegraphics{exmTreeYYg}}, \raisebox{-.25cm}{\includegraphics{exmTreeYYd}} \right]} \otimes \mathbb{Q}_{\left[ \raisebox{-.25cm}{\includegraphics{exmTreeAAd}}, \raisebox{-.25cm}{\includegraphics{exmTreeAAg}} \right]} \\ & \qquad\qquad\qquad + \;\; \mathbb{Q}_{\left[ \raisebox{-.3cm}{\includegraphics{exmCoproductDualTwin1}}, \raisebox{-.3cm}{\includegraphics{exmCoproductDualTwin2}} \right]} \otimes \mathbb{Q}_{\left[ \raisebox{-.15cm}{\includegraphics{exmTreeA}}, \raisebox{-.15cm}{\includegraphics{exmTreeA}} \right]} \;\; + \;\; \mathbb{Q}_{\left[ \raisebox{-.3cm}{\includegraphics{exmProductTwinA}}, \raisebox{-.3cm}{\includegraphics{exmProductTwinB}} \right]} \otimes 1. \end{align*}
\section{Cambrian tuples} \label{sec:tuples}
This section is devoted to a natural extension of our results on twin Cambrian trees and the Baxter-Cambrian algebra to arbitrary intersections of Cambrian congruences. Since the results presented here are straightforward generalizations of that of Sections~\ref{sec:BaxterTrees} and~\ref{sec:BaxterCambrianAlgebra}, all proofs of this section are left to the reader.
\subsection{Combinatorics of Cambrian tuples} As observed in Remark~\ref{rem:reversing}, pairs of twin Cambrian trees can as well be thought of as pairs of Cambrian trees on opposite signature whose union is acyclic. We extend this idea to arbitrary signatures. For an $\ell$-tuple~$\tuple$ and~$k \in [\ell]$, we denote by~$\tuple_{[k]}$ the $k$th element of~$\tuple$.
\begin{definition} A \defn{Cambrian $\ell$-tuple} is a $\ell$-tuple~$\tuple$ of Cambrian trees~$\tuple_{[k]}$ on the same vertex set, and whose union forms an acyclic graph. The \defn{signature} of~$\tuple$ is the $\ell$-tuple of signatures $\mathcal{E}(\tuple) \eqdef \big[ \varepsilon \big( \tuple_{[1]} \big), \dots, \varepsilon \big( \tuple_{[\ell]} \big) \big]$. Let~$\mathrm{Camb}(\mathcal{E})$ denote the set of Cambrian $\ell$-tuples of signature~$\mathcal{E}$. \end{definition}
\begin{definition} A \defn{leveled Cambrian $\ell$-tuple} is a $\ell$-tuple~$\tuple$ of leveled Cambrian trees~$\tuple_{[k]}$ with labelings~$p_{[k]}, q_{[k]}$, and such that~$q_{[k]} \circ {p_{[k]}}^{-1}$ is independent of~$k$. In other words, it is a Cambrian $\ell$-tuple endowed with a linear extension of the union of its trees. \end{definition}
For example, pairs of twin (leveled) Cambrian trees are particular (leveled) Cambrian $2$-tuples. A Cambrian $2$-tuple and a leveled Cambrian $2$-tuple with signature~$[{-}{-}{+}{-}{-}{+}{+}, {+}{+}{-}{+}{-}{-}{+}]$ are represented in \fref{fig:CambrianPair}.
\begin{figure}
\caption{A Cambrian $2$-tuple (left), and a leveled Cambrian $2$-tuple (right).}
\label{fig:CambrianPair}
\end{figure}
We now want to define an analogue of the Cambrian correspondence. For this, we need permutations recording~$\ell$ different signatures. Call \defn{$\ell$-signed permutation} a permutation table where each dot receives an~$\ell$-tuple of signs. In other words, it is an element of the wreath product of~$\mathfrak{S}$ by~$(\mathbb{Z}_2)^\ell$. For example, \[ \uptilde{\downw{2}} \uptilde{\upw{7}} \downtilde{\downw{5}} \uptilde{\downw{1}} \downtilde{\upw{3}} \uptilde{\downw{4}} \downtilde{\upw{6}} \] is a $2$-signed permutation whose signatures are marked with~$\up{\phantom{1}}/\down{\phantom{1}}$ and~$\uptilde{\phantom{1}}/\downtilde{\phantom{1}}$ respectively. For a $\ell$-signed permutation~$\tau$ and~$k \in [\ell]$, we denote by~$\tau_{[k]}$ the signed permutation where we only keep the~$k$-th signature. For example \[ \uptilde{\downw{2}} \uptilde{\upw{7}} \downtilde{\downw{5}} \uptilde{\downw{1}} \downtilde{\upw{3}} \uptilde{\downw{4}} \downtilde{\upw{6}}_{[1]} = \down{2}\up{7}\down{51}\up{3}\down{4}\up{6} \qquad\text{and}\qquad \uptilde{\downw{2}} \uptilde{\upw{7}} \downtilde{\downw{5}} \uptilde{\downw{1}} \downtilde{\upw{3}} \uptilde{\downw{4}} \downtilde{\upw{6}}_{[2]} = \simtilde{2} \simtilde{7} \downtilde{5} \simtilde{1} \downtilde{3} \simtilde{4} \downtilde{6}. \] We denote by~$\mathfrak{S}_{\pm^\ell}$ the set of all $\ell$-signed permutations and by~$\mathfrak{S}_\mathcal{E}$ (resp.~$\mathfrak{S}^\mathcal{E}$) the set of $\ell$-signed permutations with p-signatures (resp.~v-signatures)~$\mathcal{E}$. Applying the Cambrian correspondences in parallel yields a map form $\ell$-signed permutations to Cambrian $\ell$-tuples.
\begin{proposition} The map~$\Theta_\ell$ defined by~$\Theta_\ell(\tau) \eqdef \big[\Theta \big( \tau_{[1]} \big), \dots, \Theta \big( \tau_{[\ell]} \big) \big]$ is a bijection from $\ell$-signed permutations to leveled Cambrian $\ell$-tuples.
The map~$\mathbf{P}_\ell$ defined by~$\mathbf{P}_\ell(\tau) \eqdef \big[ \mathbf{P} \big( \tau_{[1]} \big), \dots, \mathbf{P} \big( \tau_{[\ell]} \big) \big]$ is a surjection from $\ell$-signed permutations to Cambrian $\ell$-tuples. \end{proposition}
For example, the Cambrian $2$-tuple and the leveled Cambrian $2$-tuple on \fref{fig:CambrianPair} are \[ \mathbf{P}_2 \left( \uptilde{\downw{2}} \uptilde{\upw{7}} \downtilde{\downw{5}} \uptilde{\downw{1}} \downtilde{\upw{3}} \uptilde{\downw{4}} \downtilde{\upw{6}} \right) \qquad\text{and}\qquad \Theta_2 \left( \uptilde{\downw{2}} \uptilde{\upw{7}} \downtilde{\downw{5}} \uptilde{\downw{1}} \downtilde{\upw{3}} \uptilde{\downw{4}} \downtilde{\upw{6}} \right). \] As in the Cambrian and Baxter-Cambrian situation, the permutations with the same $\mathbf{P}^{\upharpoonleft\!\!\downharpoonright}$-symbol define the Cambrian congruence classes on $\ell$-signed permutations.
\begin{definition} For a signature $\ell$-tuple~$\mathcal{E}$, the \defn{$\mathcal{E}$-Cambrian congruence} on~$\mathfrak{S}^\mathcal{E}$ is the intersection~$\equiv_\mathcal{E} \eqdef \bigcap_{k \in [\ell]} \equiv_{\mathcal{E}_{[k]}}$ of all $\mathcal{E}_{[k]}$-Cambrian congruences. In other words, it is the transitive closure of the rewriting rule $UacV \equiv_\mathcal{E} UcaV$ if for all~$k \in [\ell]$, there exists~$a < b_{[k]} < c$ such that~$(\mathcal{E}_{[k]})_{b_{[k]}} = +$ and~$b_{[k]}$ appears in~$U$, or~$(\mathcal{E}_{[k]})_{b_{[k]}} = -$ and~$b_{[k]}$ appears in~$V$. The \defn{Cambrian congruence} on~$\mathfrak{S}_{\pm^\ell}$ is the equivalence relation~$\equiv_\ell$ on all $\ell$-signed permutations obtained as the union of all $\mathcal{E}$-Cambrian congruences. \end{definition}
\enlargethispage{-.4cm} For example, the $[{-}{+}{-}{-}, {+}{-}{-}{-}]$-Cambrian classes are represented on \fref{fig:latticesTer}.
\begin{figure}
\caption{(Left)~The $[{-}{+}{-}{-}, {+}{-}{-}{-}]$-Cambrian classes (blue) are the intersections of the $({-}{+}{-}{-})$-Cambrian classes (red) and the $({+}{-}{-}{-})$-Cambrian classes (green). (Right) $[{-}{+}{-}{-}, {+}{-}{-}{-}]$-Cambrian (blue) and boolean (green) congruence classes on the weak order.}
\label{fig:latticesTer}
\end{figure}
\begin{proposition} Two $\ell$-signed permutations~$\tau, \tau' \in \mathfrak{S}_{\pm^\ell}$ are Cambrian congruent if and only if they have the same~$\mathbf{P}_\ell$-symbol: \[ \tau \equiv_\ell \tau' \iff \mathbf{P}_\ell(\tau) = \mathbf{P}_\ell(\tau'). \] \end{proposition}
\begin{proposition} The $\varepsilon$-Cambrian class indexed by the Cambrian $\ell$-tuple~$\tuple$ is the intersection of the $\mathcal{E}_{[k]}$-Cambrian classes indexed by~$\tuple_{[k]}$ over~$k \in [\ell]$. \end{proposition}
We now present the rotation operation on Cambrian $\ell$-tuples.
\begin{definition} \label{def:rotationTuples} Let~$\tuple$ be a Cambrian $\ell$-tuple and consider an edge~$i \to j$ of the union~$\bigcup_{k \in [\ell]} \tuple_{[k]}$. We say that the edge~$i \to j$ is \defn{rotatable} if either~$i \to j$ is an edge or $i$ and~$j$ are incomparable in each tree~$\tuple_{[k]}$ (note that $i \to j$ is an edge in at least one of these trees since it belongs to their union). If~$i \to j$ is rotatable in~$\tuple$, its \defn{rotation} transforms~$\tuple$ to the $\ell$-tuple of trees~$\tuple' \eqdef \big[ \tuple'_{[1]}, \dots, \tuple'_{[\ell]} \big]$,~where~$\tuple'_{[k]}$ is obtained by rotation of~$i \to j$ in~$\tuple_{[k]}$ if possible and~$\tuple'_{[k]} = \tuple_{[k]}$ otherwise. \end{definition}
\begin{proposition} Rotating a rotatable edge~$i \to j$ in a Cambrian $\ell$-tuple~$\tuple$ yields a Cambrian $\ell$-tuple~$\tuple'$ with the same signature. \end{proposition}
Consider the \defn{increasing rotation graph} whose vertices are $\mathcal{E}$-Cambrian tuples and whose arcs are increasing rotations~$\tuple \to \tuple'$, \textit{i.e.}~ for which~$i < j$ in Definition~\ref{def:rotationTuples}. This graph is illustrated on \fref{fig:tupleCambrian} for the signature $2$-tuple~$\mathcal{E} = [{-}{+}{-}{-}, {+}{-}{-}{-}]$.
\begin{figure}
\caption{The $[{-}{+}{-}{-}, {+}{-}{-}{-}]$-Cambrian lattice on Cambrian tuples. See also \fref{fig:latticesTer}.}
\label{fig:tupleCambrian}
\end{figure}
\begin{proposition} For any cover relation~$\tau < \tau'$ in the weak order on~$\mathfrak{S}^\mathcal{E}$, either~$\mathbf{P}_\ell(\tau) = \mathbf{P}_\ell(\tau')$ or~$\mathbf{P}_\ell(\tau) \to \mathbf{P}_\ell(\tau')$ in the increasing rotation graph. \end{proposition}
It follows that the increasing rotation graph on $\mathcal{E}$-Cambrian tuples is acyclic. We call \defn{\mbox{$\mathcal{E}$-Cambrian} poset} its transitive closure. In other words, the previous statement says that the map~$\mathbf{P}_\ell$ defines a poset homomorphism from the weak order on~$\mathfrak{S}^\mathcal{E}$ to the $\mathcal{E}$-Cambrian poset. This homomorphism is in fact a lattice homomorphism.
\begin{proposition} The~$\mathcal{E}$-Cambrian poset is a lattice quotient of the weak order on~$\mathfrak{S}^\mathcal{E}$. \end{proposition}
The $\mathcal{E}$-Cambrian lattice has natural geometric realizations, similar to the geometric realizations of the Baxter-Cambrian lattice. Namely, for a signature $\ell$-tuple~$\mathcal{E}$, the cones~$\mathrm{C}(\tuple) \eqdef \mathrm{C}(\bigcup_{k \in [\ell]} \tuple_{[k]})$ for all~$\mathcal{E}$-Cambrian tuples~$\tuple$ form (together with all their faces) a complete polyhedral fan that we call the $\mathcal{E}$-Cambrian fan. It is the common refinement of the $\mathcal{E}_{[k]}$-Cambrian fans for~$k \in [\ell]$. It is therefore the normal fan of the Minkowski sum of the associahedra~$\Asso[\mathcal{E}_{[k]}]$ for~$k \in [\ell]$. An example is illustrated on \fref{fig:MinkowskiSumsTuple}. The $1$-skeleton of this polytope, oriented in the direction of~$(n, \dots, 1)-(1, \dots, n) = \sum_{i \in [n]} (n+1-2i) \, e_i$, is the Hasse diagram of the $\mathcal{E}$-Cambrian lattice. Finally, the $\mathcal{E}$-Cambrian $\mathbf{P}_\ell$-symbol can be read geometrically~as \[ \tuple = \mathbf{P}_\ell(\tau) \iff \mathrm{C}(\tuple) \subseteq \mathrm{C}(\tau) \iff \mathrm{C}^\diamond(\tuple) \supseteq \mathrm{C}^\diamond(\tau). \]
\begin{figure}
\caption{The Minkowski sum (blue, right) of the associahedra~$\Asso[{-}{+}{-}{-}]$ (red, left) and~$\Asso[{+}{-}{-}{-}]$ (green, middle) gives a realization of the $[{-}{+}{-}{-}, {+}{-}{-}{-}]$-Cambrian lattice, represented in Figures~\ref{fig:latticesTer} and~\ref{fig:tupleCambrian}.}
\label{fig:MinkowskiSumsTuple}
\end{figure}
\subsection{Cambrian tuple Hopf Algebra} \label{subsec:CambrianTupleAlgebra}
In this section, we construct a Hopf algebra indexed by Cambrian $\ell$-tuples, similar to the Baxter-Cambrian algebra. Exactly as we needed to consider the Hopf algebra~$\mathsf{FQSym}_\pm$ on signed permutations when constructing the Cambrian algebra to keep track of the signature, we now need to consider a natural extension of~$\mathsf{FQSym}$ on $\ell$-signed permutation to keep track of the $\ell$ signatures of~$\mathcal{E}$.
The \defn{shifted shuffle product}~$\tau \,\bar\shuffle\, \tau'$ (resp.~\defn{convolution product}~$\tau \star \tau'$) of two $\ell$-signed permutations~$\tau, \tau'$ is still defined as the shifted product (resp.~convolution product) where signs travel with their values (resp.~stay at their positions). When~$\ell = 2$ and the two signatures are marked with~$\up{\phantom{1}}/\down{\phantom{1}}$ and~$\uptilde{\phantom{1}}/\downtilde{\phantom{1}}$ respectively, we have for example \begin{align*} {\color{red} \uptilde{\upw{1}}}{\color{red} \uptilde{\downw{2}}} \,\bar\shuffle\, {\color{darkblue} \downtilde{\downw{2}}}{\color{darkblue} \uptilde{\downw{3}}}{\color{darkblue} \downtilde{\upw{1}}} & = \{
{\color{red} \uptilde{\upw{1}}}{\color{red} \uptilde{\downw{2}}}{\color{darkblue} \downtilde{\downw{4}}\uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{3}}},
{\color{red} \uptilde{\upw{1}}}{\color{darkblue} \downtilde{\downw{4}}}{\color{red} \uptilde{\downw{2}}}{\color{darkblue} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{3}}},
{\color{red} \uptilde{\upw{1}}}{\color{darkblue} \downtilde{\downw{4}}\uptilde{\downw{5}}}{\color{red} \uptilde{\downw{2}}}{\color{darkblue} \downtilde{\upw{3}}},
{\color{red} \uptilde{\upw{1}}}{\color{darkblue} \downtilde{\downw{4}}\uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{3}}}{\color{red} \uptilde{\downw{2}}},
{\color{darkblue} \downtilde{\downw{4}}}{\color{red} \uptilde{\upw{1}}}{\color{red} \uptilde{\downw{2}}}{\color{darkblue} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{3}}},
{\color{darkblue} \downtilde{\downw{4}}}{\color{red} \uptilde{\upw{1}}}{\color{darkblue} \uptilde{\downw{5}}}{\color{red} \uptilde{\downw{2}}}{\color{darkblue} \downtilde{\upw{3}}},
{\color{darkblue} \downtilde{\downw{4}}}{\color{red} \uptilde{\upw{1}}}{\color{darkblue} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{3}}}{\color{red} \uptilde{\downw{2}}},
{\color{darkblue} \downtilde{\downw{4}}\uptilde{\downw{5}}}{\color{red} \uptilde{\upw{1}}}{\color{red} \uptilde{\downw{2}}}{\color{darkblue} \downtilde{\upw{3}}},
{\color{darkblue} \downtilde{\downw{4}}\uptilde{\downw{5}}}{\color{red} \uptilde{\upw{1}}}{\color{darkblue} \downtilde{\upw{3}}}{\color{red} \uptilde{\downw{2}}},
{\color{darkblue} \downtilde{\downw{4}}\uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{3}}}{\color{red} \uptilde{\upw{1}}}{\color{red} \uptilde{\downw{2}}} \}, \\ {\color{red} \uptilde{\upw{1}}}{\color{red} \uptilde{\downw{2}}} \star {\color{darkblue} \downtilde{\downw{2}}}{\color{darkblue} \uptilde{\downw{3}}}{\color{darkblue} \downtilde{\upw{1}}} & = \{
{\color{red} \uptilde{\upw{1}}}{\color{red} \uptilde{\downw{2}}}{\color{darkblue} \downtilde{\downw{4}}}{\color{darkblue} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{3}}},
{\color{red} \uptilde{\upw{1}}}{\color{red} \uptilde{\downw{3}}}{\color{darkblue} \downtilde{\downw{4}}}{\color{darkblue} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{2}}},
{\color{red} \uptilde{\upw{1}}}{\color{red} \uptilde{\downw{4}}}{\color{darkblue} \downtilde{\downw{3}}}{\color{darkblue} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{2}}},
{\color{red} \uptilde{\upw{1}}}{\color{red} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\downw{3}}}{\color{darkblue} \uptilde{\downw{4}}}{\color{darkblue} \downtilde{\upw{2}}},
{\color{red} \uptilde{\upw{2}}}{\color{red} \uptilde{\downw{3}}}{\color{darkblue} \downtilde{\downw{4}}}{\color{darkblue} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{1}}},
{\color{red} \uptilde{\upw{2}}}{\color{red} \uptilde{\downw{4}}}{\color{darkblue} \downtilde{\downw{3}}}{\color{darkblue} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{1}}},
{\color{red} \uptilde{\upw{2}}}{\color{red} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\downw{3}}}{\color{darkblue} \uptilde{\downw{4}}}{\color{darkblue} \downtilde{\upw{1}}},
{\color{red} \uptilde{\upw{3}}}{\color{red} \uptilde{\downw{4}}}{\color{darkblue} \downtilde{\downw{2}}}{\color{darkblue} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\upw{1}}},
{\color{red} \uptilde{\upw{3}}}{\color{red} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\downw{2}}}{\color{darkblue} \uptilde{\downw{4}}}{\color{darkblue} \downtilde{\upw{1}}},
{\color{red} \uptilde{\upw{4}}}{\color{red} \uptilde{\downw{5}}}{\color{darkblue} \downtilde{\downw{2}}}{\color{darkblue} \uptilde{\downw{3}}}{\color{darkblue} \downtilde{\upw{1}}} \}. \end{align*}
We denote by~$\mathsf{FQSym}_{\pm^\ell}$ the Hopf algebra with basis~$(F_\tau)_{\tau \in \mathfrak{S}_{\pm^\ell}}$ indexed by $\ell$-signed permutations and whose product and coproduct are defined by \[ \mathbb{F}_\tau \cdot \mathbb{F}_{\tau'} = \sum_{\sigma \in \tau \,\bar\shuffle\, \tau'} \mathbb{F}_\sigma \qquad\text{and}\qquad \triangle \mathbb{F}_\sigma = \sum_{\sigma \in \tau \star \tau'} \mathbb{F}_\tau \otimes \mathbb{F}_{\tau'}. \]
\begin{remark}[Cambrian algebra \textit{vs.}~ $G$-colored binary tree algebra] Checking that these product and coproduct produce a Hopf algebra is standard. It even extends to a Hopf algebra~$\mathsf{FQSym}_G$ on $G$-colored permutations for an arbitrary semigroup~$G$, see \textit{e.g.}~ \cite{NovelliThibon-coloredHopfAlgebras, BaumannHohlweg, BergeronHohlweg}. In these papers, the authors use this algebra~$\mathsf{FQSym}_G$ to defined $G$-colored subalgebras from congruence relations on permutations, see in particular~\cite{BergeronHohlweg}. Note that our construction of the Cambrian algebra and of the tuple Cambrian algebra really differs from the construction of~\cite{BergeronHohlweg} as our congruence relations depend on the signs, while their congruences do not. \end{remark}
We denote by~$\mathsf{Camb}_\ell$ the vector subspace of~$\mathsf{FQSym}_{\pm^\ell}$ generated by the elements \[ \mathbb{P}_{\tuple} \eqdef \sum_{\substack{\tau \in \mathfrak{S}_{\pm^\ell} \\ \mathbf{P}_\ell(\tau) = \tuple}} \mathbb{F}_\tau = \sum_{\tau \in \mathcal{L}\big(\bigcup\limits_{k \in [\ell]} \tuple_{[k]}\big)} \mathbb{F}_\tau, \] for all Cambrian $\ell$-tuples~$\tuple$. \renewcommand{\uptilde}[1]{\accentset{
\scalebox{.55}{$\sim$}}{#1}} \renewcommand{\downtilde}[1]{\underaccent{\,\scalebox{.55}{$\sim$}}{#1}} For example, for the Cambrian tuple of \fref{fig:CambrianPair}\,(left), we have \[ \mathbb{P}_{\left[ \raisebox{-.45cm}{\includegraphics{Tex}}, \raisebox{-.45cm}{\includegraphics{TexPair}} \right]} = \mathbb{F}_{\uptilde{\downw{2}} \uptilde{\downw{1}} \uptilde{\upw{7}} \downtilde{\downw{5}} \downtilde{\upw{3}} \uptilde{\downw{4}} \downtilde{\upw{6}}} + \mathbb{F}_{\uptilde{\downw{2}} \uptilde{\upw{7}} \uptilde{\downw{1}} \downtilde{\downw{5}} \downtilde{\upw{3}} \uptilde{\downw{4}} \downtilde{\upw{6}}} + \mathbb{F}_{\uptilde{\downw{2}} \uptilde{\upw{7}} \downtilde{\downw{5}} \uptilde{\downw{1}} \downtilde{\upw{3}} \uptilde{\downw{4}} \downtilde{\upw{6}}} + \mathbb{F}_{\uptilde{\upw{7}} \uptilde{\downw{2}} \uptilde{\downw{1}} \downtilde{\downw{5}} \downtilde{\upw{3}} \uptilde{\downw{4}} \downtilde{\upw{6}}} + \mathbb{F}_{\uptilde{\upw{7}} \uptilde{\downw{2}} \downtilde{\downw{5}} \uptilde{\downw{1}} \downtilde{\upw{3}} \uptilde{\downw{4}} \downtilde{\upw{6}}} + \mathbb{F}_{\uptilde{\upw{7}} \downtilde{\downw{5}} \uptilde{\downw{2}} \uptilde{\downw{1}} \downtilde{\upw{3}} \uptilde{\downw{4}} \downtilde{\upw{6}}}. \]
\begin{theorem} \label{thm:cambTupleSubalgebra} $\mathsf{Camb}_\ell$ is a Hopf subalgebra of~$\mathsf{FQSym}_{\pm^\ell}$. \end{theorem}
As for the Cambrian algebra, the product and coproduct of~$\mathbb{P}$-basis elements of the $\ell$-Cambrian algebra~$\mathsf{Camb}_\ell$ can be described directly in terms of combinatorial operations on Cambrian \mbox{$\ell$-tuples}.
\para{Product} The product in the $\ell$-Cambrian algebra~$\mathsf{Camb}_\ell$ can be described in terms of intervals in $\ell$-Cambrian lattices. We denote by~$\mathcal{E}\mathcal{E}' \eqdef [\mathcal{E}_{[1]}\mathcal{E}'_{[1]}, \dots, \mathcal{E}_{[\ell]}\mathcal{E}'_{[\ell]} ]$ the componentwise concatenation of two signature $\ell$-tuples~$\mathcal{E},\mathcal{E}'$. Similarly, for two Cambrian $\ell$-tuples~$\tuple, \tuple'$, we define \[ \raisebox{-6pt}{$\tuple$} \nearrow \raisebox{4pt}{$\bar\tuple'$} \eqdef \bigg[ \raisebox{-6pt}{$\tuple_{[1]}$} \nearrow \raisebox{4pt}{$\bar\tuple'_{[1]}$}, \dots, \raisebox{-6pt}{$\tuple_{[\ell]}$} \nearrow \raisebox{4pt}{$\bar\tuple'_{[\ell]}$} \bigg] \qquad\text{and}\qquad \raisebox{4pt}{$\tuple$} \nwarrow \raisebox{-6pt}{$\bar\tuple'$} \eqdef \bigg[ \raisebox{4pt}{$\tuple_{[1]}$} \nwarrow \raisebox{-6pt}{$\bar\tuple'_{[1]}$}, \dots, \raisebox{4pt}{$\tuple_{[\ell]}$} \nwarrow \raisebox{-6pt}{$\bar\tuple'_{[\ell]}$} \bigg]. \]
\begin{proposition} For any two Cambrian $\ell$-tuples~$\tuple$ and~$\tuple'$, the product~$\mathbb{P}_{\tuple} \cdot \mathbb{P}_{\tuple'}$ is given by \[ \mathbb{P}_{\tuple} \cdot \mathbb{P}_{\tuple'} = \sum_{\tuple[S]} \mathbb{P}_{\tuple[S]}, \] where~$\tuple[S]$ runs over the interval between~$\raisebox{-6pt}{$\tuple$} \nearrow \raisebox{4pt}{$\bar\tuple'$}$ and~$\raisebox{4pt}{$\tuple$} \nwarrow \raisebox{-6pt}{$\bar\tuple'$}$ in the~$\mathcal{E}(\tuple)\mathcal{E}(\tuple')$-Cambrian lattice. \end{proposition}
\begin{remark}[Multiplicative bases] \enlargethispage{.2cm} Similar to the multiplicative bases defined in Section~\ref{sec:multiplicativeBases} and Remark~\ref{rem:multiplicativeBasesBaxter}, the bases~$\mathbb{E}^{\tuple}$ and~$\mathbb{H}^{\tuple}$ defined by \[ \mathbb{E}^{\tuple} \eqdef \sum_{\tuple \le \tuple'} \mathbb{P}_{\tuple'} \qquad\text{and}\qquad \mathbb{H}^{\tuple} \eqdef \sum_{\tuple' \le \tuple} \mathbb{P}_{\tuple'} \] are multiplicative since \[ \mathbb{E}^{\tuple} \cdot \mathbb{E}^{\tuple'} = \mathbb{E}^{\raisebox{-5pt}{\scriptsize$\tuple$}\nearrow \raisebox{4pt}{\scriptsize$\bar \tuple'$}} \qquad\text{and}\qquad \mathbb{H}^{\tuple} \cdot \mathbb{H}^{\tuple'} = \mathbb{H}^{\raisebox{4pt}{\scriptsize$\tuple$}\nwarrow \raisebox{-5pt}{\scriptsize$\bar \tuple'$}}. \] The $\mathbb{E}$-indecomposable elements are precisely the Cambrian $\ell$-tuples~$\tuple$ such that all linear extensions of the union~$\bigcup_{k \in [\ell]} \tuple_{[k]}$ are indecomposable. In particular, $\tuple$ is $\mathbb{E}$-indecomposable as soon as one of the~$\tuple_{[k]}$ is $\mathbb{E}$-indecomposable, but this condition is not necessary. The $\mathbb{E}$-indecomposable $\mathcal{E}$-Cambrian tuples form an ideal of the $\mathcal{E}$-Cambrian lattice, but this ideal is not principal. \end{remark}
\para{Coproduct} A \defn{cut}~$\gamma$ of a Cambrian $\ell$-tuple~$\tuple[S]$ is a cut of the union~$\bigcup_{k \in [\ell]} \tuple[S]_{[k]}$. It defines a cut~$\gamma_{[k]}$ on each Cambrian tree~$\tuple[S]_{[k]}$. We denote by \[ A(\tuple[S], \gamma) \eqdef A(\tuple[S]_{[1]}, \gamma_{[1]}) \times \dots \times A(\tuple[S]_{[\ell]}, \gamma_{[\ell]}) \qquad\text{and}\qquad B(\tuple[S], \gamma) \eqdef B(\tuple[S]_{[1]}, \gamma_{[1]}) \times \dots \times B(\tuple[S]_{[\ell]}, \gamma_{[\ell]}). \]
\begin{proposition} For any Cambrian $\ell$-tuple~$\tuple[S]$, the coproduct~$\triangle \mathbb{P}_{\tuple[S]}$ is given~by \[ \triangle \mathbb{P}_{\tuple[S]} = \sum_{\gamma} \bigg( \sum_{\tuple[B] \in B(\tuple[S], \gamma)} \mathbb{P}_{\tuple[B]} \bigg) \otimes \bigg( \sum_{\tuple[A] \in A(\tuple[S], \gamma)} \mathbb{P}_{\tuple[A]} \bigg), \] where~$\gamma$ runs over all cuts of~$\tuple[S]$. \end{proposition}
\subsection{Dual Cambrian tuple Hopf Algebra} \label{subsec:dualCambrianTupleAlgebra}
We now consider the dual Hopf algebra of~$\mathsf{Camb}_\ell$. Again, the following statement is automatic from Theorem~\ref{thm:cambTupleSubalgebra}.
\begin{theorem} The graded dual~$\mathsf{Camb}_\ell^*$ of the $\ell$-Cambrian algebra is isomorphic to the image of~$\mathsf{FQSym}_{\pm^\ell}^*$ under the canonical projection \[ \pi : \mathbb{C}\langle A \rangle \longrightarrow \mathbb{C}\langle A \rangle / \equiv_\ell, \] where~$\equiv_\ell$ denotes the $\ell$-Cambrian congruence. The dual basis~$\mathbb{Q}_{\tuple}$ of~$\mathbb{P}_{\tuple}$ is expressed as~${\mathbb{Q}_{\tuple} = \pi(\mathbb{G}_\tau)}$, where~$\tau$ is any linear extension of~$\bigcup_{k \in [\ell]} \tuple_{[k]}$. \end{theorem}
We now describe the product and coproduct in~$\mathsf{Camb}_\ell^*$ in terms of combinatorial operations on Cambrian $\ell$-tuples. We use the definitions and notations introduced in Section~\ref{subsec:quotientAlgebra}.
\para{Product} The product in~$\mathsf{Camb}_\ell^*$ can be described using gaps and laminations similarly to Proposition~\ref{prop:productDual}. For two Cambrian trees~$\tree$ and~$\tree'$ and a shuffle~$s$ of the signatures~$\varepsilon(\tree)$ and~$\varepsilon(\tree')$, we still denote by~$\tree \,{}_s\!\backslash \tree'$ the tree described in Section~\ref{subsec:quotientAlgebra}. For two Cambrian $\ell$-tuples~$\tuple$ and~$\tuple'$, with trees of size~$n$ and~$n'$ respectively, and for a shuffle~$s$ of~$[n]$ and~$[n']$, we write \[ \tuple \,{}_s\!\backslash \tuple' \eqdef \big[ \tuple_{[1]} \,{}_s\!\backslash \tuple'_{[1]}, \dots, \tuple_{[\ell]} \,{}_s\!\backslash \tuple'_{[\ell]} \big], \] where we see~$s$ as a shuffle of the signatures~$\varepsilon(\tuple_{[k]})$ and~$\varepsilon(\tuple'_{[k]})$.
\begin{proposition} For any two Cambrian $\ell$-tuples~$\tuple, \tuple'$, the product~$\mathbb{Q}_{\tuple} \cdot \mathbb{Q}_{\tuple'}$ is given by \[ \mathbb{Q}_{\tuple} \cdot \mathbb{Q}_{\tuple'} = \sum_s \mathbb{Q}_{\tuple \,{}_s\!\backslash \tuple'}, \] where~$s$ runs over all shuffles of~$[n]$ and~$[n']$ (where~$n$ and~$n'$ denote the respective sizes of the trees of~$\tuple$ and~$\tuple'$). \end{proposition}
\para{Coproduct} The coproduct in~$\mathsf{Camb}_\ell^*$ can be described combinatorially as in Proposition~\ref{prop:coproductDual}. For a Cambrian $\ell$-tuple~$\tuple[S]$, with trees of size~$n$, and a gap~$\gamma \in \{0,\dots,n\}$, we define \[ L(\tuple[S], \gamma) = \big[ L(\tuple_{[1]}, \gamma), \dots, L(\tuple_{[\ell]}, \gamma) \big] \qquad\text{and}\qquad R(\tuple[S], \gamma) = \big[ R(\tuple_{[1]}, \gamma), \dots, R(\tuple_{[\ell]}, \gamma) \big]. \]
\begin{proposition} For any Cambrian $\ell$-tuple~$\tuple[S]$, the coproduct~$\triangle\mathbb{Q}_{\tuple[S]}$ is given~by \[ \triangle\mathbb{Q}_{\tuple[S]} = \sum_{\gamma} \mathbb{Q}_{L(\tuple[S], \gamma)} \otimes \mathbb{Q}_{R(\tuple[S], \gamma)}, \] where~$\gamma$ runs over all gaps between consecutive positions in~$[n]$ (where~$n$ denotes the size of the trees of~$\tuple$). \end{proposition}
\part{The Schr\"oder-Cambrian Hopf Algebra} \label{part:SchroderCambrian}
\section{Schr\"oder-Cambrian trees}
We already insisted on the fact that the bases of M.~Malvenuto and C.~Reutenauer's algebra on permutations, of J.-L.~Loday and M.~Ronco's algebra on binary trees, and of L.~Solomon's descent algebra correspond to the vertices of the permutahedra, of the associahedra, and of the cubes respectively. In~\cite{Chapoton}, F.~Chapoton generalized these algebras to three Hopf algebras with bases indexed by all faces of the permutahedra, of the associahedra, and of the cubes. To conclude the paper, we show that F.~Chapoton's construction extends as well to the Cambrian setting. We obtain the Schr\"oder-Cambrian Hopf algebra with basis indexed by all faces of all associahedra of C.~Hohlweg and C.~Lange. It is also a good occasion to observe relevant combinatorial properties of Schr\"oder-Cambrian trees, which correspond to the faces of these associahedra.
\subsection{Schr\"oder-Cambrian trees}
The faces of J.-L.~Loday's $n$-dimensional associahedron correspond to \defn{Schr\"oder trees} with $n+1$ leaves, \textit{i.e.}~ trees where each internal node has at least $2$ children. We first recall the Cambrian counterpart of these trees, which correspond to all faces of C.~Hohlweg and C.~Lange's associahedra (see Section~\ref{subsec:geometrySchroderTrees}). These trees were defined in~\cite{LangePilaud} as ``spines'' of dissections of polygons, see Remark~\ref{rem:dissections}.
\begin{definition} Consider a signature~$\varepsilon \in \pm^n$. For~$X \subseteq [n]$, we denote by~$X^+ \eqdef \set{x \in X}{\varepsilon_x = +}$ and~$X^- \eqdef \set{x \in X}{\varepsilon_x = -}$. A \defn{Schr\"oder $\varepsilon$-Cambrian tree} is a directed tree~$\tree$ with vertex set~$\mathrm{V}$ endowed with a vertex labeling~$p : \mathrm{V} \to 2^{[n]} \smallsetminus \varnothing$ such that \begin{enumerate}[(i)] \item the labels of~$\tree$ partition~$[n]$, \textit{i.e.}~ $v \ne w \in \mathrm{V} \Rightarrow p(v) \cap p(w) = \varnothing$ and~$\bigcup_{v \in \mathrm{V}} p(v) = [n]$; \item each vertex~$v \in \mathrm{V}$ has one incoming (resp.~outgoing) subtree~$\tree_{v,I}$ for each interval~$I$ of~${[n] \smallsetminus p(v)^-}$ (resp.~of~$[n] \smallsetminus p(v)^+$) and all labels of~$\tree_{v,I}$ are subsets of~$I$. \end{enumerate} For~$p \ge 0$ and~$\varepsilon \in \pm^n$, we denote by~$\mathrm{SchrCamb}^{\ge p}(\varepsilon)$ the set of Schr\"oder $\varepsilon$-Cambrian trees with at most~$n-p$ internal nodes, and we define~$\mathrm{SchrCamb}^{\ge p}(n) \eqdef \bigsqcup_{\varepsilon \in \pm^n} \mathrm{SchrCamb}^{\ge p}(\varepsilon)$ and $\mathrm{SchrCamb}^{\ge p} \eqdef \bigsqcup_{n \in \mathbb{N}} \mathrm{SchrCamb}^{\ge p}(n)$. They correspond to associahedron faces of dimension at least~$p$. Finally, we simply omit the superscript~$^{\ge p}$ in the previous notations to denote Schr\"oder-Cambrian trees with arbitrary many internal nodes. Note that this defines a filtration \[ \mathrm{SchrCamb} = \bigcup_{p \ge 1} \mathrm{SchrCamb}^{\ge p} \qquad\text{with}\qquad \mathrm{SchrCamb}^{\ge 0} \supset \mathrm{SchrCamb}^{\ge 1} \supset \dots \] \end{definition}
\begin{definition} A \defn{$k$-leveled Schr\"oder $\varepsilon$-Cambrian tree} is a directed tree with vertex set~$\mathrm{V}$ endowed with two labelings~$p : \mathrm{V} \to 2^{[n]} \smallsetminus \varnothing$ and~$q : \mathrm{V} \to [k]$ which respectively define a Schr\"oder $\varepsilon$-Cambrian tree and an increasing tree (meaning that $q$ is surjective and~$v \to w$ in~$\tree$ implies that~$q(v) < q(w)$). \end{definition}
A Schr\"oder-Cambrian tree and a $3$-leveled Schr\"oder-Cambrian tree are represented in \fref{fig:leveledSchroderCambrianTree}. Note that each level of a $k$-leveled Schr\"oder $\varepsilon$-Cambrian tree may contain more than one node.
\begin{figure}
\caption{A Schr\"oder-Cambrian tree (left), an increasing tree (middle), and a $3$-leveled Schr\"oder-Cambrian tree (right).}
\label{fig:leveledSchroderCambrianTree}
\end{figure}
\begin{remark}[Spines of dissections] \label{rem:dissections} \enlargethispage{-.4cm} Exactly as $\varepsilon$-Cambrian trees correspond to triangulations of the $(n+2)$-gon~$\polygon$ (see Remark~\ref{rem:triangulation}), Schr\"oder $\varepsilon$-Cambrian trees correspond to all dissections of~$\polygon$. See \fref{fig:dissection} and refer to~\cite{LangePilaud} for details.
\begin{figure}
\caption{Schr\"oder-Cambrian trees (left) and dissections (right) are dual to each other (middle).}
\label{fig:dissection}
\end{figure} \end{remark}
Remark~\ref{rem:dissections} immediately implies that the number of Schr\"oder $\varepsilon$-Cambrian trees with $k$ nodes is the number of $(n-k)$-dimensional faces of the associahedron, and is therefore independent of the signature~$\varepsilon$. An alternative proof based on generating trees is mentioned in Remark~\ref{rem:patternAvoidanceSchroder}.
\begin{proposition} \label{prop:SchroderCambrianNumbers} For any signature~$\varepsilon \in \pm^n$, the number of Schr\"oder $\varepsilon$-Cambrian trees with $k$ internal nodes is \[ \frac{1}{k+1}\binom{n+2+k}{k+1}\binom{n-1}{k+1}, \] see~\href{https://oeis.org/A033282}{\cite[A033282]{OEIS}}. \end{proposition}
\subsection{Schr\"oder-Cambrian correspondence}
We now define an analogue of the Cambrian correspondence and Cambrian $\mathbf{P}$-symbol, which will map the faces of the permutahedron to the faces of C.~Hohlweg and C.~Lange's associahedra. Recall first that the $(n-k)$-dimensional faces of the $n$-dimensional permutahedron correspond to \defn{surjections} from~$[n]$ to~$[k]$, or equivalently to \defn{ordered partitions} of~$[n]$ into $k$ parts. See \fref{fig:permutahedra}. We denote (abusively) by~$\pi^{-1}$ the ordered partition corresponding to a surjection~$\pi : [n] \to [k]$, \textit{i.e.}~ given by~$\pi^{-1} \eqdef \pi^{-1}(\{1\}) \,|\, \pi^{-1}(\{2\}) \,|\, \cdots \,|\, \pi^{-1}(\{k\})$. Conversely, we denote (abusively) by~$\lambda^{-1}$ the surjection corresponding to an ordered partition~$\lambda = \lambda_1 | \lambda_2 | \cdots | \lambda_k$, \textit{i.e.}~ such that each~$i$ belongs to the part~$\lambda_{\lambda^{-1}(i)}$. We represent graphically a surjection~$\pi : [n] \to [k]$ by the~$(k \times n)$-table with a dot in row~$\pi(j)$ in each column~$j$. Therefore, we represent an ordered partition~$\lambda \eqdef \lambda_1 | \cdots | \lambda_k$ of~$[n]$ by the~$(k \times n)$-table with a dot in row~$i$ and column~$j$ for each~$j \in \lambda_i$. See \fref{fig:insertionAlgorithmSchroder}\,(left). In this paper, we work with ordered partitions rather than surjections to match better the presentation of the previous sections: the permutations of~$[n]$ used in the previous sections have to be thought of as ordered partitions of~$[n]$ into~$n$ parts. We denote by~$\mathfrak{P}_n^{\ge p}$ the set of ordered partitions of~$[n]$ into at most~$n-p$ parts, and we set~$\mathfrak{P}^{\ge p} \eqdef \bigsqcup_{n \in \mathbb{N}} \mathfrak{P}_n^{\ge p}$. It correspond to permutahedron faces of dimension at least~$p$. As previously, we omit the superscript~$^{\ge p}$ in these notations to forget this dimension restriction.
\begin{figure}
\caption{The $3$-dimensional permutahedron~$\Perm[\protect{[4]}]$. Its $(4-k)$-dimensional faces correspond equivalently to the surjections from~$[4]$ to~$[k]$ (left), or to the ordered partitions of~$[4]$ into~$k$ parts (right). Vertices are in blue and facets in red. The reader is invited to label the edges accordingly.}
\label{fig:permutahedra}
\end{figure}
A \defn{signed ordered partition} is an ordered partition table where each dot receives a~$+$ or~$-$ sign. For a signature~$\varepsilon \in \pm^n$, we denote by~$\mathfrak{P}_\varepsilon^{\ge p}$ the set of ordered partitions of~$[n]$ into at most~$n-p$ parts signed by~$\varepsilon$, and we set \[ \mathfrak{P}_\pm^{\ge p} \eqdef \bigsqcup_{n \in \mathbb{N}, \varepsilon \in \pm^n} \mathfrak{P}_\varepsilon^{\ge p}. \] We omit again the superscript~$^{\ge p}$ in the previous notations to denote signed ordered partitions with arbitrarily many parts. This gives again a filtration \[ \mathfrak{P}_\pm = \bigcup_{p \ge 1} \mathfrak{P}_\pm^{\ge p} \qquad\text{with}\qquad \mathfrak{P}_\pm^{\ge 0} \supset \mathfrak{P}_\pm^{\ge 1} \supset \dots \]
Given such a signed ordered partition~$\lambda$, we construct a leveled Schr\"oder-Cambrian tree~$\Theta^\star(\lambda)$ as follows. As a preprocessing, we represent the table of~$\lambda$, we draw a vertical wall below the negative dots and above the positive dots, and we connect into nodes the dots at the same level which are not separated by a wall. Note that we might obtain several nodes per level. We then sweep the table from bottom to top as follows. The procedure starts with an incoming strand in between any two consecutive negative values. At each level, each node~$v$ (connected set of dots) gathers all strands in the region below and visible from~$v$ (\textit{i.e.}~ not hidden by a vertical wall) and produces one strand in each region above and visible from~$v$. The procedure finished with an outgoing strand in between any two consecutive positive values. See \fref{fig:insertionAlgorithmSchroder}.
\begin{figure}
\caption{The insertion algorithm on the signed ordered partition~$\down{125}\up{7} | \up{3}\down{4} | \up{6}$.}
\label{fig:insertionAlgorithmSchroder}
\end{figure}
\begin{proposition}[\cite{LangePilaud}] The map~$\Theta^\star$ is a bijection from signed ordered partitions to leveled Schr\"oder-Cambrian trees. \end{proposition}
We define the $\mathbf{P}^\star$-symbol of a signed ordered partition~$\lambda$ as the Schr\"oder-Cambrian tree~$\mathbf{P}^\star(\lambda)$ defined by~$\Theta^\star(\lambda)$. Note that an ordered partition of~$[n]$ into $k$ parts is sent to a Schr\"oder-Cambrian tree with at least~$k$ internal nodes, since some levels can be split into several nodes. In other words, the fibers of the Schr\"oder-Cambrian $\mathbf{P}^\star$-symbol respect the filtrations~$(\mathfrak{P}_\pm^{\ge p})_{p \in \mathbb{N}}$ and~$(\mathrm{SchrCamb}^{\ge p})_{p \in \mathbb{N}}$, in the sense that \[ (\mathbf{P}^\star)^{-1} \big( \mathrm{SchrCamb}^{\ge p} \big) \subseteq \mathfrak{P}_\pm^{\ge p}. \]
The following characterization of the fibers of the map~$\mathbf{P}^\star$ is immediate from the description of the Schr\"oder-Cambrian correspondence. For a Schr\"oder-Cambrian tree~$\tree$, we write~$i \to j$ in~$\tree$ if the node of~$\tree$ containing~$i$ is below the node of~$\tree$ containing~$j$, and~$i \sim j$ in~$\tree$ if~$i$ and~$j$ belong to the same node of~$\tree$. We say that~$i$ and~$j$ are \defn{incomparable} in~$\tree$ when~$i \not\to j$, $j \not\to i$, and~$i \not\sim j$.
\begin{proposition} \label{prop:mergeLinearExtensions} For any Schr\"oder $\varepsilon$-Cambrian tree~$\tree$ and signed ordered partition~$\lambda \in \mathfrak{P}_\varepsilon$, we have~$\mathbf{P}^\star(\lambda) = \tree$ if and only if~$i \sim j$ in~$\tree$ implies $\lambda^{-1}(i) = \lambda^{-1}(j)$ and $i \to j$ in~$\tree$ implies $\lambda^{-1}(i) < \lambda^{-1}(j)$. In other words, $\lambda$ is obtained from a linear extension of~$\tree$ by merging parts which label incomparable vertices of~$\tree$. \end{proposition}
\begin{example} When~$\varepsilon = ({+})^n$, the Schr\"oder-Cambrian tree~$\mathbf{P}^\star(\lambda)$ is the increasing tree of~$\lambda^{-1}$. Here, the \defn{increasing tree}~$\mathrm{IT}(\pi)$ of a surjection~$\pi = \pi^{(1)} 1 \pi^{(2)} 1 \dots 1 \pi^{(p)}$ is defined inductively by grafting the increasing trees~$\mathrm{IT}(\pi^{(1)}), \dots, \mathrm{IT}(\pi^{(p)})$ from left to right on a bottom root labeled by~$1$. Similarly, when~$\varepsilon = ({-})^n$, the Schr\"oder-Cambrian tree~$\mathbf{P}^\star(\lambda)$ is the decreasing tree of~$\lambda^{-1}$. Here, the \defn{decreasing tree}~$\mathrm{DT}(\pi)$ of a surjection~$\pi = \pi^{(1)} k \pi^{(2)} k \dots k \pi^{(p)}$ from~$[n]$ to~$[k]$ is defined inductively by grafting the decreasing trees~$\mathrm{DT}(\pi^{(1)}), \dots, \mathrm{DT}(\pi^{(p)})$ from left to right on a top root labeled by~$k$. See \fref{fig:constantSignsSchroder}.
\begin{figure}
\caption{The insertion procedure produces increasing Schr\"oder trees when the signature is constant positive (left) and decreasing Schr\"oder trees when the signature is constant negative (right).}
\label{fig:constantSignsSchroder}
\end{figure} \end{example}
\begin{remark}[Schr\"oder-Cambrian correspondence on dissections] Similar to Remark~\ref{rem:CambrianCorrespondenceTriangulations}, we can describe the map~$\mathbf{P}^\star$ on the dissections of the polygon~$\polygon$. Namely the dissection dual to the Schr\"oder-Cambrian tree~$\mathbf{P}^\star(\lambda)$ is the union of the paths~$\pi_0, \dots, \pi_n$ where~$\pi_i$ is the path between vertices~$0$ and~$n+1$ of~$\polygon$ passing through the vertices in the symmetric difference~$\varepsilon^{-1}(-) \,\triangle\, \big( \bigcup_{j \in [i]} \lambda_j \big)$. \end{remark}
\subsection{Schr\"oder-Cambrian congruence}
As for the Cambrian congruence, the fibers of the Schr\"oder-Cambrian $\mathbf{P}^\star$-symbol define a congruence relation on signed ordered partitions, which can be expressed by rewriting rules.
\begin{definition} For a signature~$\varepsilon \in \pm^n$, the Schr\"oder $\varepsilon$-Cambrian congruence is the equivalence relation on~$\mathfrak{P}_\varepsilon$ defined as the transitive closure of the rewriting rules \[ U | \b{a} | \b{c} | V \equiv^\star_\varepsilon U | \b{a}\b{c} | V \equiv^\star_\varepsilon U | \b{c} | \b{a} | V, \] where $\b{a}, \b{c}$ are parts while~$U, V$ are sequences of parts of~$[n]$, and there exists~$\b{a} < b < \b{c}$ such that~$\varepsilon_b = +$ and~$b \in \bigcup U$, or~$\varepsilon_b = -$ and~$b \in \bigcup V$. The Schr\"oder-Cambrian congruence is the equivalence relation~$\equiv^\star$ on~$\mathfrak{P}_{\pm}$ obtained as the union of all Schr\"oder $\varepsilon$-Cambrian congruences. \end{definition}
For example, $\down{125}\up{7} | \up{3}\down{4} | \up{6} \equiv^\star \down{12} | \down{5}\up{7} | \up{3}\down{4} | \up{6} \equiv^\star \down{5}\up{7} | \down{12} | \up{3}\down{4} | \up{6} \not\equiv^\star \down{5}\up{7} | \up{3}\down{4} | \down{12} | \up{6}$.
\begin{proposition} Two signed ordered partitions~$\lambda, \lambda' \in \mathfrak{P}_{\pm}$ are Schr\"oder-Cambrian congruent if and only if they have the same $\mathbf{P}^\star$-symbol: \[ \lambda \equiv^\star \lambda' \iff \mathbf{P}^\star(\lambda) = \mathbf{P}^\star(\lambda'). \] \end{proposition}
\begin{proof} It boils down to observe that two consecutive parts~$\b{a}$ and~$\b{c}$ of an ordered partition~$U | \b{a} | \b{c} | V$ in a fiber~$(\mathbf{P}^\star)^{-1}(\tree)$ can be merged to~$U | \b{a}\b{c} | V$ and even exchanged to~$U | \b{c} | \b{a} | V$ while staying in~$(\mathbf{P}^\star)^{-1}(\tree)$ precisely when they belong to distinct subtrees of a node of~$\tree$. They are therefore separated by the vertical wall below (resp.~above) a value~$b$ with~$\b{a} < b < \b{c}$ and such that~$\varepsilon_b = -$ and~$b \in V$ (resp.~$\varepsilon_b = +$ and~$b \in U$). \end{proof}
\subsection{Weak order on ordered partitions and Schr\"oder-Cambrian lattices}
In order to define the Schr\"oder counterpart of the Cambrian lattice, we first need to extend the weak order on permutations to all ordered partitions. This was done by D.~Krob, M.~Latapy, J.-C.~Novelli, H.~D.~Phan and~S.~Schwer in~\cite{KrobLatapyNovelliPhanSchwer}. See also~\cite{BoulierHivertKrobNovelli} for representation theoretic properties of this order and \cite{PalaciosRonco} for an extension to all finite Coxeter systems.
\begin{definition} The \defn{coinversion map}~${\coinv(\lambda) : \binom{[n]}{2} \to \{-1,0,1\}}$ of an ordered partition~$\lambda \in \mathfrak{P}_n$ is the map defined for~$i < j$ by \[ \coinv(\lambda)(i,j) = \begin{cases} -1 & \text{if } \lambda^{-1}(i) < \lambda^{-1}(j), \\ \phantom{-}0 & \text{if } \lambda^{-1}(i) = \lambda^{-1}(j), \\ \phantom{-}1 & \text{if } \lambda^{-1}(i) > \lambda^{-1}(j). \end{cases} \] It is also called the \defn{inversion map} of the surjection~$\lambda^{-1}$. \end{definition}
\begin{definition} There are two natural poset structures on~$\mathfrak{P}_n$: \begin{itemize}
\item The \defn{refinement poset}~$\subseteq$ defined by~$\lambda \subseteq \lambda'$ if~$|\coinv(\lambda)(i,j)| \ge |\coinv(\lambda')(i,j)|$ for all~${i < j}$. It is isomorphic to the face lattice of the permutahedron~$\Perm$, and respects the filtration~$(\mathfrak{P}_n^{\ge p})_{p \in [n]}$. \item The \defn{weak order}~$\le$ defined by~$\lambda \le \lambda'$ if~$\coinv(\lambda)(i,j) \le \coinv(\lambda')(i,j)$ for all~$i < j$. \end{itemize} \end{definition}
\enlargethispage{.3cm} These two posets are represented in \fref{fig:latticesOrderedPartitions} for~$n = 3$.
\begin{figure}
\caption{The refinement poset (left) and the weak order (right) on~$\mathfrak{P}_3$.}
\label{fig:latticesOrderedPartitions}
\end{figure}
Note that the restriction of the weak order to~$\mathfrak{S}_n$ is the classical weak order on permutations, which is a lattice. This property was extended to the weak order on~$\mathfrak{P}_n$ in~\cite{KrobLatapyNovelliPhanSchwer}.
\begin{proposition}[\cite{KrobLatapyNovelliPhanSchwer}] The weak order~$<$ on the set of ordered partitions~$\mathfrak{P}_n$ is a lattice. \end{proposition}
In the following statement and throughout the remaining of the paper, we define for~$X, Y \subset \mathbb{N}$ \[ X \ll Y \; \iff \; \max(X) < \min(Y) \; \iff \; x < y \text{ for all~$x \in X$ and~$y \in Y$.} \]
\begin{proposition}[\cite{KrobLatapyNovelliPhanSchwer}] The cover relations of the weak order~$<$ on~$\mathfrak{P}_n$ are given by \begin{gather*} \lambda_1 | \cdots | \lambda_i | \lambda_{i+1} | \cdots | \lambda_k \;\; < \;\; \lambda_1 | \cdots | \lambda_i\lambda_{i+1} | \cdots | \lambda_k \qquad\text{if } \lambda_i \ll \lambda_{i+1}, \\ \lambda_1 | \cdots | \lambda_i\lambda_{i+1} | \cdots | \lambda_k \;\; < \;\; \lambda_1 | \cdots | \lambda_i | \lambda_{i+1} | \cdots | \lambda_k \qquad\text{if } \lambda_{i+1} \ll \lambda_i. \end{gather*} \end{proposition}
We now extend the Cambrian lattice on Cambrian trees to a lattice on all Schr\"oder-Cambrian trees. For the constant signature~$\varepsilon = (-)^n$, the order considered below was already defined by P.~Palacios and M.~Ronco in~\cite{PalaciosRonco}, although its lattice structure (Proposition~\ref{prop:SchroderCambrianLattice}) was not discussed in there.
\begin{definition} Consider a Schr\"oder $\varepsilon$-Cambrian tree~$\tree$, and an edge~$e=\{v,w\}$ of~$\tree$. We denote by~$\tree/e$ the tree obtained by contracting~$e$ in~$\tree$. It is again a Schr\"oder $\varepsilon$-Cambrian tree. We say that the contraction is \defn{increasing} if~$p(u) \ll p(v)$ and \defn{decreasing} if~$p(v) \ll p(u)$. Otherwise, we say that the contraction is \defn{non-monotone}. \end{definition}
\begin{definition} There are two natural poset structures on~$\mathrm{SchrCamb}_\varepsilon$: \begin{itemize} \item The \defn{contraction poset}~$\subseteq$ defined as the transitive closure of the relation~$\tree \subseteq \tree/e$ for any Schr\"oder $\varepsilon$-Cambrian tree~$\tree$ and edge~$e \in \tree$. It is isomorphic to the face lattice of the associahedron~$\Asso$, and respects the filtration~$(\mathrm{SchrCamb}_\varepsilon^{\ge p})_{p \in [n]}$. \item The \defn{Schr\"oder $\varepsilon$-Cambrian poset}~$<$ defined as the transitive closure of the relation~$\tree < \tree/e$ (resp.~$\tree/e < \tree$) for any Schr\"oder $\varepsilon$-Cambrian tree~$\tree$ and any edge~$e \in \tree$ defining an increasing (resp.~decreasing) contraction. \end{itemize} \end{definition}
These two posets are represented in \fref{fig:SchroderCambrianLattices} for the signature~${+}{-}{-}$. Observe that there are two non-monotone contractions. Note also that the restriction of the Schr\"oder $\varepsilon$-Cambrian poset~$<$ to the $\varepsilon$-Cambrian trees is the $\varepsilon$-Cambrian lattice.
\begin{figure}
\caption{The contraction poset (left) and the Schr\"oder $({+}{-}{-})$-Cambrian poset (right) on Schr\"oder $({+}{-}{-})$-Cambrian trees.}
\label{fig:SchroderCambrianLattices}
\end{figure}
\begin{proposition} The map~$\mathbf{P}^\star$ defines a poset homomorphism from the weak order on~$\mathfrak{P}_\varepsilon$ to the Schr\"oder $\varepsilon$-Cambrian poset on~$\mathrm{SchrCamb}(\varepsilon)$. \end{proposition}
\begin{proof} Let~$\lambda < \lambda'$ be a cover relation in the weak order on~$\mathfrak{P}_\varepsilon$. Assume that~$\lambda'$ is obtained by merging the parts~$\lambda_i \ll \lambda_{i+1}$ of~$\lambda$ (the other case being symmetric). Let~$u$ denote the rightmost node of~$\mathbf{P}^\star(\lambda)$ at level~$i$, and~$v$ the leftmost node of~$\mathbf{P}^\star(\lambda)$ at level~$i+1$. If~$u$ and~$v$ are not comparable, then~$\mathbf{P}^\star(\lambda) = \mathbf{P}^\star(\lambda')$. Otherwise, there is an edge~$u \to v$ in~$\mathbf{P}^\star(\lambda)$ and~$\mathbf{P}^\star(\lambda')$ is obtained by the increasing contraction of~$u \to v$ in~$\mathbf{P}^\star(\lambda)$. \end{proof}
\begin{proposition} \label{prop:SchroderCambrianLattice} For any signature~$\varepsilon \in \pm^n$, the Schr\"oder $\varepsilon$-Cambrian poset on Schr\"oder $\varepsilon$-Cambrian trees is a lattice quotient of the weak order on ordered partitions of~$[n]$. \end{proposition}
This proposition is proved by the following two lemmas, similar to N.~Reading's approach~\cite{Reading-CambrianLattices}.
\begin{lemma} The Schr\"oder $\varepsilon$-Cambrian classes are intervals of the weak order. \end{lemma}
\begin{proof} Let~$\tree$ be a Schr\"oder $\varepsilon$-Cambrian tree, with vertex labeling~$p : \mathrm{V} \to 2^{[n]} \smallsetminus \varnothing$. Consider a linear extension of~$\tree$, \textit{i.e.}~ an ordered partition~$\lambda$ whose parts are the labels of~$\tree$ and such that~$p(v)$ appears before~$p(w)$ for~$v \to w$ in~$\tree$. If~$v$ and~$w$ are two incomparable nodes of~$\tree$, then either~$p(v) \ll p(w)$ or~$p(w) \ll p(v)$ since they are separated by a wall. By successive exchanges, there exists a linear extension~$\lambda_{\min}$ (resp.~$\lambda_{\max}$) of~$\tree$ such that~$p(v)$ appears before (resp.~after) $p(w)$ for any two incomparable nodes~$v$ and~$w$ such that~$p(v) \ll p(w)$. By construction, the $(i,j)$-entries of the coinversion tables of~$\lambda_{\min}$ and~$\lambda_{\max}$ are given for~$i < j$ by \[ \coinv(\lambda_{\min})(i,j) = \begin{cases} 1 & \text{if $j \to i$ in $\tree$,} \\ 0 & \text{if $i \sim j$ in $\tree$,} \\ -1 & \text{otherwise,} \end{cases} \qquad\text{and}\qquad \coinv(\lambda_{\max})(i,j) = \begin{cases} -1 & \text{if $i \to j$ in $\tree$,} \\ 0 & \text{if $i \sim j$ in $\tree$,} \\ 1 & \text{otherwise.} \end{cases} \] It follows that the fiber of~$\tree$ under~$\mathbf{P}^\star$ is the weak order interval~$[\lambda_{\min}, \lambda_{\max}]$. \end{proof}
\begin{lemma} Let~$\lambda$ and~$\lambda'$ be two signed ordered partitions from distinct Schr\"oder $\varepsilon$-Cambrian classes~$C$ and~$C'$. If~$\lambda < \lambda'$ then~$\min(C) < \min(C')$ and~$\max(C) < \max(C')$ (all in weak order). \end{lemma}
\begin{proof} We prove the result for maximums, the proof for the minimums being similar. We first observe that we can assume that $\lambda'$ covers~$\lambda$ in weak order, so that there exists a position~$i$ such that either~$\lambda'_i = \lambda_i \cup \lambda_{i+1}$ and~$\lambda_i \ll \lambda_{i+1}$, or $\lambda_i = \lambda'_i \cup \lambda'_{i+1}$ and~$\lambda'_{i+1} \ll \lambda'_i$. The proof then works by induction on the weak order distance between~$\lambda$ and~$\max(C)$. If~$\lambda = \max(C)$, the result is immediate as~$\max(C) = \lambda < \lambda' \le \max(C')$. Otherwise, consider an ordered partition~$\mu$ in~$C$ covering~$\lambda$ in weak order. There exists a position~$j \ne i$ such that~$\mu_j = \lambda_j \cup \lambda_{j+1}$ and~$\lambda_j \ll \lambda_{j+1}$, or $\lambda_j = \mu_j \cup \mu_{j+1}$ and~$\mu_{j+1} \ll \mu_j$. We now distinguish four cases, according to the relative positions of~$i$ and~$j$: \begin{enumerate}[(1)]
\item If~$|i - j| > 1$, then the local changes from~$\lambda$ to~$\lambda'$ at position~$i$ and from~$\lambda$ to~$\mu$ at position~$j$ are independent. Define~$\mu'$ to be the ordered partition obtained from~$\lambda$ by performing both local changes at~$i$ and at~$j$. We then check that~$\lambda' \equiv^\star \mu'$ since any witness for the equivalence~$\lambda \equiv^\star \mu$ is also a witness for the equivalence~$\lambda' \equiv^\star \mu'$. Moreover,~$\mu < \mu'$. \item Otherwise, the local changes at~$i$ and~$j$ are not independent anymore. We therefore need to treat various cases separately, depending on whether the local changes from~$\lambda$ to~$\lambda'$ and from~$\lambda$ to~$\mu$ are merging or splitting, and on the respective positions of these local changes. In all cases below, $\b{a}, \b{b}, \b{c}$ are parts of~$[n]$ such that~$\b{a} \ll \b{b} \ll \b{c}$, while~$U, V$ are sequences of parts of~$[n]$.
\begin{itemize}
\item If~$\lambda = U | \b{a} | \b{b} | \b{c} | V$, $\lambda' = U | \b{a}\b{b} | \b{c} | V$, and~$\mu = U | \b{a} | \b{b}\b{c} | V$, then define~$\mu' \eqdef U | \b{a}\b{b}\b{c} | V$. Any witness for the Schr\"oder-Cambrian congruence~$\lambda \equiv^\star \mu$ is also a witness for the Schr\"oder-Cambrian congruence~$\lambda' \equiv^\star \mu'$. Moreover, we have~$\mu < \mu'$ since~$\b{a} \ll \b{bc}$. The same arguments yield the same conclusions in the following cases:
\item[--] if~$\lambda = U | \b{a} | \b{b} | \b{c} | V$, $\lambda' = U | \b{a} | \b{b}\b{c} | V$, and~$\mu = U | \b{a}\b{b} | \b{c} | V$, then define~$\mu' \eqdef U | \b{a}\b{b}\b{c} | V$.
\item[--] if~$\lambda = U | \b{a}\b{b}\b{c} | V$, $\lambda' = U | \b{c} | \b{a}\b{b} | V$, and~$\mu = U | \b{b}\b{c} | \b{a} | V$, then define~$\mu' \eqdef U | \b{c} | \b{b} | \b{a} | V$.
\item[--] if~$\lambda = U | \b{a}\b{b}\b{c} | V$, $\lambda' = U | \b{b}\b{c} | \b{a} | V$, and~$\mu = U | \b{c} | \b{a}\b{b} | V$, then define~$\mu' \eqdef U | \b{c} | \b{b} | \b{a} | V$.
\item If~$\lambda = U | \b{a} | \b{b}\b{c} | V$, $\lambda' = U | \b{a}\b{b}\b{c} | V$, and~$\mu = U | \b{a} | \b{c} | \b{b} | V$, then define~$\mu' \eqdef U | \b{c} | \b{a}\b{b} | V$. Any witness for the Schr\"oder-Cambrian congruence~$\lambda \equiv^\star \mu$ is also a witness for the Schr\"oder-Cambrian congruence~$\lambda' \equiv^\star \mu'$. Moreover~$\mu < \mu'$ by comparison of the coinversion tables. The same arguments yield the same conclusions in the case:
\item[--] if~$\lambda = U | \b{a}\b{b} | \b{c} | V$, $\lambda' = U | \b{a}\b{b}\b{c} | V$, and~$\mu = U | \b{b} | \b{a} | \b{c} | V$, then define~$\mu' \eqdef U | \b{b}\b{c} | \b{a} | V$.
\item If~$\lambda = U | \b{a} | \b{b}\b{c} | V$, $\lambda' = U | \b{a} | \b{c} | \b{b} | V$, and~$\mu = U | \b{a}\b{b}\b{c} | V$, then define~$\mu' \eqdef U | \b{c} | \b{a}\b{b} | V$. Let~$d$ be a witness for the Schr\"oder-Cambrian congruence~$\lambda \equiv^\star \mu$, that is, $\b{a} < d < \b{bc}$ and either~$\varepsilon_d = -$ and~$d \in V$, or~$\varepsilon_d = +$ and~$d \in U$. Then~$d$ is also a witness for the Schr\"oder-Cambrian congruences~$\lambda' = U | \b{a} | \b{c} | \b{b} | V \equiv^\star U | \b{a}\b{c} | \b{b} | V \equiv^\star U | \b{c} | \b{a} | \b{b} | V \equiv^\star U | \b{c} | \b{a}\b{b} | V = \mu'$. Moreover, we have~$\mu < \mu'$ since~$\b{ab} \ll \b{c}$. The same arguments yield the same conclusions in the case:
\item[--] if~$\lambda = U | \b{a}\b{b} | \b{c} | V$, $\lambda' = U | \b{b} | \b{a} | \b{c} | V$, and~$\mu = U | \b{a}\b{b}\b{c} | V$, then define~$\mu' \eqdef U | \b{b}\b{c} | \b{a} | V$.
\end{itemize} \end{enumerate} In all cases, we have~$\lambda \equiv^\star \mu < \mu' \equiv^\star \lambda'$. Since~$\mu$ is closer to~$\max(C)$ than~$\lambda$, we obtain that~$\max(C) < \max(C')$ by induction hypothesis. The proof for minimums is identical. \end{proof}
\begin{remark}[Extremal elements and pattern avoidance] \label{rem:patternAvoidanceSchroder} Since the Schr\"oder-Cambrian classes are generated by rewriting rules, their minimal elements are precisely the ordered partitions avoiding the patterns~$\b{c} | \b{a} \text{ -- } \down{b}$ and~$\up{b} \text{ -- } \b{c} | \b{a}$, while their maximal elements are precisely the ordered partitions avoiding the patterns~$\b{a} | \b{c} \text{ -- } \down{b}$ and~$\up{b} \text{ -- } \b{a} | \b{c}$. This enables us to construct a generating tree for these permutations. Similar arguments as in Section~\ref{subsec:CambrianClasses} could thus provide an alternative proof of Proposition~\ref{prop:SchroderCambrianNumbers}. \end{remark}
\subsection{Canopy}
We define the canopy of a Schr\"oder-Cambrian tree using the same observation as for Cambrian trees: in any Sch\"oder-Cambrian tree, the numbers $i$ and~$i+1$ appear either in the same label, or in two comparable labels.
\begin{definition} The \defn{canopy} of a Schr\"oder-Cambrian tree~$\tree$ is the sequence~$\mathbf{can}^\star(\tree) \in \{-,0,+\}$ defined by \[ \mathbf{can}^\star(\tree)_i = \begin{cases} - & \text{if~$i$ appears above~$i+1$ in~$\tree$,} \\ 0 & \text{if~$i$ and~$i+1$ appear in the same label in~$\tree$,} \\ + & \text{if~$i$ is below~$i+1$ in~$\tree$.} \end{cases} \] \end{definition}
For example, the canopy of the Schr\"oder-Cambrian tree of \fref{fig:leveledSchroderCambrianTree}\,(left) is~$0{+}0{-}{+}{-}$. The following statement provides an immediate analogue of Proposition~\ref{prop:commutativeDiagram} in the Schr\"oder-Cambrian setting. We define the \defn{recoil} sequence of an ordered partition~$\lambda \in \mathfrak{P}_n$ as~${\mathbf{rec}^\star(\lambda) \in \{-,0,+\}^{n-1}}$, where~${\mathbf{rec}^\star(\lambda)_i = \coinv(\lambda)(i,i+1)}$.
\begin{proposition} \label{prop:commutativeDiagramSchroder} The maps~$\mathbf{P}^\star$, $\mathbf{can}^\star$, and~$\mathbf{rec}^\star$ define the following commutative diagram of lattice homomorphism \[ \begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=1.5em,column sep=5em,minimum width=2em]
{
\mathfrak{P}_\varepsilon & & \{-,0,+\}^{n-1} \\
& \mathrm{SchrCamb}(\varepsilon) & \\
};
\path[->>]
(m-1-1) edge node [above] {$\mathbf{rec}^\star$} (m-1-3)
edge node [below] {$\mathbf{P}^\star\quad$} (m-2-2.west)
(m-2-2.east) edge node [below] {$\quad\mathbf{can}^\star$} (m-1-3); \end{tikzpicture} \] \end{proposition}
\fref{fig:latticesQua}\,(left) illustrates this proposition for the signature~${+}{-}{-}$.
\begin{figure}\label{fig:latticesQua}
\end{figure}
\subsection{Geometric realizations} \label{subsec:geometrySchroderTrees}
\enlargethispage{.1cm} We close this section with the geometric motivation of Schr\"oder-Cambrian trees. More details can be found in~\cite{LangePilaud}. We still denote by~$e_1, \dots, e_n$ the canonical basis of~$\mathbb{R}^n$ and by~$\mathbb{H}$ the hyperplane of~$\mathbb{R}^n$ orthogonal to~$\sum e_i$. We define the \defn{incidence cone}~$\mathrm{C}(\tree)$ and the \defn{braid cone}~$\mathrm{C}^\diamond(\tree)$ of a directed tree~$\tree$ with vertices labeled by subsets of~$[n]$ as \begin{align*} \mathrm{C}(\tree) & \eqdef \cone\set{e_i-e_j}{\text{for all } i \to j \text{ or } i \sim j \text{ in } \tree} \quad\text{and} \\ \mathrm{C}^\diamond(\tree) & \eqdef \set{\b{x} \in \mathbb{H}}{x_i \le x_j \text{ for all } i \to j \text{ or } i \sim j \text{ in } \tree}. \end{align*} These two cones lie in the space~$\mathbb{H}$ and are polar to each other. Note that if~$\tree$ has~$k$ nodes, then~$\dim(\mathrm{C}^\diamond(\tree)) = k-1$. For an ordered partition~${\lambda \in \mathfrak{P}_n}$, we denote by~$\mathrm{C}(\lambda)$ and~$\mathrm{C}^\diamond(\lambda)$ the incidence and braid cone of the chain~$\lambda_1 \to \dots \to \lambda_n$. Finally, for a vector~$\chi \in \{-, 0, +\}^{n-1}$, we denote by~$\mathrm{C}(\tau)$ and~$\mathrm{C}^\diamond(\tau)$ the incidence and braid cone of the oriented path~$1 - \dots - n$, where~$i \to i+1$ if~$\chi_i = +$, $i \leftarrow i+1$ if~$\chi_i = -$, and $i$ and~$i+1$ are merged to the same~node~if~$\chi_i = 0$.
As explained in Section~\ref{subsec:geometricRealizations}, the collections of cones \[ \set{\mathrm{C}^\diamond(\lambda)}{\lambda \in \mathfrak{P}_n}, \qquad \set{\mathrm{C}^\diamond(\tree)}{\tree \in \mathrm{SchrCamb}_\varepsilon}, \qquad\text{and}\qquad \set{\mathrm{C}^\diamond(\chi)}{\chi \in \{-,0,+\}^{n-1}} \] form complete simplicial fans, which are the normal fans of the classical permutahedron~$\Perm$, of C.~Hohlweg and C.~Lange's associahedron~$\Asso$, and of the parallelepiped~$\Para$ respectively. See Figures~\ref{fig:latticesQua}\,(right) and~\ref{fig:permutahedraAssociahedraCubes} for $2$- and $3$-dimensional examples of these polytopes.
The incidence and braid cones also characterize the maps~$\mathbf{P}^\star$, $\mathbf{can}^\star$, and~$\mathbf{rec}^\star$ as follows \begin{gather*} \tree = \mathbf{P}^\star(\lambda) \iff \mathrm{C}(\tree) \subseteq \mathrm{C}(\lambda) \iff \mathrm{C}^\diamond(\tree) \supseteq \mathrm{C}^\diamond(\lambda), \\ \chi = \mathbf{can}^\star(\tree) \iff \mathrm{C}(\chi) \subseteq \mathrm{C}(\tree) \iff \mathrm{C}^\diamond(\chi) \supseteq \mathrm{C}^\diamond(\tree), \\ \chi = \mathbf{rec}^\star(\lambda) \iff \mathrm{C}(\chi) \subseteq \mathrm{C}(\lambda) \iff \mathrm{C}^\diamond(\chi) \supseteq \mathrm{C}^\diamond(\lambda). \end{gather*}
Finally, the weak order, the Schr\"oder-Cambrian lattice and the boolean lattice correspond to the lattice of faces of the permutahedron~$\Perm$, associahedron~$\Asso$ and parallelepiped~$\Para$, oriented in the direction~$(n, \dots, 1) - (1, \dots, n) = \sum_{i \in [n]} (n+1-2i) \, e_i$.
\section{Schr\"oder-Cambrian Hopf Algebra} \label{sec:SchroderCambrianAlgebra}
In this section, we define the Schr\"oder-Cambrian Hopf algebra~$\mathsf{SchrCamb}$, extending simultaneously the Cambrian Hopf algebra and F.~Chapoton's Hopf algebra on Schr\"oder trees~\cite{Chapoton}. We construct the algebra~$\mathsf{SchrCamb}$ as a subalgebra of a signed version of F.~Chapoton's Hopf algebra on ordered partitions~\cite{Chapoton}. We then also consider the dual algebra~$\mathsf{SchrCamb}^*$ as a quotient of the dual Hopf algebra on signed ordered partitions.
\subsection{Shuffle and convolution products on signed ordered partitions}
We define here a natural analogue of the shifted shuffle and convolution products of Section~\ref{subsec:products} on ordered partitions. Equivalent definitions in the world of surjections can be found in~\cite{Chapoton}. Here, we stick to ordered partitions to match our presentation of the Cambrian algebra in Section~\ref{sec:CambrianAlgebra}.
We first define two restrictions on ordered partitions. Consider an ordered partition~$\mu$ of~$[n]$ into~$k$ parts. As already mentioned earlier, we represent graphically~$\mu$ by the $(k \times n)$-table with a dot in row~$i$ and column~$j$ for each~$j \in \mu_i$. For~${I \subseteq [k]}$, we let~$n_I \eqdef |\set{j \in [n]}{\exists \, i \in I, \; j \in \mu_i}|$ and we denote by~$\mu_{|I}$ the ordered partition of~$[n_I]$ into $|I|$ parts whose table is obtained from the table of~$\mu$ by deleting all rows not in~$I$ and standardizing to get a~$(|I| \times n_I)$-table. Similarly, for~$J \subseteq [n]$, we let~$k_J \eqdef |\set{i \in [k]}{\exists \, j \in J, \; j \in \mu_i}|$ and we denote by~$\mu^{|J}$ the ordered partition of~$[|J|]$ into~$k_J$ parts whose table is obtained from the table of~$\mu$ by deleting all columns not in~$J$ and standardizing to get a~$(k_J \times |J|)$-table. These restrictions are illustrated in \fref{fig:restrictionOrderedPartition}.
\begin{figure}\label{fig:restrictionOrderedPartition}
\end{figure}
We define the \defn{shifted concatenation}~$\lambda \bar\lambda'$, the \defn{shifted shuffle product}~$\lambda \,\bar\shuffle\, \lambda'$, and the \defn{convolution product}~$\lambda \star \lambda'$ of two (unsigned) ordered partitions~$\lambda \in \mathfrak{P}_n$ and~$\lambda' \in \mathfrak{P}_{n'}$ as \begin{align*} \lambda \bar\lambda' & \eqdef \lambda_1 \,|\, \cdots \,|\, \lambda_k \,|\, n + \lambda'_1 \,|\, \cdots \,|\, n + \lambda'_{k'}, \qquad \text{where } n + \lambda'_i \eqdef \set{n+j}{j \in \lambda'_i}\\
\lambda \,\bar\shuffle\, \lambda' & \eqdef \set{\mu \in \mathfrak{P}_{n+n'}}{\mu^{|\{1, \dots, n\}} = \lambda \text{ and } \mu^{|\{n+1, \dots, n+n'\}} = \lambda'}, \\
\text{and}\qquad \lambda \star \lambda' & \eqdef \set{\mu \in \mathfrak{P}_{n+n'}}{\mu_{|\{1, \dots, n\}} = \lambda \text{ and } \mu_{|\{n+1, \dots, n+n'\}} = \lambda'} \end{align*}
For example, \begin{align*} {\color{red} 1} | {\color{red} 2} \,\bar\shuffle\, {\color{darkblue} 2} | {\color{darkblue} 31} = \{ &
{\color{red} 1} | {\color{red} 2} | {\color{darkblue} 4} | {\color{darkblue} 53}, \;
{\color{red} 1} | {\color{red} 2}{\color{darkblue} 4} | {\color{darkblue} 53}, \;
{\color{red} 1} | {\color{darkblue} 4} | {\color{red} 2} | {\color{darkblue} 53}, \;
{\color{red} 1} | {\color{darkblue} 4} | {\color{red} 2}{\color{darkblue} 53}, \;
{\color{red} 1} | {\color{darkblue} 4} | {\color{darkblue} 53} | {\color{red} 2}, \;
{\color{red} 1}{\color{darkblue} 4} | {\color{red} 2} | {\color{darkblue} 53}, \;
{\color{red} 1}{\color{darkblue} 4} | {\color{red} 2}{\color{darkblue} 53}, \\[-.1cm] & {\color{red} 1}{\color{darkblue} 4} | {\color{darkblue} 53} | {\color{red} 2}, \;
{\color{darkblue} 4} | {\color{red} 1} | {\color{red} 2} | {\color{darkblue} 53}, \;
{\color{darkblue} 4} | {\color{red} 1} | {\color{red} 2}{\color{darkblue} 53}, \;
{\color{darkblue} 4} | {\color{red} 1} | {\color{darkblue} 53} | {\color{red} 2}, \;
{\color{darkblue} 4} | {\color{red} 1}{\color{darkblue} 53} | {\color{red} 2}, \;
{\color{darkblue} 4} | {\color{darkblue} 53} | {\color{red} 1} | {\color{red} 2} \}, \\[.2cm] {\color{red} 1} | {\color{red} 2} \star {\color{darkblue} 2} | {\color{darkblue} 31} = \{ &
{\color{red} 1} | {\color{red} 2} | {\color{darkblue} 4} | {\color{darkblue} 53}, \;
{\color{red} 1} | {\color{red} 3} | {\color{darkblue} 4} | {\color{darkblue} 52}, \;
{\color{red} 1} | {\color{red} 4} | {\color{darkblue} 3} | {\color{darkblue} 52}, \;
{\color{red} 1} | {\color{red} 5} | {\color{darkblue} 3} | {\color{darkblue} 42}, \;
{\color{red} 2} | {\color{red} 3} | {\color{darkblue} 4} | {\color{darkblue} 51}, \\[-.1cm] & {\color{red} 2} | {\color{red} 4} | {\color{darkblue} 3} | {\color{darkblue} 51}, \;
{\color{red} 2} | {\color{red} 5} | {\color{darkblue} 3} | {\color{darkblue} 41}, \;
{\color{red} 3} | {\color{red} 4} | {\color{darkblue} 2} | {\color{darkblue} 51}, \;
{\color{red} 3} | {\color{red} 5} | {\color{darkblue} 2} | {\color{darkblue} 41}, \;
{\color{red} 4} | {\color{red} 5} | {\color{darkblue} 2} | {\color{darkblue} 31} \}. \end{align*}
Graphically, the table of the shifted concatenation~$\lambda \bar\lambda'$ contains the table of~$\lambda$ as the bottom left block and the table of~$\lambda'$ as the top right block. The tables in the shifted shuffle product~$\lambda \,\bar\shuffle\, \lambda'$ (resp.~in the convolution product~$\lambda \star \lambda'$) are obtained by shuffling the rows (resp.~columns) of the table of~$\lambda \bar\lambda'$. See \fref{fig:shuffleConvolutionOrderedPartitions}.
\begin{figure}
\caption{The table of the shifted concatenation~$\lambda\bar\lambda'$ (left) has two blocks containing the tables of the ordered partitions~$\lambda = 1 | 2$ and~$\lambda' = 2 | 31$. Elements of the shifted shuffle product~$\lambda \,\bar\shuffle\, \lambda'$ (middle) and of the convolution product~${\lambda \star \lambda'}$~(right) are obtained by shuffling respectively the rows and columns of the table of~$\lambda\bar\lambda'$.}
\label{fig:shuffleConvolutionOrderedPartitions}
\end{figure}
\begin{remark} \begin{enumerate}[(i)] \item Note that the shifted shuffle and convolution products are compatible with the filtration~$(\mathfrak{P}_n^{\ge p})_{p \in [n]}$: \[ \mathfrak{P}_n^{\ge p} \,\bar\shuffle\, \mathfrak{P}_{n'}^{\ge p'} \subseteq \mathfrak{P}_{n+n'}^{\ge p+p'} \qquad\text{and}\qquad \mathfrak{P}_n^{\ge p} \star \mathfrak{P}_{n'}^{\ge p'} \subseteq \mathfrak{P}_{n+n'}^{\ge p+p'}. \] \item By projection on the quotient~$\mathfrak{P} / \mathfrak{P}^{\ge 1} \simeq \mathfrak{S}$, the (signed) shifted shuffle and convolution products coincide with the description of Section~\ref{subsec:products}. \item The shifted product of ordered partitions preserves intervals of the weak order. Namely, \[ [\lambda, \mu] \,\bar\shuffle\, [\lambda', \mu'] = [\lambda \bar\lambda', \bar\mu' \mu]. \] \end{enumerate} \end{remark}
These definitions extend to signed ordered partitions: signs travel with their values in the signed shifted shuffle product, and stay at their positions in the signed convolution product.
\subsection{Subalgebra of~$\mathsf{OrdPart}_\pm$} \label{subsec:SchroderCambrianAlgebra}
We denote by~$\mathsf{OrdPart}_\pm$ the Hopf algebra with basis~$(\mathbb{F}_\lambda)_{\lambda \in \mathfrak{P}_{\pm}}$ and whose product and coproduct are defined by \[ \mathbb{F}_\lambda \cdot \mathbb{F}_{\lambda'} = \sum_{\mu \in \lambda \,\bar\shuffle\, \lambda'} \mathbb{F}_\mu \qquad\text{and}\qquad \triangle \mathbb{F}_\mu = \sum_{\mu \in \lambda \star \lambda'} \mathbb{F}_\lambda \otimes \mathbb{F}_{\lambda'}. \] Note that the Hopf algebra~$\mathsf{FQSym}_\pm$ is isomorphic to the quotient~$\mathsf{OrdPart}_\pm / \mathsf{OrdPart}_\pm^{\ge 1}$. Note also that the unsigned version of~$\mathsf{OrdPart}_\pm$ is the dual of the algebra~$\mathsf{WQSym}$ of word quasi-symmetric functions (also denoted~$\mathsf{NCQSym}$ for non-commutative quasi-symmetric functions), see~\cite{BergeronZabrocki, NovelliThibon-trigebres}.
\begin{remark} The proof that~$\mathsf{OrdPart}_\pm$ is indeed a Hopf algebra is left to the reader: it consists in translating F.~Chapoton's proof~\cite{Chapoton} from surjections to signed ordered partitions. In fact, F.~Chapoton's Hopf algebras on faces of the permutahedra, associahedra, and cubes could be decorated by an arbitrary group, similar to the constuctions in~\cite{NovelliThibon-coloredHopfAlgebras, BaumannHohlweg, BergeronHohlweg}. Once again the main point here is that the Schr\"oder-congruence relations depend on the decoration. \end{remark}
We denote by~$\mathsf{SchrCamb}$ the vector subspace of~$\mathsf{OrdPart}_\pm$ generated by the elements \[ \mathbb{P}_{\tree} \eqdef \sum_{\substack{\lambda \in \mathfrak{P}_\pm \\ \mathbf{P}^\star(\lambda) = \tree}} \mathbb{F}_\lambda \] for all Schr\"oder-Cambrian trees~$\tree$. For example, for the Schr\"oder-Cambrian tree of \fref{fig:leveledSchroderCambrianTree}\,(left), we have \[
\mathbb{P}_{\!\!\!\!\includegraphics{TexSchroder}} = \mathbb{F}_{\down{12}|\down{5}\up{7}|\up{3}\down{4}|\up{6}} + \mathbb{F}_{\down{125}\up{7}|\up{3}\down{4}|\up{6}} + \mathbb{F}_{\down{5}\up{7}|\down{12}|\up{3}\down{4}|\up{6}}. \] Note that the Hopf algebra~$\mathsf{Camb}$ is isomorphic to the quotient~$\mathsf{SchrCamb}_\pm / \mathsf{SchrCamb}_\pm^{\ge 1}$.
\begin{theorem} \label{thm:SchrCambSubalgebra} $\mathsf{SchrCamb}$ is a Hopf subalgebra of~$\mathsf{OrdPart}_\pm$. \end{theorem}
\begin{proof} Similar to the proof of Theorem~\ref{thm:cambSubalgebra}. \end{proof}
As for the Cambrian algebra, the product and coproduct of~$\mathbb{P}$-basis elements of the Schr\"oder-Cambrian algebra~$\mathsf{SchrCamb}$ can be described directly in terms of combinatorial operations on Schr\"oder-Cambrian trees.
\para{Product} The product in the Schr\"oder Cambrian algebra~$\mathsf{SchrCamb}$ can again be described in terms of intervals in the Sch\"oder-Cambrian lattice. Given two Schr\"oder-Cambrian trees~$\tree, \tree'$, we denote by~$\raisebox{-6pt}{$\tree$}\nearrow \raisebox{4pt}{$\bar \tree'$}$ the Schr\"oder $\varepsilon(\tree)\varepsilon(\tree')$-Cambrian tree obtained by grafting the rightmost outgoing leaf of~$\tree$ on the leftmost incoming leaf of~$\tree$ and shifting all labels of~$\tree'$. We define similarly~$\raisebox{4pt}{$\tree$} \nwarrow \raisebox{-6pt}{$\bar \tree'$}$.
\begin{proposition} For any Schr\"oder-Cambrian trees~$\tree, \tree'$, the product~$\mathbb{P}_{\tree} \cdot \mathbb{P}_{\tree'}$ is given by \[ \mathbb{P}_{\tree} \cdot \mathbb{P}_{\tree'} = \sum_{\tree[S]} \mathbb{P}_{\tree[S]}, \] where~$\tree[S]$ runs over the interval between~$\raisebox{-6pt}{$\tree$}\nearrow \raisebox{4pt}{$\bar \tree'$}$ and~$\raisebox{4pt}{$\tree$} \nwarrow \raisebox{-6pt}{$\bar \tree'$}$ in the Schr\"oder $\varepsilon(\tree)\varepsilon(\tree')$-Cambrian lattice. \end{proposition}
\begin{proof} Similar to that of Proposition~\ref{prop:product}. \end{proof}
For example, we can compute the product \[ \begin{array}{@{}c@{${} = {}$}c@{+}c@{+}c@{}@{}c} \mathbb{P}_{\!\!\includegraphics{exmProductSchroderA}} \cdot \mathbb{P}_{\includegraphics{exmProductSchroderB}} & \multicolumn{3}{l}{\mathbb{F}_{\down{1}\up{2}} \cdot \big( \mathbb{F}_{\up{1} | \down{3} | \down{2}\up{4}} + \mathbb{F}_{\up{1}\down{3} | \down{2}\up{4}} + \mathbb{F}_{\down{3} | \up{1} | \down{2}\up{4}} \big)} \\[-.3cm] & \begin{pmatrix} \quad \mathbb{F}_{\down{1}\up{2} | \up{3} | \down{5} | \down{4}\up{6}} + \mathbb{F}_{\down{1}\up{2} | \up{3}\down{5} | \down{4}\up{6}} \\ + \; \mathbb{F}_{\down{1}\up{2} | \down{5} | \up{3} | \down{4}\up{6}} + \mathbb{F}_{\down{1}\up{2}\down{5} | \up{3} | \down{4}\up{6}} \\ + \; \mathbb{F}_{\down{5} | \down{1}\up{2} | \up{3} | \down{4}\up{6}} \end{pmatrix} & \begin{pmatrix} \quad \mathbb{F}_{\down{1}\up{2}\up{3} | \down{5} | \down{4}\up{6}} \\ + \; \mathbb{F}_{\down{1}\up{2}\up{3} \down{5} | \down{4}\up{6}} \\ + \; \mathbb{F}_{\down{5} | \down{1}\up{2}\up{3} | \down{4}\up{6}} \end{pmatrix} & \begin{pmatrix} \quad \mathbb{F}_{\up{3} | \down{1}\up{2} | \down{5} | \down{4}\up{6}} + \mathbb{F}_{\up{3} | \down{1}\up{2}\down{5} | \down{4}\up{6}} + \mathbb{F}_{\up{3} | \down{5} | \down{1}\up{2} | \down{4}\up{6}} \\ + \; \mathbb{F}_{\up{3}\down{5} | \down{1}\up{2} | \down{4}\up{6}} + \mathbb{F}_{\down{5} | \up{3} | \down{1}\up{2} | \down{4}\up{6}} + \mathbb{F}_{\up{3} | \down{5} | \down{1}\up{2}\down{4}\up{6}} \\ + \; \mathbb{F}_{\up{3}\down{5} | \down{1}\up{2}\down{4}\up{6}} + \mathbb{F}_{\down{5} | \up{3} | \down{1}\up{2}\down{4}\up{6}} + \mathbb{F}_{\up{3} | \down{5} | \down{4}\up{6} | \down{1}\up{2}} \\ + \; \mathbb{F}_{\up{3}\down{5} | \down{4}\up{6} | \down{1}\up{2}} + \mathbb{F}_{\down{5} | \up{3} | \down{4}\up{6} | \down{1}\up{2}} \end{pmatrix} \\[.8cm] & \mathbb{P}_{\!\!\includegraphics{exmProductSchroder1}} & \mathbb{P}_{\!\!\includegraphics{exmProductSchroder2}} & \mathbb{P}_{\!\!\includegraphics{exmProductSchroder3}} & . \end{array} \]
\para{Coproduct} The coproduct in the Schr\"oder Cambrian algebra~$\mathsf{SchrCamb}$ can again be described in terms of cuts. A \defn{cut} of a Schr\"oder-Cambrian tree~$\tree[S]$ is a set~$\gamma$ of edges such that any geodesic vertical path in~$\tree[S]$ from a down leaf to an up leaf contains precisely one edge of~$\gamma$. We denote again by~$A(\tree[S],\gamma)$ and~$B(\tree[S],\gamma)$ the two Schr\"oder-Cambrian forests above and below~$\gamma$ in~$\tree[S]$.
\begin{proposition} For any Schr\"oder-Cambrian tree~$\tree[S]$, the coproduct~$\triangle \mathbb{P}_{\tree[S]}$ is given by \[ \triangle \mathbb{P}_{\tree[S]} = \sum_{\gamma} \bigg( \prod_{\tree \in B(\tree[S],\gamma)} \mathbb{P}_{\tree} \bigg) \otimes \bigg( \prod_{\tree' \in A(\tree[S], \gamma)} \mathbb{P}_{\tree'} \bigg), \] where~$\gamma$ runs over all cuts of~$\tree[S]$. \end{proposition}
\begin{proof} Similar to that of Proposition~\ref{prop:coproduct}. \end{proof}
For example, we can compute the coproduct \[ \begin{array}{@{}c@{${} = {}$}c@{$\;+\;$}c@{$\;+\;$}c@{$\;+\;$}c@{$\;+\;$}c@{}} \triangle \mathbb{P}_{\includegraphics{exmProductSchroderB}} & \multicolumn{5}{l}{\triangle \big( \mathbb{F}_{\up{1} | \down{3} | \down{2}\up{4}} + \mathbb{F}_{\up{1}\down{3} | \down{2}\up{4}} + \mathbb{F}_{\down{3} | \up{1} | \down{2}\up{4}} \big)} \\ & 1 \otimes \begin{pmatrix} \quad \mathbb{F}_{\up{1} | \down{3} | \down{2}\up{4}} \\ + \; \mathbb{F}_{\up{1}\down{3} | \down{2}\up{4}} \\ + \; \mathbb{F}_{\down{3} | \up{1} | \down{2}\up{4}} \end{pmatrix} & \mathbb{F}_{\up{1}} \otimes \mathbb{F}_{\down{2} | \down{1}\up{3}} & \mathbb{F}_{\down{1}} \otimes \mathbb{F}_{\up{1} | \down{2}\up{3}} & \begin{pmatrix} \quad \mathbb{F}_{\up{1} | \down{2}} \\ + \; \mathbb{F}_{\up{1}\down{2}} \\ + \; \mathbb{F}_{\down{2} | \up{1}} \end{pmatrix} \otimes \mathbb{F}_{\down{1}\up{2}} & \begin{pmatrix} \quad \mathbb{F}_{\up{1} | \down{3} | \down{2}\up{4}} \\ + \; \mathbb{F}_{\up{1}\down{3} | \down{2}\up{4}} \\ + \; \mathbb{F}_{\down{3} | \up{1} | \down{2}\up{4}} \end{pmatrix} \otimes 1 \\[.8cm] & 1 \otimes \mathbb{P}_{\!\includegraphics{exmProductSchroderB}} & \mathbb{P}_{\includegraphics{exmTreeY}} \otimes \mathbb{P}_{\!\!\includegraphics{exmTreeT}} & \mathbb{P}_{\includegraphics{exmTreeA}} \otimes \mathbb{P}_{\!\includegraphics{exmTreeG}} & \big( \mathbb{P}_{\!\includegraphics{exmTreeY}} \cdot \mathbb{P}_{\includegraphics{exmTreeA}} \big)\otimes \mathbb{P}_{\includegraphics{exmProductSchroderA}} & \mathbb{P}_{\!\includegraphics{exmProductSchroderB}} \otimes 1. \end{array} \]
\para{Matriochka algebras} To conclude, we connect the Schr\"oder-Cambrian algebra to F.~Chapoton's algebra on faces of the cubes defined in~\cite{Chapoton}. We call trilean Hopf algebra the Hopf subalgebra~$\mathsf{Tril}$ of~$\mathsf{OrdPart}_\pm$ generated by the elements \[ \mathbb{X}_\chi \eqdef \sum_{\substack{\lambda \in \mathfrak{P}_\pm \\ \mathbf{rec}^\star(\lambda) = \chi}} \mathbb{F}_\lambda \] for all~$\chi \in \{-,0,+\}^{n-1}$. The commutative diagram of Proposition~\ref{prop:commutativeDiagramSchroder} ensures that \[ \mathbb{X}_\chi = \sum_{\substack{\tree \in \mathrm{SchrCamb} \\ \mathbf{can}^\star(\tree) = \chi}} \mathbb{P}_{\tree}, \] and thus that~$\mathsf{Tril}$ is a subalgebra of~$\mathsf{SchrCamb}$. In other words, the Schr\"oder-Cambrian algebra is sandwiched between the signed ordered partitions algebra and the trilean algebra~$\mathsf{Tril} \subset \mathsf{SchrCamb} \subset \mathsf{OrdPart}_\pm$.
\subsection{Quotient algebra of~$\mathsf{OrdPart}_\pm^*$} \label{subsec:dualSchroderCambrianAlgebra}
We switch to the dual Hopf algebra~$\mathsf{OrdPart}_\pm^*$ with basis $(\mathbb{G}_\lambda)_{\lambda \in \mathfrak{P}_\pm}$ and whose product and coproduct are defined by \[ \mathbb{G}_\lambda \cdot \mathbb{G}_{\lambda'} = \sum_{\mu \in \lambda \star \lambda'} \mathbb{G}_\mu \qquad\text{and}\qquad \triangle \mathbb{G}_\mu = \sum_{\mu \in \lambda \,\bar\shuffle\, \lambda'} \mathbb{G}_\lambda \otimes \mathbb{G}_{\lambda'}. \] Note that the unsigned version of~$\mathsf{OrdPart}_\pm^*$ is the algebra~$\mathsf{WQSym}$ of word quasi-symmetric functions (also denoted~$\mathsf{NCQSym}$ for non-commutative quasi-symmetric functions), see~\cite{BergeronZabrocki, NovelliThibon-trigebres}.The following statement is automatic from Theorem~\ref{thm:SchrCambSubalgebra}.
\begin{theorem} The graded dual~$\mathsf{SchrCamb}^*$ of the Schr\"oder-Cambrian algebra is isomorphic to the image of~$\mathsf{OrdPart}_\pm^*$ under the canonical projection \[ \pi : \mathbb{C}\langle A \rangle \longrightarrow \mathbb{C}\langle A \rangle / \equiv, \] where~$\equiv$ denotes the Schr\"oder-Cambrian congruence. The dual basis~$\mathbb{Q}_{\tree}$ of~$\mathbb{P}_{\tree}$ is expressed as~$\mathbb{Q}_{\tree} = \pi(\mathbb{G}_\lambda)$, where~$\lambda$ is any ordered partition such that~$\mathbf{P}^\star(\lambda) = \tree$. \end{theorem}
Similarly as in the previous section, we can describe combinatorially the product and coproduct of $\mathbb{Q}$-basis elements of~$\mathsf{SchrCamb}^*$ in terms of operations on Schr\"oder-Cambrian trees.
\para{Product} We define \defn{gaps} and \defn{laminations} of Schr\"oder-Cambrian trees exactly as we did for Cambrian trees in Section~\ref{subsec:quotientAlgebra}. Note that laminations may or may not split the nodes of a Schr\"oder-Cambrian tree, see \fref{fig:exampleProductDualSchroder}\,(c) for examples. Given two Schr\"oder-Cambrian trees~$\tree$ and~$\tree'$ on~$[n]$ and~$[n']$ respectively, and a shuffle~$s$ of their signature defining multisets $\Gamma$ of gaps of~$[n]$ and $\Gamma'$ of gaps of~$[n']$, we still denote by~$\tree \,{}_s\!\backslash \tree'$ the Schr\"oder-Cambrian tree obtained by connecting the up leaves of~$\lambda(\tree,\Gamma)$ to the down leaves of the forest defined by the lamination~$\lambda(\tree',\Gamma')$. See \fref{fig:exampleProductDualSchroder}.
\begin{figure}\label{fig:exampleProductDualSchroder}
\end{figure}
\begin{proposition} For any Schr\"oder-Cambrian trees~$\tree, \tree'$, the product~$\mathbb{Q}_{\tree} \cdot \mathbb{Q}_{\tree'}$ is given by \[ \mathbb{Q}_{\tree} \cdot \mathbb{Q}_{\tree'} = \sum_s \mathbb{Q}_{\tree \,{}_s\!\backslash \tree'}, \] where~$s$ runs over all shuffles of the signatures of~$\tree$ and~$\tree'$. \end{proposition}
\begin{proof} Similar to that of Proposition~\ref{prop:productDual}. \end{proof}
For example, we can compute the product \[ \hspace*{-1cm}\begin{array}{@{}ccc@{$\;+\;$}c@{$\;+\;$}c@{$\;+\;$}c@{$\;+\;$}c@{$\;+\;$}c@{$\;+\;$}c@{$\;+\;$}c@{}} \mathbb{Q}_{\includegraphics{exmProductSchroderA}} \cdot \mathbb{Q}_{\includegraphics{exmProductSchroderB}} & = & \multicolumn{8}{l}{\mathbb{G}_{\down{1}\up{2}} \cdot \mathbb{G}_{\up{1}\down{3} | \down{2}\up{4}}} \\[-.2cm] & = & \quad \mathbb{G}_{\down{1}\up{2} | \up{3}\down{5} | \down{4}\up{6}} & \mathbb{G}_{\down{1}\up{3} | \up{2}\down{5} | \down{4}\up{6}} & \mathbb{G}_{\down{1}\up{4} | \up{2}\down{5} | \down{3}\up{6}} & \mathbb{G}_{\down{1}\up{5} | \up{2}\down{4} | \down{3}\up{6}} & \mathbb{G}_{\down{1}\up{6} | \up{2}\down{4} | \down{3}\up{5}} & \mathbb{G}_{\down{2}\up{3} | \up{1}\down{5} | \down{4}\up{6}} & \mathbb{G}_{\down{2}\up{4} | \up{1}\down{5} | \down{3}\up{6}} & \mathbb{G}_{\down{2}\up{5} | \up{1}\down{4} | \down{3}\up{6}} \\ & & + \; \mathbb{G}_{\down{2}\up{6} | \up{1}\down{4} | \down{3}\up{5}} & \mathbb{G}_{\down{3}\up{4} | \up{1}\down{5} | \down{2}\up{6}} & \mathbb{G}_{\down{3}\up{5} | \up{1}\down{4} | \down{2}\up{6}} & \mathbb{G}_{\down{3}\up{6} | \up{1}\down{4} | \down{2}\up{5}} & \mathbb{G}_{\down{4}\up{5} | \up{1}\down{3} | \down{2}\up{6}} & \mathbb{G}_{\down{4}\up{6} | \up{1}\down{3} | \down{2}\up{5}} & \multicolumn{2}{@{}l}{\mathbb{G}_{\down{5}\up{6} | \up{1}\down{3} | \down{2}\up{4}}} \\[.2cm] & = & \quad \mathbb{Q}_{\!\includegraphics{exmProductSchroder4}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder5}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder6}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder7}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder8}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder9}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder10}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder11}} \\ & & + \; \mathbb{Q}_{\!\includegraphics{exmProductSchroder12}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder13}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder14}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder15}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder16}} & \mathbb{Q}_{\!\includegraphics{exmProductSchroder17}} & \multicolumn{2}{@{}l}{\mathbb{Q}_{\!\includegraphics{exmProductSchroder18}} \; .} \end{array} \]
\para{Coproduct} For a gap~$\gamma$ of a Schr\"oder-Cambrian tree~$\tree[S]$, we still denote by~$L(\tree[S],\gamma)$ and~$R(\tree[S],\gamma)$ the left and right Schr\"oder-Cambrian subtrees of~$\tree[S]$ when split along the path~$\lambda(\tree[S],\gamma)$.
\begin{proposition} For any Cambrian tree~$\tree[S]$, the coproduct~$\triangle\mathbb{Q}_{\tree[S]}$ is given by \[ \triangle\mathbb{Q}_{\tree[S]} = \sum_{\gamma} \mathbb{Q}_{L(\tree[S],\gamma)} \otimes \mathbb{Q}_{R(\tree[S],\gamma)}, \] where~$\gamma$ runs over all gaps between vertices of~$\tree[S]$. \end{proposition}
\begin{proof} Similar to that of Proposition~\ref{prop:coproductDual}. \end{proof}
For example, we can compute the coproduct \[ \hspace*{-1cm}\begin{array}{@{}c@{${} = {}$}c@{${} + {}$}c@{${} + {}$}c@{${} + {}$}c@{${} + {}$}c@{}} \triangle \mathbb{Q}_{\includegraphics{exmProductSchroderB}} &
\multicolumn{5}{l}{\triangle \mathbb{G}_{\up{1}\down{3} | \down{2}\up{4}}}
\\[-.2cm] & 1 \otimes \mathbb{G}_{\up{1}\down{3} | \down{2}\up{4}} & \mathbb{G}_{\up{1}} \otimes \mathbb{G}_{\down{2} | \down{1}\up{3}} & \mathbb{G}_{\up{1} | \down{2}} \otimes \mathbb{G}_{\down{1} | \up{2}} & \mathbb{G}_{\up{1}\down{3} | \down{2}} \otimes \mathbb{G}_{\up{1}} & \mathbb{G}_{\up{1}\down{3} | \down{2}\up{4}} \otimes 1 \\[.2cm] & 1 \otimes \mathbb{Q}_{\includegraphics{exmProductSchroderB}} & \mathbb{Q}_{\includegraphics{exmTreeY}} \otimes \mathbb{Q}_{\includegraphics{exmTreeT}} & \mathbb{Q}_{\includegraphics{exmTreeYAd}} \otimes \mathbb{Q}_{\includegraphics{exmTreeAYd}} & \mathbb{Q}_{\includegraphics{exmCoproductDualSchroder1}} \otimes \mathbb{Q}_{\includegraphics{exmTreeY}} & \mathbb{Q}_{\includegraphics{exmProductSchroderB}} \otimes 1. \end{array} \]
\section{Schr\"oder-Cambrian tuples}
\enlargethispage{-.8cm} As a conclusion, we just want to mention that it would also be possible to extend simultaneously the Cambrian tuple algebra and the Schr\"oder-Cambrian algebra. The objects are \defn{Schr\"oder-Cambrian $\ell$-tuples}, \textit{i.e.}~ $\ell$-tuples of Schr\"oder-Cambrian trees whose union is acyclic.
The first step is then to describe the combinatorics of these tuples: \begin{itemize} \item applying Schr\"oder-Cambrian correspondences in parallel yields a correspondence~$\Theta^\star_\ell$ between $\ell$-signed ordered partitions and leveled Schr\"oder-Cambrian $\ell$-tuples, and thus defines a surjection~$\mathbf{P}^\star_\ell$ from $\ell$-signed ordered partitions to Schr\"oder-Cambrian $\ell$-tuples; \item the fibers of~$\mathbf{P}^\star_\ell$ are intersections of Schr\"oder-Cambrian congruences, and thus define a lattice congruence of the weak order on ordered partitions; \item the $\ell$-tuples of Schr\"oder-Cambrian trees correspond to all faces of a Minkowski sum of $\ell$ associahedra of~\cite{HohlwegLange}. \end{itemize} An interesting combinatorial problem is to count the number of $\ell$-tuples of Schr\"oder-Cambrian trees, in particular the number of Baxter-Schr\"oder-Cambrian trees.
The second step is to define as usual the Schr\"oder-Cambrian $\ell$-tuple Hopf algebra~$\mathsf{SchrCamb}_\ell$ as a subalgebra of the Hopf algebra~$\mathsf{OrdPart}_{\pm^\ell}$ of $\ell$-signed ordered partitions, and its dual~$\mathsf{SchrCamb}_\ell^*$ as a quotient of~$\mathsf{OrdPart}_{\pm^\ell}^*$. The product and coproduct both~$\mathsf{SchrCamb}_\ell$ and~$\mathsf{SchrCamb}_\ell^*$ can then directly be described by the combinatorial operations on Schr\"oder-Cambrian $\ell$-tuples, similar to the operations described in Sections~\ref{subsec:CambrianTupleAlgebra}, \ref{subsec:dualCambrianTupleAlgebra}, \ref{subsec:SchroderCambrianAlgebra} and~\ref{subsec:dualSchroderCambrianAlgebra}.
\label{sec:biblio}
\end{document} | arXiv |
\begin{document}
\title{A simple proof of asymptotic consensus in the Hegselmann-Krause and Cucker-Smale models with renormalization and delay} \author{Jan Haskovec\footnote{Computer, Electrical and Mathematical Sciences \& Engineering, King Abdullah University of Science and Technology, 23955 Thuwal, KSA. [email protected]}}
\date{}
\maketitle
\begin{abstract} We present a simple proof of asymptotic consensus in the discrete Hegselmann-Krause model and flocking in the discrete Cucker-Smale model with renormalization and variable delay. It is based on convexity of the renormalized communication weights and a Gronwall-Halanay-type inequality. The main advantage of our method, compared to previous approaches to the delay Hegselmann-Krause model, is that it does not require any restriction on the maximal time delay, or the initial data, or decay rate of the influence function. From this point of view the result is optimal. For the Cucker-Smale model it provides an analogous result in the regime of unconditonal flocking with sufficiently slowly decaying communication rate, but still without any restriction on the length of the maximal time delay. Moreover, we demonstrate that the method can be easily extended to the mean-field limits of both the Hegselmann-Krause and Cucker-Smale systems, using appropriate stability results on the measure-valued solutions. \end{abstract}
\textbf{Keywords}: Hegselmann-Krause model, asymptotic consensus, Cucker-Smale model, flocking, long-time behavior, variable delay.
\section{Introduction}\label{sec:Intro} In this paper we study asymptotic behavior of the Hegselmann-Krause \cite{HK} and Cucker-Smale \cite{CS1, CS2} models with normalized communication weights and variable time delay. The Hegselmann-Krause model describes the evolution of $N\in\mathbb{N}$ agents who adapt their opinions to the ones of other members of the group. Agent $i$'s opinion is represented by the quantity $x_i = x_i(t)\in\mathbb{R}^d$, with $d\in\mathbb{N}$ the space dimension, which is a function of time $t\geq 0$. Assuming that the agents' communication takes place subject to a variable delay $\tau=\tau(t)$, the opinions evolve according to the following system \( \label{eq:HK}
\dot x_i(t) =
\sum_{j=1}^N \psi_{ij}(t) (x_j(t-\tau(t)) - x_i(t)), \qquad i=1,\ldots,N. \)
The normalized communication weights $\psi_{ij}=\psi_{ij}(t)$ are given by \(\label{psi}
\psi_{ij}(t) := \left\{ \begin{array}{ll}
\displaystyle \frac{\psi(|x_j(t - \tau(t)) - x_i(t)|)}{\sum_{\ell\neq i} \psi(|x_\ell(t - \tau(t)) - x_i(t)|)} & \textrm{if $j \neq i$,}\\[4mm]
0 & \textrm{if $j=i$,}
\end{array} \right. \) where the nonnegative \emph{influence function} $\psi:[0,\infty)\to [0,\infty)$, also called \emph{communication rate}, measures how strongly each agent is influenced by others depending on their distance. The variable time delay $\tau=\tau(t)$ is assumed to be nonegative and uniformly bounded by some $\overline\tau>0$; this includes the generic case of the constant delay $\tau(t)\equiv\overline\tau$. The system \eqref{eq:HK} is equipped with the initial datum \(\label{IC:HK}
x_i(s) = x^0_i(s),\qquad i=1,\cdots,N, \quad s \in [-\overline\tau,0], \) with prescribed trajectories $x^0_i\in C([-\overline\tau,0])$, $ i=1,\cdots,N$.
The phenomenon of consensus finding in the context of \eqref{eq:HK} refers to the (asymptotic) emergence of one or more {opinion clusters} formed by agents with (almost) identical opinions \cite{JM}. \emph{Global consensus} is the state where all agents have the same opinion, i.e., $x_i=x_j$ for all $i,j \in\{1,\dots,N\}$. Convergence to global consensus as $t\to\infty$ for the system \eqref{eq:HK} has been proved in \cite{CPP} under a set of conditions requiring smallness of the maximal time delay in relation to the decay speed of the influence function and the fluctuation of the initial datum. However, as we shall discuss in Section \ref{sec:mam}, there is justified expectation that asymptotic global consensus is reached without any restriction on the maximal time delay $\overline\tau$, for all initial data, and only assuming that the influence function is strictly positive on $[0,\infty)$, but may decay to zero arbitrarily fast at infinity. In this paper we prove that this intuitive expectation is indeed true. As this result is essentially optimal, it closes an important gap in the theory of asymptotic behavior of the Hegselmann-Krause model with delay.
Our method of proof consists of three ingredients. First, we prove nonexpansivity of the agent group, i.e., a uniform bound on the position radius of the agents. Second, we establish a convexity argument, exploiting the following property of the renormalized weights \eqref{psi}, \( \label{psi:conv}
\sum_{j=1}^N \psi_{ij} = 1 \qquad\mbox{for all } i=1,\cdots, N. \) This implies that the terms $\sum_{j=1}^N \psi_{ij}(t) x_j(t-\tau(t))$ in \eqref{eq:HK} are convex combinations of the vectors $x_j(t-\tau(t))$. Finally, a Gronwall-Halanay-type inequality shall provide exponential decay to zero of solutions of a certain delay differential inequality.
A second goal of this paper is to study, by a slight extension of the above method, the asymptotic behavior of a variant of the Cucker-Smale model \cite{CS1, CS2} with renormalized communication weights, introduced by Motsch and Tadmor \cite{MT}. The model can be seen as a second-order version of \eqref{eq:HK} and reads, for $i=1,\cdots,N$, \begin{align}\label{eq:CS} \begin{aligned}
\dot x_i(t) &= v_i(t), \\
\dot v_i(t) & = \sum_{j=1}^N \psi_{ij}(t) (v_j(t - \tau(t)) - v_i(t)), \end{aligned} \end{align}
where the normalized communication weights $\psi_{ij}=\psi_{ij}(t)$ are again given by \eqref{psi}. The system is equipped with the initial datum \(\label{IC:CS}
x_i(s) = x^0_i(s),\quad v_i(s) = v^0_i(s), \qquad i=1,\cdots,N, \quad s \in [-\overline\tau,0], \) with prescribed phase-space trajectories $x^0_i, v^0_i\in C([-\overline\tau,0]; \mathbb{R}^d)$, $ i=1,\cdots,N$. For physical reasons it may be required that $$ x_i^0(s) = x^0_i(-\overline\tau) + \int_{-\overline\tau}^s v_i^0(\sigma) \mathrm{d}\sigma\quad\mbox{for } s \in (-\overline\tau,0],$$ but we do not pose this particular restriction here.
The Cucker-Smale model (and its variants) is in the literature considered a generic model for flocking, herding or schooling of animals with positions $x_i=x_i(t)$ and velocities $v_i=v_i(t)$. If the influence function has a heavy tail, i.e., $\int^\infty \psi(s) = \infty$, then the model exhibits the so-called \emph{unconditional flocking}, where \eqref{flocking} holds for every initial configuration, see \cite{CS1, CS2, Tadmor-Ha, Ha-Liu}. In the opposite case the flocking is \emph{conditional}, i.e., the asymptotic behavior of the system depends on the initial configuration. The Cucker-Smale model with delay was studied in \cite{Liu-Wu, EHS, HasMar, ChoiH1, ChoiH2}. In particular, in \cite{ChoiH1} the precise form \eqref{eq:CS} with the normalized communication rates \eqref{psi} was considered (with constant time delay) and asymptotic flocking was proved under a smallness condition on the delay, related to the decay properties of the influence function $\psi$ and the velocity diameter of the initial datum. In this paper we significantly improve this result by considering much less restrictive conditions that are close to optimal. In particular, we prove that if the influence function $\psi=\psi(s)$ decays sufficiently slowly at infinity, then asymptotic flocking takes place for all initial data and without restriction on the delay length. This can be seen as an analogy of the unconditional flocking result for the original Cucker-Smale model.
The paper is organized as follows. In Section \ref{sec:mam} we explain the intuitive motivation for our results, formulate the precise assumptions on the influence function $\psi=\psi(s)$ and delay $\tau=\tau(t)$, and state our main results. Their proofs for the Hegselmann-Krause model \eqref{eq:HK} are presented in Section \ref{sec:HK}, and their extension for the Cucker-Smale model \eqref{eq:CS} in Section \ref{sec:CS}. Finally, in Section \ref{sec:MF} we prove the analogues of the consensus and flocking results for the mean-field limits of \eqref{eq:HK} and \eqref{eq:CS}.
\section{Motivation, assumptions and main results}\label{sec:mam} Let us explain the intuitive expectation that {all} solutions of \eqref{eq:HK} should converge to global consensus as $t\to\infty$, regardless of the length of the delay.
For this sake, we consider the case of two agents, $N=2$, with positions $x_1(t)$, $x_2(t)$, and constant delay $\tau(t)\equiv \tau$. Then \eqref{psi} gives $\psi_{12} = \psi_{21} = 1$ and \eqref{eq:HK} reduces to the linear system \begin{align*} \begin{aligned}
\dot x_1(t) &= x_2(t - \tau) - x_1(t), \\
\dot x_2(t) &= x_1(t - \tau) - x_2(t). \end{aligned} \end{align*} Defining $w:=x_1-x_2$, we have \(
\dot w(t) = -w(t - \tau) - w(t). \label{wEq} \)
Assuming a solution of the form $w(t) = e^{\xi t}$ for some complex $\xi\in\mathbb{C}$, we obtain the characteristic equation \[
\xi = -e^{-\xi\tau} - 1. \end{eqnarray*} A simple inspection reveals that all roots $\xi$ have negative real part, which implies that all solutions $w(t)$ of \eqref{wEq} tend to zero as $t\to\infty$. I.e., we have the asymptotic consensus $\lim_{t\to\infty} x_1(t)-x_2(t) = 0$, for any value of the constant delay $\tau$.
Let us note that the situation is fundamentally different for a reaction-type delay in \eqref{eq:HK}, i.e., the system \( \label{eq:HKr}
\dot x_i(t) = \sum_{j=1}^N \psi_{ij}(t) (x_j(t-\tau(t)) - x_i(t-\tau(t))), \qquad i=1,\ldots,N. \) Then, considering again $N=2$ and constant delay $\tau(t)\equiv\tau$, we have for $w:=x_1-x_2$, \[
\dot w(t) = -2 w(t - \tau). \end{eqnarray*} Nontrivial solutions of this equation exhibit oscillations whenever $2\tau > e^{-1}$ and the amplitude of the oscillations diverges in time if $2\tau > \pi/2$, see, e.g., \cite{Smith}. The different types of asymptotic behavior of the Hegselmann-Krause system with communication-type delay \eqref{eq:HK} versus the system with reaction-type delay \eqref{eq:HKr} can be intuitively understood by noting that the instantaneous negative feedback term $-x_i(t)$ has a stabilizing effect, while the delay terms typically destabilize, and this effect becomes stronger with longer delays. Thus, the stabilizing effect of the instantaneous term $-x_i(t)$ in the right-hand side of \eqref{eq:HK} is stronger than the destabilizing effect of the delay terms $x_j(t-\tau(t))$, regardless of the maximal length of the delay $\overline\tau$. In \eqref{eq:HKr} the stabilizing effect is not present, and therefore, for large enough delays, the solutions may diverge as $t\to\infty$.
Let us now formulate the precise assumptions on the influence function and variable delay
that we adopt throughout the paper. The influence function $\psi\in C([0,\infty))$ shall satisfy \( \label{ass:psi}
0 < \psi(s) \leq 1\qquad \mbox{for all } s\geq 0. \) Note that we do not require any monotonicity properties of $\psi$.
For the variable delay function $\tau\in C([0,\infty))$ we pose the assumption \( \label{ass:tau}
0 \leq \tau(t) \leq \overline\tau \qquad \mbox{for all } t\geq 0, \) for some fixed $\overline\tau>0$. Clearly, the generic case of constant delay $\tau(t)\equiv\overline\tau$ is included in \eqref{ass:tau}. Let us stress that both the above assumptions are very minimal.
The global positivity of $\psi$ in \eqref{ass:psi} cannot be further relaxed, since universal consensus behavior cannot be expected if $\psi$ would be allowed to vanish even pointwise. The upper bound $\psi\leq 1$ can be replaced by any arbitrary positive value due to the scaling invariance of \eqref{psi}. The variable delay $\tau=\tau(t)$ is allowed to vanish on arbitrary subsets of $[0,\infty)$.
To formulate our main results, let us introduce the spatial diameter $d_x=d_x(t)$ of the agent group, \( \label{def:dX}
d_x(t) := \max_{1 \leq i,j \leq N}|x_i(t) - x_j(t)|. \) The \emph{global asymptotic consensus} is then defined as the property \( \label{def:consensus}
\lim_{t\to\infty} d_x(t) = 0. \) We note that, in general, \eqref{eq:HK} does not conserve the mean value $\frac{1}{N}\sum_{i=1}^N x_i$. Consequently, the (asymptotic) consensus vector cannot be inferred from the initial datum in a straightforward way and can be seen as an emergent property of the system.
Our main result regarding the consensus behavior of the Hegselmann-Krause model \eqref{eq:HK} is as follows.
\begin{theorem}\label{thm:HK} Let $N\geq 3$ and let the assumptions \eqref{ass:psi} on $\psi=\psi(s)$ and \eqref{ass:tau} on $\tau=\tau(t)$ be verified. Then all solutions of \eqref{eq:HK} reach global asymptotic consensus as defined by \eqref{def:consensus}. The decay of $d_x=d_x(t)$ to zero is exponential with calculable rate that improves with increasing $N$.
\end{theorem}
For the Cucker-Smale-type system \eqref{eq:CS}, we define \emph{asymptotic flocking} as the property \( \label{flocking}
\lim_{t\to\infty} d_v(t) = 0, \qquad \sup_{t\geq 0} d_x(t) < \infty, \) where the velocity diameter $d_v=d_v(t)$ of the agent group is given by \[
d_v(t) := \max_{1 \leq i,j \leq N}|v_i(t) - v_j(t)|. \end{eqnarray*} We also introduce the velocity radius $R_v=R_v(t)$, \[
R_v(t) := \max_{1 \leq i \leq N} |v_i(t)|. \end{eqnarray*} Again, we observe that \eqref{eq:CS} does not conserve the global momentum $\frac{1}{N}\sum_{i=1}^N v_i$. Finally, we introduce the quantity \( \label{def:Psi}
\Psi(r) := \min_{s\in [0,r]} \psi(s). \)
\begin{theorem}\label{thm:CS} Let $N\geq 3$ and let the assumptions \eqref{ass:psi} on $\psi=\psi(s)$ and \eqref{ass:tau} on $\tau=\tau(t)$ be verified. Moreover, assume that there exists $C\in (0,1)$ such that \( \label{ass:C}
1-C = \left( 1- \frac{N-2}{N-1} \Psi\left(\overline\tau R_v^0 + d_x^0 + \frac{d_v^0}{C} \right)\right) e^{C\overline\tau}, \) with \( \label{Rdd0}
R_v^0 := \max_{s\in [-\overline\tau,0]} R_v(s), \qquad
d_x^0 := \max_{s\in [-\overline\tau,0]} d_x(s), \qquad
d_v^0 := \max_{s\in [-\overline\tau,0]} d_v(s). \) Then the solution $(x(t),v(t))$ of \eqref{eq:CS} subject to the initial datum \eqref{IC:CS} exhibits asymptotic flocking as defined by \eqref{flocking}. Moreover, the decay of the velocity diameter $d_v=d_v(t)$ to zero as $t\to\infty$ is exponential with rate $C$. \end{theorem}
Let us note that the assumption \eqref{ass:C}, although it may seem technical, is in fact very natural in the context of the Cucker-Smale model. Indeed, for the influence function \( \label{psi:CS}
\psi(s) = \frac{1}{(1+s^2)^\beta} \qquad\mbox{for } s\geq 0, \) which was considered in the original works \cite{CS1, CS2}, assumption \eqref{ass:C} is satisfied whenever $\beta < 1/2$, regardless of the particular values of $R_v^0$, $d_x^0$ and $d_v^0$. This is precisely the setting that leads to \emph{unconditional flocking} in the original Cucker-Smale model, and $\beta>1/2$ asymptotic flocking may or may not take place, depending on the initial datum. From this point of view, our result is very close to optimal. We formulate it as the following corollary and prove it in Section \ref{sec:CS}.
\begin{corollary}\label{corr:CS} Consider the influence function given by \eqref{psi:CS} with $\beta<1/2$. Then all solutions of \eqref{eq:CS} exhibit asymptotic flocking as defined by \eqref{flocking}. \end{corollary}
The last part of the paper is devoted to the proof of asymptotic consensus and flocking in the mean-field limits of the Hegselmann-Krause model \eqref{eq:HK} and the Cucker-Smale system \eqref{eq:CS}. Letting $N\to\infty$ in \eqref{eq:HK} leads to the conservation law \( \label{mf:HK}
\partial_t f + \nabla_x\cdot (F[f] f) = 0, \) for the time-dependent probability measure $f=f(t,x)$ which describes the probability of finding an agent at time $t\geq 0$ located at $x\in\mathbb{R}^d$. The operator $F=F[f]$ is defined as \( \label{F}
F[f](t,x) := \frac{\int_{\mathbb{R}^d} \psi(|x-y|) (y-x) f(t-\tau(t),y) \mathrm{d} y}{\int_{\mathbb{R}^d} \psi(|x-y|) f(t-\tau(t),y) \mathrm{d} y}. \) The system \eqref{mf:HK}--\eqref{F} is equipped with the initial datum $f(t) = f^0(t)$, $t\in [-\overline\tau,0]$, with $f^0 \in C([-\overline\tau,0], \mathbb{P}(\mathbb{R}^d))$, where $\mathbb{P}(\mathbb{R}^d)$ denotes the set of probability measures on $\mathbb{R}^d$. We shall assume that the initial datum is uniformly compactly supported, i.e., \( \label{IC:mf:HK}
\sup_{s\in [-\overline\tau,0]} d_x[f^0(s)] < \infty, \) where the diameter $d_x[h]$ for a probability measure $h\in \mathbb{P}(\mathbb{R}^d)$ is defined as \[
d_x[h] := \sup \{ |x-y|,\, x,y\in \mathrm{supp}\, h \}. \end{eqnarray*} In Section \ref{sec:MF} we shall prove the following theorem, which is a direct consequence of a stability estimate in terms of the Monge-Kantorowich-Rubinstein distance, combined with the fact that the consensus estimates derived in Section \ref{sec:HK} are uniform with respect to the number of agents $N\in\mathbb{N}$.
\begin{theorem}\label{thm:mf:HK} Let the assumptions \eqref{ass:psi} on $\psi=\psi(s)$ and \eqref{ass:tau} on $\tau=\tau(t)$ be verified. Then all solutions $f=f(t)$ of \eqref{mf:HK} with compactly supported initial datum \eqref{IC:mf:HK} reach global asymptotic consensus in the sense \[
\lim_{t\to\infty} d_x[f(t)] = 0, \end{eqnarray*} and the decay is exponential. \end{theorem}
The mean-field limit of the Cucker-Smale system \eqref{eq:CS} is given, in its strong formulation, by the kinetic equation \( \label{mf:CS}
\partial_t g + v\cdot\nabla_x g + \nabla_v\cdot (G[g] g) = 0, \) with \( \label{G}
G[g](t,x) := \frac{\int_{\mathbb{R}^d}\int_{\mathbb{R}^d} \psi(|x-y|) (w-v) f(t-\tau(t),y,w) \mathrm{d} y\mathrm{d} w}{\int_{\mathbb{R}^d}\int_{\mathbb{R}^d} \psi(|x-y|) f(t-\tau(t),y,w) \mathrm{d} y\mathrm{d} w}. \) Here the time-dependent probability measure $g=g(t,x,v)$ describes the probability of finding an agent at time $t\geq 0$ located at $x\in\mathbb{R}^d$ with velocity $v\in\mathbb{R}^d$. The initial datum $g(t) = g^0(t)$, $t\in [-\overline\tau,0]$, with $g^0 \in C([-\tau,0], \mathbb{P}(\mathbb{R}^d\times\mathbb{R}^d))$, is again assumed to be uniformly compactly supported in the sense that there exists a compact set $K\subset \mathbb{R}^d\times\mathbb{R}^d$ such that \( \label{IC:mf:CS}
\mathrm{supp}\, g^0(t) \subset K \qquad\mbox{for all } t\in[-\overline\tau,0]. \) With a slight abuse of notation, define the position and velocity diameters of $\mathrm{supp}\, g(t)$ as \[
d_x[g(t)] := \sup \{ |x-y|,\, x,y\in \mathrm{supp}_x\, g(t) \}, \qquad
d_v[g(t)]:= \sup \{ |v-w|,\, v,w\in \mathrm{supp}_v\, g(t) \}, \end{eqnarray*} where $ \mathrm{supp}_x\, g$ and, resp., $ \mathrm{supp}_v\, g$, are projections of $ \mathrm{supp}\, g$ onto the $x$- and, resp., $v$-variables. The velocity radius is defined as \[
R_v[g(t)] := \sup \{ |w|,\, w\in \mathrm{supp}_v\, g(t) \}. \end{eqnarray*} We then have the following flocking result.
\begin{theorem}\label{thm:mf:CS} Let the assumptions \eqref{ass:psi} on $\psi=\psi(s)$ and \eqref{ass:tau} on $\tau=\tau(t)$ be verified. Moreover, assume that there exists $C\in (0,1)$ such that \( \label{ass:mf:C}
1-C = \left( 1- \Psi\left(\overline\tau R_v^0 + d_x^0 + \frac{d_v^0}{C} \right)\right) e^{C\overline\tau}, \) with $\Psi$ defined in \eqref{def:Psi} and \( \label{Rdd0}
R_v^0 := \max_{s\in [-\overline\tau,0]} R_v[g^0(s)], \qquad
d_x^0 := \max_{s\in [-\overline\tau,0]} d_x[g^0(s)], \qquad
d_v^0 := \max_{s\in [-\overline\tau,0]} d_v[g^0(s)]. \) Then the solution of \eqref{mf:CS} subject to the compactly supported initial datum \eqref{IC:mf:CS} exhibits asymptotic flocking in the sense \[
\lim_{t\to\infty} d_v[g(t)] = 0, \qquad \sup_{t\geq 0} d_x[g(t)] < \infty. \end{eqnarray*} Moreover, the decay of the velocity diameter to zero as $t\to\infty$ is exponential with rate $C$. \end{theorem}
Again, we note that if the influence function is of the form $\psi(s) = (1+s^2)^{-\beta}$, assumption \eqref{ass:mf:C} is satisfied whenever $\beta<1/2$, regardless of the values of $\overline\tau$, $R_v^0$, $d_x^0$ and $d_v^0$.
\section{Asymptotic consensus for the Hegselmann-Krause model \eqref{eq:HK}}\label{sec:HK} We define the radius of the group $R_x=R_x(t)$ by \( \label{def:diam}
R_x(t) := \max_{1\leq i\leq N} |x_i(t)|. \) The following lemma shows that the radius is bounded uniformly in time by the radius of the initial datum, defined as \( \label{def:R0}
R_x^0 := \max_{t\in [-\overline\tau,0]} R_x(t). \)
\begin{lemma}\label{lem:Rxbound} Let the initial datum $x^0\in C([-\overline\tau,0];\mathbb{R}^d)^N$ and let $R_x^0$ be given by \eqref{def:R0}. Then, along the solutions of \eqref{eq:HK}--\eqref{IC:HK}, the diameter $R_x=R_x(t)$ defined in \eqref{def:diam} satisfies \[
R_x(t) \leq R_x^0 \qquad\mbox{for all } t\geq 0. \end{eqnarray*} \end{lemma}
The proof of a slight generalization of \cite[Lemma 2.2]{ChoiH1} for the case of variable time delay. We give it here for the sake of the reader.
\begin{proof} Let us fix some $\varepsilon >0$. We shall prove that for all $t\geq 0$ \( \label{R-eps}
R_x(t) < R_x^0 + \varepsilon. \) Obviously, $R_x(0) \leq R_x^0$, so that by continuity, \eqref{R-eps} holds on the maximal interval $[0,T)$ for some $T>0$. For contradiction, let us assume that $T<+\infty$. Then we have \( \label{R-cont}
\lim_{t\to T-} R_x(t) = R_x^0 + \varepsilon. \)
However, for any $i=1,\dots,N$, we have \[
\frac12 \tot{}{t} |x_i(t)|^2 &=& \sum_{j=1}^N \psi_{ij}(t) \left[x_j(t-\tau(t))- x_i(t)\right]\cdot x_i(t) \\
&=& \sum_{j=1}^N \psi_{ij}(t) \left[ x_j(t-\tau(t))\cdot x_i(t) - |x_i(t)|^2\right]. \end{eqnarray*}
By definition, we have $|x_j(t-\tau(t))| < R_x^0 + \varepsilon$ for all $j\in\{1,\dots,N\}$ and $t < T$, so that with \eqref{psi:conv} and an application of the Cauchy-Schwarz inequality we arrive at \[
\frac12 \tot{}{t} |x_i(t)|^2 \leq \left(R_x^0 + \varepsilon\right) |x_i(t)| - |x_i(t)|^2. \end{eqnarray*}
Now, if $|x_i(t)| \neq 0$, we use the identity $\frac12\tot{|x_i(t)|^2}{t} = |x_i(t)| \tot{|x_i(t)|}{t}$
and divide the above inequality by $|x_i(t)|$. On the other hand, if $|x_i(t)| \equiv 0$ on an open subinterval of $(0,T)$, then $\tot{|x_i(t)|}{t} \equiv 0 \leq R^0_x + \varepsilon - |x_i(t)|$ on this subinterval. Thus, we obtain \[
\tot{}{t} |x_i(t)| \leq R^0_x + \varepsilon - |x_i(t)| \quad\mbox{a.e. on } (0,T), \end{eqnarray*} which implies \[
|x_i(t)| \leq \left(|x_i(0)| - (R^0_x + \varepsilon) \right)e^{-t} + R^0_x + \varepsilon \qquad\mbox{for } t < T. \end{eqnarray*}
Consequently, with $|x_i(0)| \leq R_x^0$, \[
\lim_{t \to T-} \; \max_{1 \leq i \leq N} |x_i(t)| \leq -\varepsilon e^{-T} + R_x^0 + \varepsilon < R^0_x + \varepsilon, \end{eqnarray*} which is a contradiction to \eqref{R-cont}. We conclude that, indeed, $T = \infty$, and complete the proof by taking the limit $\varepsilon\to 0$. \end{proof}
The following geometric result is based on the convexity property \eqref{psi:conv} of the renormalized communication weights \eqref{psi}.
\begin{lemma}\label{lem:geom} Let $N\geq 3$ and $\{x_1, \dots, x_N\}\subset \mathbb{R}^d$ be any set of vectors in $\mathbb{R}^d$ and denote $d_x$ its diameter,
$$d_x:=\max_{1 \leq i,j \leq N}|x_i - x_j|.$$ Fix $i, k \in \{1,2,\cdots,N\}$ such that $i\neq k$ and let $\eta^i_j \geq 0$ for all $j\in\{1,2,\cdots,N\} \setminus\{i\}$, and $\eta^k_j \geq 0$ for all $j\in\{1,2,\cdots,N\} \setminus\{k\}$, such that \[
\sum_{j\neq i} \eta^i_j = 1,\qquad \sum_{j\neq k} \eta^k_j = 1. \end{eqnarray*} Let $\mu\geq 0$ be such that \[
0 \leq \mu \leq \min \left\{ \min_{j\neq i} \eta^i_j,\; \min_{j\neq k} \eta^k_j \right\}. \end{eqnarray*} Then \( \label{est:geom}
\left| \sum_{j\neq i} \eta^i_j x_j - \sum_{j\neq k} \eta^k_j x_j \right| \leq (1-(N-2)\mu) d_x. \) \end{lemma}
\begin{proof} For $j,\ell \in\{1,2,\cdots,N\} \setminus\{i\}$ we set $\xi^i_{j\ell}:=\mu$ if $\ell\neq j$ and $\xi^i_{jj}:=1 - (N-2)\mu$. Moreover, define \( \label{xji}
\bar x_j^i := \sum_{\ell\neq i} \xi^i_{j\ell} x_\ell \qquad\mbox{for } j \in \{1,2,\cdots,N\} \setminus\{i\}. \) Then the vector $\sum_{j\neq i} \eta^i_j x_j$ can be written as the convex combination of the vectors $\bar x_j^i$, \( \label{cc1}
\sum_{j\neq i} \eta^i_j x_j = \sum_{j\neq i} \lambda_j^i \bar x_j^i \) with the coefficients \[
\lambda_j^i = \frac{\eta^i_j - \mu}{1-(N-1)\mu} \geq 0, \qquad \sum_{j\neq i} \eta^i_j = 1. \end{eqnarray*} Similarly, for $j,\ell \in\{1,2,\cdots,N\} \setminus\{k\}$ we set $\xi^k_{j\ell}:=\mu$ if $\ell\neq j$ and $\xi^k_{jj}:=1 - (N-2)\mu$. Define \( \label{xjk}
\bar x_j^k := \sum_{\ell\neq k} \xi^k_{j\ell} x_\ell \qquad\mbox{for } j \in \{1,2,\cdots,N\} \setminus\{k\}, \) and again observe that the vector $\sum_{j\neq k} \eta^k_j x_j$ can be written as the convex combination of the vectors $\bar x_j^k$, \( \label{cc2}
\sum_{j\neq k} \eta^k_j x_j = \sum_{j\neq k} \lambda_j^k \bar x_j^k \) with the coefficients \[
\lambda_j^k = \frac{\eta^k_j - \mu}{1-(N-1)\mu} \geq 0, \qquad \sum_{j\neq k} \lambda^k_j = 1. \end{eqnarray*} Obviously, due to \eqref{cc1} and \eqref{cc2}, the estimate \eqref{est:geom} is verified as soon as we prove \[
\left|\bar x_a^i - \bar x_b^k\right| \leq (1-(N-2)\mu) d_x \end{eqnarray*} for all $a\in\{1,2,\cdots,N\} \setminus\{i\}$ and $b\in\{1,2,\cdots,N\} \setminus\{k\}$. With \eqref{xji} and \eqref{xjk} we have \[
\bar x_a^i - \bar x_b^k &=& \sum_{\ell\neq i} \xi^i_{a\ell} x_\ell - \sum_{\ell\neq k} \xi^k_{b\ell} x_\ell \\
&=& \mu \sum_{\ell\neq i, a} x_\ell + (1-(N-2)\mu) x_a - \mu \sum_{\ell\neq k, b} x_\ell - (1-(N-2)\mu) x_b \\
&=& \mu (x_k-x_i) + (1-(N-1)\mu) (x_a-x_b). \end{eqnarray*} With the triangle inequality we then readily obtain \[
\left| \bar x_a^i - \bar x_b^k \right| \leq (1-(N-2)\mu) d_x. \end{eqnarray*} \end{proof}
The following Gronwall-Halanay type inequality is a generalization of \cite[Lemma 2.5]{ChoiH1} for variable time delay.
\begin{lemma}\label{lem:GH} Let $\tau=\tau(t)$ satisfy the assumptions \eqref{ass:tau}. Let $u\in C([-\overline\tau,\infty))$ be a nonnegative continuous function with piecewise continuous derivative on $(0,\infty)$, such that for some constant $a \in (0,1)$ the differential inequality is satisfied, \begin{equation}\label{diff_ineq}
\tot{}{t} u(t) \leq (1-a) u(t -\tau(t)) - u(t) \qquad\mbox{for almost all } t>0. \end{equation} Then there exists a unique solution $C\in (0,a)$ of the equation \begin{equation}\label{ass_C}
1 - C = (1-a)e^{C\overline\tau} \end{equation} and the estimate holds \( \label{Gronwall-like}
u(t) \leq \left( \max_{s\in[-\overline\tau,0]} u(s) \right) e^{-Ct} \qquad \mbox{for all } t \geq 0. \) \end{lemma}
\begin{proof} We denote \( \label{def:baru}
\bar u := \max_{s\in[-\overline\tau,0]} u(s), \qquad w(t) := \bar u e^{-Ct}, \) and for any fixed $\lambda > 1$ set \[
\mathcal{S}_\lambda := \left\{ t \geq 0 : u(s) \leq \lambda w(s) \quad \mbox{for} \quad s \in [0,t)\right\}. \end{eqnarray*} Since $0 \in \mathcal{S}_\lambda$, $T_\lambda := \sup \mathcal{S}_\lambda \geq 0$ exists. We claim that \[
T_\lambda = \infty \qquad \mbox{for any } \lambda >1. \end{eqnarray*} For contradiction, assume $T_\lambda < \infty$ for some $\lambda >1$. Then clearly $\tau(T_\lambda) > 0$, since otherwise we would have \[
\tot{}{t+} u(t) \leq \lim_{t\to T_\lambda+} (1-a) u(t -\tau(t)) - u(t) = - a u(T_\lambda) = - a \lambda w(T_\lambda) < 0, \end{eqnarray*} which contradicts the definition of $T_\lambda$. Therefore, due to the continuity of $u=u(t)$ and $\tau=\tau(t)$, there exists some $T_\lambda^* > T_\lambda$ such that $u$ is differentiable at $T_\lambda^*$ and \( \label{est_derivatives}
u(T_\lambda^*) > \lambda w(T_\lambda^*),\qquad \tot{}{t} u(T_\lambda^*) > \lambda \tot{}{t} w(T_\lambda^*),\qquad
\tau(T_\lambda^*) > T_\lambda^*-T_\lambda. \) Note that $w$ satisfies \begin{equation}\label{est_g}
w(t-\tau(t)) = e^{C\tau(t)}w(t) \quad \mbox{and} \quad \tot{}{t} w(t) = - C w(t), \end{equation} for all $t>0$. Moreover, it follows from \eqref{diff_ineq} that \begin{equation}\label{est_diff}
\tot{}{t} u(T_\lambda^*) \leq (1 - a) u(T_\lambda^* - \tau(T_\lambda^*)) - u(T_\lambda^*). \end{equation} We now consider the following two cases: \begin{itemize} \item If $T_\lambda^* - \tau(T_\lambda^*) \leq 0$, then by \eqref{def:baru} we have $u(T_\lambda^* - \tau(T_\lambda^*)) \leq \bar u$ and $u(T_\lambda^*) > \lambda w(T_\lambda^*)$ by \eqref{est_derivatives}, so that we estimate the right-hand side of \eqref{est_diff} by $$ \begin{aligned}
\tot{}{t} u(T_\lambda^*) &\leq (1-a) \bar u - u(T_\lambda^*)\cr
&< (1-a) \lambda w(0) - \lambda w(T_\lambda^*)\cr
&\leq \left((1 - a)e^{C\overline\tau} - 1\right) \lambda w(T_\lambda^*). \end{aligned} $$ For the third line we used the inequality $w(0) \leq w(T_\lambda^*)e^{C\overline\tau}$ implied by $T_\lambda^* \leq \tau(T_\lambda^*) \leq \overline\tau$. With the identities \eqref{ass_C} and \eqref{est_g}, we obtain $$
\tot{}{t} u(T_\lambda^*) < - C \lambda w(T_\lambda^*) = \lambda \tot{}{t} w(T_\lambda^*), $$ which is a contradiction to \eqref{est_derivatives}.
\item If $T_\lambda^* - \tau(T_\lambda^*) > 0$, we have $u(T_\lambda^* - \tau(T_\lambda^*)) \leq \lambda w(T_\lambda^* - \tau(T_\lambda^*))$ since, due to \eqref{est_derivatives}, $T_\lambda^* - \tau(T_\lambda^*) < T_\lambda$. Then \eqref{est_diff} gives \[
\tot{}{t} u(T_\lambda^*) &\leq& (1-a) \lambda w(T_\lambda^* - \tau(T_\lambda^*)) - \lambda w(T_\lambda^*)\\
&=& \left((1 - a)e^{C\tau(T_\lambda^*)} - 1\right) \lambda w(T_\lambda^*) \\
&\leq& \left((1 - a)e^{C\overline\tau} - 1\right) \lambda w(T_\lambda^*). \end{eqnarray*} Then, using the same argument as in the previous case, we obtain a contradiction to \eqref{est_derivatives}. \end{itemize} We conclude that, for every $\lambda>1$, $T_\lambda = \infty$ and $u(t) \leq \lambda w(t)$ for all $t\geq 0$. Passing to the limit $\lambda \to 1$ yields the claim \eqref{Gronwall-like}. \end{proof}
We are now prepared to prove Theorem \ref{thm:HK}.
\begin{proof} For the sake of legibility, let us introduce the shorthand notation $\widetilde x_j := x_j(t-\tau(t))$, while $x_j$ means $x_j(t)$.
The uniform bound on the radius $R_x=R_x(t)$ of the solution provided by Lemma \ref{lem:Rxbound} gives for all $i,j\in \{1,2,\cdots,N\}$, \[
|\widetilde x_j - x_i| \leq |\widetilde x_j| + |\widetilde x_i| \leq 2R_x^0 \qquad\mbox{for all } t\geq 0. \end{eqnarray*} Consequently, defining \( \label{def:upsi}
\underline{\psi} := \min_{s\in[0,2R_x^0]} \psi(s), \)
we have $\psi(|\widetilde x_j - x_i|) \geq \underline{\psi}$, and \( \label{psibound}
\psi_{ij} = \frac{\psi(|\widetilde x_j - x_i|)}{\sum_{\ell\neq i} \psi(|\widetilde x_\ell - x_i|)} \geq \frac{\underline{\psi}}{N-1}. \) Note that due to the assumption \eqref{ass:psi}, we have $\underline{\psi} >0$.
Due to the continuity of the trajectories $x_i=x_i(t)$, there is an at most countable system of open, mutually disjoint intervals $\{\mathcal{I}_\sigma\}_{\sigma\in\mathbb{N}}$ such that $$
\bigcup_{\sigma\in\mathbb{N}} \overline{\mathcal{I}_\sigma} = [0,\infty) $$ and for each ${\sigma\in\mathbb{N}}$ there exist indices $i(\sigma)$, $k(\sigma)$ such that $$
d_x(t) = |x_{i(\sigma)}(t) - x_{k(\sigma)}(t)| \quad\mbox{for } t\in \mathcal{I}_\sigma. $$ Then, using the abbreviated notation $i:=i(\sigma)$, $k:=k(\sigma)$, we have for every $t\in \mathcal{I}_\sigma$, \[
\frac12 \tot{}{t} d_x(t)^2 &=& (\dot x_i - \dot x_k)\cdot (x_i-x_k)\\
&=&
\left(\sum_{j\neq i} \psi_{ij} (\widetilde x_j - x_i) - \sum_{j\neq k} \psi_{kj} (\widetilde x_j - x_k) \right) \cdot (x_i-x_k)\\
&=&
\left( \sum_{j\neq i} \psi_{ij} \widetilde x_j - \sum_{j\neq k} \psi_{kj} \widetilde x_j \right) \cdot (x_i-x_k) - |x_i-x_k|^2, \end{eqnarray*} where we used the convexity property of the renormalized weights \eqref{psi:conv}. We now use \eqref{psibound} and Lemma \ref{lem:geom} with $\mu:=\frac{\underline{\psi}}{N-1}$, which gives \[
\left| \sum_{j\neq i} \psi_{ij} \widetilde x_j - \sum_{j\neq k} \psi_{kj} \widetilde x_j \right| \leq (1-(N-2)\mu) d_x(t-\tau(t)). \end{eqnarray*} Consequently, with the Cauchy-Schwartz inequality we have \[
\frac12 \tot{}{t} d_x(t)^2 \leq (1-(N-2)\mu) d_x(t-\tau) d_x(t) - d_x(t)^2, \end{eqnarray*} which implies that for almost all $t>0$, \[
\tot{}{t} d_x(t) \leq (1-(N-2)\mu) d_x(t-\tau(t)) - d_x(t). \end{eqnarray*} An application of Lemma \ref{lem:GH} with $a:=(N-2)\mu = \frac{N-2}{N-1}\underline{\psi} \in (0,1)$ gives then the exponential decay \[
d_x(t) \leq \left( \max_{s\in[-\overline\tau,0]} d_x(s) \right) e^{-Ct} \qquad \mbox{for } t \geq 0, \end{eqnarray*} where $C$ is the unique solution of \eqref{ass_C}. We note that $a$ increases with increasing $N$ (if $\underline{\psi}$ is held constant), and so does $C$. Consequently, the exponential decay rate $C$ improves with increasing $N$. \end{proof}
\section{Asymptotic flocking for the Cucker-Smale model}\label{sec:CS} The method develop in Section \ref{sec:HK} can be easily extended for the Cucker-Smale model \eqref{eq:CS}, as we demonstrate in the proof of Theorem \ref{thm:CS} below.
\begin{proof} First, note that the proof of Lemma \ref{lem:Rxbound} applies mutatis mutandis for the velocity variable in \eqref{eq:CS}, providing the uniform bound \( \label{Rvbound}
R_v(t) := \max_{1\leq i\leq N} |v_i(t)| \leq R_v^0 \qquad\mbox{for all } t\geq 0, \) with $R_v^0 := \max_{t\in [-\overline\tau,0]} R_v(t)$.
Let $C\in (0,1)$ be given by \eqref{ass:C}. With $d_v^0>0$ given by \eqref{Rdd0} and due to the continuity of $d_v=d_v(t)$, there exists some $T>0$ such that \( \label{CSbound}
\int_0^t d_v(s) \mathrm{d} s < \frac{d_v^0}{C} \qquad \mbox{for all } t<T. \) We claim that $T=\infty$. For contradiction, assume that \eqref{CSbound} holds only until some finite $T>0$. Then we have \( \label{forContr}
\int_0^T d_v(s) \mathrm{d} s = \frac{d_v^0}{C}. \) By the first equation of \eqref{eq:CS} we readily have \[
d_x(t) \leq d_x^0 + \int_0^t d_v(s) \mathrm{d} s, \end{eqnarray*} so that \eqref{CSbound} implies for all $t<T$, \[
d_x(t) \leq d_x^0 + \int_0^t d_v(s) \mathrm{d} s < d_x^0 + \frac{d_v^0}{C}. \end{eqnarray*} Moreover, using the estimate \[
|\widetilde x_j - x_j| = \left| \int_{t-\tau(t)}^t \dot x_j(s) \mathrm{d} s \right| \leq \int_{t-\tau(t)}^t |v_j(s)| \mathrm{d} s \leq \overline\tau R_v^0, \end{eqnarray*} provided by \eqref{Rvbound}, we have for any $i, j \in \{1,2,\cdots,N\}$, $i\neq j$, \[
|\widetilde x_j - x_i| \leq |\widetilde x_j - x_j| + |x_j-x_i| \leq \overline\tau R_v^0 + d_x(t) \leq \overline\tau R_v^0 + d_x^0 + \frac{d_v^0}{C}. \end{eqnarray*} Then by the definition \eqref{def:Psi} of $\Psi$, \[
\psi(|\widetilde x_j - x_i|) \geq \Psi\left( \overline\tau R_v^0 + d_x^0 + \frac{d_v^0}{C} \right), \end{eqnarray*} and by the universal bound $\psi\leq 1$, \( \label{psiboundCS}
\psi_{ij} = \frac{\psi(|\widetilde x_j - x_i|)}{\sum_{\ell\neq i} \psi(|\widetilde x_\ell - x_i|)} \geq \frac{1}{N-1} \Psi\left( \overline\tau R_v^0 + d_x^0 + \frac{d_v^0}{C} \right). \)
Similarly as in the proof of Theorem \ref{thm:HK}, we have for $t<T$ such that $d_v=|v_i-v_k|$ on some neighborhood of $t$, \( \nonumber
\frac12 \tot{}{t} d_v(t)^2 &=& \left( \sum_{j\neq i} \psi_{ij} \widetilde v_j - \sum_{j\neq k} \psi_{kj} \widetilde v_j \right) \cdot (v_i-v_k) - |v_i-v_k|^2 \\
&\leq& \left| \sum_{j\neq i} \psi_{ij} \widetilde v_j - \sum_{j\neq k} \psi_{kj} \widetilde v_j \right| d_v(t) - d_v(t)^2.
\label{ddtdv} \) We now use \eqref{psiboundCS} and Lemma \ref{lem:geom} with $\mu:=\frac{1}{N-1}\Psi\left( \overline\tau R_v^0 + d_x^0 + \frac{d_v^0}{C} \right)$, which gives \[
\left| \sum_{j\neq i} \psi_{ij} \widetilde v_j - \sum_{j\neq k} \psi_{kj} \widetilde v_j \right| \leq (1-(N-2)\mu) d_v(t-\tau(t)). \end{eqnarray*} Consequently, \eqref{ddtdv} implies that for almost all $t>0$,
\[
\tot{}{t} d_v(t) \leq (1-(N-2)\mu) d_v(t-\tau(t)) - d_v(t). \end{eqnarray*} An application of Lemma \ref{lem:GH} with \[
a := (N-2)\mu = \frac{N-2}{N-1} \psi\left( \overline\tau R_v^0 + d_x^0 + \frac{d_v^0}{C} \right), \end{eqnarray*} recalling \eqref{ass:C}, gives \( \label{decay:dv}
d_v(t) \leq d_v^0 e^{-Ct} \qquad\mbox{for all } t<T. \) But then \[
\int_0^T d_v(s) \mathrm{d} s \leq d_v^0 \int_0^T e^{-Cs} \mathrm{d} s = \frac{d_v^0}{C} \left( 1 - e^{-CT} \right) < \frac{d_v^0}{C}, \end{eqnarray*} which is a contradiction to \eqref{forContr}. Thus, we conclude that $T=\infty$, i.e., that \eqref{CSbound} holds for all $t>0$. Then also \eqref{decay:dv} holds for all $t>0$, and, moreover, $d_x=d_x(t)$ is uniformly bounded by $d_x^0 + \frac{d_v^0}{C}$. \end{proof}
Finally, we provide the proof of Corollary \ref{corr:CS}. Considering the monotone communication rate function \[
\psi(s) = \frac{1}{(1+s^2)^\beta} \end{eqnarray*} with $\beta\geq 0$, we obviously have $\Psi(r) = \psi(r)$ for all $r\geq 0$, with $\Psi(r)$ defined in \eqref{def:Psi}. Then it is straightforward to calculate \[
\lim_{C\to 0+} \left( 1- \frac{N-2}{N-1}\Psi\left(\overline\tau R_v^0 + d_x^0 + \frac{d_v^0}{C} \right)\right) e^{C\overline\tau} = 1, \end{eqnarray*} for any positive values of $R_v^0$, $d_x^0$, $d_v^0$ and $\overline\tau$. On the other hand, the above expression is strictly positive for $C=1$. Therefore, \eqref{ass:C} is solvable with some $C\in(0,1)$ as soon as \[
\tot{}{C} \left[ \left( 1- \frac{N-2}{N-1}\psi\left(\overline\tau R_v^0 + d_x^0 + \frac{d_v^0}{C} \right)\right) e^{C\overline\tau} \right]_{C=0+} < -1. \end{eqnarray*} A simple calculation reveals that this is the case if $\beta<1/2$.
\section{Consensus and flocking in the mean-field limit}\label{sec:MF} Our results for the mean-field limit systems \eqref{mf:HK}--\eqref{F} and \eqref{mf:CS}--\eqref{G} are based on the well-posedness theory in measures developed in \cite[Section 3]{ChoiH1}. In particular, existence and uniqueness of measure-valued solutions for the Cucker-Smale system \eqref{mf:CS}--\eqref{G} was proved there, together with continuous dependence on the initial datum. The proof uses the framework developed in \cite{CCR} and is based on local Lipschitz continuity of the operator $F=F[f]$ given by \eqref{F}. Without going into details, we note that the proof can be easily adapted to provide analogous results for the Hegselmann-Krause system \eqref{mf:HK}--\eqref{F}. Instead, we merely restate the stability results in Wasserstein distance for the two systems, which are essential for our proof of asymptotic consensus and flocking. For their proof we refer to \cite[Theorem 3.6]{ChoiH1}.
\begin{theorem}\label{thm:stabHK} Let $f_1, f_2 \in C([0, T];\mathbb{P}(\mathbb{R}^{d}))$ be two measure-valued solutions of \eqref{mf:HK}--\eqref{F} on the time interval $[0, T]$, subject to the compactly supported initial data $f_1^0, f_2^0 \in C([-\overline\tau, 0];\mathbb{P}_1(\mathbb{R}^{d}))$. Then there exists a constant $L=L(T)$ such that \[
\mathcal{W}_1 (f_1(t),f_2(t)) \leq L \max_{s\in[-\overline\tau,0]} \mathcal{W}_1(f^0_1(s),f^0_2(s)) \quad \mbox{for} \quad t \in [0,T], \end{eqnarray*} where $\mathcal{W}_1(f_1(t),f_2(t))$ denotes the 1-Wasserstein (or Monge-Kantorovich-Rubinstein) distance \cite{Villani} of the probability measures $f_1(t)$, $f_2(t)$. \end{theorem}
\begin{theorem}\label{thm:stabCS} Let $g_1, g_2 \in C([0, T];\mathbb{P}(\mathbb{R}^{d}\times\mathbb{R}^{d}))$ be two measure-valued solutions of \eqref{mf:CS}--\eqref{G} on the time interval $[0, T]$, subject to the compactly supported initial data $g_1^0, g_2^0 \in C([-\overline\tau, 0];\mathbb{P}_1(\mathbb{R}^{d}\times\mathbb{R}^{d}))$. Then there exists a constant $L=L(T)$ such that \[
\mathcal{W}_1 (g_1(t),g_2(t)) \leq L \max_{s\in[-\overline\tau,0]} \mathcal{W}_1(g^0_1(s),g^0_2(s)) \quad \mbox{for} \quad t \in [0,T]. \end{eqnarray*} \end{theorem}
We are now in position to provide a proof of Theorem \ref{thm:mf:HK}.
\begin{proof} Fixing an initial datum $f^0 \in C([-\overline\tau,0], \mathbb{P}(\mathbb{R}^d))$, uniformly compactly supported in the sense of \eqref{IC:mf:HK}, we construct $\{f^0_N\}_{N\in\mathbb{N}}$ a family of $N$-particle approximations of $f^0$, i.e., \[
f^0_N(s,x) := \frac{1}{N} \sum_{i=1}^N \delta(x-x^0_i(s)) \qquad\mbox{for } s\in[-\overline\tau,0], \end{eqnarray*} where the $x_i^0\in C([-\overline\tau,0];\mathbb{R}^d)$ are chosen such that \[
\max_{s\in[-\overline\tau,0]} \mathcal{W}_1(f^0_N(s),f^0(s)) \to 0 \quad\mbox{as}\quad N\to\infty. \end{eqnarray*} Denoting then $x^N_i=x^N_i(t)$ the solution of the discrete Hegselmann-Krause system \eqref{eq:HK} subject to the initial datum $x_i^0=x_i^0(s)$, $i=1,\dots,N$, the proof of Theorem \ref{thm:HK} gives exponential convergence to global consensus, i.e., \[
d_x(t) \leq \left( \max_{s\in[-\overline\tau,0]} d_x(s) \right) e^{-C_N t} \qquad \mbox{for } t \geq 0, \end{eqnarray*} with the diameter $d_x$ defined in \eqref{def:dX}, $C_N$ the unique solution of \eqref{ass_C} with $a_N := \frac{N-2}{N-1}\underline{\psi} \in (0,1)$ and $\underline{\psi}$ given by \eqref{def:upsi}. Note that $C_N$ increases with $N$ and $\lim_{N\to\infty} C_N = C$, with $C$ the unique solution of \eqref{ass_C} with $a:=\underline{\psi}$.
It is easy to check that the empirical measure \[
f^N(t,x) := \frac{1}{N} \sum_{i=1}^N \delta(x-x^N_i(t)) \end{eqnarray*} is a measure valued solution of \eqref{eq:HK}. For any fixed $T>0$, Theorem \ref{thm:stabHK} provides the stability estimate \[
\mathcal{W}_1 (f(t),f^N(t)) \leq L \max_{s\in[-\overline\tau,0]} \mathcal{W}_1 (f^0(s),f^0_N(s)) \qquad \mbox{for} \quad t \in [0,T), \end{eqnarray*} where the constant $L>0$ is independent of $N$. Thus, fixing $T>0$ and letting $N\to\infty$ implies $d_x[f(t)] = d_x(t)$ on $[0,T)$, and, consequently, \[
d_x[f(t)] \leq \left( \max_{s\in[-\overline\tau,0]} d_x[f^0(s)] \right) e^{-Ct} \qquad \mbox{for } t \in[0,T). \end{eqnarray*} We conclude by noting that $T>0$ can be chosen arbitrarily and that the constant $C$ is independent of time. \end{proof}
The proof of Theorem \ref{thm:mf:CS} is an obvious modification of the above proof, with stability provided by Theorem \ref{thm:stabCS}.
\section*{Acknowledgment} JH acknowledges the support of the KAUST baseline funds.
\end{document} | arXiv |
\begin{definition}[Definition:Clock/Hand]
The pointers on an '''analogue clock''' are known as its '''hands'''.
:320px
\end{definition} | ProofWiki |
\begin{document}
\title{On the existence of two interior-point methods for linear programming following two different paths, and it implications.}
\author{Adrien CHAN-HON-TONG}
\maketitle
\begin{center} \textbf{Abstract} \end{center}
Recently, a strong negative results has been proven for central-path log-barrier method, and more generally, for all methods relying on the central-path framework. But, a potential bypass is just to consider interior-point but not-central-path methods. In particular, this paper remarks the existence of two interior-point based methods for linear feasibility whose convergences can easily be argued different on toy linear programs. Despite those two methods are slightly slower than the state of the art, their differences prove that there may be alternatives to the central-path framework.
\section{Introduction and motivation} The central-path log-barrier method \cite{renegar1988polynomial} is the state-of-the-art of linear programming since 1988. It has even been recently improved with an efficient data-structure \cite{van2020deterministic} (which is a deterministic version of \cite{cohen2021solving}). Using \cite{van2020deterministic}, a linear program related to a matrix $A\in \mathbb{R}^{M\times N}$ with total binary size $L$ can be solved in less than $O(M^\omega L)$ operations (where $\omega$ is the exponent of matrix multiplication/inversion - this is the best known complexity assuming that the matrix is not too flat i.e. $N=O(M)$ and $M=O(N)$). This algorithm performs somehow $O(\sqrt{M}L)$ Newton steps (in details those Newton steps are converted into gradient step on a tricky energy) while maintaining a data structure allowing to perform all those $\sqrt{M}L$ matrix inversions in $M^\omega L$ operations.
Due to this low complexity, interior-point algorithms for linear programming are overwhelmingly based on central-path framework from the state-of-the-art. However the reign of this framework could be challenged by recent negative result \cite{allamigeon2018log}: the classical central-path log-barrier method is not strongly polynomial as $2^M$ steps could be required (in worse case) to deal with the curvatures of the central path (for large $L$). Even more, \cite{allamigeon2022no} proves that any algorithm whose current point makes line-move into the central-path is not strongly polynomial.
But, this negative result \cite{allamigeon2022no} only holds for methods following the central-path. So, it could be interesting to consider interior-point not-central-path algorithms for linear programming. Currently, \cite{chan2022new} makes the conjecture that the self-concordant Perceptron algorithm introduced in \cite{chan2022new} does not follow the central-path. Yet, this statement was not trivial as \cite{chan2022new} works with linear feasibility and not linear programming making it hard to see if self-concordant Perceptron algorithm is really disconnected to log-barrier central-path method.
The contribution of this paper is to compare a method inspired by central-path log-barrier method adapted to linear feasibility, and, the self-concordant Perceptron. The main result is that given a linear feasibility instances of $M$ vectors composed by only two populations, $R$ vectors with high margin and $M-R$ vector with low margin, then, the classical method will require $RL$ to converge while self-concordant Perceptron will require $(M-R)L$ ones. Thus, for large or small $R$, their behaviors is significantly different.
Those two behaviors show that those two algorithms does not follow the same path on those toy instances, and, thus, \cite{allamigeon2022no} can not directly apply to both simultaneously. Thus, despite those two methods are slower than \cite{van2020deterministic} (from a factor $M$), this comparison could open a way to bypass \cite{allamigeon2022no} negative result.
\section{Results} \subsection{Definitions and background results} \subsubsection{Linear feasibility} Linear feasibility is the problem of finding $x\in X_A=\{x\in \mathbb{R}^N, Ax>\mathbf{0}\}$ given $A\in \mathbb{R}^{M\times N}$ with the assumption that $X_A\neq \emptyset$.
As recalled in \cite{chan2022new}, given a linear feasibility solver, one is able to solve general linear programming with same complexity.
However, this requires many steps of pre-processing making this equivalence almost entirely theoretical. For example, linear feasibility solvers may fail on input instance which does not satisfy core assumption (like $X_A= \emptyset$). So, the pre-processing should take an input without assumption (could be unbounded or without admissible point) and cast it into a 16 times more constraint-variables matrix with four-times total binary size to ensure that the resulting $A$ matrix verifies $X_A\neq \emptyset$. Thus, despite it works theoretically, it is hard to compare an algorithm working with linear feasibility instance and an other working with linear program instance. This is why this paper considers linear feasibility only: an algorithm inspired by classical central-path log-barrier method is introduced to allow comparison with self-concordant Perceptron.
\subsubsection{Self-concordant theory} The two methods considered in this paper are based on self-concordant theory: there exists a class of functions on which Newton descent performs well. Due to the complexity of this theory, this paper only states two main results from the theory. But, a complete overview can be found in \cite{nemirovski2004interior}.
If $\Psi(x)$ is a self-concordant function (mainly any sum of quadratic, linear, constant and $-\log$ term), with a minimum $\Psi^*$, then, the Newton descent starting from $x_{start}$ allows to compute $x_\epsilon$ such as $\Psi(x_\epsilon)-\Psi^*\leq \varepsilon$ in $O(\Psi(x_{start})-\Psi^*+\log\log(\frac{1}{\varepsilon}))$ damped Newton steps. Each step is: \begin{itemize}
\item $\lambda_\Psi(x) \leftarrow \sqrt{(\nabla_x\Psi)^T(\nabla^2_x\Psi)^{-1}(\nabla_x\Psi)}$
\item $x \leftarrow x - \frac{1}{1+\lambda_\Psi(x)}(\nabla^2_x\Psi)^{-1}(\nabla_x\Psi)$ \end{itemize}
Importantly $\log$ will be abusively used instead of $\ln$ in this paper i.e. it will be considered that $\log'(t)=\frac{1}{t}$. Let stress that using big $O$ notations, it is still true that $O(\log(2^L)) = O(L)$.
\subsubsection{Technical lemmas}
\noindent \textbf{Lemma 1:} Let $f$ be the function such that $\forall t>0,\ f(t) = \frac{t^2}{2\times2^{2L}}-\log(t)$, then, $f(t)\geq -L$. Similarly, let $g(t) = t-\log(t+2^{-L-2})$, then, $g(t)\geq 1-2^{-L-2}>0$.
\noindent \textbf{Proof of lemma 1:}
$f$ is trivially lower bounded as $f(t) \underset{t\rightarrow 0 \ \mathrm{or} \ \infty}{\rightarrow}\infty$ so $f$ has a minimum which is reached when $\frac{t}{2^{2L}}-\frac{1}{t} = f'(t) =0$. So, $f^* = f(2^{L}) =\frac{1}{2} - \frac{1}{2}\log(2^L)>-L$.
Similarly $g(t) \underset{t\rightarrow -2^{-L-2} \ \mathrm{or} \ \infty}{\rightarrow}\infty$ so $g$ has a minimum which is reached when $1-\frac{1}{t+2^{-L-2}} = g'(t) =0$. So, $g^* = g(1-2^{-L-2}) = 1-2^{-L-2} - \log(1)=1-2^{-L-2}$.
\noindent \textbf{Lemma 2:} If $x\geq 4y>0$, then, $\log(y)-\log(x)\leq -1$
\noindent \textbf{Proof of lemma 2:}
$4y\leq x \Rightarrow \log(4y)\leq\log(x) \Rightarrow -\log(4y)\geq-\log(x) \Rightarrow \log(y)-\log(4y)\geq\log(y)-\log(x)\Rightarrow \log(y)-\log(4) - \log(y)\geq\log(y)-\log(x)\Rightarrow -1 \geq -\log(4)\geq\log(y)-\log(x)$ (as $e^1 \approx2.71)$.
\noindent \textbf{Lemma 3:} $\forall A \in \mathbb{Z}^{M\times N}$ with total binary size $L$ such that $X_A\neq \emptyset$, the problem $\underset{p \in \mathbb{R}^N, \mathbf{1}^TAp=1}{\max} \ \underset{m\in\{1,...,M\}}{\min} \ A_mp$ is well defined, and any optimal solutions $p^*$ verify $Ap^*\geq 2^{-L}\mathbf{1}$.
\noindent \textbf{Proof of lemma 3:}
$X_A\neq \emptyset \Rightarrow \exists z, Az>\mathbf{0}$, so $\mathbf{1}^TAz > \mathbf{1}^T\mathbf{0}=0$. So, $\frac{1}{\mathbf{1}^TAz}z$ exists (no division by 0) and $\mathbf{1}^TA (\frac{1}{\mathbf{1}^TAz}z)=1$ so the problem is feasible, and, there exists $p=(\frac{1}{\mathbf{1}^TAz}z)$ such that $\underset{m\in\{1,...,M\}}{\min} \ A_mp>0$. Then, the problem is trivially bounded as $M \underset{m\in\{1,...,M\}}{\min} \ A_mp \leq \mathbf{1}^TAp=1$.
Finally, some optimal solutions of this problem can be written as a linear system extracted from $A$, thus, all the corresponding solution can be written as a fraction of sub-determinant (due to Cramer rules). So $\underset{m\in\{1,...,M\}}{\min} \ A_mp^*>0$ is either $0$ (impossible as it has been proven strictly higher than 0) or at least $\frac{1}{MaxDet(A)} = 2^{-L}$.
\noindent \textbf{Lemma 4:} Similarly as lemma 3, $\forall A \in \mathbb{Z}^{M\times N}$ such that $X_A\neq \emptyset$ with total binary size $L$, the following problem is well defined $\underset{q \in \mathbb{R}^N, Aq\geq \mathbf{1}}{\min} \ q^Tq$ and the minimal value obtained is lower then $2^{L}$.
\noindent \textbf{Proof of lemma 4:}
$\forall z\in X_A, \ Az>\mathbf{0}$ so one could see that $q= \frac{1}{ \underset{m\in\{1,...,M\}}{\min} \ A_mz }z$ verifies $Aq\geq \mathbf{1}$. So the problem is feasible and the objective value is trivially bounded by 0.
Like in lemma 3, the solution of this problem is the solution of a linear system extracted from $A$, thus, all the corresponding solution can be written as a fraction of sub-determinant (due to Cramer rules). So the optimal value is at most $MaxDet(A) = 2^{L}$.
\subsection{Two self-concordant methods} This paper considers two algorithms for solving linear feasibility instance i.e. given, $A\in \mathbb{Z}^{M\times N}$ with total binary size $L$ and the assumption that $X_A=\{x\in \mathbb{R}^N, Ax\geq\mathbf{1}\}\neq \emptyset$, then, both those algorithm produce $x\in X_A$.
In addition both those algorithm produces such $x\in X_A$ in at most $O(ML)$ Newton steps.
\subsubsection{Log-barrier central-path inspired method}
\noindent \textbf{Theorem 1:} The Newton descent on $G(x) = 1^TAx - \underset{m\in\{1,...,M\}}{\sum}\ \log(A_mx - \frac{1}{4}2^{-L})$ starting on $\mathbf{0}$ terminates in $O(ML)$ steps returning $x\in X_A$.
\noindent \textbf{Stretch of the proof of theorem 1:}
$G(x) = \underset{m\in\{1,...,M\}}{\sum}g(A_mx)\geq Mg^*>0$ due to lemma 1. So, $G$ has a minimum $x^*$.
Now, let consider the following proof by contradiction: if there exists $k$, $A_kx^*\leq 0$, then, let consider $G(x^*+p^*)-G(x^*)$ where $p^*$ is any optimal solution from lemma-3 problem. Then, $G(x^*+p^*)-G(x^*) = 1^TAx^*+1^TAp^* - \underset{m\in\{1,...,M\}}{\sum}\ \log(A_mx^* - \frac{1}{4}2^{-L}+A_mp^*) - \log(A_mx^* - \frac{1}{4}2^{-L}) $
$= 1 - \log(A_kx^* - \frac{1}{4}2^{-L}+A_kp^*) + \log(A_kx^* - \frac{1}{4}2^{-L}) - \underset{m\in\{1,...,M\}\backslash \{k\} }{\sum}\ \log(A_mx^* - \frac{1}{4}2^{-L}+A_mp^*) -\log(A_mx^* - \frac{1}{4}2^{-L}) $
$< 1 - \log(A_kx^* - \frac{1}{4}2^{-L}+A_kp^*) + \log(A_kx^* - \frac{1}{4}2^{-L}) - \underset{m\in\{1,...,M\}\backslash \{k\} }{\sum}\ \log(A_mx^* - \frac{1}{4}2^{-L}) -\log(A_mx^* - \frac{1}{4}2^{-L}) $
$< 1 - \log((A_kx^* - \frac{1}{4}2^{-L})+2^{-L}) + \log(A_kx^* - \frac{1}{4}2^{-L}) \leq 0$ due to lemma 2, as $(A_kx^* - \frac{1}{4}2^{-L})+2^{-L}\geq 4 (A_kx^* - \frac{1}{4}2^{-L})$ (as $A_kx^*\leq 0$).
So, $G(x^*+p^*)<G(x^*)$ which is expected to be the minimum of $G$. This is a contradiction. Thus, the minimum $x^*$ of $G$ verifies $Ax^*>\mathbf{0}$.
Finally, Newton descent starting from $\mathbf{0}$ will produce a solution of the linear feasibility problem by being closer than $O(2^{-O(L)})$ of the minimum of $G^*$ in (at most) $O(G(\mathbf{0})-G^*+\log(L))$ steps (see 2.1.2 which recalls main point from \cite{nemirovski2004interior}). Yet, $G^*>0$ and $G(\mathbf{0})=M(L+2)$.
\noindent \textbf{Remarks of theorem 1:} in classical path-following framework, one should rather have $\theta$ instead of $\frac{1}{4}2^{-L}$ with decay of $\theta$ after several Newton steps. But this is not considered in this paper.
\subsubsection{Self-concordant Perceptron}
\noindent \textbf{Theorem 2:}
The Newton descent on $F(v) = \frac{v^TAA^Tv}{2||A||^2}-\underset{m\in\{1,...,M\}}{\sum}\ \log(v_m)$ starting on $\mathbf{1}$ terminates in $O(ML)$ steps returning $v$ with $A^Tv\in X_A$
\noindent \textbf{Stretch of the proof of theorem 2:} (The complete proof is described in \cite{chan2022new}).
The key of the theorem is to prove that $F$ is bounded. For that, one has to consider $q$ the solution of the lemma-4 problem, then, one has the two inequality:\begin{itemize}
\item $q^TA^Tv\leq \sqrt{q^Tq v^TAA^Tv}$ (Cauchy inequality)
\item $q^TA^Tv = (Aq)^Tv\geq \mathbf{1}^Tv$ (assuming $v\geq \mathbf{0}$) \end{itemize}
So, $\forall v\geq \mathbf{0}$, $\frac{v^Tv}{2^{2L}}\leq\frac{(1^Tv)^2}{||A||^2 \times q^Tq}\leq \frac{v^TAA^Tv}{||A^2||}$ (see lemma 4).
So, $F(v) \geq \underset{m\in\{1,...,M\}}{\sum} f(v_m)\geq Mf^* \geq -ML$ (from from lemma 1). So $F$ has a minimum $v^*$, and at the optimum $\nabla F=\mathbf{0}$ i.e. $\frac{AA^T}{||A||^2}v^*=\frac{1}{v^*}>\mathbf{0}$ (with $\frac{1}{v}$ being an abusive notation for the vector whose component $m$ is $\frac{1}{v_m}$).
Finally, $F(\mathbf{1}) = O(M^2)$. Thus, Newton descent starting from $\mathbf{1}$ will produce a solution of the linear feasibility problem by approximating the optimal solution of $F$ in (at most) $O(F(\mathbf{1})-F^*) = O(M^2) + O(ML) = O(ML)$ steps.
\subsection{Main result: two different convergence behaviors}
\noindent \textbf{Toy instances:}
Both theorem 1 and 2 have low practical interest as both those algorithms have time complexity larger by a factor $M$ to \cite{van2020deterministic} (more precisely they requires $\sqrt{M}$ more steps than \cite{renegar1988polynomial}, and, it has not be proven that one could lowered their complexity using \cite{van2020deterministic} data structure). Yet, the interesting point is that both algorithms tackle exactly the same problem with same complexity allowing a comparison of there behavior.
Let assume that $A$ is highly constrained (i.e. $Ax>0$ defines a very small part of the space ensuring all solutions to behave similarly) and let assume that $A$ is composed of two populations of constraints such that $\forall x \in \mathbb{R}^N, A_Rx\geq \mathbf{1} \Rightarrow A_{M\backslash R}x\geq O(2^L)\mathbf{1}$ and $\exists z\in \mathbb{R}^N, A_Rz= \frac{1}{2^{-L}}\mathbf{1}, A_{M\backslash R}z= \mathbf{1}$. In other words, $Ax>0$ defines a small set of potential solution $x$, with $R$ vectors of $A$ being support vector of the corresponding solution and the other being close to the solution (without being a trivial problem).
\noindent \textbf{Behavior of minimization of }$G$
Then, when considering $G(x)$ as $x^*\approx z$ as $A_Rz\approx \frac{1}{2^{-L}}\mathbf{1}$ and $A_{M\backslash R}z=\mathbf{1}$, it leads to $G(x^*) \approx RL$. So, the number of steps required to converge in $(M-R)L$ (because $G(0)=ML$ and $G^*\approx RL$).
So, this algorithm is \textit{fast} when $R$ is large and \textit{slow} when $R$ is small.
\noindent \textbf{Behavior of minimization of }$F$
Inversely, when considering $F(v)$, the optimal condition are that $\forall m \in \{1,...,M\}, \ \frac{A_mA^T}{||A||^2}v_m^*=\frac{1}{v^*_m}$ and $F^* = M-\underset{m}{\sum}\log(v_m^*)$. So, the more $v^*_m$ is large, the less $F^*$ and $A_mA^Tv^*$ are (because $v_m^*A_mA^Tv^*=||A||^2$, and because $F$ decreases with $v_m$).
More precisely, if there is necessarily $R$ rows such that $A_RA^Tv\approx 2^{-L}\mathbf{1}$ and $M-R$ rows such that $A_{M\backslash R}z\approx\mathbf{1}$ it means that there are $R$ large $v^*_m$, and, $M-R$ small $v_m^*$. So, $F(\mathbf{1})-F^* = O(RL)$.
So, this algorithm is \textit{fast} when $R$ is small and \textit{slow} when $R$ is large.
\noindent \textbf{Beyond converge speed}
One could then wonder if both algorithms could following the same path but with different speed depending on the situation. However, both algorithms have also different dynamic: \begin{itemize}
\item The minimization of $F$ starts with \textit{small} $v_m$ (somehow related to large $A_mA^T$) , and, all the optimization is to make some $v_k$ even larger (somehow related to make some $A_kA^T$ close to 0). In some way, the fact to terminate with $AA^Tv>\mathbf{0}$ is some kind of side effect of a primary target of minimizing $\frac{v^TAA^Tv}{v^Tv}$ while conserving positive $v$. So, $A_kA^Tv$ values which starts large and which will be large at the end, may not change a lot during the optimization.
\item Inversely, in $G$ minimization, $A_kA^Tv$ value start at 0 and value that will be low at the end will not change a lot during the optimization. Indeed, the optimization will mostly to increase the value of $A_kA^Tv$ which can be increased. \end{itemize}
So, one algorithm starts with low $A_mA^Tv$ and increases those which can be increased without modifying the others, while the other starts with high $A_mA^Tv$ and decreases those which can be decreased without modifying the others. So, those two algorithms have two different paths toward solution, despite this discussion is qualitative and not a formal proof.
\noindent \textbf{Perspectives}
Future works could focus on formalizing this behavior difference and on considering those two algorithm on hard instance from \cite{allamigeon2022no}. Yet, This paper still shows that there exists two different interior point algorithm for linear feasibility relying on the same self-concordant theory with the same complexity of $O(ML)$ Newton steps, but with strong difference in their convergences regarding the number of support vectors on high constrained instances.
\end{document} | arXiv |
On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces
Dynamic behavior and optimal scheduling for mixed vaccination strategy with temporary immunity
Siyu Liu a, , Xue Yang a,b, , Yingjie Bi a, and Yong Li a,b,,
School of Mathematics, Jilin University, Changchun 130012, China
School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China
∗ Corresponding author: Yong Li
Figure(3) / Table(1)
This paper presents an SEIRVS epidemic model with different vaccination strategies to investigate the elimination of the chronic disease. The mixed vaccination strategy, a combination of constant vaccination and pulse vaccination, is a future development tendency of disease control. Theoretical analysis and threshold conditions for eradicating the disease are given. Then we propose an optimal control problem and solve the optimal scheduling of the mixed vaccination strategy through the combined multiple shooting and collocation (CMSC) method. Theoretical results and numerical simulations can help to design the final mixed vaccination strategy for the optimal control of the chronic disease once the new vaccine comes into use.
Keywords: Mixed vaccination strategy, optimal control, biological systems, epidemic model.
Mathematics Subject Classification: Primary: 37N25, 34H05; Secondary: 34K13.
Citation: Siyu Liu, Xue Yang, Yingjie Bi, Yong Li. Dynamic behavior and optimal scheduling for mixed vaccination strategy with temporary immunity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1469-1483. doi: 10.3934/dcdsb.2018216
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Figure 1. Comparison between the constant vaccination strategy and mixed vaccination strategy with the same cost (w = 3). The red dashed line shows the constant vaccination strategy with $p = 1$. The blue solid line shows optimal mixed vaccination strategy with $p = 0.45, p_{c} = 0.2$ and $T = 5$. All the other parameters are shown in Table 1
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Figure 2. Comparison between the constant vaccination strategy and optimal mixed vaccination strategy. The red dashed line shows the constant vaccination strategy with $p = 0.85 (0.6\leq p\leq 0.85)$. The blue solid line shows optimal mixed vaccination strategy with $0.6\leq u_{1}(t)\leq 0.85, 0.1\leq u_{2}(t)\leq 0.3$ and $5\leq N\leq 10$. All the other parameters are shown in Table 1
Figure 3. Optimal mixed vaccination strategy under limited vaccinated individuals with $0.6\leq u_{1}(t)\leq 0.85, 0.1\leq u_{2}(t)\leq 0.3$ and $5\leq N\leq 10$. All the other parameters are shown in Table 1
Table 1. Parameter values
Parameter Value Source
$\mu$ $0.0143~year^{{-1}}$ [19]
$\varepsilon$ $6~year^{{-1}}$ [14]
$\alpha$ $0.0015~year^{{-1}}$ [14]
$c$ $0.05~year^{{-1}}$ Assumed
$\gamma$ $0.4055~year^{{-1}}$ Assumed
$\beta$ $0.4945$ Assumed
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\begin{document}
\author{Wen Huang} \address[Wen Huang] {Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics \\
University of Science and Technology of China,\\
Hefei, Anhui, China} \email[W. Huang]{[email protected]}
\author{Zeng Lian} \address[Zeng Lian] {College of Mathematical Sciences\\ Sichuan University\\
Chengdu, Sichuan, 610016, China} \email[Z. Lian]{[email protected], [email protected]}
\author{Xiao Ma} \address[Xiao Ma] { Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics \\
University of Science and Technology of China \\
Hefei, Anhui, China} \email[X. Ma]{[email protected]}
\author{Leiye Xu} \address[Leiye Xu] { Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics \\
University of Science and Technology of China \\
Hefei, Anhui, China} \email[L. Xu]{[email protected]}
\author{Yiwei Zhang} \address[Yiwei Zhang] { School of Mathematics and Statistics, Center for Mathematical Sciences, Hubei Key Laboratory of Engineering Modeling and Scientific Computing,\ Hua-Zhong University of Sciences and Technology \\ Wuhan 430074, China} \email[Y. Zhang]{[email protected], [email protected]}
\title[]{Ergodic optimization theory for Axiom A flows}
\thanks{Huang is partially supported by NSF of China (11431012,11731003). Lian is partially supported by NSF of China (11725105,11671279). Xu is partially supported by NSF of China (11801538, 11871188). Zhang is partially supported by NSF of China (117010200,11871262).}
\begin{abstract} In this article, we consider the weighted ergodic optimization problem Axiom A attractors of a $C^2$ flow on a compact smooth manifold. The main result obtained in this paper is that for a generic observable from function space $\mathcal C^{0,\a}$ ($\a\in(0,1]$) or $\mathcal C^1$ the minimizing measure is unique and is supported on a periodic orbit. \end{abstract}
\maketitle
\parskip 0.4cm \section{Introduction}
\noindent {\bf Context and motivation.} Ergodic optimization theory focuses on the ergodic measures on which a given observable taking an extreme ergodic average (maximum or minimum), which
has strong connection with other fields, such as Anbry-Mather theory \cite{Contreras_Mane,Mane,CIPP} in Lagrangian Mechanics; ground state theory \cite{BLL} in thermodynamics formalism and multifractal analysis; and controlling chaos \cite{HO1,OGY90,SGOY93} in control theory.
In this paper, we study the typical optimization problem in weighted ergodic optimization theory for Axiom A attractors of a $C^2$ flow on a compact smooth manifold.
For discrete time case, ergodic optimization theory has been developed broadly. Among them, Yuan and Hunt proposed an open problem in \cite[Conjecture 1.1]{YH} on 1999,
which provides a mathematical mechanism on Hunt and Ott's experimental and heuristic results in \cite{HO2,HO3} and becomes one of the fundamental questions raised in the field of ergodic optimization theory. A more general form of Yuan and Hunt's conjecture is now called the Typical Periodic Optimization Conjecture, and has attracted sustained attentions and yielded considerable results, for instances \cite{BZ,Bousch_poisson,Bousch_Walters,Bousch_note,Contreras_EO,CLT,Morris_entropy,QS}. For a more comprehensive survey for the classical ergodic optimization theory, we refer the reader to Jenkinson \cite{Jenkinson_survey,Jenkinson_newsurvey}, to Bochi \cite{Bochi}, to Baraviera, Leplaideur, Lopes \cite{BLL}, and to Garibaldi \cite{G} for a historical perspective of the development in this area. In our recent paper \cite{HLMXZ}, we extend the applicability of the theory both to a broader class of systems and to a broader class of observables, which leads to a positive answer to Yuan and Hunt's conjecture for $C^1$-observable case. To our knowledge, because of difficulties appears on both conceptual level and technical level, there is no existing result of ergodic optimization theory for flows so far, which make the results obtained in the present paper the first achievement on flows towards ergodic optimization theory.
On the other hand, as mentioned in \cite{HLMXZ}, the reason of adding the non-constant weight $\psi$ mainly lies in the studies on the zero temperature limit (or ground state) of the $(u,\psi)$-weighted equilibrium state for thermodynamics formalism (for more details, we refer readers to works \cite{BF,BCW,FH}).
\noindent{\bf Summary of the results.} To avoid unnecessary complexity, we only introduce the result in the framework of standard ergodic optimization theory. Consider a $C^2$ flow $\Phi$ on a compact smooth manifold $M$. Let $\L$ be an Axiom A attractor of $\Phi$ (detailed definition is given by Definition \ref{D:AxiomAAttractor}). For a given observable $u:M\to \mathbb R$, the ergodic averages of $u$ on $\L$ is defined by the integration of $u$ with respect to $\Phi|_\L$-ergodic measures, and the $u$-maximizing (or minimizing) measure is the measure with respect to which the ergodic average of $u$ takes maximum (or minimum) value. As a consequence of the main result (Theorem \ref{T:MainResult}) of the present paper, we have the following result:
\noindent{\bf Theorem A:} Let $(\L,\Phi)$ be an Axiom A attractor on a compact smooth Riemannian manifold $M$, then for a generic observable $u$ in function space $\mathcal C^{0,\a}(M)$ or $\mathcal C^1(M)$, the $u$-maximizing (or $u$-minimizing) measure is unique and is supported on a periodic orbit.\\
\noindent{\bf Remarks on techniques of the proof.} It seems that the proof given in \cite{HLMXZ} provides a more general mechanism in the study on ergodic optimization problems, which also shed a light on the case of flows for sure. However, the results of the present paper depend crucially on the continuous time nature of the system; that is to say, they do not follow from the properties of their time-1 maps. Therefore, we must build certain theoretical base and create certain new techniques to address issues raised in the case of flows.
We mention three differences of note between our setting and the existing literatures at both conceptual level and technical level which pervade the arguments in this paper: (1) At conceptual level, the most significant issue is that the {\em gap function} of a discrete time periodic orbit, i.e. the minimum separation of finite isolated points, is not well defined for continuous periodic orbit. Such a gap function plays a key role in the proof of \cite{HLMXZ}. (2) The presence of {\em shear}, i.e. the sliding of some orbits past other nearby orbits due to the slightly different speed at which they travel, is a typical phenomenon of continuous time systems, which causes tremendous amount of "tail estimates" throughout this paper. (3) Several main fundamental theoretical tools are not existing and need to be rebuilt from the base, such as Anosov Closing Lemma, Ma\~n\'e-Conze-Guivarc'h-Bousch's Lemma and Periodic Approximation Lemma.\\
\noindent{\bf Structure of this paper.} In Section \ref{S:Setting}, we set up the theoretic model and state the main results; In Section \ref{S:ProAxiomA}, we state (without proofs) some well known properties of Axiom A attractors, and some theoretical tools including Anosov Closing Lemma, Ma\~n\'e-Conze-Guivarc'h-Bousch's Lemma and Periodic Approximation Lemma preparing for the proof the main results; In Section \ref{S:PfMainThm}, we give the proof of Theorem \ref{T:MainResult}, of which proving Part I) of the Theorem costs most efforts; As follows, we leave the proofs of all the technical lemmas to Section \ref{S:ProofsTechLem}. On one hand, readers may go through the main proof by assuming the validity of these technical lemmas without extra interruptions; on the other hand, these technical lemmas with their proofs may be of independent interest. Finally, we discuss the case when observables have higher regularity in Section \ref{S:HighReg} in which only some partial results are presented.
\section{main setting and results}\label{S:Setting} Let $M$ be a compact smooth Riemannian manifold with Riemannian metric $d$ and $\Phi=\{\phi_t:M\to M\}_{t\in\mathbb{R}}$ be a $C^2$ flow on $M$.
\begin{defn}\label{D:AxiomAAttractor} For $\L\subset M$, $(\L,\Phi)$ is called an Axiom A attractor if the following conditions hold: \begin{itemize} \item[A1)] $\L$ is a nonempty $\Phi$-invariant compact set. \item[A2)] There exists an $\e_0>0$ such that for any $x\in M$ with $d(x,\L)<\e_0$ $$\lim_{t\to\infty}d\left(\phi_t(x),\L\right)=0.$$ \item[A3)] There exist $\l_0>0$, $C_0>1$ and a continuous splitting of tangential spaces of $M$ restricted on $\L$, $T_xM=E^u_x\oplus E^c_x\oplus E^s_x\ \forall \ x\in \L$, such that the following hold \begin{align*} &(D_M\phi_t)_x(E^\tau_x)=E^\tau_{\phi_t(x)},\ \tau=u,c,s,\ \forall t\in\mathbb R\text{ and }x\in\L,\\
&\max\left\{\|(D_M\phi_{-t})_x|_{E^u_x}\|,\|(D_M\phi_{t})_x|_{E^s_x}\|\right\}\le C_0 e^{-t\l_0},\ \forall t\in \mathbb R^+, \end{align*} where $(D_M\phi_t)_x$ is the derivative of the time-$t$ map $\phi_t$ on $x$ with respect to space variables.
\item[A4)] $\inf_{x\in \L}\left\|\frac{d\phi_t(x)}{dt}\right\|>0\text{ and }E^c_x=span\left\{\frac{d}{dt}\phi_t(x)\right\},\ \forall x\in \L$. \end{itemize} \end{defn}
Denote by $\mathcal{M}(\L,\Phi)$ the set of all $\Phi$-invariant Borel probability measures on $\L$, which is a non-empty convex and compact topological space with respect to weak$^*$ topology. Denote by $\mathcal{M}^{e}(\L,\Phi)\subset\mathcal{M}(\L,\Phi)$ the set of ergodic measures, which is the set of the extremal points of $\mathcal{M}(\L,\Phi)$. Let $u:M\to\mathbb{R}$ and $\psi:M\to\mathbb{R}^{+}$ be continuous functions. The quantity $\beta(u;\psi, \L, \Phi)$ being defined by \begin{equation}\label{equ_ratiomimimalerg} \beta(u;\psi,\L,\Phi):=\min_{\nu\in\mathcal{M}(\L,\Phi)}\frac{\int u d\nu}{\int \psi d\nu}, \end{equation} is called the \emph{ratio minimum ergodic average}, and any $\nu\in\mathcal{M}(\L,\Phi)$ satisfying $$\frac{\int u d\nu}{\int \psi d\nu}=\beta(u;\psi,\L,\Phi)$$ is called a \emph{$(u,\psi)$-minimizing measure}. Denote that $$\mathcal{M}_{\min}(u;\psi,\L,\Phi):=\left\{\nu\in\mathcal{M}(\L,\Phi):\frac{\int ud\nu}{\int \psi d\nu}=\beta(u;\psi,\L,\Phi)\right\}.$$ By compactness of $\mathcal{M}(\L,\Phi)$, and the continuity of the operator $\frac{\int ud(\cdot)}{\int \psi d(\cdot)}$, it directly follows that $\mathcal{M}_{min}(u;\psi,\L,\Phi)\neq\emptyset$, which contains at least one ergodic $(u,\psi)$-minimizing measure by ergodic decomposition.
For $\a\in (0,1]$, let $\mathcal{C}^{0,\alpha}(M)$ be the space of $\alpha$-H\"{o}lder continuous real-valued functions on $M$ endowed with the $\alpha$-H\"{o}lder norm $\|u\|_{\alpha}:=\|u\|_{0}+[u]_{\alpha}$, where $\|u\|_{0}:=\sup_{x\in M}|u(x)|$ is the super norm, and $[u]_{\alpha}:=\sup_{x\neq y}\frac{|u(x)-u(y)|}{(d(x,y))^{\alpha}}$. Also note that when $\a=1$, $C^{0,1}(M)$ becomes the collection of all real-valued Lipschitz continuous functions, and $[u]_{1}$ becomes the minimum Lipschitz constant of $u$. Additionally, denote by $\mathcal C^{1,0}(M)$ the Banach space of continuous differentiable functions on $M$ endowed with the standard $C^1$-norm.
In this paper, we consider the weighted ergodic optimization problem and derive the following result.
\begin{thm}\label{T:MainResult} Let $M$ be a compact smooth Riemannian manifold with Riemannian metric $d$ and $\Phi$ be a $C^2$ flow on $M$. Suppose that $(\L,\Phi)$ is an Axiom A attractor, then the following hold: \begin{itemize}
\item[I)] For $\a\in (0,1]$, given a $\psi\in \mathcal C^{0,\a}(M)$ with $\inf_{x\in M}\psi(x)>0$, then there exists an open and dense set $\mathfrak P\subset \mathcal C^{0,\a}(M)$ such that for any $u\in \mathfrak P$, the $(u|_\L,\psi|_\L)$-minimizing measure of $(\L,\Phi)$ is unique and is supported on a periodic orbit of $\Phi$.
\item[II)] For $\psi\in \mathcal C^{0,1}(M)$ with $\inf_{x\in M}\psi(x)>0$, there exists an open and dense set $\mathfrak P\subset \mathcal C^{1,0}(M)$ such that for any $u\in \mathfrak P$, the $(u|_\L,\psi|_\L)$-minimizing measure of $(\L,\Phi)$ is unique and is supported on a periodic orbit of $\Phi$. \end{itemize} \end{thm} {\bf We remark here that $M,\L,\Phi$ are assumed to satisfy conditions in Theorem \ref{T:MainResult} throughout the rest of this paper.}
\section{Properties of Axiom A attractors}\label{S:ProAxiomA} This section devotes to building theoretic tools as preparations for the proof of Theorem \ref{T:MainResult}. \subsection{Invariant Manifolds}\label{S:InvMani} For a point $x\in \L$ and $\epsilon>0$ the local stable and unstable sets are defined by \begin{align*} W_\epsilon^s(x)=&\{y\in M:d(\phi_t(x),\phi_t(y))\le\epsilon\ \forall t\ge0,\ d(\phi_t(x),\phi_t(y))\to0 \text{ as } t\to+\infty\},\\ W_\epsilon^u(x)=&\{y\in M:d(\phi_{-t}(x),\phi_{-t}(y))\le\epsilon\ \forall t\ge0,\ d(\phi_{-t}(x),\phi_{-t}(y))\to0 \text{ as } t\to+\infty\}. \end{align*} The following Lemma is a standard result of invariant manifolds in existing literature, of which the proof is omitted. \begin{lem}\label{L:InvMani} For any $\l_1\in (0,\l_0)$, there exists $\e_1>0$ and $C_1\ge 1$ such that for any $\e\in(0,\e_1]$, the following hold: \begin{itemize} \item[i)] $W^s_\e(x), W^u_\e(x)$ are $C^2$ embedded discs for all $x\in \L$ with $T_xW_\e^\tau(x)=E^\tau_{x}$, $\tau=u,s$; \item[ii)] $d(\phi^t(x),\phi^t(y))\le C_1e^{-t\l_1}d(x,y)$ for $y\in W^s_\e(x)$, $t\ge 0$, and\\ $d(\phi^{-t}(x),\phi^{-t}(y))\le C_1e^{-t\l}d(x,y)$ for $y\in W^u_\e(x)$, $t\ge 0$; \item[iii)] $W^s_\e(x), W^u_\e(x)$ vary continuously with respect to $x$ (in $C^1$ topology). \end{itemize} \end{lem}
By choosing the Riemannian metric, the Axiom A flow in Theorem \ref{T:MainResult} meets the following {\bf basic canonical setting} : There are positive constants $\delta, \epsilon,\beta,\lambda, C$ with $C\ge 1$ and $\d\ll\e\ll\min\{\e_0,\e_1\}$, where $\e_0$ is as in A2) of the definition of Axiom A attractors and $\e_1$ is as in Lemma \ref{L:InvMani}, such that:
\begin{itemize}
\item[(1)]
For $x,y\in M$ with $d(x,y)\le\delta$, there is a unique time $v=v(x,y)$ with $|v|\le Cd(x,y)$ such that
\begin{itemize}
\item[(a)] $W^s_\epsilon(\phi_{v}(x))\cap W^u_\epsilon(y)$ is not empty and contains only one element which is noted by $w=w(x,y)$.
\item[(b)] $d(x,y)\ge C^{-1}\max\{d(\phi_{v}(x),w),d(y,w),d(\phi_{v}(x),x), d(w,x)\}.$
\end{itemize}
\item[(2)] For $x\in M$ , $y\in W_\epsilon^u(x)$ and $t\ge 0$, $d(\phi_{-t}x,\phi_{-t}y)\le Ce^{-\lambda t}d(x,y)$,\\ For $x\in M$ , $y\in W_\epsilon^s(x)$ and $t\ge 0$, $d(\phi_tx,\phi_ty) \le Ce^{-\lambda t}d(x,y)$.
\item[(3)] For $x,y\in M$, $d(\phi_tx,\phi_ty)\le Ce^{\beta |t|}d(x,y)$ for all $t\in\mathbb{R}$. \end{itemize} \begin{rem}\label{remark-1-1}
In our following text, $\delta,\epsilon,\lambda,\beta,C$ are the positive constants as above. Additionally, for convenience, we assume $C\gg 1, 0<\delta\ll\epsilon\ll1$. Otherwise, we set a positive constant $\epsilon'$ such that $\epsilon'\ll\frac{\epsilon}{C^{10}e^{10\beta}}$. We set another positive constant $\delta'$ with $\delta'\ll\delta$ such that for any $x,y\in M$ with $d(x,y)\le \frac{C^{10}e^{10\beta+10\lambda}}{e^{\lambda}-1}\delta'$, there is an unique time $v=v(x,y)$ with $|v|\le Cd(x,y)$ such that $W^s_{\epsilon'}(\phi_{v}(x))\cap W^u_{\epsilon'}(y)$ is not empty and contains only one element. \end{rem}
\begin{rem}\label{R:BasicCanSet} For proofs and more details of Lemma \ref{L:InvMani} and the {\bf basic canonical setting}, we refer readers to \cite{PSh}, \cite{Bowen}, and \cite{Bowen-Ruelle}. The only property which is not appearing in the above references is the following inequality \begin{equation}\label{E:ShearEst}
|v(x,y)|\le Cd(x,y) \end{equation}
appearing in (1) of {\bf basic canonical setting}. We remark here that this inequality holds when $\Phi$ is $C^2$. When $\Phi$ is $C^{1+\a}$ for some $\a\in(0,1]$, the above inequality will be replaced by $|v(x,y)|\le Cd^\a(x,y)$ which is still sufficient for the proof of this paper (although necessary modifications are required). This concludes that the Theorem \ref{T:MainResult} still holds for $C^{1+\a}$ flows.
Finally, to our knowledge, there is no explicit statement equivalent to (\ref{E:ShearEst}) in existing literature. Nevertheless, (\ref{E:ShearEst}) can be proved by combining Lemma 6, Proposition 8, Proposition 9 and Lemma 13 from \cite{Lian-Young}. Since (\ref{E:ShearEst}) is intuitively natural but at the same time the proof involves considerable technical complexity, we decide not to put the detailed proof in this paper for the sake of simplicity. \end{rem}
\subsection{Anosov Closing Lemma}\label{S:AnosovClosingLem}
Let $\delta',\epsilon',\delta,\epsilon$, $\lambda,\beta,C$ be the constants as in Remark \ref{remark-1-1}. Then we have the following Lemma. \begin{lem}\label{An-1}Given $\eta\le \frac{C^{10}e^{10\beta+10\lambda}}{e^{\lambda}-1}\delta'$ and $T>0$,
if $x,y\in \L$ and continuous function $s:\mathbb{R}\to\mathbb{R}$ with $s(0)=0$ satisfy $$d(\phi_{t+s(t)}(y),\phi_t(x))\le \eta \text{ for } t\in [0,T],$$
then for all $t\in [0,T]$, the following hold:
\begin{itemize}
\item[ASh1)] $|s(t)|\le 2C\eta$;
\item[ASh2)] $d(\phi_t\phi_{v(y,x)}(y),\phi_t(x))\le C^2e^{-\lambda\min(t,T-t)}\left(d(y,x)+d\left(\phi_{T}(y),\phi_T(x)\right)\right)$, where $v(y,x)$ is as in Remark \ref{remark-1-1} satisfying $|v(y,x)|\le Cd(x,y)$. \end{itemize}
Especially, one has that
\begin{itemize} \item[(1)] If $d(\phi_{t+s(t)}(y),\phi_t(x))\le \eta$ for all $t\ge 0,$ then
$$d(\phi_t\phi_{v(y,x)}(y),\phi_t(x))\to 0\text{ as } t\to +\infty.$$
\item[(2)] If $d(\phi_{t+s(t)}(y),\phi_t(x))\le \eta$ for all $t\in\mathbb{R},$ then $\phi_{v(y,x)}(y)=x.$
\end{itemize} \end{lem}
A segment of $\Phi$ is a curve $\mathcal{S}: [a,b]\to M:t\to\phi_t(x)$ for some $x\in M$ and real numbers $a\le b$. We denote the left endpoint of $\mathcal{S}$ by $\mathcal{S}^L=\phi_a(x)$, the right endpoint of $\mathcal{S}$ by $\mathcal{S}^R=\phi_b(x)$ and the length of $\mathcal{S}$ by $|\mathcal{S}|=b-a$. By a segment $\mathcal{S}$, if $\mathcal{S}^L=\mathcal{S}^R$, we say $\mathcal{S}$ is a periodic segment. We have the following version of Anosov Closing Lemma. \begin{lem}[Anosov Closing Lemma]\label{lemma-a}
There are positive constants $L$ and $K$ depending on the system constants only such that if segment $\mathcal{S}$ of $\Phi|_\L$ satisfy
\begin{itemize}
\item[(a)] $|\mathcal{S}|\ge K$;
\item[(b)]$d(\mathcal{S}^L, \mathcal{S}^R)\le \delta'$.
\end{itemize}
Then, there is a periodic segment $\mathcal{O}$ such that $$\left| |\mathcal{S}|-|\mathcal{O}|\right|\le Ld(\mathcal{S}^L, \mathcal{S}^R)$$ and
$$ d(\phi_t(\mathcal{O}^L),\phi_t(\mathcal{S}^L))\le Ld(\mathcal{S}^L, \mathcal{S}^R)\text{ for all } 0\le t\le \max(|\mathcal{S}|,|\mathcal{O}|).$$ \end{lem} \begin{rem}
In the following text, we also use $\mathcal{S}$ (so is $\mathcal{O}$ and $\mathcal{Q}$) to represnet the collection of points $\phi_t(\mathcal{S}^L), 0\le t\le |\mathcal{S}|$ as no confusion being caused. By Lemma \ref{An-1} and the choices of $\e$ and $\d$, $\mathcal O$ clearly belongs to $\L$. \end{rem} \subsection{Ma\~n\'e-Conze-Guivarc'h-Bousch's Lemma} For $\g\in \mathbb R\setminus\{0\}$ and continuous function $u:M\to \mathbb R$, define that \begin{equation}\label{E:u_g} u_\g(x):=\frac1\g\int_0^\g u(\phi_t(x))dt. \end{equation}
\begin{lem}[Ma\~n\'e-Conze-Guivarc'h-Bousch's Lemma]\label{MCGB}
For $0< \alpha\le 1$ and $N_0>0$, there exists a positive constant $\gamma=\gamma(\alpha)>N_0$ such that if $u\in\mathcal{C}^{0,\alpha}(M)$ satisfies $\beta(u;1,\L,\Phi|_\L)\ge0$, then there is a $v\in\mathcal{C}^{0,\alpha}(\L)$ such that $u_\gamma|_\L\ge v\circ \phi_\gamma|_\L-v.$ \end{lem}
\begin{rem}\label{rem-reveallossregularity} We remark that the key point of Lemma \ref{MCGB} lies in the fact that $v$ is chosen with the same H\"{o}lder exponent as $u$. Indeed, there were a number of weak versions of Lemma \ref{MCGB} in the setting of smooth Anosov flows without fixed points, or certain expansive non-Anosov geodesic flows, where $v$ is still H\"{o}lder, but the H\"{o}lder exponent is less than $\alpha$ (for details, see \cite{Lopes-Rosas-Ruggiero,Lopes-Thieullen,Pollicott-Sharp}). \end{rem}
By using Lemma \ref{MCGB}, we have the following Lemma. \begin{lem}\label{revael} For $0<\alpha\le 1$, there exists large $\gamma=\gamma(\alpha)$ such that, for $u\in\mathcal{C}^{0,\alpha}(M)$ and strictly positive $\psi\in\mathcal{C}^{0,\alpha}(M)$, there is a $v\in\mathcal{C}^{0,\alpha}(\L)$ such that
\begin{itemize}
\item[(1)] $u_\gamma|_\L-v\circ \phi_\gamma|_\L+v-\beta(u;\psi,\L,\Phi)\psi_\gamma|_\L\ge 0;$
\item[(2)] $Z_{u,\psi}\subset\left\{x\in \L:\left(u_\gamma|_\L+v\circ \phi_\gamma|_\L-v-\beta(u;\psi,\L,\Phi)\psi_\gamma|_\L\right)(x)=0\right\},$
\end{itemize} where $Z_{u,\psi}=\overline{\cup_{\mu\in\mathcal{M}_{min}(u;\psi,\L,\Phi)}supp(\mu)}.$ \end{lem}
\begin{rem}\label{rem-reveal}For convenience, in the following text, if we need to use Lemma \ref{revael}, we use $\bar u$ to represent
$u_\gamma|_\L+v\circ \phi_\gamma|_\L-v-\beta(u;\psi,\L,\Phi)\psi_\gamma|_\L$ for short. Then, $\bar u\ge 0$ and $Z_{u,\psi}\subset\{x\in{\L}:\bar u(x)=0\}.$ \end{rem}
\subsection{Periodic Approximation}\label{S:PeriodicApp} For $\alpha\in(0,1]$, $Z\subset M$ and a segment $\mathcal{S}$ of $\Phi$, we define the {\it $\alpha$-deviation } of $\mathcal{S}$ with respect to $Z$ by
\[d_{\alpha,Z}(\mathcal{S})=\int_0^{|\mathcal{S}|}d^\alpha\left(\phi_t\left(\mathcal{S}^L\right),Z\right)dt.\] For $P\ge 0$, using $\mathcal{O}^P_\L$ denote the collection of periodic segments in $\L$ with length not larger than $P$. Now we have the following version of Quas and Bressaud's periodic approximation Lemma. \begin{lem}\label{lemma-2}
Let $Z\subset \L$ be a $\Phi$-invariant compact subset of $\L$. Then, for all $\alpha\in(0,1],k\ge 0$,
\[\lim_{P\to+\infty}P^k\min_{\mathcal{S}\in \mathcal{O}^P_\L}d_{\alpha,Z}(\mathcal{S})=0.\] \end{lem}
\section{Proof of Theorem \ref{T:MainResult}. }\label{S:PfMainThm} This section contains two main subsections \ref{S:PfPartI} and \ref{S:ProofPartIIMainThm} corresponding to the proofs of Part I and Part II of Theorem \ref{T:MainResult} respectively. Indeed, Proposition \ref{prop-2} in Subsection \ref{S:MainProp} plays the key role in the proof of Theorem \ref{T:MainResult}, based on which the Part I result follows immediately and the Part II result follows in a straightforward way with the help of an approximation lemma. We also note that throughout the whole section $\delta',\epsilon',\delta,\epsilon,\lambda,\beta,C$ are the fixed constants as in Remark \ref{remark-1-1}.
\subsection{Proof of Part I) of Theorem \ref{T:MainResult}}\label{S:PfPartI} This section mainly contains three parts: \ref{S:Locking}, \ref{S:GoodPerOrb} and \ref{S:MainProp}. As mentioned above, Proposition \ref{prop-2} in Subsection \ref{S:MainProp} is the key to prove the main theorem of this paper, proving which is the main aim of Section \ref{S:PfPartI}. While Section \ref{S:Locking} and \ref{S:GoodPerOrb} are devoted to building notions, tools and results as preparations for the proof of Proposition \ref{prop-2}, of which Section \ref{S:Locking} investigate the basic properties of periodic orbits and Section \ref{S:GoodPerOrb} constructs periodic orbits with "good shapes".
\subsubsection{Locking Property of Periodic Segments}\label{S:Locking}
In this subsection, we show that periodic segments have locking property in some sense.
For $0<\eta\le\delta$, the {\it $\eta$-disk} of $x$ is defined by \begin{equation}\label{E:StableCylinder} \mathcal D(x,\eta)=\{y\in \L: d(x,y)\le\eta,W^s_\epsilon(x)\cap W^u_\epsilon(y)\neq\emptyset\}. \end{equation}
$\mathcal D(x,\eta)$ has the following properties: \begin{itemize}
\item[(a)] $W^s_\eta(x)\subset \mathcal D(x,\eta)$ and $\phi_{t}(W^s_\eta(x))\subset \mathcal D(\phi_{t}(x),Ce^{-\lambda t}\eta)$ for $t\ge 0$;
\item[(b)] $W^u_\eta(x)\subset \mathcal D(x,\eta)$ and $\phi_{t}(W^u_\eta(x))\subset \mathcal D(\phi_{t}(x),Ce^{\lambda t}\eta)$ for $t\le 0$;
\item[(c)]$\phi_t(\mathcal D(x,\eta))\subset \mathcal D\left(\phi_t(x),Ce^{\beta |t|}\eta\right)$ for $t\in\mathbb{R}$ satisfying $Ce^{\beta |t|}\eta<\delta$.
\item[(d)] for $\eta\le\frac{\delta}{C}$ and $x,y\in \L$ with $d(x,y)\le \eta$, there exists a unique time $v=v(x,y)$ with $|v|\le Cd(x,y)$ such that $y\in \mathcal D(\phi_v(x),\delta)$. In fact, $v$ is the one given by the {\bf basic canonical setting}. \end{itemize} Now we define $D:\L\times \L\to[0,+\infty)$ by \begin{equation}\label{E:DMetric} D(x,y)=\left\{\begin{array}{ll} \delta', & \text{ if } y\notin \mathcal D(x,\delta'),\\ d(x,y), & \text{ if } y\in \mathcal D(x,\delta'). \end{array} \right. \end{equation}
By a periodic segment $\mathcal{O}$ of $\Phi|_\L$, we define the {\it gap } of $\mathcal{O}$ by \begin{equation}\label{E:GapFunc}
D(\mathcal{O})=\min_{x\in\mathcal{O},0<t<|\mathcal{O}|_{\min}}D(x,\phi_t(x)), \end{equation}
where $|\mathcal O|_{\min}=\min\{t>0|\phi_t(x)=x,x\in \mathcal O\}$. In \cite{HLMXZ}, the gap function of periodic orbits of a discrete time system plays an irreplaceable role in identifying the optimal periodic measures. Here, the gap of a periodic segment is an analogue of the gap function for discrete periodic orbits, which reflects the geometric characteristic of the periodic segment. Indeed, the definition of the gap of periodic segment is an conceptual enhancement, which originates the study of the ergodic optimization theory for the flow case.
Note that A4) of Definition \ref{D:AxiomAAttractor} implies that $D(\mathcal{O})>0$ automatically, therefore in the rest of this section we keep in mind that $D(\mathcal{O})>0$ without extra explanation.
Firstly, we present several technical Lemmas.
\begin{lem}\label{M-1}Let $\mathcal{O}$ be a periodic segment of $\Phi|_\L$. If $x,y\in \mathcal{O}$ satisfy $d(x,y)< \frac{D(\mathcal{O})}{C}$, then $\phi_v(x)=y$ where $v=v(x,y)$. \end{lem} \begin{proof} Let $\delta',\epsilon',\delta,\epsilon,\lambda,\beta,C$ be the constants as in Remark \ref{remark-1-1}. Then,
$$d(x,y)\le \frac{D(\mathcal{O})}{C}<\delta'\ll\delta.$$
Hence, by the {\bf basic canonical setting}, there is a constant $v=v(x,y)$ such that
$$y\in \mathcal D(\phi_v(x),Cd(x,y))\subset \mathcal D(\phi_v(x),\delta).$$
If $\phi_v(x)\neq y$, then
$$D(\mathcal{O})\le d(\phi_v(x),y)\le Cd(x,y)<D(\mathcal{O}),$$
which is impossible. Thus, $\phi_v(x)= y$. This ends the proof. \end{proof}
By a periodic segment $\mathcal{O}$ of $\Phi$, the periodic measure $\mu_{\mathcal{O}}$ is defined by $$\mu_{\mathcal{O}}=\frac{1}{|\mathcal{O}|}\int_0^{|\mathcal{O}|}\delta_{\phi_t(\mathcal{O}^L)}dt.$$
By an ergodic measure $\mu\in\mathcal{M}^e(\L,\Phi|_\L)$, a point $x\in M$ is called a generic point of $\mu$ if the following holds $$\lim_{T\to+\infty}\frac{1}{T}\int_0^Tf(\phi_t(x))dt=\int fd\mu\text{ for all } f\in \mathcal{C}(M).$$ The following Lemma shows that periodic segments have locking property in some sense. \begin{lem}\label{M-3}
Let $\mathcal{O}$ be a periodic segment of $\Phi|_\L$ and $x\in M$. If \begin{equation}\label{E:M-3-1} d(\phi_t(x),\mathcal{O})\le\frac{D(\mathcal{O})}{4C^2e^\beta}\text{ for all }0\le t\le T, \end{equation}
then there is a $y\in\mathcal{O}$ such that
$$d(\phi_t(x),\phi_t(y))\le C d(\phi_t(x),\mathcal{O}) \text{ for all } 0\le t\le T.$$ Especially, if $d(\phi_t(x),\mathcal{O})\le\frac{D(\mathcal{O})}{4C^2e^\beta}$ for all $t\ge 0$, then $x$ is a generic point of $\mu_{\mathcal{O}}.$ \end{lem}
\begin{proof} Let $\delta',\epsilon',\delta,\epsilon,\lambda,\beta,C$ be the constants as in Remark \ref{remark-1-1}. Take a positive constant $\theta\ll \delta$ such that $d(\phi_t(z),z)\le\frac{D(\mathcal{O})}{4C}$ for all $|t|\le \theta$ and $z\in M$. By assumption (\ref{E:M-3-1}), there are $y_t'\in\mathcal{O}$ such that $$d(\phi_t(x),y_t')=d(\phi_t(x),\mathcal{O})\le\frac{D(\mathcal{O})}{4C^2e^\beta}\ll\delta\text{ for all } 0\le t\le T.$$
Let $y_t=\phi_{v(y_t',\phi_t(x))}(y_t')$ for $0\le t\le T$. Then
\begin{align}\label{em-1}
\phi_{t}(x)\in \mathcal D\left(y_{t},Cd\left(\phi_t(x),\mathcal{O}\right)\right)\subset \mathcal D\left(y_{t},\frac{D(\mathcal{O})}{4Ce^\beta}\right)\text{ for all }0\le t\le T.
\end{align}
Fix $t_1,t_2\in [0,T]$ with $|t_1-t_2|\le\theta$. Then, by \eqref{em-1},
\begin{align*}d(y_{t_1},y_{t_2})&\le d(y_{t_1},\phi_{t_1}(x))+d(\phi_{t_1}(x),\phi_{t_2}(x))+d(\phi_{t_2}(x),y_{t_2})\\
&\le \frac{D(\mathcal{O})}{4Ce^\beta}+\frac{D(\mathcal{O})}{4C}+\frac{D(\mathcal{O})}{4Ce^\beta}<\frac{D(\mathcal{O})}{C}.
\end{align*}
Thus, by Lemma \ref{M-1}, there is a constant $\tau$ satisfying $|\tau|\le Cd(y_{t_1},y_{t_2})\ll\delta$ such that
$$\phi_\tau(y_{t_1})=y_{t_2}.$$
By the uniqueness of $v$ given in the {\bf basic canonical setting} and the smallness of both $\t$ and $|\tau|$, one has $\tau=t_2-t_1$. Hence,
$$y_{t_2}=\phi_{t_2-t_1}(y_{t_1})\text{ for all }t_1,t_2\in [0,T]\text{ with } |t_1-t_2|\le\theta.$$
Therefore, $y_t=\phi_t(y_0)$ for all $t\in[0,T]$ by induction, which implies that
$$d(\phi_t(y_0),\phi_t(x))=d(y_t,\phi_t(x))\le Cd(y_t',\phi_t(x))=Cd(\phi_t(x),\mathcal{O})\text{ for all }0\le t\le T.$$
Thus, $y=y_0$ is the point as required.
Now we assume that $d(\phi_t(x),\mathcal{O})\le\frac{D(\mathcal{O})}{4C^2e^\beta}$ for all $t\ge 0$. Then by the arguments above, there is $y\in \mathcal O$ such that $$d(\phi_t(x),\phi_t(y))\le \frac{D(\mathcal{O})}{4Ce^\beta}\le \delta' \text{ for all }t\ge 0.$$ Then by Lemma \ref{An-1}, we have $$d(\phi_t\phi_v(x),\phi_t(y))\to 0\text{ as }t\to+\infty,$$ where $v=v(x,y)$. Then $x$ must be a generic point of $\mu_\mathcal{O}$. This ends the proof. \end{proof}
\begin{lem}\label{M-0} There is positive constant $\tau_0$ such that for any $x\in \L$
$$\phi_t(x)\notin \mathcal D(x,\delta)\text{ for all }0<|t|\le\tau_0.$$ \end{lem}
\begin{proof} Let $\delta',\epsilon',\delta,\epsilon,\lambda,\beta,C$ be the constants as in Remark \ref{remark-1-1}. Take $\tau_0>0$ small enough such that $d(\phi_t(z),z)\le\frac{\delta'}{C^2}$ for all $|t|\le \tau_0$ and $z\in M$. Suppose that there is an $x\in \L$ and $0<|\tau|\le\tau_0$ such that $\phi_\tau(x)\in \mathcal D(x,\delta)$. Note $$w=W^s_\epsilon(x)\cap W^u_\epsilon(\phi_\tau(x)).$$
Then, by (1)(b) of the {\bf basic canonical setting}, one has that
$$\max\left\{d(w,x),d(w,\phi_\tau(x))\right\}\le Cd(x,\phi_\tau(x))\le \frac{\delta'}{C}.$$
Then for $t\ge 0$, $$d(\phi_t(w),\phi_t(x))\le Ce^{-\lambda t}d(w,x)\le Cd(w,x)\le \frac{\delta'} C,$$
and for $t< 0$,
$$d(\phi_t(w),\phi_t(x))\le d(\phi_t(w),\phi_t\phi_\tau(x))+d(\phi_\tau\phi_t(x),\phi_t(x))\le \frac{\delta'}{C}+\frac{\delta'}{C^2}\le \delta',$$
where we used $w=W^s_\epsilon(x)\cap W^u_\epsilon(\phi_\tau(x))$ and the selection of $\tau$.
Then by Lemma \ref{An-1}, there is a constant $l$ with $|l|\ll\delta$ such that $w=\phi_l(x)=\phi_{l-\tau}\phi_\tau(x)$. It is clear that at least one of $l$ and $l-\tau$ is not zero since $\tau\neq 0$. Without loss of any generality, we assume that $l\neq 0$, then
$\{x,\phi_l(x)\}\subset W^s_\epsilon(x)$ (otherwise $\{\phi_l(x),\phi_\tau(x)\}\subset W^u_\e(\phi_\tau(x))$).
Thus
$$W_\epsilon^s(x)\cap W^u_\epsilon(\phi_l(x))\neq\emptyset\text{ and }W_\epsilon^s(x)\cap W^u_\epsilon(x)\neq\emptyset,$$
which is impossible by the uniqueness of function $v$ given in the {\bf basic canonical setting} and A4) of Definition \ref{D:AxiomAAttractor}. This ends the proof. \end{proof} \begin{rem}\label{rem-2-4}
Lemma \ref{M-0} provides a lower bound $\tau_0$ of the periods of periodic segments.
\end{rem} \begin{rem}
We say a periodic segment $\mathcal{O}$ is {\it pure} if $\phi_t(y)\neq y$ for all $y\in\mathcal{O}$ and $0<t<|\mathcal{O}|$. By Lemma \ref{M-0}, a periodic segment $\mathcal{O}$ is pure if and only if $|\mathcal O|=|\mathcal O|_{\min}$. \end{rem}
\subsubsection{Good periodic orbits}\label{S:GoodPerOrb} In this subsection, we mainly demonstrate that for a given compact invariant set, there exist periodic segments being closed enough as well as with reasonable large gap, which are the candidates to support certain minimizing measures. \begin{prop}\label{P:GoodPer} For any $\a\in(0,1]$, a given $\tilde L>0$ and $\Phi$-forward-invariant non-empty subset $Z\subset \L$ (i.e. $\phi_t(Z)\subset Z,\ \forall t\ge 0$), there exists a periodic segment $\mathcal O$ of $\Phi_{\L}$ such that \begin{align}\label{E:GapCond0}
\frac{D^\alpha(\mathcal{O})}{d_{\alpha,Z}(\mathcal{O})}> \tilde L. \end{align} \end{prop} \begin{proof}
Fix $0<\alpha\le 1$, and recall that $\delta',\epsilon',\delta,\epsilon,\lambda,\beta,C$ are as in Remark \ref{remark-1-1}, and $K$, $L$ are as in Lemma \ref{lemma-a}. For the sake of convenience, we assume that $K\ge L$ additionally. By Lemma \ref{lemma-2}, for any $k\in \mathbb N$, there exists a periodic segment $\mathcal O_0$ of $\Phi|_\L$ with period $P_0$ large enough such that \begin{equation}\label{E:GP1} d_{\a,Z}(\mathcal O_0)<P_0^{-k}<<\d'. \end{equation} We remark here that the period of a periodic segment is always assumed to be the {\bf MINIMUM} period, which will avoid unnecessary complexity without harming the argument.
If $D^\a(\mathcal O_0)>\tilde Ld_{\a,Z}(\mathcal O_0)$, the proof is done. Otherwise, one has that \begin{equation}\label{E:GP2} D^\a(\mathcal O_0)\le\tilde L d_{\a,Z}(\mathcal O_0)<\tilde LP_0^{-k}. \end{equation} Since $P_0,k$ can be chosen as large as needed, one can request that $D(\mathcal O_0)<\d'$. Therefore, by definition of $D(\mathcal O_0)$ (see (\ref{E:GapFunc})), there exist $y\in \mathcal O_0$ and $t_0\in(0,P_0)$ such that $$\phi_{t_0}(y)\in \mathcal D(y,D(\mathcal O_0)).$$ Split the periodic segment $\mathcal O_0$ into two segments which are noted by \begin{align*} &\mathcal Q^1_0:[0,t_0]\to\L:t\to \phi_t(y);\\ &\mathcal Q^2_0:[t_0,P_0]\to\L:t\to\phi_t(y). \end{align*} We choose the segment with smaller length and note it by $\mathcal Q_0$. Then either $\mathcal Q_0^L\in \mathcal D(\mathcal Q^R,\d')$ or $\mathcal Q_0^R\in \mathcal D(\mathcal Q^L,\d')$. It is clear that in either case $$d(\mathcal Q_0^L,\mathcal Q_0^R)\le \d'\text{ and }d_{\a,Z}(\mathcal Q_0)\le d_{\a,Z}(\mathcal O_0).$$
Next, we will estimate the increment of orbit deviation after orbit splitting. We will employ different discussions for two different situations according to the length of the segment for which we set $3K$ as a landmark.
\noindent{\bf Case 1.} If the following condition holds \begin{equation}\label{E:LargePerCond}
|\mathcal Q_0|>3K, \end{equation} also note that $d(\mathcal Q_0^L,\mathcal Q_0^R)\le \d'$, then Lemma \ref{lemma-a} is applicable here, by which one has that there exists a periodic segment $\mathcal O_1$ such that \begin{equation}\label{E:O_1-1}
\left|\left|\mathcal Q_0\right|-\left|\mathcal O_1\right|\right|\le Ld(\mathcal Q_1^L,\mathcal Q_1^R)\le LD(\mathcal O_0)<L\tilde LP_0^{-k}, \end{equation} \begin{equation}\label{E:O_1-2}
d\left(\phi_t\left(\mathcal Q_0^L\right),\phi_t\left(\mathcal O_1^L\right)\right)\le Ld(\mathcal Q_0^L,\mathcal Q_0^R)\ \forall 0<t<\max\{|\mathcal Q_0|,|\mathcal O_1|\}. \end{equation}
Since $K\ge L$ and $\d'<<1$, (\ref{E:O_1-1}) together with the assumption $|\mathcal Q_0|>3K$ implies that \begin{equation}\label{E:O_1-3}
|\mathcal O_1|\le \frac43 |\mathcal Q_0|\le \frac23|\mathcal O_0|. \end{equation}
\begin{align*}\begin{split}
d_{\a,Z}(\mathcal O_1)=&\int_0^{|\mathcal O_1|}d^\a\left(\phi_t(\mathcal O_1^L),Z\right)dt \\
=&\int_0^{|\mathcal O_1|}d^\a\left(\phi_t\phi_{v(\mathcal O^L_1,\mathcal Q^L_0)}(\mathcal O_1^L),Z\right)dt \\
\le&\int_0^{|\mathcal Q_0|}d^\a\left(\phi_t\phi_{v(\mathcal O^L_1,\mathcal Q^L_0)}(\mathcal O_1^L),Z\right)dt \\
&+\int_{\left||\mathcal Q_0|-|\mathcal O_1|\right|}^{|\mathcal Q_0|+\left||\mathcal Q_0|-|\mathcal O_1|\right|}d^\a\left(\phi_t\phi_{v(\mathcal O^L_1,\mathcal Q^L_0)}(\mathcal O_1^L),Z\right)dt\\
\le&\int_0^{|\mathcal Q_0|}d^\a\left(\phi_t\phi_{v(\mathcal O^L_1,\mathcal Q^L_0)}(\mathcal O_1^L),\phi_t(\mathcal Q_0^L)\right)dt\quad\quad \quad\quad\quad{\bf \left(Int(a)\right)}\\
&+\int_0^{|\mathcal Q_0|}d^\a\left(\phi_t(\mathcal Q^L_0),Z\right)dt\quad\quad \quad\quad\quad\quad\quad\quad\quad\ \quad{\bf \left(Int(b)\right)}\\
&+\int_0^{|\mathcal Q_0|}d^\a\left(\phi_{\left||\mathcal Q_0|-|\mathcal O_1|\right|}\phi_t\phi_{v(\mathcal O^L_1,\mathcal Q^L_0)}(\mathcal O_1^L),Z\right)dt\quad\quad{\bf \left(Int(c)\right)}. \end{split} \end{align*} By applying Ash2) of Lemma \ref{An-1}, one has that \begin{equation}\label{E:Int(a)}
{\bf Int(a)}\le \int_0^{|\mathcal Q_0|}2C^2e^{-\l\min(t,|\mathcal Q_0|-t)}Ld^\a\left(\mathcal Q_0^L,\mathcal Q_0^R\right) dt\le \frac{2(2C^2L)^\a}{\l\a}d^\a(\mathcal Q_0^L,\mathcal Q_0^R). \end{equation} By definition, one has that \begin{equation}\label{E:Int(b)} {\bf Int(b)}=d_{\a,Z}(\mathcal Q_0). \end{equation} By (3) of the {\bf basic canonical setting}, one has that \begin{align}\begin{split}\label{E:Int(c)}
{\bf Int(c)}= &\int_0^{|\mathcal Q_0|}d^\a\left(\phi_{\left||\mathcal Q_0|-|\mathcal O_1|\right|}\phi_t\phi_{v(\mathcal O^L_1,\mathcal Q^L_0)}(\mathcal O_1^L),\phi_{\left||\mathcal Q_0|-|\mathcal O_1|\right|}Z\right)dt\\
\le&\left(Ce^{\b\left||\mathcal Q_0|-|\mathcal O_1|\right|}\right)^\a\int_0^{|\mathcal Q_0|}d^\a\left(\phi_t\phi_{v(\mathcal O^L_1,\mathcal Q^L_0)}(\mathcal O_1^L),Z\right)dt\\
\le&\left(Ce^{\b\left||\mathcal Q_0|-|\mathcal O_1|\right|}\right)^\a\left({\bf Int(a)+Int(b)}\right). \end{split} \end{align}
By taking $P_0$ and $k$ large, one is able to make $\left||\mathcal Q_0|-|\mathcal O_1|\right|<1$. Therefore, one has the following simplified estimate \begin{equation}\label{E:DaZO1} d_{\a,Z}(\mathcal O_1)\le L_1 d^\a(\mathcal Q_0^L,\mathcal Q_0^R)+L_2 d_{\a,Z}(\mathcal Q_0)\le \hat L d_{\a,Z}(\mathcal O_0), \end{equation} where \begin{align*} L_1&=\left(C^\a e^{\a\b}+1\right)\frac{2(2C^2L)^\a}{\l\a};\\ L_2&=C^\a e^{\a\b}+1;\\ \hat L&=L_1\tilde L+L_2=\left(C^\a e^{\a\b}+1\right)\left(\frac{2(2C^2L)^\a}{\l\a}\tilde L+1\right). \end{align*}
Once $d_{\a,Z}(\mathcal O_1)>\tilde L d_{\a,Z}(\mathcal O_1)$, $\mathcal O_1$ is the periodic segment as required, the splitting process stops. Otherwise, repeat the operation above as long as it is doable. Note that such a process will stop at a finite time, since the operation above will reduce the period of periodic segment at least $\frac13$ by (\ref{E:O_1-3}). Therefore, under the assumption that the operation above is always doable, the process will end on an periodic segment $\mathcal O_m$ for some $m\in\mathbb N\cup\{0\}$, which either satisfies the requirement of this Proposition or $|\mathcal O_m|\ge 3K$ while $|\mathcal Q_m|<3K$. In either cases, one has that $$m\le \frac{\log P_0-\log(3K)}{\log 1.5}+1,$$ and $$d_{\a,Z}(\mathcal O_i)\le \hat L^id_{\a,Z}(\mathcal O_0),\ \forall 1\le i\le m.$$ In order to make each operation above doable, one needs that $$D(\mathcal O_i)<\d',\ \forall 1\le i\le m-1,$$ which can be done by assuming the largeness of $P_0$ and $k$ in advance. To be precise, one can take \begin{equation}\label{E:CondPk1} k>\frac{\log \hat L}{\log 1.5}\text{ and }P_0^{\frac{\log \hat L}{\log 1.5}-k}<\frac{(\d')^\a}{\tilde L\hat L}, \end{equation}
where the second inequality above implies that for all $0\le i\le m-1$
$$\tilde Ld_{\a,Z}(\mathcal O_i)\le P_0^{-k}\tilde L\hat L^{m}<(\d')^\a,$$
which ensures the existence of $\mathcal O_{i+1}$ and $ \tilde L d_{\a,Z}(\mathcal O_m)<(\d')^\a$.
\noindent{\bf Case 2.} We will deal with the case that $|\mathcal Q_m|\le 3K$, which is the counterpart of the case when (\ref{E:LargePerCond}) holds. We will show that by rearranging extra largeness of $P_0$ and $k$, one can make $\mathcal O_m$ satisfy the requirement of Proposition \ref{P:GoodPer}. We will prove this by contradiction.
Before going to further discussion, we should note first that the union of all periodic orbits of $\Phi|_\L$ with period $\le 4K$ is a nonempty compact subset of $\L$, which is denoted by $Per_{4K}$. Once $Z\cap Per_{4K}\neq \emptyset$ Proposition \ref{P:GoodPer} holds automatically; otherwise, there exists $\s>0$ such that
\begin{equation}\label{E:4KSeparation}
d(x,Z)>\s\ \forall x\in Per_{4K}.
\end{equation}
Suppose that
\begin{equation}\label{E:FakeCond}
D^\a(\mathcal O_m)\le \tilde L d_{\a,Z}(\mathcal O_m)<(\d')^\a.
\end{equation}
When
\begin{equation}\label{E:Kto3K}
K\le |\mathcal Q_m|\le3K,
\end{equation}
by the exactly same argument as on $\mathcal Q_0$, one has that there exists a periodic segment $\mathcal O_{m+1}$ such that
$$|\mathcal O_{m+1}|\le 4K\text{ and }d_{\a,Z}(\mathcal O_{m+1})\le \hat L d_{\a,Z}(\mathcal O_m)\le \hat L^{m+1}d_{\a,Z}(\mathcal O_0)<\hat L^{m+1}P_0^{-k}.$$
By choosing $P_0$ and $k$ large enough one can make $d_{\a,Z}(\mathcal O_{m+1})<\s$ which implies an contradiction with (\ref{E:4KSeparation}). Therefore (\ref{E:FakeCond}) and (\ref{E:Kto3K}) can not hold simultaneously for $P_0$ and $k$ large enough.
When
\begin{equation}\label{E:0toK}
|\mathcal Q_m|< K,
\end{equation}
Lemma \ref{lemma-a} is not applicable directly. For sake of convenience, note $l=|\mathcal Q_m|$. By (\ref{E:FakeCond}), $\mathcal{Q}_m^L\in D(\mathcal{Q}_m^R,\delta')$ or $\mathcal{Q}_m^R\in D(\mathcal{Q}_m^L,\delta')$. Then by Lemma \ref{M-0}, $l>\tau_0$. Let $q$ be the integer such that $$K\le ql\le 2K\text{ and thus } 2\le q\le \left[\frac{2K}{\tau_0}\right].$$ Note $$\mathcal{S}_i:[0,l]\to M: t\to \phi_{il+t}(\mathcal{Q}_m^L)\text{ for }i=0,1,2,\cdots, q-1,$$ and $$\mathcal{S}:[0,ql]\to M: t\to \phi_{t}(\mathcal{Q}_m^L).$$ Then, $$d(\mathcal{S}_0^L,\mathcal{S}_0^R)=d(\mathcal{Q}_m^L,\mathcal{Q}_m^R)$$ $$d(\mathcal{S}_1^L,\mathcal{S}_1^R)=d(\phi_l(\mathcal{S}_0^L),\phi_l(\mathcal{S}_0^R))\le Ce^{\beta l} d(\mathcal{S}_0^L,\mathcal{S}_0^R)\le Ce^{\beta K} d(\mathcal{Q}_m^L,\mathcal{Q}_m^R) ,$$ $$\dots$$ $$d(\mathcal{S}_{q-1}^L,\mathcal{S}_{q-1}^R)\le (Ce^{\beta K} )^{q-1}d(\mathcal{Q}_m^L,\mathcal{Q}_m^R).$$ Therefore, \begin{equation*} d(\mathcal{S}^L,\mathcal{S}^R)\le \sum_{i=0}^{q-1}(Ce^{\beta K} )^{i}d(\mathcal{Q}_m^L,\mathcal{Q}_m^R)\le \frac{(Ce^{\beta K} )^{\left[\frac{2K}{\tau_0}\right]}-1}{Ce^{\beta K} -1} D(\mathcal O_m), \end{equation*} which together with (\ref{E:FakeCond}) implies that \begin{equation}\label{E:SGap} d(\mathcal{S}^L,\mathcal{S}^R)\le \frac{(Ce^{\beta K} )^{\left[\frac{2K}{\tau_0}\right]}-1}{Ce^{\beta K} -1} \left(\tilde Ld_{\a,Z}(\mathcal O_m)\right)^{\frac1\a}\le \frac{(Ce^{\beta K} )^{\left[\frac{2K}{\tau_0}\right]}-1}{Ce^{\beta K} -1} \left(\tilde L\hat L^mP_0^{-k}\right)^{\frac1\a}. \end{equation}
By taking $P_0$ and $k$ large enough, one can make $d(\mathcal S^L,\mathcal S^R)<\d'$. Also note that $|\mathcal S|\ge K$, then Lemma \ref{lemma-a} is applicable to $\mathcal S$. Therefore, there exists a periodic segment $\mathcal{O}_*$ such that $|\mathcal{O}_*|\le |\mathcal{S}|+Ld(\mathcal{S}^L,\mathcal{S}^R)\le 3K$ and \begin{align}\label{E-11}
d\left(\phi_t\phi_{v(\mathcal{O}_*^L,\mathcal{S}^L)}(\mathcal{O}_*^L),\phi_t(\mathcal{S}^L)\right)\le L\frac{(Ce^{\beta K} )^{\left[\frac{2K}{\tau_0}\right]}-1}{Ce^{\beta K} -1} \left(\tilde L\hat L^mP_0^{-k}\right)^{\frac1\a}\ \forall 0\le t\le |\mathcal{S}|, \end{align} where the right hand side of the above inequality can be make smaller than $\frac13\s$ by taking $P_0$ and $k$ large enough. On the other hand, $$d_{\alpha,Z}(\mathcal{S}_0)=d_{\alpha,Z}(\mathcal{Q}_m)$$ $$d_{\alpha,Z}(\mathcal{S}_1)=\int_0^ld^\alpha(\phi_{l+t}(\mathcal{S}_0^L),Z)\le (Ce^{\beta l})^\alpha d_{\alpha,Z}(\mathcal{S}_0)\le (Ce^{\beta K})^\alpha d_{\alpha,Z}(\mathcal{Q}_m),$$ $$\dots$$ $$d_{\alpha,Z}(\mathcal{S}_{q-1})\le(Ce^{\beta K})^{(q-1)\alpha} d_{\alpha,Z}(\mathcal{Q}_m).$$ Thus, \begin{align} \begin{split}\label{E:SSeparation} d_{\alpha,Z}(\mathcal{S})&=\sum_{i=0}^{q-1}d_{\alpha,Z}(\mathcal{S}_{i})\le \sum_{i=0}^{q-1}(Ce^{\beta K})^{i\alpha} d_{\alpha,Z}(\mathcal{Q}_m)\\ &\le\sum_{i=0}^{[\frac{2K}{\tau_0}]-1}C^{i\a}e^{i\beta K\a}d_{\alpha,Z}(\mathcal{O}_{m})\\ &\le \sum_{i=0}^{[\frac{2K}{\tau_0}]-1}C^{i\a}e^{i\beta K\a} \hat L^{m}d_{\alpha,Z_{u}}(\mathcal{O}_0)\\ &\le \frac{(Ce^{\beta K} )^{\left[\frac{2K}{\tau_0}\right]\a}-1}{\left(Ce^{\beta K}\right)^\a -1} \hat L^mP_0^{-k}, \end{split} \end{align}
which can be make smaller than $\frac13\s$ by taking $P_0$ and $k$ large enough. Since $|\mathcal{S}|\ge K>1$, there is a point $t_*\in[0,|\mathcal{S}|]$ such that $$d(\phi_{t_*}(\mathcal{S}^L),Z)\le \frac{\sigma}{3}.$$ Therefore, by \eqref{E-11}, by taking $P_0$ and $k$ large enough, we have $$d(\phi_{t_*}\phi_{v_*}(\mathcal{O}_*^L),Z)\le d\left(\phi_{t_*}\phi_{v(\mathcal{O}_*^L,\mathcal{S}^L)}(\mathcal{O}_*^L),\phi_{t_*}(\mathcal{S}^L))+d(\phi_{t_*}(\mathcal{S}^L),Z\right)\le \frac{2}{3}\s<\sigma$$
which contradicts with (\ref{E:4KSeparation}) as $K\le |\mathcal S|\le 2K$.\\ Hence (\ref{E:FakeCond}) can not hold for large enough $P_0$ and $k$. This ends the proof. \end{proof}
Here, we remark that there is no fixed point under the setting of this paper by A4) of Definition \ref{D:AxiomAAttractor}, thus the orbit splitting process cannot stop at a fixed point, which is the main difference comparing to the discrete time case in \cite{HLMXZ}. The Case 2. above is mainly taking care this issue.
\subsubsection{Main Proposition}\label{S:MainProp} In this subsection, we state and prove our main proposition. For a continuous function $u$ and a segment $\mathcal S$ of $\Phi$, define the integration of $u$ along $\mathcal S$ with time interval $[a,b]$ and starting point $x$ by the following \begin{equation}\label{E:IntSeg} \langle \mathcal S,u\rangle:=\int_a^bu\left(\phi_t\left(\mathcal S^L\right)\right)dt, \end{equation} also recall that the definition of $u_\g$ for $\g>0$ by $$u_\g(x)=\frac1{\g}\int_0^\g u\left(\phi_t(x)\right)dt.$$ Now we have the following proposition. \begin{prop}\label{prop-2}
Given $0<\varepsilon\le 1$, $0< \alpha\le 1$, a strictly positive function $\psi\in\mathcal{C}^{0,\alpha}(M)$ and $u\in \mathcal{C}^{0,\alpha}(M)$, if a periodic segment $\mathcal{O}$ of $\Phi|_\L$ satisfies the following comparison condition
\begin{align}\label{condition}
\frac{D^\alpha(\mathcal{O})}{d_{\alpha,Z_{u,\psi}}(\mathcal{O})}>
\left(\frac{4C^{3\alpha}(\|\bar u\|_\alpha+10\varepsilon+\frac{ \|\bar u\|_0 + 1 }{\psi_{min} }\|\psi_\gamma\|_\alpha) }{\lambda\alpha\varepsilon \tau_0}+1+\frac{1}{\tau_0}\right) \frac{\|\bar u\|_\alpha\|\psi\|_0}{\psi_{min}}\cdot\frac{ 100(4C^3e^{2\beta})^\alpha}{\varepsilon},
\end{align}
where $\bar u$ is defined in Remark \ref{revael} and $\tau_0$ is the constant in Lemma \ref{M-0},
then the periodic measure $$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+\varepsilon d^\alpha(\cdot,\mathcal{O}) +h;\psi,\L,\Phi),$$ where $h\in\mathcal C^{0,\a}(M)$ satisfying $\|h\|_\alpha<10\varepsilon$ and
$$\|h\|_0<\min\left\{\frac{\frac{\varepsilon}{2}\left(\frac{D(\mathcal{O})}{4C^3e^{2\beta}}\right)^\alpha}{\left(\frac{4C^{3\alpha}(\|\bar u\|_\alpha+10\varepsilon+\frac{ \|\bar u\|_0 + 1 }{\psi_{min} }\|\psi_\gamma\|_\alpha) }{\lambda\alpha\varepsilon}+|\mathcal{O}|+1\right) \frac{\|\psi\|_0+\psi_{min}}{\psi_{min}}},1\right\}.$$ \end{prop} \begin{proof}Fix $\varepsilon, \alpha,\mathcal{O},\psi,u,h$ as in the Proposition, $\delta',\epsilon',\delta,\epsilon,\lambda,\beta,C$ as in Remark \ref{remark-1-1}, $\bar u,Z_{u,\psi},\gamma,\psi_\gamma$ as in Remark \ref{rem-reveal}, $\tau_0$ as in Lemma \ref{M-0}. Note $$G=\bar u+\varepsilon d^\alpha(\cdot,\mathcal{O})+h-a_\mathcal{O}\psi_\gamma$$ where
\begin{align*}
\begin{split}
a_{\mathcal{O}}&:=\frac{\left\langle \mathcal{O},\bar u+\varepsilon d^\alpha(\cdot,\mathcal{O})+h\right\rangle }{\left\langle \mathcal{O},\psi_\gamma\right\rangle}=\frac{\left\langle \mathcal{O},\bar u+h\right\rangle }{\left\langle \mathcal{O},\psi_\gamma\right\rangle}.
\end{split}\end{align*}
By straightforward computation, one has that
\begin{align}\label{X-1}
\begin{split}
\left| a_{\mathcal O}\right|\le\frac{\left\langle \mathcal{O},\|\bar u\|_\alpha d^\alpha(\cdot,Z_{u,\psi})+h\right\rangle}{\left\langle \mathcal{O},\psi_{min}\right\rangle}\le \frac{\|\bar u\|_\alpha d_{\alpha,Z_{u,\psi}}(\mathcal{O})}{| \mathcal{O}|\psi_{min}}+\frac{\|h\|_0}{\psi_{min}},
\end{split}
\end{align}
where we used $\bar u|_{Z_{u,\psi}}=0$.
Notice that for all $\mu\in\mathcal{M}(\L,\Phi)$
\begin{align*}\frac{\int\left( u+\varepsilon d^\alpha(\cdot,\mathcal{O}) +hd\right)\mu}{\int \psi d\mu}&=\frac{\int \left(u_\gamma+\varepsilon d^\alpha(\cdot,\mathcal{O}) +h\right)d\mu}{\int \psi_\gamma d\mu}\\
&=\frac{\int \left(\bar u+\varepsilon d^\alpha(\cdot,\mathcal{O}) +h\right)d\mu}{\int \psi_\gamma d\mu}+\beta(u;\psi,\L,\Phi)\\
&=\frac{\int Gd\mu}{\int \psi d\mu}+a_{\mathcal{O}}+\beta(u;\psi,\L,\Phi).
\end{align*}
Then, in order to show that $\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+\varepsilon d^\alpha(\cdot,\mathcal{O})+h;\psi,\L,\Phi)$,
it is enough to show that $\mu_{\mathcal{O}}\in\mathcal{M}_{min}(G;\psi_\gamma,\L,\Phi)$. Since $\psi$ is strictly positive and $\int Gd\mu_\mathcal{O}=0$, it is enough to show that
\begin{equation}\label{E:PositiveIntG}
\int Gd\mu\ge0\text{ for all }\mu\in\mathcal{M}^e(\L,\Phi).
\end{equation} Define a compact set $\mathcal R\subset M$ by
\[\mathcal R=\left\{y\in M:d(y,\mathcal{O})\le\left(\frac{|a_{\mathcal{O}}|\|\psi\|_0+\|h\|_0}{\varepsilon}\right)^{\frac{1}{\alpha}}\right\}.\]
We have the following Claim.
\noindent{\it {\bf Claim 1.} $\mathcal R$ contains all $x\in M$ with $G(x)\le0$. }
\begin{proof}[Proof of Claim 1.]
Given $x\in M\setminus \mathcal R$, we are to show that $G(x)>0$. Note that
\begin{align}\label{X-2}
\begin{split}\bar u+h-a_{\mathcal{O}}\psi_\gamma&\ge-|a_{\mathcal{O}}|\|\psi_\gamma\|_0-\|h\|_0.
\end{split}
\end{align}
where we used $\bar u\ge0$ and $\|\psi_\gamma\|_0\le \|\psi\|_0$. Then
\begin{align*}G(x)&=\bar u(x)+\varepsilon d^\alpha(x,\mathcal{O})+h(x)-a_{\mathcal{O}}\psi_\gamma\\
&\ge \varepsilon d^\alpha(x,\mathcal{O})-|a_{\mathcal{O}}|\|\psi\|_0-\|h\|_0\\
&> \varepsilon\cdot\left(\left(\frac{|a_{\mathcal{O}}|\|\psi\|_0+\|h\|_0}{\varepsilon}\right)^{\frac{1}{\alpha}}\right)^\alpha-|a_{\mathcal{O}}|\|\psi\|_0-\|h\|_0\\
&=0.
\end{align*}
This ends the proof of Claim 1.
\end{proof} Define a compact set $\mathcal R'\subset M$ by
\[\mathcal R'=\left\{y\in M:d(y,\mathcal{O})\le\left(\frac{2(|a_{\mathcal{O}}|\|\psi\|_0+\|h\|_0)}{\varepsilon}\right)^{\frac{1}{\alpha}}\right\}.\]
It is easy to see that $\mathcal R$ is in the interior of $\mathcal R'$ and the following holds because of (\ref{condition}), (\ref{X-1}) and the range of $\|h\|_0$
\begin{equation}\label{E:R'Est}
d(y,\mathcal O)\le\left(\frac{2(a_{\mathcal{O}}\|\psi\|_0+\|h\|_0)}{\varepsilon}\right)^{\frac{1}{\alpha}}\le\frac{D(\mathcal{O})}{4C^3e^{2\beta}},\ \forall y\in \mathcal R'.
\end{equation}
By Claim 1, there is a constant $\tau$ with $0<\tau<1$ such that $G(\phi_t(x))>0$ for all $x\in M\setminus \mathcal R'$ and $|t|\le\tau$.
Now we claim the following assertion:\\
{\it {\bf Claim 2.} If $z\in M$ is not a generic point of $\mu_\mathcal{O}$, then there is $m\ge\tau$ such that
$$\int_{0}^mG(\phi_t(z))dt>0.$$}
Next we prove the Proposition by assuming the validity of Claim 2, while the proof of Claim 2. is left to the end of this section. For a given ergodic measure $\mu\in\mathcal{M}^e(\L,\Phi)$, if $\mu=\mu_{\mathcal{O}}$, (\ref{E:PositiveIntG}) obviously holds. Otherwise, let $z$ be a generic point of $\mu$, thus $z$ is not a generic point of $\mu_{\mathcal{O}}$. Therefore, by Claim 2, there is a $t_1\ge\tau$ such that \[\int_0^{t_1} G(\phi_t(z))dt>0.\]
Since $\phi_{t_1}(z)$ is still not a generic point of $\mu_{\mathcal{O}}$, by using claim 2 agian, one has a $t_2\ge t_1+\tau$ such that \[\int_{t_1}^{t_2} G(\phi_t(z))dt>0.\]
By repeating the above process, one has $0\le t_1<t_2<t_3<\cdots$ with all gaps not less than $\tau$ such that
\[\int_{t_i}^{t_{i+1}} G(\phi_t(z))dt>0\text{ for } i=0,1,2,3,\cdots,\]
where we assign $t_0=0$. Therefore
\begin{align*}
\int Gd\mu&=\lim_{l\to+\infty}\frac{1}{l}\int_{0}^{l}G(\phi_t(z))dt\\
&= \lim_{i\to+\infty}\frac{1}{t_i}\left(\int_{t_0}^{t_1}G(\phi_t(z))dt+\int_{t_1}^{t_2}G(\phi_t(z))dt+\cdots+\int_{t_{i-1}}^{t_i}G(\phi_t(z))dt\right)\\
&\ge 0.
\end{align*}
Thus, $\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+\varepsilon d^\alpha(\cdot,\mathcal{O})+h;\psi,\L,\Phi)$. This ends the proof.
\end{proof}
\begin{rem}\label{remark} It is not difficult to see that for any $\varepsilon'>\varepsilon$, $\mu_\mathcal{O}$ is the unique measure in $\mathcal{M}_{min}(u+\varepsilon' d^\alpha(\cdot,\mathcal{O})+h;\psi,\L,\Phi)$ whenever $\|h\|_\alpha<10\varepsilon$ and
$\|h\|_0$ is sufficiently small. The Proposition shows that there is an open set of $\mathcal{C}^{0,\alpha}(M)$ near $u$ such that these $\alpha$-H\"older functions in the open set has the same unique minimizing measure with respect to $\psi$ being supported on a periodic orbit. \end{rem}
\begin{proof}[Proof of Claim 2.]
If $z\notin \mathcal R'$, just take $m=\tau$,
we have nothing to prove since $G(\phi_t(z))>0$ for all $|t|\le\tau$.
Therefore, one needs only to consider the case that $z\in \mathcal R'$. Also note that, since $z$ is not a generic point of $\mu_\mathcal{O}$, Lemma \ref{M-3} implies that the following inequality
$$d(\phi_t (z),\mathcal{O})\le\frac{D(\mathcal{O})}{4C^2e^\beta}$$
{\bf CANNOT} hold for all $t\ge 0$. Thus there is an $m_1>0$ such that $$d(\phi_{m_1} (z),\mathcal{O})>\frac{D(\mathcal{O})}{4C^2e^\beta}.$$
Let $m_2>0$ be the smallest time such that
\begin{align}\label{X-3}
d(\phi_{m_2} (z),\mathcal{O})=\frac{D(\mathcal{O})}{4C^2e^\beta},
\end{align}
where the existence of such $m_2$ is ensured by (\ref{E:R'Est}) and the continuity of the flow. Then, by (3) of the {\bf basic canonical setting}, one has the following
$$d(\phi_{m_2-t}(z),\mathcal{O})>\frac{D(\mathcal{O})}{4C^3e^{2\beta}},\ \forall 0< t\le 1,$$
which together with (\ref{E:R'Est}) implies that
\begin{align}\label{Y-2}
\phi_{m_2-t}(z)\notin \mathcal R'\text{ for all }0< t\le 1.
\end{align} Thus
\begin{align}\label{X-5}
\begin{split}\int_{m_2-1}^{m_2}G(\phi_{t}(z))dt&=\int_{m_2-1}^{m_2} \bar{u}(\phi_{t}(z))+\varepsilon d^\alpha(\phi_{t}(z),\mathcal{O})+h(\phi_{t}(z))-a_{\mathcal{O}}\psi_\gamma(\phi_{t}(z))dt\\
&\ge \int_{m_2-1}^{m_2}\varepsilon d^\alpha(\phi_{t}(z),\mathcal{O})-|a_{\mathcal{O}}|\|\psi\|_0-\|h\|_0dt\\
&\ge\varepsilon\cdot\left(\frac{D(\mathcal{O})}{4C^3e^{2\beta}}\right)^\alpha-|a_{\mathcal{O}}|\|\psi\|_0-\|h\|_0,\\
\end{split}
\end{align}
where we used \eqref{X-2}.
Now since $\mathcal R'$ is compact, there is an $m_3$ which is the largest time such that $0\le m_3\le m_2$
and $\phi_{t}(z)\in \mathcal R'$.
By \eqref{Y-2}, it is clear that $m_3\le m_2-1$.
Then by Claim 1, for all $m_3<t<m_2-1$ \begin{align}\label{X-12}
G(\phi_t(z))>0,\end{align}
where we used the fact $\mathcal R\subset \mathcal R'$. On the other hand, since $m_3<m_2$, by the choice of $m_2$ (see \eqref{X-3}), one has that
$$d(\phi_t(z),\mathcal{O})\le \left(\frac{2(a_{\mathcal{O}}\|\psi\|_0+\|h\|_0)}{\varepsilon}\right)^{\frac{1}{\alpha}}<\frac{D(\mathcal{O})}{4C^2e^\beta}\text{ for all } 0\le t\le m_3.$$ Therefore, by Lemma \ref{M-3}, there is $y_0\in\mathcal{O}$ such that
$$d(\phi_t(z),\phi_t(y_0))\le C\left(\frac{2(a_{\mathcal{O}}\|\psi\|_0+\|h\|_0)}{\varepsilon}\right)^{\frac{1}{\alpha}}\le \delta'\text{ for all }t\in[0,m_3].$$
By using Lemma \ref{An-1}, we have for all $0\le t\le m_3$,
\begin{align*}
d(\phi_{t}\phi_{v(y_0,z)}(y_0),\phi_t(z))\le 2C^2e^{-\lambda\min(t,m_3-t)}C\left(\frac{2(a_{\mathcal{O}}\|\psi\|_0+\|h\|_0)}{\varepsilon}\right)^{\frac{1}{\alpha}}.
\end{align*} Hence,
\begin{align*}
\begin{split}
&\int_{0}^{ m_3}d^\alpha(\phi_{t}(z),\phi_{t}\phi_{v(y_0,z)}(y_0))dt\\
\le&\int_{0}^{ m_3}\left(2C^3\left(\frac{2\left(|a_{\mathcal{O}}|\|\psi\|_0+\|h\|_0\right)}{\varepsilon}\right)^{\frac{1}{\alpha}}\left(e^{-\lambda t}+e^{-\lambda(m_3-t)}\right)\right)^\alpha dt\\
\le&\frac{4C^{3\alpha}}{\lambda\alpha}\cdot \frac{|a_{\mathcal{O}}|\|\psi\|_0+\|h\|_0}{\varepsilon}.
\end{split}
\end{align*}
Therefore,
\begin{align*}
\begin{split}&\int_{0}^{ m_3}G(\phi_{t}(z))-G(\phi_{t}\phi_v(y_0))dt\\
=&\int_{0}^{ m_3}\bar{u}(\phi_{t}(z))+\varepsilon d^\alpha(\phi_{t}(z),\mathcal{O})+h(\phi_{t}(z))-\bar{u}(\phi_{t+v}(y_0))-h(\phi_{t+v}(y_0))\\
&-a_{\mathcal{O}}\big(\psi_\gamma(\phi_{t}(z))
-\psi_\gamma(\phi_{t+v}(y_0))\big)dt\\
\ge&\int_{0}^{ m_3}\bar{u}(\phi_{t}(z))-\bar{u}(\phi_{t+v}(y_0))+h(\phi_{t}(z))-h(\phi_{t+v}(y_0))\\
&-a_{\mathcal{O}}\big(\psi_\gamma(\phi_{t}(z))
-\psi_\gamma(\phi_{t+v}(y_0))\big)dt\\
\ge& -(\|\bar u\|_\alpha+\|h\|_\alpha+|a_\mathcal{O}|\|\psi_\gamma\|_\alpha)\int_{0}^{ m_3}d^\alpha(\phi_{t}(z),\phi_{t+v}(y_0))dt\\
\ge& - (\|\bar u\|_\alpha+\|h\|_\alpha+|a_\mathcal{O}|\|\psi_\gamma\|_\alpha) \cdot \frac{4C^{3\alpha}}{\lambda\alpha}\cdot \frac{|a_{\mathcal{O}}|\|\psi\|_0+\|h\|_0}{\varepsilon},
\end{split}
\end{align*}
where we write $v$ short for $v(y_0,z)$. Also note that
\begin{align*}
|a_\mathcal{O}|
=\left|\frac{\left\langle \mathcal{O},\bar u+h\right\rangle }{\left\langle \mathcal{O},\psi_\gamma\right\rangle}\right|\le \frac{ \|\bar u\|_0 + \|h\|_0 }{\psi_{min} }\le \frac{ \|\bar u\|_0 + 1}{\psi_{min} }.
\end{align*}
Thus, one has that
\begin{align}\label{X-6}
\begin{split}&\int_{0}^{ m_3}G(\phi_{t}(z))-G(\phi_{t}\phi_v(y_0))dt\\
\ge& - (\|\bar u\|_\alpha+\|h\|_\alpha+\frac{ \|\bar u\|_0 + 1 }{\psi_{min} }\|\psi_\gamma\|_\alpha) \cdot \frac{4C^{3\alpha}}{\lambda\alpha}\cdot \frac{|a_{\mathcal{O}}|\|\psi\|_0+\|h\|_0}{\varepsilon},
\end{split}
\end{align}
Rewrite $m_3=p|\mathcal{O}|+r$ for some nonnegative integer $p$ and real number $0\le r\le |\mathcal{O}|$. By applying \eqref{X-2} and $\int Gd\mu_\mathcal{O}=0$, one has that
\begin{align}\label{X-7}
\begin{split}
\int_{0}^{ m_3}G(\phi_t\phi_v(y_0))dt&=\int_{m_3-r}^{m_3}G(\phi_t\phi_v(y_0))dt\ge -|\mathcal{O}|\cdot (|a_{\mathcal{O}}|\|\psi\|_0+\|h\|_0).\\
\end{split}
\end{align}
Combining \eqref{X-1}, \eqref{X-5}, \eqref{X-12}, \eqref{X-6} and \eqref{X-7}, we have
\begin{align*}
&\int_{0}^{ m_2}G(\phi_t(z))dt\ge \int_{0}^{ m_3}G(\phi_t(z))dt+\int_{m_2-1}^{ m_2}G(\phi_t(z))dt &\text{by }(\ref{X-12})\\
=& \int_{0}^{ m_3}\left(G(\phi_t(z))-G(\phi_{t+v}(y_0))\right)dt\\
&+\int_{0}^{ m_3}G(\phi_{t+v}(y_0))dt+\int_{m_2-1}^{ m_2}G(\phi_t(z))dt\\
\ge&-(\|\bar u\|_\alpha+\|h\|_\alpha+\frac{ \|\bar u\|_0 + 1}{\psi_{min} }\|\psi_\gamma\|_\alpha) \cdot \frac{4C^{3\alpha}}{\lambda\alpha}\cdot \frac{|a_{\mathcal{O}}|\|\psi\|_0+\|h\|_0}{\varepsilon}&\text{by }(\ref{X-6})\\
&-|\mathcal{O}|\cdot (|a_{\mathcal{O}}|\|\psi\|_0+\|h\|_0)&\text{by }(\ref{X-7})\\
&+\varepsilon\left(\frac{D(\mathcal{O})}{4C^3e^{2\beta}}\right)^\alpha-|a_{\mathcal{O}}|\|\psi\|_0-\|h\|_0&\text{by }(\ref{X-5})\\
\ge&\varepsilon\left(\frac{D(\mathcal{O})}{4C^3e^{2\beta}}\right)^\alpha\\
&-\left(\frac{4C^{3\alpha}(\|\bar u\|_\alpha+\|h\|_\alpha+\frac{ \|\bar u\|_0 + 1 }{\psi_{min} }\|\psi_\gamma\|_\alpha) }{\lambda\alpha\varepsilon|\mathcal{O}|}+1+\frac{1}{|\mathcal{O}|}\right) \frac{\|\bar u\|_\alpha\|\psi\|_0}{\psi_{min}} d_{\alpha,Z_{u,\psi}}(\mathcal{O})\\
&-\left(\frac{4C^{3\alpha}(\|\bar u\|_\alpha+\|h\|_\alpha+\frac{ \|\bar u\|_0 + 1}{\psi_{min} }\|\psi_\gamma\|_\alpha) }{\lambda\alpha\varepsilon}+|\mathcal{O}|+1\right) \frac{\|\psi\|_0+\psi_{min}}{\psi_{min}} \|h\|_0&\text{by }(\ref{X-1})\\
\ge&\varepsilon\left(\frac{D(\mathcal{O})}{4C^3e^{2\beta}}\right)^\alpha\\
&-\left(\frac{4C^{3\alpha}(\|\bar u\|_\alpha+10\varepsilon+\frac{ \|\bar u\|_0 + 1 }{\psi_{min} }\|\psi_\gamma\|_\alpha) }{\lambda\alpha\varepsilon \tau_0}+1+\frac{1}{\tau_0}\right) \frac{\|\bar u\|_\alpha\|\psi\|_0}{\psi_{min}} d_{\alpha,Z_{u,\psi}}(\mathcal{O})\\
&-\left(\frac{4C^{3\alpha}(\|\bar u\|_\alpha+10\varepsilon+\frac{ \|\bar u\|_0 + 1}{\psi_{min} }\|\psi_\gamma\|_\alpha) }{\lambda\alpha\varepsilon}+|\mathcal{O}|+1\right) \frac{\|\psi\|_0+\psi_{min}}{\psi_{min}} \|h\|_0\\
>&0,
\end{align*}
where we used Remark \ref{rem-2-4} and condition \eqref{condition}. Therefore, $m=m_2$ is the time as required since $m_2\ge 1\ge \tau$. This ends the proof of Claim 2.
\end{proof}
\subsection{Proof of Part II) of Theorem \ref{T:MainResult}}\label{S:ProofPartIIMainThm} Firstly, we state a technical result on function approximation, which plays a key role in proving Proposition \ref{prop-3}. Proposition \ref{prop-3} can be viewed as a $C^1$-version of Proposition \ref{prop-2} which implies the part II) of Theorem \ref{T:MainResult}. \begin{thm}[\cite{GW79}]\label{thm-G} Let $M$ be a smooth compact manifold. Then $\mathcal{C}^\infty(M)\cap \mathcal{C}^{0,1}(M)$ is Lip-dense in $\mathcal{C}^{0,1}(M)$. \end{thm}
\begin{rem}In this theorem, {\it $\mathcal{C}^\infty(M)\cap \mathcal{C}^{0,1}(M)$ is Lip-dense in $\mathcal{C}^{0,1}(M)$} means that for any $g_1\in \mathcal{C}^{0,1}(M)$ and $\varepsilon>0$ there is corresponding $g_2\in \mathcal{C}^\infty(M)$ such that $\|g_1-g_2\|_0<\varepsilon$ and $\|g_2\|_1<\varepsilon+\|g_1\|_1$. Especially, $\|D_Mg_2\|_0< \varepsilon+\|g_1\|_1$, where $D_Mg$ is the derivative of function $g$ with respect to space variables. \end{rem} \begin{prop}\label{prop-3}
Given $0<\varepsilon\le 1$, a strictly positive $\psi\in\mathcal{C}^{1,0}(M)$ and $u\in \mathcal{C}^{1,0}(M)$, if a periodic segment $\mathcal{O}$ of $\Phi|_\L$ satisfies the following comparison condition
\begin{align*}
D(\mathcal{O})>
\Big(\frac{4C^{3}(\|\bar u\|_1+10\varepsilon+\frac{ \|\bar u\|_0 + 1 }{\psi_{min} }\|\psi_\gamma\|_1) }{\l\varepsilon \tau_0}+1+\frac{1}{\tau_0}\Big) \frac{\|\bar u\|_1\|\psi\|_0}{\psi_{min}}\cdot\frac{ 400C^3e^{2\beta}}{\varepsilon}\cdot d_{1,Z_{u,\psi}}(\mathcal{O}),
\end{align*}
where $\bar u$ is defined in Remark \ref{revael} and $\tau_0$ is the constant in Lemma \ref{M-0}, then there is a function $w\in C^\infty(M)$ with $\|w\|_0<2\varepsilon\cdot diam(M)$ and $\|D_Mw\|_0<2\varepsilon$ such that the probability measure $$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+w+h;\psi,\L,\Phi),$$ where $h$ is any $C^1$ function with $\|D_Mh\|_0<5\varepsilon$ and
$$\|h\|_0<\frac{1}{2}\cdot\min\left\{\frac{\frac{\varepsilon}{2}\left(\frac{D(\mathcal{O})}{4C^3e^{2\beta}}\right)}{\left(\frac{4C^{3}(\|\bar u\|_1+10\varepsilon+\frac{ \|\bar u\|_0 + 1 }{\psi_{min} }\|\psi_\gamma\|_1) }{\lambda\varepsilon}+|\mathcal{O}|+1\right) \frac{\|\psi\|_0+\psi_{min}}{\psi_{min}}},1\right\}.$$ \end{prop} \begin{proof}By Theorem \ref{thm-G}, there exists a function $ w\in C^\infty$ such that
$$\|D_Mw\|_0< \|\varepsilon d(\cdot,\mathcal{O})\|_1+\varepsilon\le 2\varepsilon$$
and
$$\| w-\varepsilon d(\cdot,\mathcal{O})\|_0<\min\left(\frac{H}{2},\varepsilon\cdot diam(M)\right),$$
where $$H=\min\left\{\frac{\frac{\varepsilon}{2}\left(\frac{D(\mathcal{O})}{4C^3e^{2\beta}}\right)}{\left(\frac{4C^{3}(\|\bar u\|_1+10\varepsilon+\frac{ \|\bar u\|_0 + 1 }{\psi_{min} }\|\psi_\gamma\|_1) }{\lambda\varepsilon}+|\mathcal{O}|+1\right) \frac{\|\psi\|_0+\psi_{min}}{\psi_{min}}},1\right\}.$$ Next we show that $w$ is the function as required. Note that
$$u+w+h=u+\varepsilon d(\cdot,\mathcal{O})+(w-\varepsilon d(\cdot,\mathcal{O})+h).$$
Notice that, $$\|w-\varepsilon d(\cdot,\mathcal{O})+h\|_1\le\|D_Mw\|_0+\|\varepsilon d(\cdot,\mathcal{O})\|_1+\|h\|_1\le 2\varepsilon+\varepsilon+5\varepsilon<10\varepsilon,$$
and
$$\|w-\varepsilon d(\cdot,\mathcal{O})+h\|_0\le\|w-\varepsilon d(\cdot,\mathcal{O})\|_0+\|h\|_0< \frac{H}{2}+\frac{H}{2}=H.$$
Then by Proposition \ref{prop-2}, we have that $\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+w+h;\psi,\L,\Phi).$ Additionally,
$$\|w\|_0<\|\varepsilon d(\cdot,\mathcal{O})\|_0+\varepsilon\cdot diam(M)\le 2\varepsilon\cdot diam(M).$$
This ends the proof. \end{proof}
\begin{rem}\label{remark-2} Let $\widetilde w\in\mathcal{C}^{1,0}(M)$ be such that $\|\widetilde{w}\|_{1,0}<\varepsilon$, $\widetilde{w}|_\mathcal{O}=0$ and $\widetilde w|_{M\setminus\mathcal{O}}>0$. Then $\mu_\mathcal{O}$ is the unique measure in $\mathcal{M}_{min}(u+\widetilde w+w+h;\psi,\L,\Phi)$ whenever $\|h\|_1<5\varepsilon$ and
$\|h\|_0$ is sufficiently small. The Proposition shows that there is an open set of $\mathcal{C}^{1,0}({M})$ near $u$ such that functions in the open set have the same unique minimizing measure with respect to $\psi$ and the measure supports on a periodic orbit. \end{rem}
\section{Proofs of Technical Lemmas}\label{S:ProofsTechLem} Note that throughout this section, $\d,\e,\l,\b,C,\e',\d'$ are same as the ones in Remark \ref{remark-1-1}. \subsection{Proof of Lemma \ref{An-1}}\label{S:LemAn-1}
\begin{proof}We put a small positive constant $\tau$ with $\tau\ll1$ such that $|s(t_1)-s(t_2)|\le \eta$ for all $|t_1-t_2|\le\tau$ and $t_1,t_2\in[0,T].$ Since $\eta\le \frac{C^{10}e^{10\beta+10\lambda}}{e^{\lambda}-1}\delta'$, for all $0\le t\le T,$ there exists $r(t)$ with $|r(t)|<C\eta$ such that
\begin{align}\label{l-1}
w(\phi_{t+s(t)+r(t)}(y),\phi_{t}(x))=W^s_{\epsilon'}(\phi_{t+s(t)+r(t)}(y))\cap W^u_{\epsilon'}(\phi_{t}(x)).
\end{align}
Then for $t'\in[-\tau,\tau]$ and $t\in[\tau,T-\tau]$,
$$\phi_{t'}(w( \phi_{t+s(t)+r(t)}(y),\phi_{t}(x)))\in W^s_{\epsilon'}(\phi_{t+s(t)+r(t)+t'}(y))\cap W^u_{\epsilon'}(\phi_{t+t'}(x)).$$
On the other hand, one has that
$$w(\phi_{t+t'+s(t+t')+r(t+t')}(y),\phi_{t+t'}(x))= W^s_{\epsilon'}(\phi_{t+t'+s(t+t')+r(t+t')}(y))\cap W^u_{\epsilon'}(\phi_{t+t'}(x)).$$
Since $$|(t+t'+s(t+t')+r(t+t'))-(t+s(t)+r(t)+t')|\le (2C+1)\eta\ll\d,$$
by the uniqueness of $v(\phi_{t+s(t)+r(t)+t'}y,\phi_{t+t'}x)$ given by (1) of the {\bf basic canonical setting}, one has that
$$t+t'+s(t+t')+r(t+t')=t+s(t)+r(t)+t',$$
and
$$w(\phi_{t+t'+s(t+t')+r(t+t')}(y),\phi_{t+t'}(x))=\phi_{t'}(w( \phi_{t+s(t)+r(t)}(y),\phi_{t}(x))),$$
for all $t'\in[-\tau,\tau]$ and $t\in[\tau,T-\tau]$.
Since $\tau$ can be taken arbitrarily small, one has the following by induction
\begin{align}\label{l-2}
s(t)+r(t)=s(\tau)+r(\tau)=s(0)+r(0)=r(0)=v(y,x),\ \forall t\in[0,T].
\end{align}
Thus $$|s(t)|\le |r(t)|+|r(0)|\le 2C\eta,$$
and for all $t\in[\tau,T-\tau]$ and $t'\in[-\tau,\tau]$
$$w(\phi_{t+t'+v(y,x)}(y),\phi_{t+t'}(x))=\phi_{t'}(w(\phi_{t+v(y,x)}(y),\phi_{t}(x))),$$
which implies that
\begin{align}\label{l-3} w(\phi_{t+v(y,x)}(y),\phi_{t}(x))=\phi_{t}(w(\phi_{v(y,x)}(y),x))\ \forall t\in[0,T].
\end{align}
Now, we prove Ash 2). Note $w=w(\phi_{v(y,x)}(y),x)=W_{\epsilon'}^s(\phi_{v(y,x)}(y))\cap W_{\epsilon'}^u(x)$. Then by \eqref{l-1}, \eqref{l-2} and \eqref{l-3}, one has
$$\phi_t(w)=W^s_{\epsilon'}(\phi_{t}\phi_{v(y,x)}(y))\cap W^u_{\epsilon'}(\phi_{t}(x))\text{ for all }t\in[0,T].$$
Thus, for all $t\in[0,T]$, by (b) of the {\bf basic canonical setting},
$$d(\phi_t(w),\phi_{t}\phi_{v(y,x)}(y))<Cd(\phi_{t}(x),\phi_{t}(y))\text{ and }d(\phi_t(w),\phi_{t}(x))<Cd(\phi_{t}(x),\phi_{t}(y)).$$
Therefore
$$d(\phi_t(w),\phi_t(x))\le Ce^{-\lambda(T-t)}d(\phi_T(w),\phi_T(x))\le C^2e^{-\lambda(T-t)}d(\phi_T(x),\phi_{T}(y)),$$
where we used $w\in W_{\epsilon'}^u(x)$ and
$$d(\phi_t(w),\phi_t\phi_{v(y,x)}(y))\le Ce^{-\lambda(t)}d(w,\phi_{v(y,x)}(y))\le C^2e^{-\lambda(t)}d(x,y).$$
where we used $w\in W_{\epsilon'}^s(\phi_{(y,x)}v(y))$. By summing up, we have
$$d(\phi_t\phi_{v(y,x)}(y),\phi_t(x))\le C^2e^{-\lambda\min(t,T-t)}(d(x,y)+d(\phi_T(x),\phi_T(y)))\text{ for } 0\le t\le T.$$
Now we assume that $d(\phi_{t+s(t)}(y),\phi_t(x))\le \eta$ for all $t\ge 0,$ then by the arguments above.
We have for all $t\ge 0$,
\begin{align*}d(\phi_t\phi_{v(y,x)}(y),\phi_t(x))&\le C^2e^{-\lambda\min(t,2t-t)}(d(x,y)+d(\phi_{2t}(x),\phi_{2t}(y)))\\
&\le 2C^2\eta e^{-\lambda\min(t,2t-t)}\to 0\text{ as }t\to+\infty.
\end{align*}
This ends the proof. \end{proof} \subsection{Proof of Lemma \ref{lemma-a}}\label{S:Lemma-a}
\begin{proof}We partially follow Bowen's arguments in \cite{Bowen}. Firstly we fix a constant $K\gg C$ with $2C^2e^{-\lambda K}\ll1$ and a segment $\mathcal{S}$ as in Lemma \ref{lemma-a}. We let $\tau=|\mathcal{S}|$ and $\eta=d(\mathcal{S}^L,\mathcal{S}^R)$. Then $\eta<\delta'$ and $2Ce^{-\lambda\tau}\ll1$. Therefore, we have the following claim.
\noindent{\it {\bf Claim A. } For the segment $\mathcal S$ in Lemma \ref{lemma-a}, there is a $y\in \L$ and a continuous function $\hat{s}:\mathbb{R}\to \mathbb{R}$ with $\hat{s}(0)=0$ and $Lip(\hat{s})\le\frac{2C\eta}{\tau}$ such that $d(\phi_{i\tau+t_1+s(i\tau+t_1)}(y), \phi_{t_1}(\mathcal{S}^L))\le L_1\eta$ for all $t_1\in[0,\tau]$ and $i\in\mathbb{Z}$ where $L_1=2C^2(\frac{2}{e^\lambda-1}+e^{\beta}+2)$.}
Since the proof of Claim A is long, we postpone the proof of Claim A to the next subsection. Let $y\in M$ and $\hat{s}:\mathbb{R}\to \mathbb{R}$ be as in Claim A. We divide the following proof into two steps.
\noindent {\it {\bf Step 1. } At first, we show that $y$ is a periodic point.}
By Claim A,
\begin{align*}
d(\phi_{t+\hat{s}(t)}(y),\phi_{t+\tau+\hat{s}(t+\tau)}(y))\le 2L_1\eta\text{ for } t\in\mathbb{R}.
\end{align*}
Since $Lip(\hat{s})\ll 1$ and $\hat s(0)=0$, $g(t)=t+\hat{s}(t)$ is a homomorphism of $\mathbb{R}$ onto itself, the above inequality can be rewritten as the following
\begin{align*}
d(\phi_{t}(y),\phi_{g^{-1}(t)+\tau+\hat{s}(g^{-1}(t)+\tau)}(y))\le 2L_1\eta\text{ for } t\in\mathbb{R}.
\end{align*}
We note $$y'=\phi_{g^{-1}(0)+\tau+\hat{s}(g^{-1}(0)+\tau)}(y)$$ and $$s(t)=g^{-1}(t)+\hat{s}(g^{-1}(t)+\tau)-g^{-1}(0)-\hat{s}(g^{-1}(0)+\tau)-t.$$ Then
$$d(\phi_{t}(y),\phi_{t+s(t)}(y'))\le 2L_1\eta\text{ for } t\in\mathbb{R} \text{ and } s(0)=0.$$
Therefore, by Lemma \ref{An-1}, one has
$$\phi_{v_2}(y')=y \text{ and }|v_2|\le 2CL_1\eta,$$
where $v_2=v(y',y)$. Thus
$$\phi_{g^{-1}(0)+\tau+\hat{s}(g^{-1}(0)+\tau)+v_2}(y)=y.$$
Notice that $g^{-1}(0)=0$ since $g(0)=0$. Thus,
$$|g^{-1}(0)+\hat{s}(g^{-1}(0)+\tau)+v_2|\le |\hat{s}(\tau)|+|v_2|\le (2C+2CL_1)\eta\ll \tau.$$ Therefore, $y$ is a periodic point.
\noindent{{\bf Step 2.} There is a periodic segment $\mathcal{O}$ such that $$ ||\mathcal{S}|-|\mathcal{O}||\le Ld(\mathcal{S}^L, \mathcal{S}^R)$$ and
$$ d(\phi_t(\mathcal{O}^L),\phi_t(\mathcal{S}^L))\le Ld(\mathcal{S}^L, \mathcal{S}^R)\text{ for all } 0\le t\le \max(|\mathcal{S}|,|\mathcal{O}|),$$
where $L=2CL_2+L_2$ and $L_2=2C^3L_1+C^4L_1.$ }
By Claim A,
\begin{align*}
d(\phi_{t+\hat{s}(t)}(y),\phi_{t}(\mathcal{S}^L))\le L_1\eta\text{ for } t\in[0,\tau].
\end{align*}
By Lemma \ref{An-1},
for $t\in [0,\tau],$ $|\hat{s}(t)|\le 2CL_1\eta$ and
\begin{align}\label{eq-ano-1}\begin{split}
d(\phi_t\phi_{v_1}(y),\phi_t(\mathcal{S}^L))&\le C^2e^{-\lambda\min(t,\tau-t)}(d(y,\mathcal{S}^L)+d(\phi_\tau(y),\phi_\tau(\mathcal{S}^L)))\\
&\le C^2(d(y,\mathcal{S}^L)+d(\phi_{\tau+\hat{s}(\tau)}(y),\phi_\tau(\mathcal{S}^L))+d(\phi_{\tau+\hat{s}(\tau)}(y),\phi_{\tau}(y)))\\
&\le L_2\eta,
\end{split}\end{align}
where $v_1=v(y,\mathcal S^L)$. Now we put $y^*=\phi_{v_1}y$ and we have a periodic segment,
$$\mathcal{O}:[0,\tau+g^{-1}(0)+\hat{s}(g^{-1}(0)+\tau)+v_2]\to M: t\to \phi_t(y^*),$$
where $v_2$ is as in Step 1.
It is clear that
$$||\mathcal{S}|-|\mathcal{O}||\le |g^{-1}(0)+\hat{s}(g^{-1}(0)+\tau)+v_2|\le L_2\eta.$$
If $|\mathcal{O}|\le|\mathcal{S}|$, by \eqref{eq-ano-1},
$$d(\phi_t(y^*),\phi_t(\mathcal{S}^L))\le L_2\eta,\text{ for }t\in[0,\max(|\mathcal{S}|,|\mathcal{O}|)],$$
where we used $\tau=\max(|\mathcal{S}|,|\mathcal{O}|).$\\
If $|\mathcal{O}|>|\mathcal{S}|$, by \eqref{eq-ano-1},
$$d(\phi_t(y^*),\phi_t(\mathcal{S}^L))\le L_2\eta,\text{ for }t\in[0,|\mathcal{S}|],$$
and for $t\in(|\mathcal{S}|,|\mathcal{O}|]$,
\begin{align*}d(\phi_t(y^*),\phi_t(\mathcal{S}^L))&\le d(\phi_t(y^*),\phi_\tau(y^*))+d(\phi_\tau(y^*),\phi_\tau(\mathcal{S}^L))+d(\phi_\tau(\mathcal{S}^L),\phi_t(\mathcal{S}^L))\\
&\le L\eta,\end{align*}
where $L=2CL_2+L_2$.
This ends the proof since $L_2\le L$. \end{proof} \subsubsection{Proof of Claim A}
\begin{proof} Recall that $\mathcal{S}$ is a segment of $\Phi|_\L$ with $|\mathcal{S}|=\tau\ge K$ and $d(\mathcal{S}^L,\mathcal{S}^R)=\eta<\delta'$ where $K$ satisfies $2C^2e^{-\lambda K}\ll 1$. We define $x_{-k}, \zeta_{-k}$ recursively
for $k\ge 0$ by $$x_0=\mathcal{S}^R, \zeta_{0}=0$$ and
$$ \zeta_{-k-1}=v(\phi_{-\tau}(x_{-k}),\mathcal{S}^R), x_{-k-1}=W_{\epsilon'}^s(\phi_{-\tau+\zeta_{-k-1}}(x_{-k}))\cap W_{\epsilon'}^u(\mathcal{S}^R) \text{ for }k=1,2,\cdots.$$
We have the following two assertions.
\noindent {\it {\bf Assertion 1.} $x_{-k}$ and $\zeta_{-k}$ are well defined and $d(x_{-k},\mathcal{S}^R)\le 2C\eta$ for $k\ge 0.$}
\begin{proof}
In the case $k=0$, it is obviously true. Now assume that we have $\zeta_{-k},x_{-k}$ and $d(x_{-k},\mathcal{S}^R)\le 2C\eta$.
Then
\begin{align}\begin{split}\label{E:XkSR}
d(\phi_{-\tau}(x_{-k}),\mathcal{S}^R)&\le d(\phi_{-\tau}(x_{-k}),\phi_{-\tau}(\mathcal{S}^R))+d(\mathcal{S}^R,\phi_{-\tau}(\mathcal{S}^R))\\
&\le Ce^{-\lambda \tau}\cdot 2C\eta+\eta\\
&\le 2\eta,
\end{split}
\end{align}
where we used $x_{-k}\in W_{\epsilon'}^u(\mathcal{S}^R)$.
Since $2\eta\le 2\delta'$, $x_{-k-1}$ is well defined as well as $\zeta_{-k-1}$, and moreover one has that
\begin{align*}
d(x_{-k-1},\mathcal{S}^R)\le 2C\eta.
\end{align*}
This ends the proof.
\end{proof}
By (\ref{E:XkSR}), we have that
\begin{align}\label{eq-an-1}
|\zeta_k|\le 2C\eta\ll 1.
\end{align}
Next we denote $x^{(-k)}=\phi_{k\tau-\sum_{i=0}^{k}\zeta_{-i}}(x_{-k})$ for $k\ge 0$. For $k\in\mathbb{N}$, we define $s^*_{-k}:\mathbb{R}\to\mathbb{R}$ by
\[s^*_{-k}(t)=\left\{\begin{array}{ll}
\zeta_0, & \text{ if } t> 0,\\
\sum_{i=0}^{l-1}\zeta_{-i}, & \text{ if } -l\tau< t\le-(l-1)\tau, l\in\{1,2,\cdots,k\},\\
\sum_{i=0}^{k}\zeta_{-i}, & \text{ if } t\le-k\tau.
\end{array}
\right.\]
\noindent {\bf Assertion 2.} {\it There exists a constant $L_0$ such that for $t=-j\tau-t_0$ satisfying $ t_0\in[0,\tau)$ and $j\in\{0,1,\cdots,k-1\}$, the following holds} $$d\left(\phi_{t+s^*_{-k}(t)}\left(x^{(-k)}\right),\phi_{-t_0}\left(\mathcal{S}^R\right)\right)\le L_0\eta.$$
\begin{proof}We fix $t=-j\tau-t_0$ for some $j\in\{0,1,\cdots,k-1\}$ and $ t_0\in[0,\tau)$. Since $x_{-j}\in W^{u}_{\epsilon'}(\mathcal{S}^{R}) $, we have
\begin{align}\label{eq-an-2}
d\left(\phi_{-t_0}\left(x_{-j}\right),\phi_{-t_0}\left(\mathcal{S}^R\right)\right)\le Ce^{-\lambda t_0}d\left(x_{-j},\mathcal{S}^R\right)\le 2C^2\eta.
\end{align}
Note that $\tau-\zeta_{-j-1}-t_0\ge -1$ and $x_{-j-1}\in W^{s}_{\epsilon'}(\phi_{-\tau +\zeta_{-j-1}}(x_{-j}))\cap W^{u}_{\epsilon'}(\mathcal{S}^{R})$, we have
\begin{align}\label{eq-an-3}
\begin{split}
&d\left(\phi_{\tau-\zeta_{-j-1}-t_0}\left(x_{-j-1}\right),\phi_{-t_0}\left(x_{-j}\right)\right)\\
=&d\left(\phi_{\tau-\zeta_{-j-1}-t_0}\left(x_{-j-1}\right),\phi_{\tau-\zeta_{-j-1}-t_0}\phi_{-\tau+\zeta_{-j-1}}\left(x_{-j}\right)\right)\\
\le& e^{\beta }d\left(x_{-j-1},\phi_{-\tau+\zeta_{-j-1}}\left(x_{-j}\right)\right)\\
\le&2C^2e^{\beta}\eta,
\end{split}\end{align}
where we used (2) and (3) of the {\bf basic canonical setting} for the case $\tau-\zeta_{-j-1}-t_0> 0$ and $\tau-\zeta_{-j-1}-t_0\le 0$, respectively.
Note that $|\zeta_{-l}|\ll 1, \tau\gg 1$ and $t_0\in[0, \tau)$, i.e., $\tau-\zeta_{-j-1}-t_0\geq -1$ and $\tau -\zeta_{-i} >\tau-1>1$. Since $x_{-(k-l)}\in W^{s}_{\epsilon'}(\phi_{-\tau+\zeta_{-(k-l)}}(x_{-(k-j-1)}))\cap W^{u}_{\epsilon'}(\mathcal{S}^{R})$, we have
\begin{align}\begin{split}\label{eq-an-4}
&\sum_{l=0}^{k-j-2}d\left(\phi_{(k-j-l-1)\tau-\sum_{i=j+1}^{k-l}\zeta_{-i}-t_0}\left(x_{-(k-l)}\right),\phi_{(k-j-l-2)\tau-\sum_{i=j+1}^{k-l-1}\zeta_{-i}-t_0}\left(x_{-(k-l-1)}\right)\right)\\
&\le \sum_{l=0}^{k-j-2} Ce^{-\lambda\left(\left(k-j-l\right)\tau-\sum_{i=j+1}^{k-l}\zeta_{-i}-t_0\right)}d\left(x_{-(k-l)},\phi_{-\tau+\zeta_{-(k-l)}}\left(x_{-(k-j-1)}\right)\right)\\
&\le \sum_{l=0}^{k-j-2} Ce^{-\lambda\left(\left(k-j-l\right)\tau-\sum_{i=j+1}^{k-l}\zeta_{-i}-t_0\right)} 4C\eta\\
&\le 4C^2\eta e^{-\lambda (2\tau - \zeta_{-j-1}-\zeta_{-j-2}-t_0)}\sum_{l=0}^{k-l-1}e^{-\lambda\cdot (\tau -1 )\cdot l}\\
&\le 4C^2\eta e^{-\lambda(\tau-2)}\frac{1}{1-e^{-\lambda}}\\
&=4C^2\eta \frac{1}{e^{\lambda}-1},
\end{split} \end{align}
where we used 1b) of the {\bf basic canonical setting}. Combining \eqref{eq-an-2}, \eqref{eq-an-3} and \eqref{eq-an-4}, we have that for $t=-j\tau-t_0$
\begin{align*}
&d\left(\phi_{t+s^*_{-k}(t)}\left(x^{(-k)}\right),\phi_{-t_0}\left(\mathcal{S}^R\right)\right)\\
=&d\left(\phi_{-j\tau+\sum_{i=0}^{j}\zeta_{-i}-t_0}\left(x^{(-k)}\right),\phi_{-t_0}\left(\mathcal{S}^R\right)\right)\\
=&d\left(\phi_{(k-j)\tau-\sum_{i=j+1}^{k}\zeta_{-i}-t_0}\left(x_{-k}\right),\phi_{-t_0}\left(\mathcal{S}^R\right)\right)\\
\le&\sum_{l=0}^{k-j-2}d\left(\phi_{(k-j-l-1)\tau-\sum_{i=j+1}^{k-l}\zeta_{-i}-t_0}\left(x_{-(k-l)}\right),\phi_{(k-j-l-2)\tau-\sum_{i=j+1}^{k-l-1}\zeta_{-i}-t_0}\left(x_{-(k-l-1)}\right)\right)\\
&+d(\phi_{\tau-\zeta_{-j-1}-t_0}(x_{-j-1}), \phi_{-t_0}(x_{-j}))+d\left(\phi_{-t_0}(x_{-j}),\phi_{-t_0}(\mathcal{S}^R)\right)\\
\le &L_0\eta,
\end{align*}
where $L_0=2C^2(\frac{2}{e^\lambda-1}+e^{\beta}+1)$. This ends the proof of Assertion 2.
\end{proof}
Now for $k\in\mathbb{N}$, we define $\bar{s}_{-k}:\mathbb{R}\to\mathbb{R}$ by
\[\bar{s}_{-k}(t)=\left\{\begin{array}{ll}
\zeta_0, & \text{ if } t> 0,\\
\sum_{i=0}^{l-1}\zeta_{-i}-\frac{t+(l-1)\tau}{\tau}\zeta_{-l}, & \text{ if } -l\tau< t\le -(l-1)\tau, l\in\{1,2,\cdots,k\},\\
\sum_{i=0}^{k}\zeta_{-i}, & \text{ if } t\le -k\tau.
\end{array}
\right.\]
It is clear that $\bar{s}_{-k}$ is Lipschitz continuous with
$$Lip(\bar{s}_{-k})\le \frac{\max_{i\in\{0,1,2,\cdots,k\}}|\zeta_i|}{\tau}\le \frac{2C\eta}{\tau},$$
and
$$|\bar{s}_{-k}(t)-s^*_{-k}(t)|\le\max_{i\in\{0,1,2,\cdots,k\}}|\zeta_i|\le 2C\eta.$$
Therefore, by {Assertion 2}, when $t=-j\tau-t_0$ for some $j\in\{0,1,2,\cdots,k-1\}$ and $ t_0\in[0,\tau)$, one has that
\begin{align}\begin{split}\label{E:BarSEst}
&d\left(\phi_{t+\bar{s}_{-k}(t)}\left(x^{(-k)}\right),\phi_{-t_0}\left(\mathcal{S}^R\right)\right)\\
\le& d\left(\phi_{t+\bar{s}_{-k}(t)}\left(x^{(-k)}\right),\phi_{t+s^*_{-k}(t)}\left(x^{(-k)}\right)\right)+d\left(\phi_{t+s^*_{-k}(t)}\left(x^{(-k)}\right),\phi_{-t_0}\left(\mathcal{S}^R\right)\right)\\
\le& L_1\eta,
\end{split}\end{align}
where $L_1=L_0+2C^2.$
Now for $k\in\mathbb{N}$, we define ${s}_{-k}:\mathbb{R}\to\mathbb{R}$ by
$${s}_{-k}(t)=\bar{s}_{-2k}(t-k\tau)-\sum_{i=0}^{k}\zeta_{-i}.$$
It is clear that ${s}_{-k}(0)=0$.
On the other hand, we note $y_k=\phi_{-\tau k+\sum_{i=0}^{k}\zeta_{-i}}(x^{(-2k)})$.
Thus, when $t=-j\tau-t_0$ for some $j\in\{-k,-k+1,\cdots,k-1\}$ and $ t_0\in[0,\tau)$, (\ref{E:BarSEst}) implies that
\begin{align*}
d\left(\phi_{t+{s}_{-k}(t)}(y_k),\phi_{-t_0}\left(\mathcal{S}^R\right)\right)=d\left(\phi_{t-\tau k+\bar{s}_{-2k}(t-\tau k)}\left(x^{(-2k)}\right),\phi_{-t_0}\left(\mathcal{S}^R\right)\right)\le L_1\eta.
\end{align*}
Notice that $s_{-k}$ are Lipschitz with $Lip(s_{-k})\le \frac{2C\eta}{\tau}\ll \eta$ for all $k\in\mathbb{N}$. Applying the Ascoli-Azel\'a theorem, there exists a subsequence $(s_{-k_i})_{i=1}^{+\infty}$ that converges to a Lipschitz continuous function $\hat{s}:\mathbb{R}\to\mathbb{R}$ with $Lip(\hat{s})\le \frac{2C\eta}{\tau}\ll \eta$ and $\hat {s}(0)=0$. Without losing any generality, we assume that $y_{k_i}\to y$ as $i\to+\infty$. By the continuity, if $t=-j\tau-t_0$ for some $j\in\mathbb{Z}$ and $ t_0\in[0,\tau)$, then
\begin{align*}
d(\phi_{t+\hat{s}(t)}(y),\phi_{-t_0}(\mathcal{S}^R))\le L_1\eta.
\end{align*}
That is to say, if $t=-(j+1)\tau+(\tau-t_0)$ for some $j\in\mathbb{Z}$ and $ t_0\in[0,\tau)$, then
\begin{align*}
d\left(\phi_{t+\hat{s}(t)}(y),\phi_{\tau -t_0}(\mathcal{S}^L)\right)\le L_1\eta.
\end{align*}
Note that $y_{k_i}\in \L$ for each $i\in\mathbb N$, thus $y\in\L$.
Let $t_1=\tau -t_0$, then the proof of Claim A is completed. \end{proof}
\subsection{Proof of Lemma \ref{MCGB}}\label{S:ProofMCGB} In this section, we mainly prove a version of the so called Ma\~n\`e-Conze-Guivarc'h-Bousch's Lemma. The proof partially follows Bousch's arguments in \cite{Bousch_Mane}.
\subsubsection{Integration along segment} Recall that, for a continuous function $u$ and a segment $\mathcal{S}$ of $\Phi$, the integration of $u$ along $\mathcal{S}$ is defined by $$\left\langle\mathcal{S},u\right\rangle:=\int_{a}^bu(\phi_t(x))dt.$$ \begin{lem}\label{Bou-2}
Let $u:M\to\mathbb{R}$ be an $\alpha$-H\"older function with $\beta(u;1,\L,\Phi)\ge 0$. Then for a segment $\mathcal{S}$ of $\Phi|_\L$ satisfying $|\mathcal{S}|\ge K$ and $d(\mathcal{S}^L,\mathcal{S}^R)\le\delta'$, the following holds $$\left\langle\mathcal{S},u\right\rangle\ge-K_1d^\alpha(\mathcal{S}^L, \mathcal{S}^R),$$
where $K_1=\frac{(CL)^\alpha}{\lambda\alpha}\|u\|_\alpha+L\|u\|_0$. \end{lem}
\begin{proof}Since $d(\mathcal{S}^L,\mathcal{S}^R)\le\delta'$ and $|\mathcal S|\ge K$, by Anosov Closing Lemma, there exists a periodic segment $\mathcal{O}$ of $\Phi|_\L$ such that
$$ ||\mathcal{S}|-|\mathcal{O}||\le Ld(\mathcal{S}^L, \mathcal{S}^R)$$ and
$$ d(\phi_t(\mathcal{O}^L),\phi_t(\mathcal{S}^L))\le Ld(\mathcal{S}^L, \mathcal{S}^R)\text{ for all } 0\le t\le \max(|\mathcal{S}|,|\mathcal{O}|).$$ Therefore, Letting $v=v(\mathcal{O}^L,\mathcal{S}^L)$ as in Lemma \ref{An-1}, one has that \begin{align*}
\left\langle\mathcal{S},u\right\rangle-\left\langle\mathcal{O},u\right\rangle&=\int_0^{|\mathcal{O}|}u(\phi_t(\mathcal{S}^L))-u(\phi_t\phi_v(\mathcal{O}^L))dt+\int_{|\mathcal{O}|}^{|\mathcal{S}|}u(\phi_t(\mathcal{S}^L))dt\\
&\ge -\|u\|_\alpha\int_0^{|\mathcal{O}|}d^\alpha(u(\phi_t(\mathcal{S}^L)),u(\phi_t\phi_v(\mathcal{O}^L)))dt-\|u\|_0 ||\mathcal{S}|-|\mathcal{O}||\\
&\ge -\|u\|_\alpha\int_0^{|\mathcal{O}|}\big(Ce^{-\lambda\min(t,T-t)}Ld(\mathcal{S}^L, \mathcal{S}^R)\big)^\alpha dt-\|u\|_0Ld(\mathcal{S}^L, \mathcal{S}^R)\\
&\ge -\left(\frac{(CL)^\alpha}{\lambda\alpha}\|u\|_\alpha+L\|u\|_0\right)d^\alpha(\mathcal{S}^L, \mathcal{S}^R), \end{align*} where we used the assumption $0<\alpha\le 1$ and $0<d(\mathcal{S}^L, \mathcal{S}^R)<\delta'\ll1$. Then the Lemma is immediately from the fact $\left\langle\mathcal{O},u\right\rangle\ge 0$ since $\beta(u;1,\L,\Phi)\ge 0$. This ends the proof. \end{proof}
\begin{lem}\label{Bou-3}
Let $\mathcal{P}$ be a finite partition of $M$ with diameter smaller than $\delta'$ and $u:M\to\mathbb{R}$ be an $\alpha$-H\"older function with $\beta(u;1,\L,\Phi)\ge 0$. Then for a given segment $\mathcal{S}$ of $\Phi|_\L$, the following holds
\[\left\langle\mathcal{S},u\right\rangle\ge-K_2\delta'^\alpha,\]
where $K_2=\sharp \mathcal P\cdot\left(\frac{K\|u\|_0}{\delta'^\alpha}+K_1\right)$ and $K$, $K_1$ are as in Lemma \ref{lemma-a} and \ref{Bou-2} respectively.
\end{lem}
\begin{proof}For $x\in M$, denote $\mathcal{P}(x)$ the element in $\mathcal{P}$ which contains $x$. Assume $|\mathcal{S}|=(n-1)K+r$ for some $n\ge 1$ and $0\le r<K$. Let $t_i=iK$ for $0\le i\le n-1$ and $t_n=|\mathcal{S}|$.
We define the function $w:\mathbb{N}\to[0,n]\cap\mathbb{N}$ inductively by letting
\[w(0)=0\]
\[w(k)=\min\{\eta(w(k-1))+1,n\}.\]
where $\eta: [0, n-1]\cap\mathbb{N}\to[0, n-1]\cap\mathbb{N}$ is the function that maps each $i$ to the largest $j\in[0,n-1]\cap\mathbb{N}$ such that $\mathcal{P}(\phi_{t_i}(\mathcal{S}^L))=\mathcal{P}(\phi_{t_{j}}(\mathcal{S}^L))$. Let $s\ge 0$ be the smallest integer for which $\eta(w(s))=n-1$. Then $\mathcal{P}(\phi_{t_{w(i)}}(\mathcal{S}^L))\neq\mathcal{P}(\phi_{t_{w(j)}}(\mathcal{S}^L))$ for $0\le i<j\le s$ which implies $s\le\sharp\mathcal{P}$. For $0\le j\le s$, we have two cases:
If $\eta(w(j))=w(j)$
\begin{align}\label{B-1}
\int_{t_{w(j)}}^{t_{\eta(w(j))}}u(\phi_t(x))dt=0\text{ and }\int_{t_{\eta(w(j))}}^{t_{\eta(w(j))+1}}u(\phi_t(x))dt\ge -K\|u\|_0.
\end{align}
If $\eta(w(j))>w(j)$, by using Lemma \ref{Bou-2},
\begin{align}\label{B-2}
\int_{t_{w(j)}}^{t_{\eta(w(j))}}u(\phi_t(x))dt\ge-\left(\frac{(CL)^\alpha}{\lambda\alpha}\|u\|_\alpha+L\|u\|_0\right)\delta'^\alpha, \end{align}
where we use the fact $d(\phi_{t_i}(\mathcal{S}^L),\phi_{t_{j}}(\mathcal{S}^L))<\delta'$ since $\mathcal{P}(\phi_{t_i}(\mathcal{S}^L))=\mathcal{P}(\phi_{t_{j}}(\mathcal{S}^L))$.
On the other hand, as in \eqref{B-1},
\begin{align}\label{B-3}
\int_{t_{\eta(w(j))}}^{t_{\eta(w(j))+1}}u(\phi_t(\mathcal{S}^L))dt\ge -K\|u\|_0.
\end{align}
Combining \eqref{B-1}, \eqref{B-2} and \eqref{B-3}, one has
\begin{align*}
&\left\langle\mathcal{S},u\right\rangle=\sum_{j=0}^{s-1} \int_{t_{w(j)}}^{t_{\eta(w(j))}}+\int_{t_{\eta(w(j))}}^{t_{\eta(w(j))+1}}u(\phi_t(\mathcal{S}^L))dt\\
\ge&-s\left(K\|u\|_0+\left(\frac{(CL)^\alpha}{\lambda\alpha}\|u\|_\alpha+L\|u\|_0\right)\delta'^\alpha\right)\\
\ge&-\sharp \mathcal P\cdot\left(K\|u\|_0+\left(\frac{(CL)^\alpha}{\lambda\alpha}\|u\|_\alpha+L\|u\|_0\right)\delta'^\alpha\right),
\end{align*} which completes the proof. \end{proof}
In the following, we deal with the so called shadowing property for two finite time segments, which will allow one to use one segment to shadow two segments of which the ending point of one segment is close to the beginning point of the other. Let $\mathcal S_1$ and $\mathcal S_2$ be two segments of $\Phi|_\L$, suppose that $$d(\mathcal S_1^R,\mathcal S_2^L)<\d'.$$
Then there exist $v(\mathcal S_2^L,\mathcal S_1^R)$ and $w(\mathcal S_2^L,\mathcal S_1^R)=W^s_{\e'}(\phi_{v(\mathcal S_2^L,\mathcal S_1^R)}(\mathcal S_2^L))\cap W^u_{\e'}(\mathcal S_1^R)$. Define a new segment $\mathcal S_1*\mathcal S_2:\left[-|\mathcal S_1|,|\mathcal S_2|-v(\mathcal S_2^L,\mathcal S_1^R)\right]$ by letting \begin{equation}\label{E:S1*S2}
\mathcal S_1*\mathcal S_2(t)=\phi_t\left(w(\mathcal S_2^L,\mathcal S_1^R)\right)\ \forall t\in \left[-|\mathcal S_1|,|\mathcal S_2|-v(\mathcal S_2^L,\mathcal S_1^R)\right]. \end{equation} We remark here that the definition of $\mathcal S_1*\mathcal S_2$ above is not the unique way for describing the shadowing property. Nevertheless, it is the most convenient way for the rest of the proof. \begin{lem}\label{Bou-4}
Given $0<\alpha\le 1$ and a large constant $\gamma=\gamma(\alpha)\gg 1$ satisfying that $2C^{2\alpha}e^{-\frac{\gamma\alpha\lambda}{2}}\ll 1$, when two segments $\mathcal{S}_1$ and $\mathcal{S}_2$ of $\Phi|_\L$ satisfy the following
$$d(\mathcal{S}_1^R,\mathcal{S}_2^L)\le \delta'\text{ and } \min\{|\mathcal{S}_1|,|\mathcal{S}_2|\}\ge \gamma,$$ then for all $u\in \mathcal{C}^{0,\alpha}(M)$,
$$\frac{\left|\left\langle\mathcal{S}_1*\mathcal{S}_2,u\right\rangle-\left\langle\mathcal{S}_1,u\right\rangle-\left\langle\mathcal{S}_2,u\right\rangle\right|}{d^\a(\mathcal{S}_1^R,\mathcal{S}_2^L)-d^\a((\mathcal{S}_1*\mathcal{S}_2)^R,\mathcal{S}_2^R)-d^\a((\mathcal{S}_1*\mathcal{S}_2)^L,\mathcal{S}_1^L)}\le K_3,$$
where $K_3=\frac{C\|u\|_0+\frac{2C^{2\alpha}\|u\|_\alpha}{\lambda\alpha}}{1-2C^{2\alpha}e^{-{(\gamma-1)\alpha\lambda}}}$ and the denominator of the left side of the above inequality is always positive by the choice of $\g$. \end{lem} \begin{proof}Fix $\alpha,\gamma,u,\mathcal{S}_1,\mathcal{S}_2$ as in this Lemma. Note $v=v(\mathcal{S}_2^L,\mathcal{S}_1^R)$, $w=w\left(\mathcal{S}_2^L,\mathcal{S}_1^R\right)$ and
$$\widetilde{\mathcal{S}}_2:[0,|\mathcal{S}_2|-v]:t\to\phi_{t+v}(\mathcal{S}_2^L).$$ Thus, we have
\begin{align*}
&\left|\left\langle\mathcal{S}_1*\mathcal{S}_2,u\right\rangle-\left\langle\widetilde{\mathcal{S}}_2,u\right\rangle-\left\langle\mathcal{S}_1,u\right\rangle\right|\\
=&\left|\int_0^{|\mathcal{S}_2|-v}u(\phi_{t}(w))-u(\phi_{t}\phi_v(\mathcal{S}_2^L))dt+\int_0^{|\mathcal{S}_1|}u(\phi_{-t}(w))-u(\phi_{-t}(\mathcal{S}_1^R))dt\right|\\
\le&\int_0^{|\mathcal{S}_2|-v}\|u\|_\alpha d^\alpha(\phi_{t}(w),\phi_{t}\phi_{v}(\mathcal{S}_2^L))dt+\int_0^{|\mathcal{S}_1|}\|u\|_\alpha d^\alpha(\phi_{-t}(w),\phi_{-t}(\mathcal{S}_1^R))dt\\
\le &\int_0^{|\mathcal{S}_2|-v}\|u\|_\alpha (Ce^{-\lambda t})^\alpha d^\a(w,\mathcal S_2^L)dt+\int_0^{|\mathcal{S}_1|}\|u\|_\alpha (Ce^{-\lambda t})^\alpha d^\a(w,\mathcal S_1^R)dt\\
\le&2\|u\|_\alpha\frac{C^{2\alpha}}{\lambda\alpha}d^\a(\mathcal S_1^R,\mathcal S_2^L),
\end{align*}
and
\begin{align*}
\left|\left\langle\widetilde{\mathcal{S}}_2,u\right\rangle-\left\langle\mathcal{S}_2,u\right\rangle\right|\le \|u\|_0|v|\le \|u\|_0Cd(\mathcal{S}_1^R,\mathcal{S}_2^L)\le \|u\|_0Cd^\a(\mathcal{S}_1^R,\mathcal{S}_2^L).
\end{align*}
Therefore
\begin{equation}\label{E:ShadowEst}
\left|\left\langle\mathcal{S}_1*\mathcal{S}_2,u\right\rangle-\left\langle\mathcal{S}_1,u\right\rangle-\left\langle\mathcal{S}_2,u\right\rangle\right|\le \left(C\|u\|_0+\frac{2C^{2\alpha}\|u\|_\alpha}{\lambda\alpha}\right)d^\a(\mathcal{S}_1^R,\mathcal{S}_2^L)
\end{equation}
On the other hand, one has that
$$d^\a((\mathcal{S}_1*\mathcal{S}_2)^L,\mathcal{S}_2^L)\le C^\a e^{-\a\lambda(\gamma-v)}d^\a(w,\mathcal S_2^L)\le C^{2\a}e^{-\a\lambda(\gamma-1)}d^\a(\mathcal{S}_1^R,\mathcal{S}_2^L),$$
and $$d^\a((\mathcal{S}_1*\mathcal{S}_2)^R,\mathcal{S}_2^R)\le C^\a e^{-\a\lambda\gamma}d^\a(w,\mathcal S_1^R)\le C^{2\a}e^{-\a\lambda\gamma}d^\a(\mathcal{S}_1^R,\mathcal{S}_2^L),$$
which combining with (\ref{E:ShadowEst}) and the choice of $\g$ implies what needed, thus accomplish the proof.
\end{proof} \subsubsection{Proof of Lemma \ref{MCGB}}\label{S:PfMCGB} Before the main proof, we first state a technical Lemma which can be deduced from the Lemma 1.1 of \cite{Bousch_Mane}. \begin{lem}\label{lemma-21}Given $0<\alpha\le1,A>0, \gamma\in\mathbb{R}$ and a continuous function $u:M\to\mathbb{R}$, the following are equivalent
\begin{itemize}
\item[(1).] For all $n\ge 1$ and $x_i\in M, {i\in\mathbb{Z}/n\mathbb{Z}}$,
\begin{align}\label{2-12-1}\sum_{i\in\mathbb{Z}/n\mathbb{Z}}u(x_i)+A\sum_{i\in\mathbb{Z}/n\mathbb{Z}}d^\alpha(\phi_\gamma x_i,x_{i+1})\ge 0.\end{align}
\item[(2).] There exists an $\alpha$-H\"older function $v:M\to\mathbb{R}$ with $\|v\|_\alpha\le A$ such that $u\ge v\circ \phi_\gamma-v$.
\end{itemize} \end{lem} Now we prove Lemma \ref{MCGB}.
\begin{proof}Let $K_1,K_2,K_3$ be the constants as in Lemmas \ref{Bou-2}, \ref{Bou-3} and \ref{Bou-4}. We fix a $\g>N_0$ satisfying the condition in Lemma \ref{Bou-4} and a large number $Q$ such that \[Q>\max\{K_1,K_2,K_3\}.\]
For $n\ge 1$, we note $i^{(n)}=i+n\mathbb{Z}\in\mathbb{Z}/n\mathbb{Z}$ for $i\in[0,n-1]\cap\mathbb{Z}$. Now we fix an integer $n\ge 1$ and points $x_{i^{(n)}}\in \L,i^{(n)}\in\mathbb{Z}/n\mathbb{Z}$. Note $$\mathcal{S}_{i^{(n)}}: [0,\g]\to \L:t\to\phi_t(x_{i^{(n)}})\text{ for }i^{(n)}\in\mathbb{Z}/n\mathbb{Z},$$ $$\mathcal{L}^{(n)}=\{\mathcal{S}_{i^{(n)}},i^{(n)}\in\mathbb{Z}/n\mathbb{Z}\}$$ and \begin{align*} \Sigma^{(n)}=\sum_{i^{(n)}\in\mathbb{Z}/n\mathbb{Z}}\left\langle\mathcal{S}_{i^{(n)}},u\right\rangle+Q\sum_{i^{(n)}\in\mathbb{Z}/n\mathbb{Z}}d^\alpha(\mathcal{S}_{i^{(n)}}^R,\mathcal{S}_{i^{(n)}+1^{(n)}}^L). \end{align*} If there is some $j^{(n)}\in \mathbb{Z}/n\mathbb{Z}$ such that $d(\mathcal{S}_{j^{(n)}}^R,\mathcal{S}_{j^{(n)}+1^{(n)}}^L)<\delta'$, just take \[\mathcal{S}_{1^{(n-1)}}=\mathcal{S}_{j^{(n)}}*\mathcal{S}_{j^{(n)}+1^{(n)}}\text{ and }\mathcal{S}_{i^{(n-1)}}=\mathcal{S}_{j^{(n)}+i^{(n)}}\text{ for }i=2,3,\cdots n-1\] $$\mathcal{L}^{(n-1)}=\{\mathcal{S}_{i^{(n-1)}},i^{(n-1)}\in\mathbb{Z}/(n-1)\mathbb{Z}\}$$ and \begin{align*} \Sigma^{(n-1)}=\sum_{i^{(n-1)}\in\mathbb{Z}/(n-1)\mathbb{Z}}\left\langle\mathcal{S}_{i^{(n-1)}},u\right\rangle+Q\sum_{i^{(n-1)}\in\mathbb{Z}/n\mathbb{Z}}d^\alpha(\mathcal{S}_{i^{(n-1)}}^R,\mathcal{S}_{i^{(n-1)}+1^{(n-1)}}^L). \end{align*} Note that by Lemma \ref{Bou-4} \begin{align*} &\Sigma^{(n)}-\Sigma^{(n-1)}\\
\ge&-\left| \left\langle\mathcal{S}_{j^{(n)}}*\mathcal{S}_{j^{(n)}+1^{(n)}},u\right\rangle-\left\langle\mathcal{S}_{j^{(n)}},u\right\rangle-\left\langle\mathcal{S}_{j^{(n)}+1^{(n)}},u\right\rangle\right|\\ &+Qd^\alpha\left(\mathcal{S}_{j^{(n)}}^R,\mathcal{S}_{(j+1)^{(n)}}^L\right)\\ &-Q\left(d^\alpha\left(\mathcal{S}_{j^{(n)}}^L,(\mathcal{S}_{j^{(n)}}*\mathcal{S}_{(j+1)^{(n)}})^L\right)-d^\alpha\left(\mathcal{S}_{(j+1)^{(n)}}^R,(\mathcal{S}_{j^{(n)}}*\mathcal{S}_{(j+1)^{(n)}})^R\right)\right)\\ &\ge 0. \end{align*} That is \begin{align}\label{eq213} \Sigma^{(n)}\ge\Sigma^{(n-1)}. \end{align} Repeat the above process until $\mathcal{L}^{(1)}$ with $d(\mathcal{S}_{1^{(1)}}^R,\mathcal{S}_{1^{(1)}}^L)<\delta'$ {\bf OR} some $m\in[1,n]\cap\mathbb{N}$ with \[d\left(\mathcal{S}_{j^{(m)}}^R,\mathcal{S}_{(j+1)^{(m)}}^L\right)\ge\delta'\text{ for all } j\in\mathbb{Z}/m\mathbb{Z}.\] In the case that the process ends at $\mathcal{L}^{(1)}$ with $d(\mathcal{S}_{1^{(1)}}^R,\mathcal{S}_{1^{(1)}}^L)<\delta'$. We have by Lemma \ref{Bou-2} that \begin{align}\label{eq-1}\begin{split} \Sigma^{(1)}&=\left\langle\mathcal{S}_{1^{(1)}},u\right\rangle+Qd^\alpha(\mathcal{S}_{1^{(1)}}^R,\mathcal{S}_{1^{(1)}}^L)\\ &\ge -K_1 d^\alpha(\mathcal{S}_{1^{(1)}}^R,\mathcal{S}_{1^{(1)}}^L)+Qd^\alpha(\mathcal{S}_{1^{(1)}}^R,\mathcal{S}_{1^{(1)}}^L)\\ &\ge 0. \end{split} \end{align} In the case that the process ends at some $m\in[1,n]\cap\mathbb{N}$ with \[d\left(\mathcal{S}_{j^{(m)}}^R,\mathcal{S}_{(j+1)^{(m)}}^L\right)\ge\delta'\text{ for all } j^{(m)}\in\mathbb{Z}/m\mathbb{Z}.\] We have by Lemma \ref{Bou-3} that \begin{align}\label{eq-2}\begin{split} \Sigma^{(m)}&=\sum_{i^{(m)}\in\mathbb{Z}/m\mathbb{Z}}\left\langle\mathcal{S}_{i^{(m)}},u\right\rangle+Q\sum_{i^{(m)}\in\mathbb{Z}/m\mathbb{Z}}d^\alpha(\mathcal{S}_{i^{(m)}}^R,\mathcal{S}_{i^{(m)}+1^{(m)}}^L)\\ &\ge-mK_2\delta'^\alpha+mQ\delta'^\alpha\\ &\ge 0. \end{split} \end{align} Combining the inequality \eqref{eq-1}, \eqref{eq-2} and the fact $\Sigma^{(n)}\ge \Sigma^{(n-1)}\ge \Sigma^{(n-2)}\ge\cdots$ by \eqref{eq213}, one has \[\Sigma^{(n)}\ge0.\] Then \[\sum_{i^{(n)}\in\mathbb{Z}/n\mathbb{Z}}u_\gamma(x_{i^{(n)}})+\frac{Q}{\gamma}\sum_{i^{(n)}\in\mathbb{Z}/n\mathbb{Z}}d^\alpha(\mathcal{S}^R_{i^{(n)}},\mathcal{S}^L_{i^{(n)}+1^{(n)}})=\frac{\Sigma^{(n)}}{\gamma}\ge 0.\]
By Lemma \ref{lemma-21}, there is an $\alpha$-H\"older function $v$ on $\L$ with $\|v\|_\alpha \le \frac{Q}{\gamma}$ such that $$u_\gamma|_\L\ge v\circ \phi_\gamma|_\L-v.$$This ends the proof. \end{proof} Finally , we give the proof of Lemma \ref{revael}. \begin{proof}[Proof of Lemma \ref{revael}] (1). By Lemma \ref{MCGB}, we only need to show that
\[\int u-\beta(u;\psi,\L,\Phi)\psi d\mu\ge 0\text{ for all } \mu\in\mathcal{M}(\Phi|_\L).\]
It is immediately from the fact
\[\frac{\int ud\mu}{\int \psi d\mu}\ge\beta(u;\psi,\L,\Phi)\text{ for all } \mu\in\mathcal{M}(\Phi|_\L)\]
since $\psi$ is strictly positive.
(2). Given a probability measure $\mu\in\mathcal{M}_{min}(u;\psi,\L,\Phi)$, one has
\[\int u_\gamma+v\circ \phi_\gamma-v-\beta(u;\psi,\L,\Phi)\psi_\gamma d\mu=\int u-\beta(u;\psi,\L,\Phi)\psi d\mu=0.\]
Combining (1) and the fact $u_\gamma|_\L+v\circ \phi_\gamma|_\L-v-\beta(u;\psi,\L,\Phi)\psi_\gamma|_\L$ is continuous on $\L$, one has
\[supp(\mu)\subset \{x\in \L:(u_\gamma+v\circ \phi_\gamma-v-\beta(u;\psi,\L,\Phi)\psi_\gamma)|_\L(x)=0\}.\]
Therefore, \[Z_{u,\psi}\subset\left\{x\in \L:(u_\gamma|_\L+v\circ \phi_\gamma|_\L-v-\beta(u;\psi,\L,\Phi)\psi_\gamma|_\L)(x)=0\right\}.\] This ends the proof. \end{proof}
\subsection{Proof of Lemma \ref{lemma-2}}\label{S:ProofPerApp} In this section, we mainly prove the periodic approximation. The proof partially follows the arguments in \cite{BQ}. \subsubsection{Joining of segments} Recall that the definition of the joining of two segments $\mathcal S_1\star \mathcal S_2$ are given by (\ref{E:S1*S2}), we give some properties of jointed segments.
\begin{lem}\label{lem-Q-2}If two segments $\mathcal{S}_1$ and $\mathcal{S}_2$ satisfy $|\mathcal{S}_1|\ge 1$ and $d(\mathcal{S}_1^R,\mathcal{S}_2^L)\le \delta'$, then \begin{itemize}
\item[(1).]$\max_{x\in \mathcal{S}_1*\mathcal{S}_2}d(x, \mathcal{S}_1\cup\mathcal{S}_2)\le C^3d(\mathcal{S}_1^R,\mathcal{S}_2^L);$
\item[(2).] $|\mathcal{S}_1|+|\mathcal{S}_2|-1\le|\mathcal{S}_1*\mathcal{S}_2|\le |\mathcal{S}_1|+|\mathcal{S}_2|+1.$
\end{itemize} \end{lem} \begin{proof}(1). Note $v=v(\mathcal{S}_2^L,\mathcal{S}_1^R)$, $w=w(\mathcal{S}_2^L,\mathcal{S}_1^R)$ and
$$\widetilde{\mathcal{S}}_2:[v,|\mathcal{S}_2|]:t\to\phi_t(\mathcal{S}_2^L).$$
Then for $t\in[-|\mathcal{S}_1|,0]$,
$$d(\phi_t(w),\mathcal{S}_1)\le d(\phi_t(w),\phi_t(\mathcal{S}_1^R))\le Ce^{\lambda t}d(w,\mathcal{S}_1^R)\le C^2e^{\lambda t}d(\mathcal{S}_1^R,\mathcal{S}_2^L)\le C^2d(\mathcal{S}_1^R,\mathcal{S}_2^L),$$
where we used $w\in W_\epsilon^u(\mathcal{S}_1^R).$ For $t\in [0,|\mathcal{S}_2|-v]$,
$$d(\phi_t(w),\widetilde{\mathcal{S}}_2)\le d(\phi_t(w),\phi_t(w)\phi_v(\mathcal{S}_2^L))\le Ce^{-\lambda t}d(w,\phi_v(\mathcal{S}_2^L))\le C^2e^{-\lambda t}d(\mathcal{S}_1^R,\mathcal{S}_2^L).$$
where we used $w\in W_\epsilon^s(\phi_v(\mathcal{S}_2^L)).$ Thus, for $t\in [0,|\mathcal{S}_2|-v]$,
\begin{align*}
d(\phi_t(w),\mathcal{S}_2)&\le d(\phi_t(w),\widetilde{\mathcal{S}}_2)+\max_{x\in \widetilde{\mathcal{S}}_2}d(\mathcal{S}_2,\widetilde{\mathcal{S}}_2)\\
&\le C^2e^{-\lambda t}d(\mathcal{S}_1^R,\mathcal{S}_2^L)+d(\mathcal{S}_2^R,\phi_v(\mathcal{S}_2^R))\\
&\le C^3d(\mathcal{S}_1^R,\mathcal{S}_2^L),
\end{align*}
where we used $C\gg1$ and 1b) of the {\bf basic cononical setting}. Thus, by summing up,
$$\max_{x\in \mathcal{S}_1*\mathcal{S}_2}d(x, \mathcal{S}_1\cup\mathcal{S}_2)=
\max_{t\in[-|\mathcal{S}_1|,|\mathcal{S}_2|-v]} d(\phi_t(w), \mathcal{S}_1\cup\mathcal{S}_2)\le C^3d(\mathcal{S}_1^R,\mathcal{S}_2^L).$$
(2). One has
$$|\mathcal{S}_1*\mathcal{S}_2|=|\mathcal{S}_1|+|\mathcal{S}_2|-v\ge |\mathcal{S}_1|+|\mathcal{S}_2|-Cd(\mathcal{S}_1^R,\mathcal{S}_2^L)\ge |\mathcal{S}_1|+|\mathcal{S}_2|-C\delta'>|\mathcal{S}_1|+|\mathcal{S}_2|-1,$$ where we used the assumption $\delta'\ll\frac{1}{C}.$
On the other hand, one also has that
$$|\mathcal{S}_1*\mathcal{S}_2|=|\mathcal{S}_1|+|\mathcal{S}_2|-v\le |\mathcal{S}_1|+|\mathcal{S}_2|+C\delta'\le |\mathcal{S}_1|+|\mathcal{S}_2|+1.$$
This ends the proof. \end{proof}
\begin{lem}\label{lem-Q-3}There exists a large constant $P_0>1$ such that if two segments $\mathcal{S}_1$ and $\mathcal{S}_2$ satisfy $|\mathcal{S}_1|\ge P_0$, $|\mathcal{S}_2|\ge P_0$ and $d(\mathcal{S}_1^R,\mathcal{S}_2^L)\le \delta'$, then
$$d(\mathcal{S}_1^R,\mathcal{S}_2^L)\ge 2d(\mathcal{S}_1^L,(\mathcal{S}_1*\mathcal{S}_2)^L)+2d((\mathcal{S}_1*\mathcal{S}_2)^R,\mathcal{S}_2^R).$$ \end{lem} \begin{proof} First we fix a large constant $P_0\gg 1$ such that $C^2e^{-\lambda(P_0-1)}+C^2e^{-\lambda P_0}<\frac{1}{2}$. Fix two segments $\mathcal{S}_1$ and $\mathcal{S}_2$ as in Lemma. Note $v=v(\mathcal{S}_2^L,\mathcal{S}_1^R)$ and $w=w(\mathcal{S}_2^L,\mathcal{S}_1^R)$. Then
\begin{align*}
d(\mathcal{S}_1^L,(\mathcal{S}_1*\mathcal{S}_2)^L)&= d(\phi_{-|\mathcal{S}_1|}(\mathcal{S}_1^R),\phi_{-|\mathcal{S}_1|}(w))\\
&\le Ce^{-\lambda|\mathcal{S}_1|}d(\mathcal{S}_1^R,w)\\
&\le C^2e^{-\lambda P_0}d(\mathcal{S}_1^R,\mathcal{S}_2^L),
\end{align*}
where we used $w\in W_\epsilon^u(\mathcal{S}_1^R).$ On the other hand,
\begin{align*}
d(\mathcal{S}_2^R,(\mathcal{S}_1*\mathcal{S}_2)^R)&= d(\phi_{|\mathcal{S}_2|-v}\phi_v(\mathcal{S}_2^L),\phi_{|\mathcal{S}_2|-v}(w))\\
&\le Ce^{-\lambda(|\mathcal{S}_2|-v)}d(\phi_v(\mathcal{S}_2^L),w)\\
&\le C^2e^{-\lambda (P_0-1)}d(\mathcal{S}_1^R,\mathcal{S}_2^L),
\end{align*}
where we used $w\in W_\epsilon^u(\phi_v(\mathcal{S}_2^L)).$ By assumption, we have
\begin{align*} d(\mathcal{S}_1^L,(\mathcal{S}_1*\mathcal{S}_2)^L)+d(\mathcal{S}_2^R,(\mathcal{S}_1*\mathcal{S}_2)^R)&\le C^2e^{-\lambda(P_0-1)}d(\mathcal{S}_1^R,\mathcal{S}_1^L)+C^2e^{-\lambda P_0}d(\mathcal{S}_1^R,\mathcal{S}_1^L)\\
&\le \frac{1}{2}d(\mathcal{S}_1^R,\mathcal{S}_1^L).
\end{align*}
This ends the proof. \end{proof} \subsubsection{Periodic approximation} For integer $n\ge 1$, let $\Sigma_n=\{0,1,2,\cdot,n-1\}^{\mathbb{N}}$ and $\sigma$ be a shift on $\Sigma_n$. Assume $F$ is a subset of $\bigcup_{i\ge 1}\{0,1,2,\cdot,n-1\}^{i}$, then the subshift with forbidden $F$ is denoted by $(Y_F,\sigma)$ where $$Y_F=\left\{x\in \{0,1,2,\cdot,n-1\}^{\mathbb{N}},w\text{ does not appear in } x\text{ for all }w\in F\right\}.$$ The following lemma is Lemma 5 of \cite{BQ}, which will be used later. \begin{lem}[\cite{BQ}]\label{lemma-1} Suppose that $(Y,\sigma)$ is a shift of finite type $($with forbidden words of length 2$)$ with $M$ symbols and entropy $h$. Then $(Y,\sigma)$ contains a periodic point of period at most $1+Me^{(1-h)}$. \end{lem}
Now we are ready to prove Lemma \ref{lemma-2}, which partially follow the argument in \cite{BQ}. \begin{proof}[ Proof of Lemma \ref{lemma-2}]
Fix a positive constant $\delta''\ll \frac{\delta'}{C^{10}e^\beta}$. Let $\mathcal{P}=\{B_1,B_2,\cdots,B_m\}$ be a finite partition of $\L$ with diameter smaller than $\delta''$. For $x\in \L$, $\widehat{ x}\in\{1,2,3,\cdots,m\}^{\mathbb{N}}$ is defined by
\[\widehat{x}(n)=j\text{ if } \phi_n(x)\in B_j, n=0,1,2,\cdots.\]
Denote $\widehat{Z}=\{\widehat{x}: x\in Z\}$ and $W_n$ the collection of
length $n$ string that appears in $\widehat{Z}$. One has
\[\sharp W_n=K_ne^{nh}\]where $h=h_{top}(\overline{\widehat{Z}},\sigma)$ and $K_n$ grows at a subexponential rate. Let
\[Y_n=\{y_0y_1y_2\cdots\in W_n^\mathbb{N}:y_iy_{i+1}\in W_{2n}\text{ for all }i\in\mathbb{N}\}\]
and $(Y_n,\sigma_n)$ is the 1-step shift of finite type on $W_n$.
From Lemma \ref{lemma-1}, the shortest periodic orbit in $Y_n$ is at most $1+K_ne^{nh}e^{1-nh}=1+eK_n$. Denote one of the shortest periodic orbit in $Y_n$ by $z_0z_1z_2\cdots z_{p_n-1}z_0z_1z_2\cdots$ for some $p_n\le 1+eK_n$ and $z_i\in W_n, i=0,1,2,\cdots,p_n-1$. For $i=0,1,2,\cdots, p_n-1$, there is $x_i\in Z$ such that the leading $2n$ string of $\widehat{x_i}$ is $z_iz_{i+1}$ (we note $z_{p_n}=z_{0}$, $x_{p_n}=x_{0}$, $\mathcal{S}_{p_n}=\mathcal{S}_{0},\cdots$). Choose segments $\mathcal{S}_i$ by
\[\mathcal{S}_i:[\frac{n}{2},\frac{3n}{2}+v_i]\to M:t\to \phi_t(x_i)\text{ for }i=0,1,2,\cdots,p_n-1,\]
where $v_i=v(\phi_n(x_i),x_{i+1})$. We have the following Claim.
\noindent{\it {\bf Claim Q1. } $d(\mathcal{S}_i^R,\mathcal{S}_{i+1}^L)\le 2C^2e^{-\frac{n\lambda}{2}}\delta''$ for $i=0,1,2,\cdots,p_n-1$.} \begin{proof}[Proof of Claim Q1.] Note that the leading $2n$ string of $\widehat{x_i}$ is $z_iz_{i+1}$ and leading $n$ string of $\widehat{x_{i+1}}$ is $z_{i+1}$, which means $$\mathcal{P}(\phi_{n+j}(x_i))=\mathcal{P}(\phi_{j}(x_{i+1}))\text{ for }j=0,1,\cdots,n-1.$$
Therefore, $$d(\phi_{n+j}(x_i),\phi_{j}(x_{i+1}))<\delta''\text{ for }j=0,1,\cdots,n-1.$$ Thus
$$d(\phi_{n+t}(x_{i}),\phi_{t}(x_{i+1}))<Ce^\beta\delta''<\delta'\text{ for }t\in[0,n].$$
Then by Lemma \ref{An-1}, we have
\begin{align*}d\left(\phi_{\frac{3n}{2}+v_i}(x_{i}),\phi_{\frac{n}{2}}(x_{i+1})\right)&\le C^2e^{-\frac{n\lambda}{2}}(d(\phi_n(x_i),x_{i+1})+d(\phi_{2n}(x_i),\phi_{n}(x_{i+1})))\\
&\le 2C^2e^{-\frac{n\lambda}{2}}\delta''.
\end{align*}
This ends the proof of { Claim Q1}. \end{proof}
Now we define segments $\overline{\mathcal{S}}_i$ recursively for $0\le i\le p_n-1$ by $\overline{\mathcal{S}}_0:=\mathcal{S}_0$ and $$\overline{\mathcal{S}}_i:=\overline{\mathcal{S}}_{i-1}*\mathcal{S}_i\text{ for }1\le i\le p_n-1.$$ Based on Claim Q1, we have the following claim.
\noindent{\it {\bf Claim Q2. }There is a positive integer $N$ such that for any $n\ge N$, one has \begin{itemize}
\item[(1).] $\overline{\mathcal{S}}_i$ is well defined for $0\le i\le p_n-1;$
\item[(2).] $d(\overline{\mathcal{S}}_{i}^R, \mathcal{S}_{i+1}^L)\le 4C^2p_ne^{-\frac{n\lambda}{2}}\delta''<\delta'$ for $0\le i\le p_n-2;$
\item[(3).] $ d(\overline{\mathcal{S}}_{p_n-1}^R, \overline{\mathcal{S}}_{p_n-1}^L)\le 2C^2p_ne^{-\frac{n\lambda}{2}}\delta''<\min\left\{\delta',\frac1L\right\}$, where $L$ is as in Lemma \ref{lemma-a};
\item[(4).] $(n-1)p_n\le |\overline{\mathcal{S}}_{p_n-1}|\le (n+1)p_n;$
\item[(5).] $\max_{x\in \overline{\mathcal{S}}_{p_n-1}}d(x, Z)\le C^4p_n^2e^{-\frac{n\lambda}{2}}\delta''.$ \end{itemize}}
\begin{proof}[Proof of Claim Q2.] Since $p_n$ grows at a subexponential rate, we can take $N$ large enough such that
\begin{align}\label{Q2-1}
N>P_0\text{ and }4p_nC^2e^{-\frac{n\lambda}{2}}\delta''< \min\left\{\delta',\frac1L\right\}\text{ for all }n\ge N,
\end{align}
where $P_0$ is the constant as in Lemma \ref{lem-Q-3}.
For $0\le i\le p_n-2$, we define
$$\chi(i)=d(\overline{\mathcal{S}}_i^R,\mathcal{S}_{i+1}^L)+d(\mathcal{S}_{i+1}^R,\mathcal{S}_{i+2}^L)+\cdots+d(\mathcal{S}_{p_n-2}^R,\mathcal{S}_{p_n-1}^L)+d(\mathcal{S}_{p_n-1}^R,\overline{\mathcal{S}}_i^L).$$
By Claim Q1,
$$\chi(0)\le 2C^2p_ne^{-\frac{n\lambda}{2}}\delta''.$$
Now we are to show that $\chi(i)$ and $\overline{\mathcal{S}}_i$ are well defined, which satisfy that
$$\chi(i)\le \delta'\text{ and }|\overline{\mathcal{S}}_i|>P_0\text{ for }i=0, 1, 2, \cdots,p_n-2.$$ These are clearly true for $i=0$. Now we assume that these are true for some $i\in\{0,1,2\cdots,p_n-2\}$. Then for $i+1$, since $\chi(i)\le \delta'$, one has $d(\overline{\mathcal{S}}_i^R,\mathcal{S}_{i+1}^L)\le \delta'$.
Thus, we can join $\overline{\mathcal{S}}_i$ and $\mathcal{S}_{i+1}$ by letting
$$\overline{\mathcal{S}}_{i+1}=\overline{\mathcal{S}}_i*\mathcal{S}_{i+1}.$$
It is clear that $|\overline{\mathcal{S}}_{i+1}|>P_0$ by Lemma \ref{lem-Q-2} (2).
On the other hand, by triangle inequality and Lemma \ref{lem-Q-3}, one has that
\begin{align}\begin{split}\label{Q-315}
&\chi(i)-\chi(i+1)\\
=&d(\overline{\mathcal{S}}_i^R,\mathcal{S}_{i+1}^L)+d(\mathcal S_{i+1}^R,\mathcal S_{i+2}^L)-d(\overline{\mathcal{S}}_{i+1}^R,\mathcal{S}_{i+2}^L)+d(\mathcal{S}_{p_n-1}^R,\overline{\mathcal{S}}_i^L)-d(\mathcal{S}_{p_n-1}^R,\overline{\mathcal{S}}_{i+1}^L)\\
\ge& d(\overline{\mathcal{S}}_i^R,\mathcal{S}_{i+1}^L)- d(\overline{\mathcal{S}}_i^L,\overline{\mathcal{S}}_{i+1}^L)-d(\overline{\mathcal{S}}_{i+1}^R,\mathcal{S}_{i+1}^R)\\
\ge&\frac12 d(\overline{\mathcal{S}}_i^R,\mathcal{S}_{i+1}^L).
\end{split}\end{align}
Therefore, $\chi(i+1)\le \chi(i)\le \delta'$. By induction, we can finish the proof of (1).
By \eqref{Q-315}, one has
$$d(\overline{\mathcal{S}}_{i}^R, \mathcal{S}_{i+1}^L)\le2 \chi(i)\le \cdots\le 2\chi(0)\le 4C^2p_ne^{-\frac{n\lambda}{2}}\delta''\text{ for }i=0,1,2\cdots, p_n-2,$$
and we can deduce the following by triangle inequality:
$$d(\overline{\mathcal{S}}_{p_n-1}^R, \overline{\mathcal{S}}_{p_n-1}^L)\le\chi(p_n-2)\le \cdots\chi(0)\le 2C^2p_ne^{-\frac{n\lambda}{2}}\delta''.$$
This ends the proof of (2) as well as (1), (3), and (4) follows from Lemma \ref{lem-Q-2} (2).
Now we denote $D_i:=\overline{\mathcal{S}}_i\bigcup\cup_{j=i+1}^{p_n-1}\mathcal{S}_j$ for $i=0,1,2,\cdots,p_n-1.$ Then by Lemma \ref{lem-Q-2} (1),
$$\max_{x\in D_{i+1}}d(x,D_i)\le \max_{x\in \overline{\mathcal{S}}_i*\mathcal{S}_{i+1}}d(x, \overline{\mathcal{S}}_i\cup\mathcal{S}_{i+1})\le C^3d(\mathcal{S}_i^R,\mathcal{S}_{i+1}^L)\le 2C^5p_ne^{-\frac{n\lambda}{2}}\delta''.$$
Therefore, by triangle inequality and the fact that $D_0\subset Z$,
$$\max_{x\in \overline{\mathcal{S}}_{p_n-1}}d(x, Z)\le \max_{x\in D_0}d(x, Z)+\sum_{i=0}^{p_n-2}\max_{x\in D_{i+1}}d(x,D_i)\le 2C^5p_n^2e^{-\frac{n\lambda}{2}}\delta''.$$
This ends the proof of Claim Q2. \end{proof}
Recall that $P_0$ is the constant as in Lemma \ref{lem-Q-3}, $K, L$ are the constants as in Lemma \ref{lemma-a} and $N$ is the constant as in Claim Q2. We fix an integer $n>\max(P_0, K, N)+1$ and let $\overline{\mathcal{S}}_{p_n-1}$ be the segment as in Claim Q2. Then by Claim Q2 (3),
$$ |\overline{\mathcal{S}}_{p_n-1}|>K \text{ and } d(\overline{\mathcal{S}}_{p_n-1}^R, \overline{\mathcal{S}}_{p_n-1}^L)\le\delta'.$$ Applying the Anosov Closing Lemma, we have a periodic segment $\mathcal{O}_n$ such that \begin{align}\label{e-Q-1}
\left||\overline{\mathcal{S}}_{p_n-1}|-|\mathcal{O}_n|\right|\le Ld\left(\overline{\mathcal{S}}_{p_n-1}^L,\overline{\mathcal{S}}_{p_n-1}^R\right)
\end{align} and
\begin{align}\label{e-Q-2} d\left(\phi_t(\mathcal{O}_n^L),\phi_t(\overline{\mathcal{S}}_{p_n-1}^L)\right)\le Ld(\overline{\mathcal{S}}_{p_n-1}^L, \overline{\mathcal{S}}_{p_n-1}^R)\ \forall t\in \left[0, \max\left(|\overline{\mathcal{S}}_{p_n-1}|,|\mathcal{O}_n|\right)\right]. \end{align}
We claim the following:
{\it {\bf Claim Q3. }$\max_{x\in \mathcal{O}_n}d(x, Z)\le (2C^3Lp_n+C^4p_n^2)e^{-\frac{n\lambda}{2}}\delta''.$} \begin{proof}[Proof of Claim Q3]
If $|\mathcal{O}|\le |\overline{\mathcal{S}}_{p_n-1}|$, by \eqref{e-Q-2},
$$\max_{x\in \mathcal{O}}d(x, \overline{\mathcal{S}}_{p_n-1})\le Ld\left(\overline{\mathcal{S}}_{p_n-1}^L,\overline{\mathcal{S}}_{p_n-1}^R\right).$$
If $|\mathcal{O}|> |\overline{\mathcal{S}}_{p_n-1}|$, note $t_*=\min(t, |\overline{\mathcal{S}}_{p_n-1}|)$ for $t\in [0,|\mathcal{O}|]$. Then, by (\ref{e-Q-1}) and (\ref{e-Q-2}), one has that for $t\in [0,|\mathcal{O}|]$
\begin{align*}
d\left(\phi_t(\mathcal{O}^L), \overline{\mathcal{S}}_{p_n-1}\right)&\le d\left(\phi_{t}(\mathcal{O}^L), \phi_{t_*}(\mathcal{O}^L)\right)+d\left(\phi_{t_*}(\mathcal{O}^L),\phi_{t_*}(\overline{\mathcal{S}}_{p_n-1}^L)\right)\\
&\le CLd\left(\overline{\mathcal{S}}_{p_n-1}^L,\overline{\mathcal{S}}_{p_n-1}^R\right)+Ld\left(\overline{\mathcal{S}}_{p_n-1}^L,\overline{\mathcal{S}}_{p_n-1}^R\right)\\
&\le C^2Ld\left(\overline{\mathcal{S}}_{p_n-1}^L,\overline{\mathcal{S}}_{p_n-1}^R\right),
\end{align*}
where $C\gg 1$ as in Remark \ref{remark-1-1}.
Therefore, $$\max_{x\in \mathcal{O}}d\left(x, \overline{\mathcal{S}}_{p_n-1}\right)=\max_{t\in [0,|\mathcal{O}|]}d\left(\phi_t(\mathcal{O}^L), \overline{\mathcal{S}}_{p_n-1}\right)\le C^2Ld\left(\overline{\mathcal{S}}_{p_n-1}^L,\overline{\mathcal{S}}_{p_n-1}^R\right).$$
Combining with (3), (5) of Claim Q2, we have
\begin{align*}\max_{x\in \mathcal{O}}d(x, Z)&\le \max_{x\in \mathcal{O}}d\left(x, \overline{\mathcal{S}}_{p_n-1}\right)+\max_{x\in \overline{\mathcal{S}}_{p_n-1}}d(x, Z)\\
&\le C^2L\cdot 2C^2p_ne^{-\frac{n\lambda}{2}}\delta''+C^5p_n^2e^{-\frac{n\lambda}{2}}\delta''\\
&= (2C^4Lp_n+C^5p_n^2)e^{-\frac{n\lambda}{2}}\delta''.
\end{align*} This ends the proof of Claim Q3. \end{proof}
By {Claim Q3} and (\ref{e-Q-1}), we have \begin{align*}
d_{\alpha,Z}(\mathcal{O}_n)&\le |\mathcal{O}_n|\left(\max_{x\in \mathcal{O}}d(x, Z)\right)^\alpha\\
&\le \left(|\overline{\mathcal{S}}_{p_n-1}|+ Ld\left(\overline{\mathcal{S}}_{p_n-1}^L,\overline{\mathcal{S}}_{p_n-1}^R\right)\right)\cdot\left((2C^4Lp_n+C^5p_n^2)e^{-\frac{n\lambda}{2}}\delta''\right)^\alpha\\ &\le H_ne^{-\frac{n\alpha\lambda}{2}}, \end{align*} where $H_n= \big((2C^4Lp_n+C^5p_n^2)\delta''\big)^\alpha\cdot((n+1)p_n+ 1)$. Note that $H_n$ grows at a subexponential rate as $n$ increases as $p_n$ does. Hence
\[\limsup_{P\to+\infty}P^k\min_{\mathcal{O}\in \mathcal{O}^P_\L}d_{\alpha,Z}(\mathcal{O})\le \limsup_{n\to+\infty}((n+1)p_n+1)^k\cdot H_ne^{-\frac{n\alpha\lambda}{2}}=0,\]
where we used the fact that $p_n, H_n$ grow at a subexponential rate as $n$ increases and $|\mathcal{O}_n|\le np_n+1$.
The proof of Lemma \ref{lemma-2} is completed. \end{proof}
\section{Further discussions on the case of $C^{s,\a}$-observables}\label{S:HighReg} For $s\in\mathbb{N}$, $0\le\alpha\le 1$ and a strictly positive function $\psi$ on $M$, $Per^*_{s,\alpha}(M,\psi)$ is defined as the collection of $C^{s,\alpha}$-continuous functions on $M$, such that for each $u\in Per^*_{s,\alpha}(M,\psi)$, $\mathcal{M}_{min}(u;\psi,\L,\Phi)$ contains at least one periodic measure. And $Loc_{s,\alpha}(M,\psi)$ is defined by \begin{align*}Loc_{s,\alpha}(M,\psi):=\{&u\in Per^{*}_{s,\alpha}(M,\psi):\text{ there is } \varepsilon>0\text{ such that }\\
&\mathcal{M}_{min}(u+h;\psi,\L,\Phi)=\mathcal{M}_{min}(u;\psi,\L,\Phi)\text{ for all }\|h\|_{r,\alpha}<\varepsilon \}.\end{align*} In the case $s\ge1$ and $\alpha>0$ or $s\ge 2$, we do not have analogous result like Proposition \ref{prop-2}. However, we have the following weak version. \begin{prop}\label{prop-4}
Let $\mathcal{O}$ be a periodic segment of $\Phi|_\L$ with $D(\mathcal{O})>0$ and $u\in \mathcal{C}(M)$ with $u|_\mathcal{O}=0$ and $u|_{M/\mathcal{O}}>0$. Then there exists a constant $\varrho>0$ such that the probability measure $$\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+h;\psi,\L,\Phi),$$ where $h$ is any $\mathcal{C}^{0,1}(M)$ function with $\|h\|_1<\varrho$. \end{prop}
\begin{rem}\label{remark-3} As in Remark \ref{remark-2}, for $s\in\mathbb{N}$ and $0\le \alpha\le 1$, we let $\widetilde w\in\mathcal{C}^{s,\a}(M)$ such that $\|\widetilde{w}\|_{s,\alpha}<\varepsilon$, $\widetilde{w}|_\mathcal{O}=0$ and $\widetilde w|_{M\setminus\mathcal{O}}>0$. Then $\mu_\mathcal{O}$ is the unique measure in $\mathcal{M}_{min}(u+\widetilde w+h;\psi,\L,\Phi)$ whenever $\|h\|_{s,\alpha}<\varrho$. The Proposition shows that there is an open subset of $\mathcal{C}^{s,\alpha}({M})$ near $u$ such that functions in the open set have the same unique minimizing measure with respect to $\psi$ and the probability measure supports on a periodic orbit. \end{rem} By using Remark \ref{remark-3}, we have the following result. \begin{thm}
$Loc_{s,\alpha}(M,\psi)$ is an open dense subset of $Per^*_{s,\alpha}(M,\psi)$ w.r.t. $\|\cdot\|_{s,\alpha}$ for integer $s\ge 1$ and real number $0\le\alpha\le 1$. \end{thm}
\begin{proof}Given $s\ge 1$ and $0\le \alpha\le 1$. The openness is clearly true. We prove $Loc_{s,\alpha}(M,\psi)$ is dense in $Per^*_{s,\alpha}(M,\psi)$ w.r.t. $\|\cdot\|_{s,\alpha}$. Since
$$\int ud\mu=\int \bar ud\mu\text{ for all }\mu\in\mathcal{M}(\Phi|_\L),$$ we have $\mathcal{M}_{min}(u;\psi,\L,\Phi)=\mathcal{M}_{min}(\bar u;\psi,\L,\Phi)$. Then the theorem follows from Remark \ref{remark-3} immediately. \end{proof} \subsection{Proof of Proposition \ref{prop-4}} Now we finish the proof of Proposition \ref{prop-4}. \begin{proof}[Proof of Proposition \ref{prop-4}]
Let $\mathcal{O}$ be a periodic segment of $\Phi|_\L$ and $u\in \mathcal{C}(M)$ with $u|_\mathcal{O}=0$ and $u|_{M/\mathcal{O}}>0$. For $0\le \rho\le D(\mathcal{O})$, we note $\theta( \rho)=\min\{u(x):d(x,\mathcal{O})\ge \rho, x\in M\}.$ It is clear that $\theta(0)=0$, $\theta(\rho)>0$ for $ \rho\neq 0$ and $\theta$ is non-decreasing. Since $D(\mathcal{O})>0$ by assumption, there are two constants $\rho_1,\rho_2$ satisfy \begin{align}\label{bbb-0} 0<\rho_1<\rho_2<\frac{D(\mathcal{O})}{4C^3e^{2\beta}}. \end{align}
Next we will show that $\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+h;\psi, \Lambda,\Phi)$ for all $h\in C^{0,1}(M)$ with $\|h\|_1<\varrho$,
where the constant $\varrho$ is positive and \begin{align}\label{b-0}
\varrho<\frac{1}{2}\min\left\{\frac{\psi_{min}\theta(\rho_1)}{1+\psi_{min}},\frac{\theta(\frac{D(\mathcal{O})}{4C^2e^{2\beta}})}{(1+\frac{\|\psi\|_1}{\psi_{min}}) \cdot \frac{4C^{3}\rho_2}{\lambda}+|\mathcal{O}|\cdot \frac{1+\psi_{min}}{\psi_{min}}+\frac{1+\psi_{min}}{\psi_{min}}}\right\}. \end{align}
Now we fix a function $h$ as above. Note $G=u+h-a_\mathcal{O}\psi$ where
\begin{align}\label{b-1}
\begin{split}
a_{\mathcal{O}}&=\frac{\left\langle \mathcal{O},u+h\right\rangle }{\left\langle \mathcal{O},\psi\right\rangle}\le\frac{\|h\|_0}{\psi_{min}}.
\end{split}
\end{align}
Then $\frac{\int Gd\mu}{\int \psi d\mu}=\frac{\int u+hd\mu}{\int \psi d\mu}-a_\mathcal{O}.$
Therefore, to show that $\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+h;\psi,\L,\Phi)$, it is enough to show that
\[\int Gd\mu\ge0\text{ for all }\mu\in\mathcal{M}^e(\Phi|_\L),\]
where we used the assumption $\psi$ is strictly positive and the fact $\int Gd\mu_\mathcal{O}=0$. Now we let
$Area_1:=\{y\in M:d(y,\mathcal{O})\le \rho_1\}.$ We have the following claim.
\noindent{\it {\bf Claim F1.} $Area_1$ contains all $x\in M$ with $G(x)\le0$. } \begin{proof}[Proof of Claim F1]
For $x\notin Area_1$, we have
\begin{align*}G(x)&= u(x)+h(x)-a_{\mathcal{O}}\psi\ge \theta(\rho_1)-\|h\|_0-a_{\mathcal{O}}\|\psi\|_0\ge\theta(\rho_1)-\frac{1+\psi_{min}}{\psi_{min}}\|h\|_0>0.
\end{align*}
where we used \eqref{b-0} and \eqref{b-1}. This ends the proof of Claim F1.\end{proof}
Note $Area_2=\{y\in M:d(y,\mathcal{O})\le \rho_2\}.$ It is clear that $Area_1$ is in the interior of $Area_2$. Thus, $d(Area_1,M\setminus Area_2)>0$. Therefore, by Claim F1, we can fix a constant $0<\tau<1$ such that $G(\phi_t(x))>0$ for all $x\in M\setminus Area_2$ and $|t|\le\tau$.
\noindent{\it {\bf Claim F2.} If $z\in \L$ is not a generic point of $\mu_\mathcal{O}$, then there is $m\ge\tau$ such that $\int_{0}^mG(\phi_t(z))dt>0$.}
Next we prove Proposition \ref{prop-4} by assuming the validity of {Claim F2}, proof of which is left to the next subsection. Same as the argument at the beginning of the proof, it is enough to show that for all $\mu\in\mathcal{M}^e(\Phi|_\L)$
\[\int Gd\mu\ge0.\]
Given $\mu\in\mathcal{M}^e(\Phi|_\L)$, in the case $\mu=\mu_{\mathcal{O}}$, it is obviously true. In the case $\mu\neq\mu_{\mathcal{O}}$, just let $z$ be a generic point of $\mu$. Note that $z$ is not a generic point of $\mu_{\mathcal{O}}$. By Claim F2, we have $t_1\ge\tau$ such that \[\int_0^{t_1} G(\phi_t(z))dt>0.\]
Note that $\phi_{t_1}(z)$ is still not a generic point of $\mu_{\mathcal{O}}$. Apply Claim F2 again, we have $t_2\ge t_1+\tau$ such that \[\int_{t_1}^{t_2} G(\phi_t(z))dt>0.\]
By repeating the above process, we have $0\le t_1<t_2<t_3<\cdots$ with the gap not less than $\tau$ such that
\[\int_{t_i}^{t_{i+1}} G(\phi_t(z))dt>0\text{ for } i=0,1,2,3,\cdots,\]
where $t_0=0$. Therefore,
\begin{align*}\int Gd\mu&=\lim_{m\to+\infty}\frac{1}{m}\int_{0}^{m}G(\phi_t(z))dt\\
&= \lim_{i\to+\infty}\frac{1}{t_i}\left(\int_{t_0}^{t_1}G(\phi_t(z))dt+\int_{t_1}^{t_2}G(\phi_t(z))dt+\cdots+\int_{t_{i-1}}^{t_i}G(\phi_t(z))dt\right)\\
&\ge 0.
\end{align*}
That is, $\mu_{\mathcal{O}}\in\mathcal{M}_{min}(u+h;\psi,\L,\Phi)$. This ends the proof of the Proposition. \end{proof}
\subsection{Proof of Claim F2} Assume that $z$ is not a generic point of $\mu_\mathcal{O}$, if $z\notin Area_2$, let $m=\tau$,
we have nothing to prove since $G(\phi_t(z))>0$ for all $|t|\le\tau$. Now we assume that $z\in Area_2$. In the case that $d(\phi_t (z),\mathcal{O})<\frac{D(\mathcal{O})}{4C^2e^\beta}$ for all $t\ge 0$, by Lemma \ref{M-3}, $z$ is a generic point of $\mu_\mathcal{O}$, which contradicts to our assumption.
Hence, there must be some $m_1>0$ such that $$d(\phi_{m_1} (z),\mathcal{O})\ge\frac{D(\mathcal{O})}{4C^2e^\beta}.$$
We can assume $m_2>0$ be the smallest time such that
\begin{align}\label{b-2}
d(\phi_{m_2} (z),\mathcal{O})\ge\frac{D(\mathcal{O})}{4C^2e^\beta}.
\end{align}
The existence of $m_2$ is ensured by \eqref{bbb-0}. Then for $0\le t\le 1$,
\[d(\phi_{m_2-t}(z),\mathcal{O})\ge\frac{D(\mathcal{O})}{4C^3e^{2\beta}}.\]
Then
\begin{align}\label{b-12}
\begin{split}\int_{m_2-1}^{m_2}G(\phi_{t}(z))dt&=\int_{m_2-1}^{m_2} {u}(\phi_{t}(z))+h(\phi_{t}(z))-a_{\mathcal{O}}\psi(\phi_{t}(z))dt\\
&\ge\int_{m_2-1}^{m_2} \theta\left(\frac{D(\mathcal{O})}{4C^3e^{2\beta}}\right)-\|h\|_0- a_{\mathcal{O}}\|\psi\|_0dt\\
&\ge\theta\left(\frac{D(\mathcal{O})}{4C^3e^{2\beta}}\right)- \frac{1+\psi_{min}}{\psi_{min}}\|h\|_0.\\
\end{split}
\end{align}
where we used \eqref{b-1} and the definition of $\theta(\cdot)$.
On the other hand, one has that
$\frac{D(\mathcal{O})}{4C^3e^{2\beta}}>\rho_2$, which implies that
\begin{align}\label{b-4}
\phi_{m_2-t}(z)\notin Area_2\text{ for all }0\le t\le 1.
\end{align}
Since $Area_2$ is compact, we can take $m_3$ the largest time with $0\le m_3\le m_2$ such that
\[\phi_{m_3}(z)\in Area_2,\]
where we use the assumption $z\in Area_2$.
By \eqref{b-4}, it is clear that $m_3<m_2-1$.
Then by Claim F1 and the fact that $Area_1\subset Area_2$, \begin{align}\label{b-5}
G(\phi_t(z))>0\text{ for all }m_3<t<m_2-1.\end{align}
Since $m_3<m_2$, one has by \eqref{b-2} that
$$d(\phi_t(z),\mathcal{O})<\frac{D(\mathcal{O})}{4C^2e^\beta}<\delta'\text{ for all } 0\le t\le m_3.$$ Therefore, by Lemma \ref{M-3}, there is $y_0\in\mathcal{O}$ such that
$$d(\phi_t(z), \phi_t(y_0))\le Cd(\phi_t(z),\mathcal{O})\le \frac{D(\mathcal{O})}{4Ce^\beta}\text{ for all }t\in[0,m_3].$$
Also notice that
$$d(z,y_0)\le C\rho_2\text{ and }d(\phi_{m_3}(z),\phi_{m_3}(y_0))\le C\rho_2,$$
where we used $z,\phi_{m_3}(z)\in Area_2$.
By using Lemma \ref{An-1}, we have for all $0\le t\le m_3$,
\begin{align*}
d(\phi_t(z),\phi_t\phi_v(y_0))&\le C^2e^{-\lambda\min(t,m_3-t)}(d(z,y_0)+d(\phi_{m_3}(z),\phi_{m_3}(y_0)))\\
&\le 2C^3e^{-\lambda\min(t,m_3-t)}\rho_2,
\end{align*}
where $v=v(y_0,z)$.
Hence,
\begin{align*}
\begin{split}\int_{0}^{ m_3}d(\phi_{t}(z),\phi_{t}\phi_v(y_0))dt&\le\int_{0}^{ m_3}2C^3\rho_2(e^{-\lambda t}+e^{-\lambda(m_3-t)}) dt\le\frac{4C^{3}\rho_2}{\lambda}.
\end{split}
\end{align*}
Since $u(\phi_{t}(y_0))=0$ for all $t\in\mathbb{R}$ and $u\ge 0$, one has
\begin{align}\label{b-6} \begin{split}&\int_{0}^{ m_3}G(\phi_{t}(z))-G(\phi_{t+v}(y_0))dt\\ =&\int_{0}^{ m_3}{u}(\phi_{t}(z))+h(\phi_{t}(z))-{u}(\phi_{t+v}(y_0))-h(\phi_{t+v}(y_0))-a_\mathcal{O}(\psi(\phi_t(z))-\psi(\phi_{t+v}(y_0)))dt\\ \ge&\int_{0}^{ m_3}h(\phi_{t}(z))-h(\phi_{t+v}(y_0))-a_\mathcal{O}(\psi(\phi_t(z))-\psi(\phi_{t+v}(y_0)))dt\\
\ge& -(\|h\|_1+|a_\mathcal{O}|\|\psi\|_1)\int_{0}^{ m_3}d(\phi_{t}(z),\phi_{t+v}(y_0))dt\\
\ge& - (\|h\|_1+\frac{\|h\|_0\|\psi\|_1}{\psi_{min}}) \cdot \frac{4C^{3}\rho_2}{\lambda}\\
\ge& - \|h\|_1(1+\frac{\|\psi\|_1}{\psi_{min}}) \cdot \frac{4C^{3}\rho_2}{\lambda}. \end{split} \end{align}
By assuming that $m_3=p|\mathcal{O}|+q$ for some nonnegative integer $p$ and real number $0\le q\le |\mathcal{O}|$, one has by \eqref{b-2} that
\begin{align}\label{b-7}
\begin{split}
\int_{0}^{ m_3}G(\phi_{t+v}(y_0))dt&=\int_{m_3-q}^{m_3}G(\phi_{t+v}(y_0))dt\ge -|\mathcal{O}|\cdot \frac{1+\psi_{min}}{\psi_{min}}\|h\|_0,
\end{split}
\end{align}
where we used $\int Gd\mu_\mathcal{O}=0$.
Combining \eqref{b-1}, \eqref{b-12}, \eqref{b-5}, \eqref{b-6} and \eqref{b-7}, we have
\begin{align*}&\ \ \ \int_{0}^{ m_2}G(\phi_t(z))dt\\ &\ge \int_{0}^{ m_3}G(\phi_t(z))dt+\int_{m_2-1}^{ m_2}G(\phi_t(z))dt\\ &= \int_{0}^{ m_3}G(\phi_t(z))-G(\phi_{t+v}(y_0))dt+\int_{0}^{ m_3}G(\phi_{t+v}(y_0))dt+\int_{m_2-1}^{ m_2}G(\phi_t(z))dt\\
&\ge - \|h\|_1(1+\frac{\|\psi\|_1}{\psi_{min}}) \cdot \frac{4C^{3}\rho_2}{\lambda}-|\mathcal{O}|\cdot \frac{1+\psi_{min}}{\psi_{min}}\|h\|_0
+\theta\left(\frac{D(\mathcal{O})}{4C^3e^{2\beta}}\right)- \frac{1+\psi_{min}}{\psi_{min}}\|h\|_0\\
&=\theta\left(\frac{D(\mathcal{O})}{4C^3e^{2\beta}}\right)- \left((1+\frac{\|\psi\|_1}{\psi_{min}}) \cdot \frac{4C^{3}\rho_2}{\lambda}+|\mathcal{O}|\cdot \frac{1+\psi_{min}}{\psi_{min}}+\frac{1+\psi_{min}}{\psi_{min}}\right)\|h\|_1\\ &>0, \end{align*}
where we used assumption \eqref{b-0}. Therefore, $m=m_2$ is the time as required since $m_2\ge 1> \tau$ by \eqref{b-5}. This completes the proof of Claim F2.
\end{document} | arXiv |
An algorithm for constructing integral row stochastic matrices
Asma Ilkhanizadeh manesh
Department of Mathematics Vali-e-Asr University of Rafsanjan P.O. Box: 7713936417, Rafsanjan, Iran
Let $\textbf{M}_{n}$ be the set of all $n$-by-$n$ real matrices, and let $\mathbb{R}^{n}$ be the set of all $n$-by-$1$ real (column) vectors. An $n$-by-$n$ matrix $R=[r_{ij}]$ with nonnegative entries is called row stochastic, if $\sum_{k=1}^{n} r_{ik}$ is equal to 1 for all $i$, $(1\leq i \leq n)$. In fact, $Re=e$, where $e=(1,\ldots,1)^t\in \mathbb{R}^n$. A matrix $R\in \textbf{M}_{n}$ is called integral row stochastic, if each row has exactly one nonzero entry, $+1$, and other entries are zero. In the present paper, we provide an algorithm for constructing integral row stochastic matrices, and also we show the relationship between this algorithm and majorization theory.
Eigenvalue
Majorization
Integral row stochastic
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[2] C. Bebeacua, T. Mansour, A. Postnikov, S. Severini, On the X-rays of permutations, Electron. Notes Discrete Math., 20, (2005), 193-203.
[3] R.A. Brualdi, G. Dahl, Constructing (0, 1)-matrices with given line sums and a zero block, in: G.T. Herman, A. Kuba (Eds.), Advances in Discrete Tomography and Its Applications, Birkhuser, Boston, (2007), 113-123.
[4] R.A. Brualdi, E. Fritscher, Hankel and Toeplitz X-rays of permutations, Linear Algebra Appl., 449, (2014), 305-380.
[5] G. Dahl, L-rays of permutation matrices and doubly stochastic matrices, Linear Algebra Appl., 480, (2015), 127-143.
[6] A. Ilkhanizadeh Manesh, Sglt-Majorization on Mn;m and its linear preservers, J. Mahani Mathematical Reserch Center, 7, (2018), 57-125.
[7] A. Ilkhanizadeh Manesh, Right gut-Majorization on Mn;m, Electron. J. Linear Algebra, 31, (2016), 13-26.
[8] A. Mohammadhasani and M. Radjabalipour, The structure of linear operators strongly preserving majorizations of matrices, Electron. J. Linear Algebra, 15, (2006), 266-272.
[9] G.T. Herman, A. Kuba (Eds.), Discrete Tomography Foundations, Algorithms, and Applications, Appl. Numer. Harmon. Anal., Birkhuser, Basel, (1999).
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Volume 11, Issue 1 - Serial Number 21
Winter and Spring 2022
Revise Date: 21 November 2021
Accept Date: 03 December 2021
PDF Download: 8
Ilkhanizadeh manesh, A. (2022). An algorithm for constructing integral row stochastic matrices. Journal of Mahani Mathematical Research Center, 11(1), 65-73. doi: 10.22103/jmmrc.2021.13883.1089
Asma Ilkhanizadeh manesh. "An algorithm for constructing integral row stochastic matrices". Journal of Mahani Mathematical Research Center, 11, 1, 2022, 65-73. doi: 10.22103/jmmrc.2021.13883.1089
Ilkhanizadeh manesh, A. (2022). 'An algorithm for constructing integral row stochastic matrices', Journal of Mahani Mathematical Research Center, 11(1), pp. 65-73. doi: 10.22103/jmmrc.2021.13883.1089
Ilkhanizadeh manesh, A. An algorithm for constructing integral row stochastic matrices. Journal of Mahani Mathematical Research Center, 2022; 11(1): 65-73. doi: 10.22103/jmmrc.2021.13883.1089 | CommonCrawl |
\begin{document}
\begin{abstract} The quantum disc is used to define a noncommutative analogue of a dense coordinate chart and of left-invariant vector fields on quantum SU(2). This yields two twisted Dirac operators for different twists that are related by a gauge transformaton and have bounded twisted commutators with a suitable algebra of differentiable functions on quantum SU(2). \end{abstract}
\title{Twisted Dirac operator on quantum SU(2) in disc coordinates}
\section{Introduction} Connes' noncommutative differential geometry \cite{C-buch,C-ihes} provides in particular a geometric approach to the construction of K-homology classes of a
$C^*$-algebra ${\mathcal{A}}$: for the commutative $C^*$-algebra of continuous functions on a compact smooth manifold, the phase $F:=\frac{D}{|D|}$ of an elliptic first order differential operator $D$ on a vector bundle defines such a class, and for a noncommutative algebra, the fundamental task is to represent ${\mathcal{A}}$ on a Hilbert space $\hH$ and to find a self-adjoint operator $D$ that has compact resolvent (so it is ``very'' unbounded) but at the same time has bounded commutators with the elements of ${\mathcal{A}}$.
In classical geometry, equivariant differential operators on Lie groups provide examples that can be described purely in terms of representation theory, so since the discovery of quantum groups, many attempts were made to apply Connes' programme to these $C^*$-algebras. The fact that their representation theory resembles that of their classical counterparts so closely allows one indeed to define straightforwardly an analogue say of the Dirac operator on a compact simple Lie group, but it turns out to have unbounded commutators with the elements of ${\mathcal{A}}$.
Many solutions to this conundrum were found and studied, focusing on various approaches and motivations ranging from index theory \cite{DLSSV1, DLSSV2, C-index, CP, KRS, NT, NT1} over the theory of covariant differential calculi \cite{H,S} to the Baum-Connes conjecture \cite{V-BCc}. However, it seems fair to say that there is still no sufficient general understanding of how Connes' machinery applies to algebras obtained by deformation quantisation in general and quantum groups in particular.
The aim of the present note is to use the fundamental example of $\mathrm{SU}(2)$ for discussing yet another mechanism for obtaining bounded commutators. In a nutshell, the idea is to have a representation of ${\mathcal{A}} \otimes {\mathcal{A}}^\mathrm{op}$ on $\hH$ and to use differential operators $D$ with ``coefficients'' in ${\mathcal{A}}^\mathrm{op}$ to achieve bounded commutators with ${\mathcal{A}}$. Our starting point is a noncommutative analogue of a dense coordinate chart on $\mathrm{SU}(2)$ that is compatible with the symplectic foliation of the quantised Poisson manifold $\mathrm{SU}(2)$. The noncommutative analogue is obtained by replacing a complex unit disc by the quantum disc. We use a quantised differential calculus on this chart to define quantisations of left invariant vector fields that act on the function algebra by twisted derivations. This is where it becomes necessary to consider coefficients from ${\mathcal{A}}^\mathrm{op}$.
We then build two twisted Dirac operators using these twisted derivations and show that they are related by a gauge transformation that arises from a rescaling of the volume form. A fruitful direction of further research might be to investigate the spectral and homological properties of these and similar operators.
\section{The Dirac operator on $\mathrm{SU}(2)$} The $C^*$-algebra ${\mathcal{A}}$ we are going to consider is a strict deformation quantisation of the algebra of continuous complex-valued functions on the Lie group $$ \mathrm{SU}(2)=\left\{ \left( \begin{array}{rr} \alpha\ & \beta \\ -\bar\beta\ & \bar\alpha \\ \end{array} \right) \mid \alpha,\beta\in\mathbb{C},\ \alpha\bar \alpha + \beta \bar\beta = 1 \right\} $$ that we identify as usual with $\mathbb{S}^3 \subset \mathbb{C}^2$, identifying the above matrix with $(\alpha,\beta)$.
We denote by $$ X_0 := \left( \begin{array}{rr} \mathrm{i}\ & 0 \\ 0\ & -\mathrm{i} \\ \end{array} \right),\quad X_1 := \left( \begin{array}{rr} 0\ & 1 \\ -1\ & 0 \\ \end{array} \right), \quad X_2:= \left( \begin{array}{rr} 0\ & \mathrm{i} \\ \mathrm{i}\ & 0 \\ \end{array} \right) $$ the standard generators of the Lie algebra $\mathrm{su}(2)$ of $\mathrm{SU}(2)$, and, by a slight abuse of notation, also the corresponding left invariant vector fields on $\mathrm{SU}(2)$.
In this section, we describe the Dirac operator $D$ of $\mathrm{SU}(2)$ in local coordinates that are adapted to the quantisation process. To define the local coordinates, consider the map \[\label{lc}
\bar \mathbb{D} \times \mathbb{S}^1 \rightarrow
\mathbb{S}^3,\quad
(z,v) \mapsto
(z,\sqrt{1-z \bar z}{\hspace{1pt}} v), \]
where $\mathbb{D}:=\{ z\in \mathbb{C} : |z|<1\}$ is the open unit disc, $\bar \mathbb{D}$ is its closure, and $\mathbb{S}^1=\partial\mathbb{D}$ is its boundary. Restricting the map to $(\mathbb{D} \times \mathbb{S}^1)\setminus(\mathbb{D} \times\{-1\}) \cong \mathbb{D} \times (-\pi, \pi)$ defines a dense coordinate chart \[\label{x}
\mathbf{x} \colon \mathbb{D} \times (-\pi, \pi)
\, \longrightarrow \, \mathbb{S}^3, \quad \mathbf{x}(z,t) :=
(z, \sqrt{1-z\bar z}\, \mathrm{e}^{\mathrm{i} t}) \] that is compatible with the standard differential structure on $\mathrm{SU}(2) \cong \mathbb{S}^3$. The pull-back of the bi-invariant volume form on $\mathbb{S}^3$ assigns a measure to $\mathbb{D} \times (-\pi,\pi)$ and the resulting Hilbert space of $L_2$-functions will be denoted by $\hH$.
We write $f(z, t):= f\circ \mathbf{x}(z,t)$ for functions $f$ on $\mathbb{S}^3$ and thus identify these with continuous functions on $\bar\mathbb{D} \times [-\pi,\pi]$ satisfying the boundary conditions \[\label{CSU}
f(u, t) = f(u, 0),\quad
f(z,-\pi)=
f(z,\pi)\quad \forall \ u \in \mathbb{S}^1,\ z \in \mathbb{D}, \
t\in [-\pi, \pi]. \] Let $\Gamma^{(1)}(\mathrm{SU}(2))$ denote the set of $C^{(1)}$-functions (continuously differentiable ones) on $\bar \mathbb{D} \times [-\pi, \pi] $ satisfying \eqref{CSU}. The corresponding functions on $\mathbb{S}^3$ are not necessarily $C^{(1)}$, but absolutely continuous, and can therefore be considered as belonging to the domain of the first order differential operators $X_0$, $X_1$ and $X_2$. Here, derivations are understood to be taken in the weak sense. Therefore we may consider $$ H:= -\mathrm{i} X_0, \quad E:= \half (X_1 - \mathrm{i} X_2), \quad F:= - \half (X_1 +\mathrm{i} X_2) $$
as first order differential operators on $\hH$ with domain $\Gamma^{(1)}(\mathrm{SU}(2))$.
A direct calculation shows that these operators take in the parametrisation \eqref{x} the following form:
\begin{align} H &= z \dz - \bar z \dbz + \mathrm{i} \dt, \label{H} \\ E &= - \sqrt{1-\bar z z} \,\mathrm{e}^{-\mathrm{i} t} \dbz -
\mbox{$\frac{\mathrm{i}}{2}$}{\hspace{1pt}} \mbox{$\frac{z}{\sqrt{1-\bar z z }}$}{\hspace{1pt}} \mathrm{e}^{-\mathrm{i} t} {\hspace{1pt}} \dt, \label{E} \\ F &= \sqrt{1-\bar z z} \,\mathrm{e}^{\mathrm{i} t} \dz -
\mbox{$\frac{\mathrm{i}}{2}$}{\hspace{1pt}} \mbox{$\frac{\bar z}{\sqrt{1-\bar z z }}$}{\hspace{1pt}} \mathrm{e}^{\mathrm{i} t} {\hspace{1pt}} \dt . \label{F} \end{align}
Since $\mathrm{SU}(2)$ is a Lie group, its tangent bundle is trivial and hence admits a trivial spin structure. We consider $\Gamma^{(1)}(S) := \Gamma^{(1)}(\mathrm{SU}(2))\oplus \Gamma^{(1)}(\mathrm{SU}(2))$ as a vector space of differentiable sections (in the weak sense) of the associated spinor bundle. The Dirac operator with respect to the bi-invariant metric on $\mathrm{SU}(2)$ is then given by the closure of $$ D:=\left( \begin{array}{cc} H-2\ & E \\ F\ & -H-2 \\ \end{array} \right)\ : \ \Gamma^{(1)}(S) \subset \hH\oplus \hH \ \longrightarrow\ \hH\oplus \hH, $$ see e.g.~\cite{F}.
\section{A representation of quantum SU(2) by multiplication operators}
The quantised coordinate ring of $\mathrm{SU}(2)$ at $q \in (0,1)$ is the universal unital *-al\-ge\-bra $\Aq$ containing elements $a,c$ such that \begin{eqnarray*} &&
ac=qca,\quad
ac^*=qc^*a,\quad
cc^*=c^*c,\\ &&
aa^*+q^2cc^*=1,\quad a^*a+ cc^*=1. \end{eqnarray*}
It admits a faithful Hilbert space representation $\rho$ on $ \ell_2(\N)\otimes \ell_2(\Z)$ given on orthonormal bases $\{e_n\}_{n\in{\mathbb{N}}}\subset \ell_2(\N)$ and $\{b_k\}_{k\in\mathbb{Z}}\subset \ell_2(\Z)$ by \begin{align} & \rho(a) (e_n\otimes b_k) = \sqrt{1 - q^{2n}} {\hspace{1pt}} e_{n-1} \otimes b_k, \label{a} \\ & \rho(c) (e_n\otimes b_k) = q^n {\hspace{1pt}} e_{n} \otimes b_{k-1}. \label{c} \end{align} The norm closure of the *-algebra generated by $ \rho(a), \rho(c)\in B( \ell_2(\N)\otimes \ell_2(\Z))$ is isomorphic to $C(\mathrm{SU}_q(2))$, the universal C*-algebra of $\Aq$, see e.g.\ \cite{MNW}.
The starting point of this paper is a quantum counterpart to the chart \eqref{lc}. To define it, let $z\in B(\ell_2(\N))$ and $u\in B(\ell_2(\Z))$ be given by \[ \label{zu}
z{\hspace{1pt}} e_n := \sqrt{1 - q^{2n}} {\hspace{1pt}} e_{n-1}, \ \ n\in\mathbb{N}, \qquad u{\hspace{1pt}} b_k:= b_{k-1},\ \ k\in\mathbb{Z}, \] and set \[\label{y} y := \sqrt{1- z^*z}\in B(\ell_2(\N)). \] Then $y$ is a positive self-adjoint trace class operator on $\ell_2(\N)$ acting by $$ y{\hspace{1pt}} e_n = q^n {\hspace{1pt}} e_n $$ and satisfying the relations \[ \label{zy}
z y = q y z, \quad y z^* = q z^* y. \] Note that one can now rewrite Equations \eqref{a} and \eqref{c} as \[ \label{zyu} \rho(a) = (z\otimes 1){\hspace{1pt}} (e_n \otimes b_n), \qquad \rho(c) = (y\otimes u) {\hspace{1pt}} (e_n \otimes b_n). \]
The bilateral shift $u$ generates a commutative C*-subalgebra of $B(\ell_2(\Z))$ which is isomorphic to $C(\mathbb{S}^1)$. The operator $z\in B(\ell_2(\N))$ satisfies the defining relation of the quantum disc algebra $\cO(\oqd)$, \[
zz^* - q^2 z^*z = 1-q^2. \label{qdr} \] It is known \cite{KL} that the universal C*-algebra of the quantum disc $\cO(\oqd)$, generated by a single generator and its adjoint satisfying \eqref{qdr}, is isomorphic to the Toeplitz algebra ${\mathcal{T}}$ which can also be viewed as the C*-subalgebra of $B(\ell_2(\N))$ generated by the unilateral shift \[ \label{so}
s{\hspace{1pt}} e_n = e_{n+1} , \quad n\in\mathbb{N}. \] Moreover, the bounded operator $z$ defined in \eqref{zu} also generates the Toeplitz algebra ${\mathcal{T}}\subset B(\ell_2(\N))$, and the so-called symbol map of the Toeplitz extension \cite{V} \begin{equation} \label{ext} \xymatrix{
0\;\ar[r]&\; {\mathcal{K}}(\ell_2(\N)) \;\ar@{^{(}->}[r] &\; {\mathcal{T}} \;\ar[r]^{\!\!\tau} &\; C(\mathbb{S}^1) \;\ar[r] &\,0} \end{equation} can be given by $\tau(z)=\tau(s)=u$, where $u$ denotes the unitary generator of $C(\mathbb{S}^1)$ and ${\mathcal{K}}(\ell_2(\N))$ stands for the C*-algebra of compact operators on $\ell_2(\N)$.
Returning to the map \eqref{lc}, observe that $\mathrm{SU}(2)\cong \mathbb{S}^3$ is homeomorphic to the topological quotient of $\bar \mathbb{D} \times \mathbb{S}^1$ given by shrinking the circle $\mathbb{S}^1$ to a point on the boundary of $\bar \mathbb{D}$. This can be visualised by the following push-out diagram:
\begin{align*}\mbox{ } \\[-18pt]
\xymatrix@=3mm@R=1mm{ &&\ \mathbb{S}^3&&\\
\chi_1(z,v):=(z, \sqrt{1-z \bar z}{\hspace{1pt}} v) \;&&&&\; \chi_2(u):=(u,0)\\ &\bar\mathbb{D}\times \mathbb{S}^1 \ar[ruu]^{\chi_1} &&\mathbb{S}^1 \ar[luu]_{\chi_2}& \\ (\iota,{\mathrm{id}})(u,v):=(u,v)\;&&&&\;\mathrm{pr}_1(u,v):=u.\\ &&\mathbb{S}^1 \times \mathbb{S}^1 \ar[ruu]_{\mathrm{pr}_1}\ar[luu]^{(\iota,{\mathrm{id}})}&&
} \end{align*}
Applying the functor that assigns to a topological space the algebra of continuous functions, we obtain a pull-back diagram of C*-algebras. Quantum SU(2) is now obtained by replacing in this pull-back diagram the C*-algebra $C(\bar \mathbb{D})$ by the Toeplitz algebra $C(\bar\mathbb{D}_q):={\mathcal{T}}$, regarded as the algebra of continuous functions on the quantum disc. The restriction map $\iota^*: C(\bar \mathbb{D})\longrightarrow C(\mathbb{S}^1)$, \,$\iota^*(f)=f\!\!\upharpoonright_{\mathbb{S}^1}$, is replaced by the symbol map $\tau: {\mathcal{T}}\longrightarrow C(\mathbb{S}^1)$ from \eqref{ext}. The resulting pull-back diagram has the following structure: \begin{equation*} \xymatrix{ \nonumber & \makebox[48pt][c]{ $P\,:=\,\big(C(\bar{\mathbb{D}}_q)\otimes C(\mathbb{S}^1)\big) \hspace{0pt}{\underset{(\pi_1,\pi_2)}{\times}}\hspace{0pt} C(\mathbb{S}^1) \hspace{74pt}\mbox{ }$} \ar[dl]_{\mathrm{pr}_1} \ar[dr]^{\mathrm{pr}_2}& \\ C(\bar{\mathbb{D}}_q) \otimes C(\mathbb{S}^1) \ar[rd]_{\pi_1:=\tau {\hspace{1pt}} \otimes{\hspace{1pt}} {\mathrm{id}}} & & C(\mathbb{S}^1) \ar[ld]^{\pi_2:={\mathrm{id}}{\hspace{1pt}} \otimes{\hspace{1pt}} 1}\\
& C(\mathbb{S}^1)\otimes C(\mathbb{S}^1)& } \end{equation*} Note that $ (t\otimes f,g)\in P$ if and only if $\tau(t)\otimes f {\hspace{-1pt}}={\hspace{-1pt}} g \otimes 1 \in C(\mathbb{S}^1)\otimes \mathbb{C}{\hspace{1pt}} 1\subset C(\mathbb{S}^1)\otimes C(\mathbb{S}^1)$. Since $\mathbb{C}{\hspace{1pt}} 1 \cong \mathbb{C}(\{\text{pt}\})$, the interpretation of $\tau(t)\otimes f = g \otimes 1$ is that, whenever we evaluate $t\in C(\bar{\mathbb{D}}_q)$ on the boundary, the circle $\mathbb{S}^1$ in $\bar{\mathbb{D}}_q\times \mathbb{S}^1$ collapses to a point.
Moreover, it can be shown \cite[Section 3.2]{HW} that $\mathrm{pr}_1$ yields an isomorphism of \mbox{C*-algebras} $P\cong \mathrm{pr}_1(P) \cong C(\mathrm{SU}_q(2)) \subset B( \ell_2(\N)\otimes \ell_2(\Z))$ such that $\rho(a)=$ \mbox{$\mathrm{pr}_1((z\otimes 1, u))$} and $\rho(c)= \mathrm{pr}_1( (y\otimes u, 0))$, see \eqref{zyu}.
Viewing $C(\mathrm{SU}_q(2))$ as a subalgebra of $C(\bar{\mathbb{D}}_q) \otimes C(\mathbb{S}^1)$ allows us to construct the following faithful Hilbert space representation of the C*-algebra $C(\mathrm{SU}_q(2))$ in which it acts by multiplication operators on a noncommutative function algebra. This leads to an interpretation as an algebra of integrable functions on the quantum space $\mathbb{D}_q \times \mathbb{S}^1 $. First note that, since $z^*z= 1-y^2$ and $zz^*= 1- q^{2}y^2$, any element $p\in \cO(\oqd)$ can be written as $$ p=\sum_{n=0}^N z^{*n} p_n(y) + \sum_{n=1}^M p_{-n} (y){\hspace{1pt}} z^{n} , \quad N,M\in\mathbb{N}, $$ with polynomials $p_n$ and $p_{-n}$. Using the functional calculus of the self-adjoint operator $y$ with spectrum $\mathrm{spec}(y)= \{q^n: n\in\mathbb{N}\} \cup \{0\}$, we define $$ \F(\oqd):= \left\{\sum_{n=0}^N z^{*n} f_n(y) + \sum_{n=1}^M f_{-n} (y){\hspace{1pt}} z^{n} : \ N,M\in\mathbb{N}, \ \
f_k \in L_\infty(\mathrm{spec}(y))\right\}. $$ Using the commutation relations $$ z f(y) = f(qy) z, \quad f(y) z^* = z^* f(qy), \quad f \in L_\infty(\mathrm{spec}(y)), $$ one easily verifies that $\F(\oqd)$ is a *-algebra. Let $s$ denote the unilateral shift operator on $\ell_2(\N)$ from Equation \eqref{so}. For all functions $f \in L_\infty(\mathrm{spec}(y))$, it satisfies the commutation relations $$ s^* f(y) = f(qy) s^*, \quad f(y) s = s f(qy). $$
Writing $z$ in its polar decomposition $z=s^*{\hspace{1pt}} |z| = s^* \sqrt {1-y^2}$, one sees that $$ \F(\oqd)= \left\{\sum_{n=0}^N s^n f_n(y) + \sum_{n=1}^M f_{-n} (y) s^{*n} : \ N,M\in\mathbb{N}, \ \ f_k \in L_\infty(\mathrm{spec}(y))\right\}. $$ Since $y^\alpha e_n = q^{\alpha n} e_n$, the operator $y^\alpha$ is trace class for all $\alpha >0$. Therefore the positive functional \[ \label{Tr} \int_{\mathbb{D}_q}(\,\cdot\,)\,\mathrm{d} \mu_\alpha \ :\ \F(\oqd) \,\longrightarrow\, \mathbb{C}, \qquad \int_{\mathbb{D}_q}\! f \,\mathrm{d} \mu_\alpha := (1-q) \mathrm{Tr}_{\ell_2(\N)}(fy^\alpha ) . \] is well defined. Explicitly, it is given by $$
\int_{\mathbb{D}_q} \left( \sum_{n=0}^N s^n f_n(y) + \sum_{n=1}^M f_{-n} (y) s^{*n} \right) {\hspace{-1pt}} \mathrm{d} \mu_\alpha
= (1-q) \sum_{n\in\mathbb{N}} f_0(q) q^{\alpha n} . $$ Using $ \mathrm{Tr}_{\ell_2(\N)}(s^n s^{*k} f(y)y) =0$ if $k\neq n$, one easily verifies that it is faithful. In terms of the Jackson integral $\int_0^1 f(y)\mathrm{d}_q y= (1-q) \sum_{n\in\mathbb{N}} f(q^n)q^n$, we can write \begin{align} \nonumber
&\int_{\mathbb{D}_q} \left( \sum_{n=0}^N s^n f_n(y) + \sum_{n=1}^M f_{-n} (y) s^{*n}\! \right) {\hspace{-1pt}} \mathrm{d} \mu_\alpha \\
&\qquad \qquad \quad = \int_0^1\! \int_{-\pi}^{\pi}\sum_{n=0}^N \mathrm{e}^{\mathrm{i} n\phi} f_n(y) + \sum_{n=1}^M f_{-n} (y) {\hspace{1pt}} \mathrm{e}^{-\mathrm{i} n\phi}
{\hspace{1pt}} \mathrm{d} \phi\, y^{\alpha -1} \mathrm{d}_q y. \label{measure} \end{align}
Note that the commutation relation between $y^\alpha $ and functions from $\F(\oqd)$ can be expressed by the automorphism $\sigma^\alpha : \F(\oqd) \longrightarrow \F(\oqd)$ given by \[ \label{s} \sigma^\alpha (s) = q^{- \alpha } {\hspace{1pt}} s,\quad \sigma^\alpha(s^*) = q^\alpha {\hspace{1pt}} s^*, \quad
\sigma^\alpha (f(y)) = f(y), \ \ f \in L_\infty(\mathrm{spec}(y)), \] where $\alpha\in\mathbb{R}$. Then, for all $h,g\in \F(\oqd)$, $$ g {\hspace{1pt}} y^\alpha = y^\alpha {\hspace{1pt}} \sigma^\alpha (g), $$ and therefore \begin{align} \nonumber \int_{\mathbb{D}_q} \! gh \,\mathrm{d}\mu_\alpha &= (1{\hspace{-1pt}} -{\hspace{-1pt}} q) \mathrm{Tr}_{\ell_2(\N)}(ghy^\alpha) = (1{\hspace{-1pt}} -{\hspace{-1pt}} q) \mathrm{Tr}_{\ell_2(\N)}(\sigma^\alpha (h) g y^\alpha ) \\ &= \int_{\mathbb{D}_q} \!\sigma^\alpha(h)g \,\mathrm{d}\mu_\alpha. \label{mod} \end{align} Note that we also have \[ \label{astar}
(\sigma^\alpha( f))^* = \sigma^{-\alpha}(f^*), \quad f \in \F(\oqd). \]
We use the faithful positive functional $\int_{\mathbb{D}_q} (\,\cdot\,) \, \mathrm{d} \mu_\alpha$ to define an inner product on $\F(\oqd)$ by $$ \ip{f}{g} := \int_{\mathbb{D}_q} f^* g \,\mathrm{d}\mu_\alpha . $$ The Hilbert space closure of $\F(\oqd)$ will be denoted by $L_2(\mathbb{D}_q,\mu_\alpha)$. Left multiplication with functions $x\in \F(\oqd)$ defines a faithful *-representation of $\F(\oqd)$ on
$L_2(\mathbb{D}_q,\mu_\alpha)$ since
$$
\ip{xf}{g} = \int_{\mathbb{D}_q} f^* x^*g \,\mathrm{d}\mu_\alpha = \ip{f}{x^*g}.
$$ Observe that $\F(\oqd)$ leaves the subspace \[ \label{F0} {\mathcal{F}}_0(\mathbb{D}_q):= \left\{\sum_{n=0}^N s^n f_n(y) + \sum_{n=1}^M f_{-n} (y) s^{*n} \in \F(\oqd) : \
\mathrm{supp}(f_k) \text{ is finite} \right\} \] of $L_2(\mathbb{D}_q,\mu_\alpha)$ invariant. Since ${\mathcal{F}}_0(\mathbb{D}_q)$ contains an orthonormal basis (see \cite[Proposition 1]{W}), it is dense in $L_2(\mathbb{D}_q,\mu_\alpha)$. We extend $\F(\oqd)$ by the unbounded element $y^{-1}$ and define $\mathcal{O}^+(\mathbb{D}_q)$ as the *-algebra generated by the operators $y^{-1}$ and all $f\in \F(\oqd)$, considered as operators on ${\mathcal{F}}_0(\mathbb{D}_q)$. Furthermore, let $\mathcal{O}^+(\mathbb{D}_q)^{\mathrm{op}}$ denote the *-algebra obtained from $\mathcal{O}^+(\mathbb{D}_q)$ by replacing the multiplication with the opposite one, i.e.\ $a\cdot b:= ba$.
Then we obtain
a representation of $\mathcal{O}^+(\mathbb{D}_q)^{\mathrm{op}}$ on ${\mathcal{F}}_0(\mathbb{D}_q) \subset L_2(\mathbb{D}_q,\mu_\alpha)$ by right multiplication,
\[ \nonumber
a^{\mathrm{op}}f:= fa, \quad a\in \mathcal{O}^+(\mathbb{D}_q)^{\mathrm{op}}, \ \ f \in {\mathcal{F}}_0(\mathbb{D}_q).
\]
Clearly, this representation commutes with the operators of $\mathcal{O}^+(\mathbb{D}_q)$, as these act by left multiplication.
However, it is not a *-representation.
More precisely, \eqref{mod} and \eqref{astar} give
$$
\ip{x^{\mathrm{op}}f}{g} = \ip{fx}{g} = \int_{\mathbb{D}_q} x^* f^* g\,\mathrm{d}\mu_\alpha = \int_{\mathbb{D}_q} f^* g \sigma^{-\alpha} (x^*)\,\mathrm{d}\mu_\alpha =
\ip{f}{(\sigma^\alpha(x)^*)^{\mathrm{op}}g},
$$ therefore \[ \label{opstar} (x^{\mathrm{op}})^* = (\sigma^\alpha(x)^*)^{\mathrm{op}} . \]
Note that $y>0$ and $\sigma(y) =y$ imply that the multiplication operators $y^{\beta}$ and $(y^{\beta})^{\mathrm{op}} = (y^{\mathrm{op}})^{\beta} $, $\beta \in \mathbb{R}$, determine well defined (unbounded) self-adjoint operators on $L_2(\mathbb{D}_q,\mu_\alpha)$.
Next we use the isomorphism $\ell_2(\Z)\cong L_2(\mathbb{S}^1)$ given by $b_n := \frac{1}{\sqrt{2\pi}} \mathrm{e}^{\mathrm{i} n t}$ and identify $u$ from \eqref{zu} with the multiplication operator $uf(t):= \mathrm{e}^{\mathrm{i} t} f(t)$. In this way we obtain a faithful *-representation $\tilde\rho : \Aq \longrightarrow B(L_2(\mathbb{D}_q,\mu_\alpha) \otimes L_2(\mathbb{S}^1))$ by multiplication operators. On generators, it is given by $$ \tilde\rho(a)(f\otimes g) := zf\otimes g, \quad \tilde\rho(c) := yf\otimes ug. $$ The closure of the image of $\tilde\rho$ is again isomorphic to $C(\mathrm{SU}_q(2))$.
\section{Quantised differential calculi}
Taking as its domain the absolutely continuous functions $AC(\mathbb{S}^1)$ with the weak derivative in $L_2(\mathbb{S}^1)$, the partial derivative $ \mathrm{i} \dt$ becomes a self-adjoint operator on $L_2(\mathbb{S}^1)$ satisfying the Leibniz rule $$
\mathrm{i} \dt (\varphi{\hspace{1pt}} g) = (\mathrm{i} \mbox{$\frac{\partial \varphi }{\partial t}$})g + \varphi (\mathrm{i} \dt g) ,
\quad \varphi\in C^{(1)}(\mathbb{S}^1), \ \ g\in \mathrm{dom}( \mathrm{i} \dt) . $$
We consider a first order differential *-calculus $\mathrm{d} : \cO(\oqd) \longrightarrow \Omega(\mathbb{D}_q)$, where $\Omega(\mathbb{D}_q)= \mathrm{d} z{\hspace{1pt}} \cO(\oqd) + \mathrm{d} z^* \cO(\oqd)$ with $\cO(\oqd)$-bimodule structure given by $$ \mathrm{d} z{\hspace{1pt}} z^* = q^2 z^*{\hspace{1pt}} \mathrm{d} z ,\ \ \mathrm{d} z^* {\hspace{1pt}} z = q^{-2} z{\hspace{1pt}} \mathrm{d} z^* ,\ \ \mathrm{d} z {\hspace{1pt}} z =q^{-2} z \mathrm{d} z , \ \ \mathrm{d} z^* {\hspace{1pt}} z^* =q^{2} z^* \mathrm{d} z^*, $$ see \cite{KS} for definitions and background on differential calculi. With $\sigma^\alpha$ from \eqref{s}, it follows that $$ \mathrm{d} z {\hspace{1pt}} f = \sigma^{-2}(f) {\hspace{1pt}} \mathrm{d} z,\quad \mathrm{d} z^* {\hspace{1pt}} f = \sigma^{-2}(f) {\hspace{1pt}} \mathrm{d} z^*,\quad f\in \cO(\oqd). $$ We define partial derivatives $\dz$ and $\dbz$ by $$ \mathrm{d}(f) = \mathrm{d} z {\hspace{1pt}} \dz(f) +\mathrm{d} z^*{\hspace{1pt}} \dbz(f) , \quad f\in \cO(\oqd). $$ Recall that $y^2= 1- z^* z$ and $zz^* -z^* z= (1-q^2) y^2$ by \eqref{qdr} and \eqref{y}. Using $$ 1 = \dz(z) = \mbox{$\frac{-1}{1-q^2}$} {\hspace{1pt}} y^{-2} [z^*,z], \quad
1 = \dbz(z^*) = \mbox{$\frac{1}{1-q^2}$} {\hspace{1pt}} y^{-2} [z,z^*] , $$ the Leibniz rule for the commutator and $y^{-2} p= \sigma^{2}(p) y^{-2}$ for all $p\in \cO(\oqd)$, one verifies by direct calculations on monomials $z^n z^{*m}$ that \[ \nonumber \dz(p) = \mbox{$\frac{-1}{1-q^2}$} {\hspace{1pt}} y^{-2} [z^*,p], \quad
\dbz(p) = \mbox{$\frac{1}{1-q^2}$} {\hspace{1pt}} y^{-2} [z,p] , \quad p\in \cO(\oqd). \] We extend the partial derivatives $\dz$ and $\dbz$ to $$ \F^{(1)}(\oqd):= \{ f\in\F(\oqd): y^{-2}[z^*, f] \in \F(\oqd), \ \ y^{-2} [z, f] \in\F(\oqd) \} $$ by setting \[ \nonumber \dz (f) := \mbox{$\frac{-1}{1-q^2}$} {\hspace{1pt}} y^{-2} [z^*,f], \quad
\dbz(f) := \mbox{$\frac{1}{1-q^2}$} {\hspace{1pt}} y^{-2} {\hspace{1pt}} [z,f], \quad f\in \F^{(1)}(\oqd). \] Note that $\cO(\oqd)\subset \F^{(1)}(\oqd)$. By the spectral theorem for functions in $y=y^*$, one readily proves that $\cO(\oqd)$ is dense in $L_2(\mathbb{D}_q,\mu_\alpha)$. Thus $\dz$ and $\dbz$ are densely defined linear operators on $L_2(\mathbb{D}_q,\mu_\alpha)$.
Moreover, it is easily seen that the automorphism $\sigma^\alpha$ from \eqref{s} preserves $\F^{(1)}(\oqd)$. For instance, $y^{-2}[z^*,\sigma^\alpha(s^n f(y))] = q^{-\alpha n} y^{-2}[z^*,s^n f(y)]\in \F(\oqd)$ for all $s^n f(y) \in \F^{(1)}(\oqd)$. Similarly one shows that $\F^{(1)}(\oqd)$ is a *-algebra. For example, \begin{align*} y^{-2} [ z^*, fg] & = y^{-2} [ z^*, f]g + y^{-2} f y^{-2}y^{-2}[ z^*,g] \\ &= y^{-2} [ z^*, f]g + \sigma^2(f ){\hspace{1pt}} y^{-2}[ z^*,g]\in \F(\oqd) \end{align*} and $$ y^{-2} [ z^*, f^*] =- (y^{-2} y^{2} [ z, f] y^{-2})^* = -q^2 (y^{-2} [z, \sigma^2(f)])^* \in \F(\oqd) $$ for $f,g\in\F^{(1)}(\oqd)$.
\section{Twisted derivations}
\subsection{Twist: $\pmb{\sigma^1}$} \mbox{ }\\[4pt] \noindent Our aim is to replace the first order differential operators $H$, $E$ and $F$ from \eqref{H}-\eqref{F} by appropriate noncommutative versions. First we consider $q$-analogues of the operators $\sqrt{1-\bar z z} \dz $ and $\sqrt{1-\bar z z} \dbz $ and define $T_i: \F^{(1)}(\oqd) \longrightarrow \F(\oqd)$, \,$i=1,2$, by \[ \label{defT1T2}
T_1f:= y \dz f = \mbox{$\frac{-1}{1-q^2}$} {\hspace{1pt}} y^{-1} [z^*,f], \quad
T_2f:= y \dbz f = \mbox{$\frac{1}{1-q^2}$} {\hspace{1pt}} y^{-1} [z,f]. \]
Observe that $T_1$ and $T_2$ satisfy a twisted Leibniz rule: For all $f,g\in \F^{(1)}(\oqd)$, \begin{align}\nonumber T_1(fg) &= \mbox{$\frac{-1}{1-q^2}$} {\hspace{1pt}} y^{-1}[z^*,fg] = \mbox{$\frac{-1}{1-q^2}$} {\hspace{1pt}} y^{-1}([z^*, f] g + f y y^{-1} [z^*, g]) \\ &= (T_1 f)g + \sigma^{1}{\hspace{-1pt}} (f){\hspace{1pt}} T_1g \label{T1} \end{align} and similarly \[ \label{T2} T_2(fg) = (T_2 f)g + \sigma^{1}{\hspace{-1pt}} (f){\hspace{1pt}} T_2g. \]
Setting $\hat T_1 := T_1 \otimes 1$, $\hat T_2 := T_2 \otimes 1$ and $\hat \sigma^1 := \sigma^1 \otimes 1$, we get
\[ \label{T1T2} \hat T_1(\phi \psi ) = (\hat T_1 \phi)\psi + \hat\sigma^{1}{\hspace{-1pt}} (\phi){\hspace{1pt}} \hat T_1\psi, \quad \hat T_2(\phi \psi ) = (\hat T_2 \phi)\psi + \hat\sigma^{1}{\hspace{-1pt}} (\phi){\hspace{1pt}} \hat T_2\psi \] for all $\phi, \psi \in \F^{(1)}(\oqd) \otimes C^{(1)}(\mathbb{S}^1)$ by \eqref{T1} and \eqref{T2}.
Next consider the operator $\hat T_0:= y^{-1} \dt$ on the domain $\mathrm{dom}(y^{-1}) \otimes C^{(1)}(\mathbb{S}^1)$ in $L_2(\mathbb{D}_q,\mu_\alpha) \otimes L_2(\mathbb{S}^1)$. Note that, for all $f,g\in \F(\oqd)$ with $g\in\mathrm{dom}(y^{-1})$, one has $y^{-1}fg= \sigma^1(f)y^{-1} g\in L_2(\mathbb{D}_q,\mu_\alpha)$, hence $fg\in\mathrm{dom}(y^{-1})$. Now, for all $\varphi ,\, \xi \in C^{(1)}(\mathbb{S}^1)$,
\begin{align} \nonumber \hat T_0(fg\otimes \varphi\xi) &= y^{-1} fg\otimes \dt ( \varphi \xi ) \nonumber \\ &= y^{-1} fg\otimes (\dt \varphi)\xi + y^{-1} fy{\hspace{1pt}} y^{-1} g\otimes \varphi\,(\dt \xi) \nonumber \\ &= \big(\hat T_0(f\otimes \varphi)\big) (g\otimes\xi) + (\sigma^1(f) \otimes \varphi) \big(\hat T_0 (g\otimes\xi)\big).
\nonumber \end{align} Therefore, for all $\phi, \psi \in \F(\oqd) \otimes C^{(1)}(\mathbb{S}^1)$ with $\psi \in \mathrm{dom} (y^{-1} \otimes 1)$, we have \[ \label{T0} \hat T_0 (\phi \psi) = (\hat T_0\phi) \psi + \hat\sigma^1(\phi) {\hspace{1pt}} ( \hat T_0\psi). \] As a consequence, $\hat T_0,$ $\hat T_1$ and $\hat T_2$ satisfy the same twisted Leibniz rule.
In the definition of the Dirac operator, we will multiply $\hat T_0$, $\hat T_1$ and $\hat T_2$ with multiplication operators from the opposite algebra. The following lemma shows that these multiplication operators do not change the twisted Leibniz rule. Our aim is to prove that the Dirac operator has bounded twisted commutators with functions of an appropriate *-algebra, where the twisted commutator of densely defined operators $T$ on $L_2(\mathbb{D}_q,\mu_\alpha)\bar\otimes L_2(\mathbb{S}^1)$ with $\phi\in \F(\oqd) \otimes C(\mathbb{S}^1)$ is defined by $$ [T, \phi]_{\sigma^1} := T \phi - \sigma^1(\phi) T. $$
The purpose of the following lemma is to clarify the setup for the algebraic manipulations to be carried out, and to ensure that these make sense in their Hilbert space realisation.
\begin{lem} \label{lem1} Let ${\mathcal{A}}$ be a unital *-algebra, $\int : {\mathcal{A}} \rightarrow \mathbb{C}$ a faithful positive functional, $\hH$ the Hilbert space closure of ${\mathcal{A}}$ with respect to the inner product $\ip{a}{b}:=\int a^*b$, and assume that left multiplication by an element in ${\mathcal{A}}$ defines a bounded operator on $\hH$. Let $T$ be a densely defined linear operator on $\hH$, ${\mathcal{A}}^1\subset {\mathcal{A}}$ a *-sub\-algebra and $\mathcal{D}\subset \hH$ a dense subspace such that $\mathcal{D} + {\mathcal{A}}^1\subset\mathrm{dom}(T)$, $T({\mathcal{A}}^1)\subset{\mathcal{A}}$, and $T$ satisfies the twisted Leibniz rule $$ T(f \psi) = (T f)\psi + \sigma(f){\hspace{1pt}} T\psi,\quad f\in {\mathcal{A}}^1, \ \ \psi\in \mathcal{D} + {\mathcal{A}}^1 $$ for an automorphism $\sigma:{\mathcal{A}}\rightarrow{\mathcal{A}}$. Assume finally that $\hat x$ is a densely defined linear operator on $\hH$ with $\mathcal{D}\subset \mathrm{dom}(\hat x)$, and that $f\psi \in \mathcal{D}$ and $\hat x f \psi = f\hat x \psi$ hold for all $f\in{\mathcal{A}}$ and $\psi\in \mathcal{D}$. Then \[ \label{Tfg} \hat x T(fg) = \hat x (Tf) g + \sigma(f) \hat x T(g), \quad f,g\in {\mathcal{A}}^1 \] as operators on $\mathcal{D}$ and $$ [\hat x T, f]_{\sigma} \psi = \hat x (Tf)\psi, \quad f\in {\mathcal{A}}^1, \ \ \psi \in \mathcal{D}. $$ \end{lem}
\begin{proof} The only slightly nontrivial statement is that each term in the following algebraic computations is well-defined as an operator on the domain $\mathcal{D}$: Let $\psi \in\mathcal{D}$ and $f,g\in {\mathcal{A}}^1$. From the twisted Leibniz rule, we get \begin{align*} ( T(fg) )\psi &= T(fg\psi) - \sigma(fg) T\psi = (T f)(g\psi) + \sigma(f) T(g\psi) - \sigma(f) \sigma(g) T\psi \\ &= (T f)(g\psi) + \sigma(f) (T g)\psi . \end{align*} Since $\sigma(f )\in {\mathcal{A}}$ for all $f \in {\mathcal{A}}$, it follows that $$ \hat x( T(fg) )\psi = \big(\hat x (T f)g + \sigma(f){\hspace{1pt}} \hat x (T g)\big)\psi , $$ which proves \eqref{Tfg}. As $ \hat x{\hspace{1pt}} T (f \psi) = \hat x{\hspace{1pt}} (Tf) \psi + \hat x {\hspace{1pt}} \sigma(f){\hspace{1pt}} T \psi = \hat x{\hspace{1pt}} (Tf) \psi + \sigma(f)\hat x{\hspace{1pt}} T \psi , $ we also have $[\hat x{\hspace{1pt}} T, f]_{\sigma} \psi = \hat x{\hspace{1pt}} (Tf) \psi + \sigma(f){\hspace{1pt}} \hat x {\hspace{1pt}} T \psi - \sigma(f){\hspace{1pt}} \hat x{\hspace{1pt}} T\psi = \hat x{\hspace{1pt}} ( T f) \psi $. \end{proof}
By \eqref{T1T2} and \eqref{T0}, the lemma applies in particular to the operators $\hat T_0$, $\hat T_1$ and $\hat T_2$ with ${\mathcal{A}}:= \F(\oqd) \otimes C(\mathbb{S}^1)$, ${\mathcal{A}}^1:=\F^{(1)}(\oqd) \otimes C^{(1)}(\mathbb{S}^1)$, $\hH:=L_2(\mathbb{D}_q,\mu _\alpha) \bar\otimes L_2(\mathbb{S}^1)$ (where integration on $\mathbb{S}^1$ is taken with respect to the Lebesgue measure), the automorphism $\hat \sigma^1$ and the operators $\hat x $ coming from $\mathcal{O}^+(\mathbb{D}_q)^\mathrm{op}$. As the dense domain, we may take \[\label{defbereich3} \mathcal{D}:= {\mathcal{F}}_0(\mathbb{D}_q) \otimes C^{(1)}(\mathbb{S}^1). \]
\subsection{Twist: $\pmb{\sigma^2}$} \label{sec2}\mbox{ }\\[4pt] \noindent First we show that $\dz$ and $\dbz$ satisfy a twisted Leibniz rule for the automorphism $\sigma^2$. Let $f,g\in \F^{(1)}(\oqd)$. Then \begin{align}\label{S1} \dz(fg) &=
\mbox{$\frac{-1}{1-q^2}$}
{\hspace{1pt}} y^{-2}[z^*{\hspace{-1pt}},fg]
=
\mbox{$\frac{-1}{1-q^2}$}
{\hspace{1pt}} y^{-2}([z^*{\hspace{-1pt}}, f] g
+ f y^2 y^{-2} [z^*{\hspace{-1pt}}, g])
\nonumber\\ &= (\dz f)g + \sigma^{2}{\hspace{-1pt}} (f){\hspace{1pt}} \dz g \end{align} and similarly \[ \label{S2} \dbz(fg) = (\dbz f)g + \sigma^{2}{\hspace{-1pt}} (f){\hspace{1pt}} \dbz g. \]
Setting $\hat S_1 := \dz \otimes 1$, $\hat S_2 := \dbz \otimes 1$ and $\hat \sigma^2 := \sigma^2 \otimes 1$, we get for all $\phi, \psi \in \F^{(1)}(\oqd) \otimes C^{(1)}(\mathbb{S}^1)$ \[ \nonumber \hat S_1(\phi \psi ) = (\hat S_1 \phi)\psi + \hat\sigma^{2}{\hspace{-1pt}} (\phi){\hspace{1pt}} \hat S_1\psi, \quad \hat S_2(\phi \psi ) = (\hat S_2 \phi)\psi + \hat\sigma^{2}{\hspace{-1pt}} (\phi){\hspace{1pt}} \hat S_2\psi \] by \eqref{S1} and \eqref{S2}.
Next consider the operator $\hat S_0:= y^{-2} \dt$ on the domain $\mathrm{dom}(y^{-2}) \otimes C^{(1)}(\mathbb{S}^1)$ in $L_2(\mathbb{D}_q,\mu_\alpha) \otimes L_2(\mathbb{S}^1)$. Again $fg\in\mathrm{dom}(y^{-2})$ for all $f\in \F(\oqd)$ and $g\in\mathrm{dom}(y^{-2})$ since $y^{-2}fg= \sigma^2(f)y^{-2} g\in L_2(\mathbb{D}_q,\mu_\alpha)$. Now, for all $\varphi ,\, \xi \in C^{(1)}(\mathbb{S}^1)$,
\begin{align} \nonumber \hat S_0(fg\otimes \varphi\xi) &= y^{-2} fg\otimes \dt ( \varphi \xi ) \nonumber \\ \nonumber &= y^{-2} fg\otimes (\dt \varphi)\xi + y^{-2} fy^2{\hspace{1pt}} y^{-2} g\otimes \varphi\,(\dt \xi) \\ &= \big(\hat S_0(f\otimes \varphi)\big) (g\otimes\xi) + (\sigma^2(f) \otimes \varphi) \big(\hat S_0 (g\otimes\xi)\big). \nonumber \end{align} Therefore, for all $\phi, \psi \in \F(\oqd) \otimes C^{(1)}(\mathbb{S}^1)$ with $\psi \in \mathrm{dom} (y^{-1} \otimes 1)$, we have \[ \nonumber \hat S_0 (\phi \psi) = (\hat S_0\phi) \psi + \hat\sigma^2(\phi) {\hspace{1pt}} ( \hat S_0\psi). \] As a consequence, $\hat S_0,$ $\hat S_1$ and $\hat S_2$ satisfy the same twisted Leibniz rule for the twist $\hat\sigma^2$, and so do $x_i^\mathrm{op} S_i$ for $x_i^\mathrm{op}\in\{y^{\mathrm{op}}, (y^{\mathrm{op}})^2, z^\mathrm{op}, z^{*\mathrm{op}} \}$ by Lemma \ref{lem1}.
\section{Adjoints}
\subsection{$\pmb{\alpha = 2}$}\mbox{ }\\[4pt] \noindent Set $\alpha = 2$ in \eqref{Tr}, $$ \mathcal{D}_0:=\{ f\in \F^{(1)}(\oqd)\cap \mathrm{dom}( y^{-1})\cap \mathrm{dom}((y^{-1})^\mathrm{op}) : T_1(f), T_2(f) \in \mathrm{dom}((y^{-1})^\mathrm{op})\} $$ and \[\label{defbereich} \mathcal{D}:= \mathcal{D}_0 \otimes C^{(1)}(\mathbb{S}^1) \;\subset \; L_2(\mathbb{D}_q,\mu_\alpha) \;\bar\otimes \;L_2(\mathbb{S}^1). \] It follows from ${\mathcal{F}}_0(\mathbb{D}_q)\subset \mathcal{D}_0$ that $\mathcal{D}$ is dense in $L_2(\mathbb{D}_q,\mu_\alpha) \;\bar\otimes \;L_2(\mathbb{S}^1)$. Let $T_1$ and $T_2$ be the operators from \eqref{defT1T2} with domain $\mathcal{D}_0$. Using \eqref{zy} and the trace property, we compute for all $f,g\in \mathcal{D}_0$, \begin{align*} & \ip{ T_1 f}{g} = -\mbox{$\frac{1-q}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)}( f^*z y^{-1} g y^2 - z f^* y^{-1} g y^2 )\\ & =- \mbox{$\frac{1-q}{1-q^2}$}\,\mathrm{Tr}_{\ell_2(\N)}( q^{-1} f^*y^{-1} z g y^2 - q^{-2} f^* y^{-1} g z y^2) \\ & =\mbox{$\frac{(1-q)(q^{-2}-q^{-1}) }{1-q^2}$}\,\mathrm{Tr}_{\ell_2(\N)}( f^*y^{-1} g z y^2)
- \mbox{$\frac{(1-q)q^{-1}}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)}( f^*y^{-1} z g y^2 - f^* y^{-1} g z y^2) \\ & = \ip{ f}{ (\mbox{$\frac{q^{-2}}{1+q}$} z^{\mathrm{op}} y^{-1} - q^{-1} T_2 )g}, \end{align*} therefore \[ \label{T1*} \mbox{$\frac{q^{-2}}{1+ q}$} z^{\mathrm{op}} y^{-1} - q^{-1} T_2 \ \subset \ T_1^*. \] From \eqref{opstar}, it also follows that $$ \mbox{$\frac{q}{1+ q}$} z^{*\mathrm{op}} y^{-1} - qT_1 \subset T_2^*. $$
Similarly, using $zz^* -z^* z= (1-q^2) y^2$, \begin{align*} & \ip{ (zy^{-1})^{\mathrm{op}}T_1 f}{g} = -\mbox{$\frac{1-q}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)}(y^{-1}z^* f^*z y^{-1} g y^2 - y^{-1} z^* z f^* y^{-1} g y^2 )\\ & = -\mbox{$\frac{1-q}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)}(y^{-1} (z z^* -z^*z) f^* y^{-1} g y^2 + y^{-1}z^* f^*z y^{-1} g y^2 - y^{-1} z z^* f^* y^{-1} g y^2 )\\ & = -(1-q) \, \mathrm{Tr}_{\ell_2(\N)} \big( f^* y^{-1} gy y^2 \big) -\mbox{$\frac{1-q}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)} \big(f^*y^{-1}(zg-gz) z^*y^{-1} y^2\big) \\ &= \ip{f}{\big(-\sigma^1 -(z^*y^{-1})^\mathrm{op} T_2\big)g}, \end{align*} where $\sigma^1(g) = y^{-1}g y= y^{-1} y^{\mathrm{op}} g$ for all $g\in\F(\oqd)\cap\mathrm{dom}(y^{-1})$. Hence \[ \label{T21} -\sigma^1 -(z^*y^{-1})^\mathrm{op} T_2 \ \subset \big((zy^{-1})^{\mathrm{op}}{\hspace{1pt}} T_1\big)^* . \] Since $y^{-1}$ and $ y^{\mathrm{op}} $ are self-adjoint and thus $\sigma^1$ is symmetric, we also get from the above calculations \[ \label{T12} -\sigma^1 -(zy^{-1})^\mathrm{op} T_1 \ \subset \big((z^*y^{-1})^{\mathrm{op}}{\hspace{1pt}} T_2\big)^* . \]
Recall that $\mathrm{i} \dt$ is a symmetric operator on $C^{(1)}(\mathbb{S}^1)\subset L_2(\mathbb{S}^1)$. Also, for all $\varphi \in C^{(1)}(\mathbb{S}^1)$, we have \[ \label{edt}
(\mathrm{e}^{\mathrm{i} t} \mathrm{i} \dt)^* \varphi = \mathrm{i} \dt (\mathrm{e}^{-\mathrm{i} t} {\hspace{1pt}} \varphi )
= \mathrm{e}^{-\mathrm{i} t} {\hspace{1pt}} \varphi + \mathrm{e}^{-\mathrm{i} t}{\hspace{1pt}} \mathrm{i} \dt \varphi. \] From this, \eqref{opstar} and the self-adjointness of $y^{-1}$, it follows that $$
q^{-2} z^{\mathrm{op}} y^{-1} \mathrm{e}^{-\mathrm{i} t} + q^{-2} z^{\mathrm{op}} y^{-1} \mathrm{e}^{-\mathrm{i} t} \mathrm{i} \dt \ \subset \
( z^{*\mathrm{op}} y^{-1} \mathrm{e}^{\mathrm{i} t} \mathrm{i} \dt )^*, $$ where the left-hand side and $z^{*\mathrm{op}} y^{-1} \mathrm{e}^{\mathrm{i} t} \mathrm{i} \dt $ are operators on $\mathcal{D}$. Analogously, \[ \label{zdt*}
- q^{2} z^{*\mathrm{op}} y^{-1} \mathrm{e}^{\mathrm{i} t} + q^{2} z^{*\mathrm{op}} y^{-1} \mathrm{e}^{\mathrm{i} t} \mathrm{i} \dt \ \subset \
( z^{\mathrm{op}} y^{-1} \mathrm{e}^{-\mathrm{i} t} \mathrm{i} \dt )^* . \] Now we are in a position to state the main result of this section.
\begin{prop} \label{P1} Consider the following operators on $L_2(\mathbb{D}_q,\mu_\alpha) \;\bar\otimes \;L_2(\mathbb{S}^1)$ with domain $\mathcal{D}$ defined in (\ref{defbereich}) above: \begin{align*} \hat H &:= (z y^{-1})^{\mathrm{op}}{\hspace{1pt}} y{\hspace{1pt}} \dz - (z y^{-1})^{*\mathrm{op}}{\hspace{1pt}} y{\hspace{1pt}} \dbz + {\hspace{1pt}} y^{\mathrm{op}}{\hspace{1pt}} y^{-1} {\hspace{1pt}} \mathrm{i} \dt , \\ \hat E &:= - \mathrm{e}^{-\mathrm{i} t}{\hspace{1pt}} y {\hspace{1pt}} \dbz - \mbox{$\frac{q^{-1}}{1+q}$} z^{\mathrm{op}} {\hspace{1pt}} y^{-1} {\hspace{1pt}} \mathrm{e}^{-\mathrm{i} t}{\hspace{1pt}} \mathrm{i} \dt , \\ \hat F &:= q{\hspace{1pt}} \mathrm{e}^{\mathrm{i} t}{\hspace{1pt}} y {\hspace{1pt}} \dz - \mbox{$\frac{q}{1+q}$} z^{*\mathrm{op}} {\hspace{1pt}} y^{-1} {\hspace{1pt}} \mathrm{e}^{\mathrm{i} t}{\hspace{1pt}} \mathrm{i} \dt . \end{align*} Then \,$\hat H \subset \hat H^*$, \,$\hat F\subset \hat E^*$ and \,$\hat E\subset \hat F^*$. \end{prop}
\begin{proof} Since $y^{\mathrm{op}}$, \,$y^{-1}$ and $\mathrm{i} \dt$ are commuting symmetric operators on $\mathcal{D}$, we have $y^{\mathrm{op}}{\hspace{1pt}} y^{-1} {\hspace{1pt}} \mathrm{i} \dt \subset (y^{\mathrm{op}}{\hspace{1pt}} y^{-1} {\hspace{1pt}} \mathrm{i} \dt)^*$. Now it follows from \eqref{T21} and \eqref{T12} that $$ \hat H^*\ \supset\ -\sigma^1 -(z^*y^{-1})^\mathrm{op} {\hspace{1pt}} y{\hspace{1pt}} \dbz + \sigma^1
+ (zy^{-1})^\mathrm{op} {\hspace{1pt}} y{\hspace{1pt}} \dz + {\hspace{1pt}} y^{\mathrm{op}}{\hspace{1pt}} y^{-1} {\hspace{1pt}} \mathrm{i} \dt \ = \ \hat H. $$ Furthermore, from \eqref{T1*} and \eqref{zdt*}, we obtain $$ \hat F^* \ \supset\ \mathrm{e}^{-\mathrm{i} t} \mbox{$\frac{q^{-1}}{1+ q}$} z^{\mathrm{op}} y^{-1} - \mathrm{e}^{-\mathrm{i} t} {\hspace{1pt}} y {\hspace{1pt}} \dbz - \mathrm{e}^{-\mathrm{i} t} \mbox{$\frac{q^{-1}}{1+q}$} {\hspace{1pt}} z^{\mathrm{op}} y^{-1}
- \mbox{$\frac{q^{-1}}{1+q}$} {\hspace{1pt}} z^{\mathrm{op}} y^{-1} \mathrm{e}^{-\mathrm{i} t} \mathrm{i} \dt \ = \ \hat E . $$ The last relation also shows that $\hat F^*$ is densely defined, thus $\hat F \subset \hat F^{**} \subset \hat E^*$. \end{proof}
\subsection{$\pmb{\alpha = 1}$} \mbox{ }\\[4pt] \noindent Consider now \[\label{defbereich2}
\mathcal{D}:=
\mathcal{D}_0 \otimes C^{(1)}(\mathbb{S}^1)
\;\subset \; L_2(\mathbb{D}_q,\mu_\alpha)
\;\bar\otimes \;L_2(\mathbb{S}^1) \] with $
\mathcal{D}_0:=
\F^{(1)}(\oqd)\cap
\mathrm{dom}( y^{-2})$. For all $f,g \in \F^{(1)}(\oqd)$, \begin{align*}
\ip{ y^\mathrm{op} \dz f}{g} &= -\mbox{$\frac{1-q}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)}( y f^*z y^{-2} g y - y z f^* y^{-2} g y )\\ & =- q^{-2}\,\mbox{$\frac{1-q}{1-q^2}$}\,\mathrm{Tr}_{\ell_2(\N)}( f^*y^{-2} z g y^2 - f^* y^{-2} g z y^2) \\ & = - q^{-2}\,\mbox{$\frac{1-q}{1-q^2}$}\,\mathrm{Tr}_{\ell_2(\N)}( f^* y^{-2}[ z, g] y^2 ) \\ & = \ip{ f}{ -q^{-2} y^\mathrm{op} \dbz g}, \end{align*} thus \[ \label{yopdz} -q^{-1} y^\mathrm{op} \dbz \subset (q y^\mathrm{op} \dz)^*\quad \text{and}\quad
q y^\mathrm{op} \dz \subset (q y^\mathrm{op} \dz)^{**} \subset (-q^{-1} y^\mathrm{op} \dbz)^*. \] Next, \begin{align} \nonumber
&\ip{ z^\mathrm{op} \dz f}{g} = -\mbox{$\frac{1-q}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)}( z^* f^*z y^{-2} g y - z^* z f^* y^{-2} g y )\\
&\quad = -\mbox{$\frac{1-q}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)}( q^{-1} f^*y^{-2} z g z^* y - f^* y^{-2} g z^* z y )\nonumber\\ &\quad = - \mbox{$\frac{1-q}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)}\big( q^{-1}( f^* y^{-2} z g z^* y - f^* y^{-2} g z z^* y) \nonumber\\ &\quad \mbox{ } \hspace{8pt} + q^{-1} f^* y^{-2} g z z^* y - f^* y^{-2} g z^* z y \big ) \nonumber\\
&\quad = - \ip{ f}{(q^{-1}z^{*\mathrm{op}} \dbz )g} -q^{-1} \mbox{$\frac{(1-q)^2}{1-q^2}$}\,
\mathrm{Tr}_{\ell_2(\N)}\big( f^* y^{-2} g y +q f^* y^{-2} g y^2 y\big)\label{zopdz} \end{align} and \begin{align} \nonumber
\ip{ (z^{*\mathrm{op}} \dbz ) f}{g} &= \mbox{$\frac{1-q}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)}( z f^*z^* y^{-2} g y - z z^* f^* y^{-2} g y )\\
&= \mbox{$\frac{1-q}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)}( q f^* y^{-2} z^* g z y - f^* y^{-2} g z z^* y )\nonumber\\
&= \mbox{$\frac{1-q}{1-q^2}$}\, \mathrm{Tr}_{\ell_2(\N)}\big( q( f^* y^{-2} z^* g z y - f^* y^{-2} g z^* z y)
\nonumber\\
&\hspace{12pt} + q f^* y^{-2} g z^* z y - f^* y^{-2} g z z^* y \big ) \nonumber \\
&= - \ip{ f}{(qz^{\mathrm{op}} \dz )g} - \mbox{$\frac{(1-q)^2}{1-q^2}$}\,
\mathrm{Tr}_{\ell_2(\N)}\big( f^* y^{-2} g y +q f^* y^{-2} g y^2 y\big). \label{starzopdz} \end{align} From \eqref{zopdz} and \eqref{starzopdz}, \[ \label{qzdz} \ip{ (q z^\mathrm{op} \dz - z^{*\mathrm{op}} \dbz ) f}{g} = \ip{ f}{(q z^\mathrm{op} \dz - z^{*\mathrm{op}} \dbz ) g} \] i.e., the operator $q z^\mathrm{op} \dz - z^{*\mathrm{op}} \dbz $ is symmetric. As $(y^{2})^{\mathrm{op}}{\hspace{1pt}} y^{-2} \mathrm{i} \dt$ is the product of commuting symmetric operators, \[ \label{yydt} (y^{2})^{\mathrm{op}}{\hspace{1pt}} y^{-2} \mathrm{i} \dt \ \subset\ \big((y^{2})^{\mathrm{op}}{\hspace{1pt}} y^{-2} \mathrm{i} \dt\big)^* \] is also symmetric.
By \eqref{opstar} and \eqref{edt}, $$
z^{\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{-\mathrm{i} t} + z^{\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{-\mathrm{i} t} \mathrm{i} \dt\ \subset\
( z^{*\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{\mathrm{i} t} \mathrm{i} \dt)^*. $$ Similarly, $$
- z^{*\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{\mathrm{i} t}
+ z^{*\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{\mathrm{i} t} \mathrm{i} \dt \ \subset \ ( z^\mathrm{op} y^\mathrm{op} y^{-2} \mathrm{e}^{-\mathrm{i} t} \mathrm{i} \dt)^* . $$ Since $z^{\mathrm{op}} y^\mathrm{op} \subset (z^{*\mathrm{op}} y^\mathrm{op})^*$ by \eqref{opstar}, \[ \label{ef}
\half z^{\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{-\mathrm{i} t} + z^{\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{-\mathrm{i} t} \mathrm{i} \dt\ \subset\
(- \half z^{*\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{\mathrm{i} t} + z^{*\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{\mathrm{i} t} \mathrm{i} \dt)^*. \] Analogously, using $z^{*\mathrm{op}} y^\mathrm{op} \subset (z^{\mathrm{op}} y^\mathrm{op})^*$, \[ \label{ee}
-\half z^{*\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{\mathrm{i} t} + z^{*\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{\mathrm{i} t} \mathrm{i} \dt \ \subset \
( \half z^{\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{-\mathrm{i} t} + z^\mathrm{op} y^\mathrm{op} y^{-2} \mathrm{e}^{-\mathrm{i} t} \mathrm{i} \dt)^* . \] As in the previous section, we summarise our results in a proposition.
\begin{prop} \label{P2} Let $\gamma_q$ be a non-zero real number. Consider the following operators on $L_2(\mathbb{D}_q) \;\bar\otimes \;L_2(\mathbb{S}^1)$ with domain $\mathcal{D}$ defined in (\ref{defbereich2}) above: \begin{align*} \hat H_1 &:= q z^{\mathrm{op}} \dz - z^{*\mathrm{op}} \dbz + (y^{2})^{\mathrm{op}}{\hspace{1pt}} y^{-2} \mathrm{i} \dt, \\ \hat E_1 &:= - q^{-1} \mathrm{e}^{-\mathrm{i} t}{\hspace{1pt}} y^\mathrm{op} {\hspace{1pt}} \dbz - \gamma_q z^{\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{-\mathrm{i} t} \mathrm{i} \dt
- \mbox{$\frac{\gamma_q}{2}$} z^{\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{-\mathrm{i} t} , \\ \hat F_1 &:= q{\hspace{1pt}} \mathrm{e}^{\mathrm{i} t}{\hspace{1pt}} y^\mathrm{op} {\hspace{1pt}} \dz - \gamma_q z^{*\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{\mathrm{i} t} \mathrm{i} \dt + \mbox{$\frac{\gamma_q}{2}$} z^{*\mathrm{op}} y^\mathrm{op} y^{-2} \mathrm{e}^{\mathrm{i} t} . \end{align*} Then \,$\hat H_1 \subset \hat H_1^*$, \,$\hat F_1\subset \hat E_1^*$ and \,$\hat E_1\subset \hat F_1^*$. \end{prop}
\begin{proof} $\hat H_1 \subset \hat H_1^*$ follows from \eqref{qzdz} and \eqref{yydt}, \,$\hat E_1\subset \hat F_1^*$ follows from \eqref{yopdz} and \eqref{ef}, and \,$\hat F_1\subset \hat E_1^*$ follows from \eqref{yopdz} and \eqref{ee}. \end{proof}
\section{The Dirac operator}
Classically, the left invariant vector fields $H$, $E$ and $F$ act as first order differential operators on differentiable functions on $\mathrm{SU}(2)$. In the noncommutative case, we will use the actions of $\hat H$, $\hat E$ and $\hat F$ to define an algebra of
differentiable functions.
For $\hat x_i = x_i^{\mathrm{op}} \otimes \eta_i \in \mathcal{O}^+(\mathbb{D}_q)^{\mathrm{op}} \otimes C^\infty (\mathbb{S}^1)$, $i=0,1,2$, consider the action on
$f\otimes \varphi \in \F^{(1)}(\oqd)\otimes C^{(1)}(\mathbb{S}^1)$
given by \begin{align} \nonumber (\hat x_0\hat T_0 + \hat x_1\hat T_1 &+ \hat x_2\hat T_2 )( f\otimes \varphi) \\ \label{action}
&:= y^{-1} f{\hspace{1pt}} x_0 \otimes \eta_0 {\hspace{1pt}} \dt(\varphi) + T_1(f) {\hspace{1pt}} x_1\otimes \eta_1 {\hspace{1pt}} \varphi + T_2(f){\hspace{1pt}} x_2 \otimes \eta_2 {\hspace{1pt}} \varphi, \end{align} where the right-hand side of \eqref{action} is understood as an unbounded operator on $L_2(\mathbb{D}_q,\mu_\alpha) \,\bar\otimes\, L_2(\mathbb{S}^1)$ with domain of definition containing the subspace $\mathcal{D}$ introduced in (\ref{defbereich3}). Note that the operators $\hat H$, $\hat E$ and $\hat F$ are of the form described in \eqref{action}, and that the operators $\hat x_i$ and $\hat T_i$ satisfy the assumptions of Lemma \ref{lem1}. We define \begin{align} \nonumber &\Gamma^{(1)}(\mathrm{SU}_q(2))
:= \{ \phi \in \F^{(1)}(\oqd) \otimes C^{(1)}(\mathbb{S}^1) : \\ \label{Gamma} &\hspace{70pt} \hat H(\phi),\ \hat E(\phi), \ \hat F(\phi) ,\ \hat H(\phi^*),\ \hat E(\phi^*), \ \hat F(\phi^*) \ \text{are bounded}\}. \end{align} From \eqref{T1}, \eqref{T2}, \eqref{T0} and Lemma \ref{lem1}, it follows that \[ \label{hatT} \hat T (\varphi \psi) = (\hat T\varphi) \psi + \hat\sigma^1(\varphi) {\hspace{1pt}} ( \hat T\psi), \] for all $\varphi, \psi \in \Gamma^{(1)}(\mathrm{SU}_q(2))$ and $\hat T\in \{\hat H, \hat E,\hat F\}$. In particular, $\hat T (\varphi \psi)$ is again bounded so that $\Gamma^{(1)}(\mathrm{SU}_q(2))$ is a *-algebra.
Finally note that the classical limit of $\hat H$, $\hat E$ and $\hat F$ for $q\rightarrow 1$ is formally $H$, $E$ and $F$, respectively. This will also be the case if we rescale $\hat E$ and $\hat F$ by a real number $c=c(q)$ such that $\lim_{q\rightarrow 1} c(q)=1$. Such a rescaling might be useful in later computations of the spectrum of the Dirac operator.
\begin{thm} \label{Thm1} Let $\alpha =2$. Set $\hH := (L_2(\mathbb{D}_q,\mu_\alpha) \bar\otimes L_2(\mathbb{S}^1))\oplus (L_2(\mathbb{D}_q,\mu_\alpha) \bar\otimes L_2(\mathbb{S}^1))$ and define $$ \pi :\Gamma^{(1)}(\mathrm{SU}_q(2)) \longrightarrow B(\hH), \quad \pi(\phi):= \phi \oplus \phi $$ as left multiplication operators. Then, for any $c\in\mathbb{R}$, the operator $$ D:=\left( \begin{array}{cc} \hat H-2\ & c \hat E \\ c \hat F\ & -\hat H-2 \\ \end{array} \right) $$ is symmetric on $\mathcal{D} \oplus \mathcal{D}$ with $\mathcal{D}$ from (\ref{defbereich}). Furthermore, $$ [D, \pi(\phi)]_{\hat\sigma^1} : =D{\hspace{1pt}} \pi(\phi) - \pi(\hat\sigma^1(\phi)) {\hspace{1pt}} D $$ is bounded for all $\phi\in \Gamma^{(1)}(\mathrm{SU}_q(2))$. \end{thm}
\begin{proof} $D\subset D^*$ follows from Proposition \ref{P1}. For all $\phi\in \Gamma^{(1)}(\mathrm{SU}_q(2))$ and $\psi_1\oplus\psi_2$ in the domain of $D$, \begin{align*} [D, \pi(\phi)]_{\hat\sigma^1} (\psi_1\oplus\psi_2) &= \hat H (\phi \psi_1) - \hat\sigma^1(\phi) \hat H ( \psi_1) + c(\hat E (\phi \psi_2) - \hat\sigma^1(\phi) \hat E ( \psi_2))\\
&\hspace{10pt} \oplus \, c(\hat F (\phi \psi_1) - \hat\sigma^1(\phi) \hat F ( \psi_1))
-(\hat H (\phi \psi_2) - \hat\sigma^1(\phi) \hat H ( \psi_2)) \\ & = (\hat H\phi) \psi_1 + c (\hat E \phi) \psi_2 \ \oplus \ c(\hat F \phi) \psi_1 - (\hat H\phi )\psi_2 \end{align*} by \eqref{hatT}, so $[D, \pi(\phi)]_{\sigma^1}$ is bounded by the definition of $\Gamma^{(1)}(\mathrm{SU}_q(2))$. \end{proof}
\begin{thm} \label{Thm2} Let $\hH$ and $\pi$ be defined as in Theorem \ref{Thm1} but with the measure on $\mathbb{D}_q$ given by setting $\alpha=1$ in \eqref{Tr}. Let $\hat H_1$, $\hat F_1$ and $\hat E_1$ be defined as in Proposition \ref{P2} and $\Gamma^{(1)}(\mathrm{SU}_q(2))$ as in \eqref{Gamma} with $\hat H$, $\hat F$ and $\hat E$ replaced by $\hat H_1$, $\hat F_1$ and $\hat E_1$, respectively. Then, for any $c\in\mathbb{R}$, the operator $$ D_1:=\left( \begin{array}{cc} \hat H_1-2\ & c \hat E_1 \\ c \hat F_1\ & -\hat H_1-2 \\ \end{array} \right) $$ is symmetric on $\mathcal{D} \oplus \mathcal{D}$ with $\mathcal{D}$ from (\ref{defbereich2}). Furthermore, $$ [D_1, \pi(\phi)]_{\hat\sigma^2} : =D_1{\hspace{1pt}} \pi(\phi) - \pi(\hat\sigma^2(\phi)) {\hspace{1pt}} D_1 $$ is bounded for all $\phi\in \Gamma^{(1)}(\mathrm{SU}_q(2))$. \end{thm} \begin{proof} Using the results for $\hat S_0$, \,$\hat S_1$ and \,$\hat S_2$ from Section \ref{sec2} and Proposition \ref{P2}, the proof is essentially the same as the proof of the previous theorem. \end{proof}
To view the Dirac operator of Theorem \ref{Thm2} as a deformation of the classical Dirac operator, one may choose a continuously varying positive real number $\gamma_q$ such that $\lim_{q\rightarrow 1}\gamma_q=\half$. For instance, if $\gamma_q := \frac{q}{1+q}$, then the Dirac operator of Theorem \ref{Thm2} resembles the one of Theorem \ref{Thm1}, the main difference being the additional functions (0-order differential operators) in the definitions of $\hat E_1$ and $\hat F_1$. In the classical case $q=1$, the operator $D_1$ can be obtained from the Dirac operator $D$ in Theorem \ref{Thm1} by the ``gauge transformation'' $D_1= \sqrt{y}{\hspace{1pt}} D \sqrt{y}^{-1}$.
On the other hand, if one rescales the volume form to $\mathrm{vol}_1 := \frac{1}{y}{\hspace{1pt}} \mathrm{vol}$ with a non-constant function $y$ without changing the Riemannian metric, then the Dirac operator ceases to be self-adjoint but the above gauge transformed Dirac operator will remedy the problem. To see this, let $f,g, \sqrt{y}^{-1}{\hspace{-1pt}} f,\sqrt{y}^{-1}{\hspace{-1pt}} g \in \mathrm{dom}(D)$. Then \begin{align*} &\ip{\sqrt{y}{\hspace{1pt}} D \sqrt{y}^{-1}{\hspace{-1pt}} f}{g}_{L_2(S,\frac{1}{y} \mathrm{vol})} = \int \ip{\sqrt{y}{\hspace{1pt}} D \sqrt{y}^{-1}{\hspace{-1pt}} f}{g}\frac{1}{y}{\hspace{1pt}}\mathrm{d}\mathrm{vol} = \int \ip{ D \sqrt{y}^{-1}{\hspace{-1pt}} f}{ \sqrt{y}^{-1}{\hspace{-1pt}} g} \mathrm{d}\mathrm{vol}\\ &\quad= \int \ip{ f}{ \sqrt{y} D\sqrt{y}^{-1{\hspace{-1pt}} }g} \frac{1}{y}\mathrm{d}\mathrm{vol} = \ip{f}{\sqrt{y}{\hspace{1pt}} D \sqrt{y}^{-1}{\hspace{-1pt}} g}_{L_2(S,\frac{1}{y} \mathrm{vol})}. \end{align*} For this reason and in view of \eqref{measure}, we may regard $D_1$ as the Dirac operator obtained from $D$ by rescaling the volume form $y {\hspace{1pt}} \mathrm{d}_q y{\hspace{1pt}} \mathrm{d} \phi\,\mapsto\,\mathrm{d}_q y{\hspace{1pt}} \mathrm{d} \phi$.
\end{document} | arXiv |
\begin{definition}[Definition:Range of Sequence]
Let $\sequence {x_n}_{n \mathop \in A}$ be a sequence.
The '''range of $\sequence {x_n}$''' is the set:
:$\set {x_n: n \mathop \in A}$
\end{definition} | ProofWiki |
\begin{document}
\title{Bertlmann's chocolate balls and quantum type cryptography}
\author{Karl Svozil} \email{[email protected]} \homepage{http://tph.tuwien.ac.at/~svozil} \affiliation{Institute for Theoretical Physics, Vienna University of Technology, \\ Wiedner Hauptstra\ss e 8-10/136, A-1040 Vienna, Austria}
\begin{abstract} Some quantum cryptographic protocols can be implemented with specially prepared chocolate balls, others protected by value indefiniteness cannot. Similarities and differences of cryptography with quanta and chocolate are discussed. Motivated by these considerations it is proposed to certify quantum random number generators and quantum cryptographic protocols by value indefiniteness. This feature, which derives itself from Bell- and Kochen-Specker type arguments, is only present in systems with three or more mutually exclusive outcomes. \end{abstract}
\pacs{03.67.Hk,03.65.Ud} \keywords{Quantum information, quantum cryptography, singlet states, entanglement, quantum nonlocality}
\maketitle
\section{Quantum resources for cryptography}
Quantum cryptography\footnote{ In view of the many superb presentations of quantum cryptography --- to name but a few, see Refs.~\cite{gisin-qc-rmp,arXiv:0802.4155} and \cite[Chapter~6]{mermin-04} (or, alternatively, \cite[Section 6.2]{mermin-07}), as well as \cite[Section~12.6]{nielsen-book}; apologies to other authors for this incomplete, subjective collection --- we refrain from any extensive introduction. } uses quantum resources to encode plain symbols forming some message. Thereby, the security of the code against cryptanalytic attacks to recover that message rests upon the validity of physics, giving new and direct meaning to Landauer's dictum~\cite{landauer} ``information is physical.''
What exactly are those quantum resources on which quantum cryptography is based upon? Consider, for a start, the following qualities of quantized systems: \renewcommand{(\roman{enumi})}{(\roman{enumi})} \begin{enumerate} \item randomness of certain individual events, such as the occurrence of certain measurement outcomes for states which are in a superposition of eigenstates associated with eigenvalues corresponding to these outcomes; \item complementarity, as proposed by Pauli, Heisenberg and Bohr; \item value indefiniteness, as attested by Bell, Kochen \& Specker and others (often, this property is referred to as ``contextuality''); \item interference and quantum parallelism, allowing the co-representation of classically contradicting states of information by a coherent superposition thereof; \item entanglement of two or more particles, as pointed out by Schr\"odinger, such that their state cannot be represented as the product of states of the isolated, individual quanta, but is rather defined by the {\em joint} or {\em relative} properties of the quanta involved. \end{enumerate}
The first quantum cryptographic protocols, such as the ones by Wiesner~\cite{wiesner} and Bennett \& Brassard~\cite{benn-84,benn-92}, just require complementarity and random individual outcomes. This might be perceived ambivalently as and advantage --- by being based upon only these two features --- yet also as a disadvantage, since they are not ``protected'' by Bell- or Kochen-Specker type value indefiniteness.
This article addresses two issues: a critical re-evaluation of quantum cryptographic protocols in view of quantum value indefiniteness; as well as suggestions to improve them to assure the best possible protection ``our''~\cite[p.~866]{born-26-1} present quantum theory can afford. In doing so, a toy model will be introduced which implements complementarity but still is value definite. Then it will be exemplified how to do perform ``quasi-classical'' quantum-like cryptography with these models. Finally, methods will be discussed which go beyond the quasi-classical realm.
Even nowadays it is seldom acknowledged that, when it comes to value definiteness, there definitely {\em is} a difference between two- and three-dimensional Hilbert space. This difference can probably be best explained in terms of (conjugate) bases: whereas different basis in two-dimensional Hilbert space are disjoint and separated (they merely share the trivial origin), from three dimensions onwards, they may share common elements. It is this inter-connectedness of bases and ``frames'' which supports both Gleason's and the Kochen-Specker theorem. This can, for instance, be used in derivations of the latter one in three dimensions, which effectively amount to a succession of rotations of bases along one of their elements (the original Kochen-Specker~\cite{kochen1} proof uses 117 interlinked bases), thereby creating new rotated bases, until the original base is reached. Note that certain (even dense~\cite{meyer:99}) ``dilutions'' of bases break up the possibility to interconnect, thus allowing value definiteness.
The importance of these arguments for physics is this: since in quantum mechanics the dimension of Hilbert space is determined by the number of mutually exclusive outcomes, a {\em necessary} condition for a quantum system to be protected by value indefiniteness thus is that the associated quantum system has {\em at least three} mutually exclusive outcomes; two outcomes are insufficient for this purpose. Of course, one could argue that systems with two outcomes are still protected by complementarity.
\section{Realizations of quantum cryptographic protocols}
Let us, for the sake of demonstration, discuss a concrete ``toy'' system which features complementarity but (not) value (in)definiteness. It is based on the partitions of a set. Suppose the set is given by $S=\{1,2,3,4\}$, and consider two of its equipartitions $A=\{\{1,2\},\{3,4\}\}$ and $B=\{\{1,3\},\{2,4\}\}$, as well as the usual set theoretic operations (intersection, union and complement) and the subset relation among the elements of these two partitions. Then $A$ and $B$ generate two Boolean algebras $L_A= \{\emptyset ,\{1,2\},\{3,4\},S\}$ and $L_B= \{\emptyset ,\{1,3\},\{2,4\},S\}$ which are equivalent to $2^2$; with two atoms $a_1=\{1,2\}$ \& $a_2=\{3,4\}$, as well as $b_1=\{1,3\}$ \& $b_2=\{2,4\}$ per algebra, respectively. Then, the partition logic $L_A \oplus L_B = L_{A,B} = \left\langle \{L_A,L_B\},\cap, \cup, ',\subset \right\rangle $ is obtained as a pasting construction from $L_A$ and $L_B$: only elements contribute which are in $L_A$, or in $L_B$, or in both $L_A \cap L_B$ of them (the atoms of this algebra being the elements $a_1,\ldots ,b_2$), and all common elements --- in this case only the smallest and greatest elements $\emptyset$ and $S$ --- are identified. $L_{A,B}$ ``inherits'' the operations and relations of its subalgebras (also called {\em blocks} or {\em contexts}) $L_A$ and $L_B$. This pasting construction yields a nondistributive and thus nonboolean, orthocomplemented propositional structure. Nondistributivity can quite easily be proven, as $a_1 \wedge (b_1 \vee b_2) \neq (a_1 \wedge b_1) \vee (a_1 \wedge b_2)$, since $b_1 \vee b_2=S$, whereas $a_1 \wedge b_1= a_1 \wedge b_2 =\emptyset$. Note that, although $a_1,\ldots ,b_2$ are compositions of elements of $S$, not all elements of the power set $2^S\equiv 2^4$ of $S$, such as $\{1\}$ or $\{1,2,3\}$, are contained in $L_{A,B}$.
Figure~\ref{2006-ql-nondist}(a) depicts a Greechie (orthogonality) diagram of $L_{A,B}$, which represents elements in a Boolean algebra as single smooth curves; in this case there are just two atoms (least elements above $\emptyset$) per subalgebra; and both subalgebras are not interconnected. \begin{figure}
\caption{ (a) Greechie diagram of $L_{A,B}$, consisting of two separate Boolean subalgebras $L_A$ and $L_B$; (b) two-dimensional configuration of spin-$\frac{1}{2}$ state measurements along two noncollinear directions. As there are only two mutually exclusive outcomes, the dimension of the Hilbert space is two. }
\label{2006-ql-nondist}
\end{figure}
Several realizations of this partition logic exist; among them \begin{enumerate} \item the propositional structure~\cite{birkhoff-36,svozil-ql} of spin state measurements of a spin-$\frac{1}{2}$ particle along two noncollinear directions, or of the linear polarization of a photon along two nonorthogonal, noncollinear directions. A two-dimensional Hilbert space representation of this configuration is depicted in Figure~\ref{2006-ql-nondist}(b). Thereby, the choice of the measurement direction decides which one of the two complementary spin state observables is measured; \item generalized urn models~\cite{wright,dvur-pul-svo}; in particular ones with black balls painted with two symbols having two possible values (say, ``$0$´´ and ``$1$´´) in two colors (say, ``red'' and ``green''), resulting in four types of balls --- more explicitly, carrying all variation of the symbols \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 0}}$}} \end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 1}}$}} \end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 0}}$}} \end{picture}, as well as \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 1}}$}} \end{picture} --- many copies of which are randomly distributed in an urn. Suppose the experimenter looks at them with one of two differently colored eyeglasses, each one ideally matching the colors of only one of the symbols, such that only light in this wave length passes through. Thereby, the choice of the color decides which one of the two complementary observables associated with ``red'' and ``green'' is measured. Propositions refers to the possible ball types drawn from the urn, given the information printed in the chosen color. \item initial state identification problem for deterministic finite (Moore or Mealy) automata in an unknown initial state~\cite{e-f-moore,svozil-2001-eua}; in particular ones $\left\langle S, I, O, \delta, \lambda \right\rangle$ with four internal states $S=\{1,2,3,4\}$, two input and two output states $I=O=\{0,1\}$,
an ``irreversible'' (all-to-one) transition function $\delta (s, i) =1$ for all $s\in S$, $i\in I$, and an output function ``modelling'' the state partitions by $\lambda (1,0)=\lambda (2,0) =0$, $\lambda (3,0)=\lambda (4,0) =1$, $\lambda (1,1)=\lambda (3,1) =0$, $\lambda (2,1)=\lambda (4,1) =1$. Thereby, the choice of the input symbol decides which one of the two complementary observables is measured. \end{enumerate}
Let us, for the moment, consider generalized urn models, because they allow a ``pleasant'' representation as chocolate balls coated in black foils and painted with color symbols. With the four types of chocolate balls \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 0}}$}} \end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 1}}$}} \end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 0}}$}} \end{picture}, and \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 1}}$}} \end{picture} drawn from an urn it is possible to execute the 1984 Bennett-Brassard (BB84) protocol~\cite{benn-84,benn-92} and ``generate'' a secret key shared by two parties~\cite{svozil-2005-ln1e}. Formally, this reflects (i) the random draw of balls from an urn, as well as (ii) the complementarity modeled {\em via} the color painting and the colored eyeglasses. It also reflects the possibility to embed this model into a bigger Boolean (and thus classical) algebra $2^4$ by ``taking off the eyeglasses'' and looking at both symbols of those four balls types simultaneously. The atoms of this Boolean algebra are just the ball types, associated with the four cases \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 0}}$}} \end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 1}}$}} \end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 0}}$}} \end{picture}, and \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 1}}$}} \end{picture}. The possibility of a classical embedding is also reflected in a ``sufficient'' number (i.e., by a separating, full set) of two-valued, dispersionless states $P(a_1)+P(a_2)=P(b_1)+P(b_2)=1$, with $P(x)\in \{0,1\}$. These two-valued states can also be interpreted as logical truth assignments, irrespective of whether or not the observables have been (co-)measured.
The possibility to ascribe certain ``ontic states'' interpretable as observer-independent ``omniscient elements of physical reality'' (in the sense of Einstein, Podolsky and Rosen~\cite[p.~777]{epr}, a paper which amazingly contains not a single reference) even for complementarity observables may raise some skepticism or even outright rejection, since that is not how quantum mechanics is known to perform ``at its most mind-boggling mode.'' Indeed, so far, the rant presented merely attempted to convince the reader that one can have complementarity {\em as well as} value definiteness; i.e., complementarity is not sufficient for value indefiniteness in the sense of the Bell- and Kochen-Specker argument.
Unfortunately, the two-dimensionality of the associated Hilbert space is also a feature plaguing present random number generators based on beam splitters~\cite{svozil-qct,rarity-94,zeilinger:qct,stefanov-2000}. In this respect, most of the present random number generators using beam splitters are protected only by the randomness of single outcomes as well as by complementarity, but are not by certified value indefiniteness, as guaranteed by quantum theory in its standard form~\cite{v-neumann-49}.
Their methodology should also be improved by the methods discussed below.
\section{Supporting cryptography with value indefiniteness}
Alas, quantum mechanics is more resourceful and mind-boggling than that, as it does not permit any two-valued states which may be ontologically interpretable as elements of physical reality. So we have to go further, reminding ourselves that value indefiniteness comes about only for Hilbert spaces of dimensions three and higher. There are several ways of doing this. The following options will be discussed: \begin{enumerate} \item the known protocols can be generalized to three or more outcomes; \item entangled pairs of particles~\cite{ekert91} associated with statistical value indefiniteness may be considered; \item full, nonprobabilistic value indefiniteness may be attempted, alt least counterfactually. \end{enumerate}
\subsection{Generalizations to three and more outcomes}
In constructing quantum random number generators {\em via} beam splitters which ultimately are used in cryptographic setups, it is important (i) to have full control of the particle source, and (ii) to use beam splitters with three or more output ports, associated with three- or higher-dimensional Hilbert spaces. Thereby, it is {\em not sufficient} to compose a multiport beam splitter by a succession of phase shifters and beam splitters with two output ports~\cite{rzbb,svozil-2004-analog}, based on elementary decompositions of the unitary group~\cite{murnaghan}.
Dichotomic sequences could be obtained from sequences containing more than two symbols by discarding all other symbols from that sequence~\cite{MR997340}, or by identifying the additional symbols with one (or both) of the two symbols. For standard normalization procedures and their issues, the reader is referred to Refs.~\cite{von-neumann1,elias-72,PeresY-1992,dichtl-2007,Lacharme-2008}.
One concrete realization would be a spin-$\frac{3}{2}$ particle. Suppose it is prepared in one of its four spin states, say the one associated with angular momentum $+\frac{3}{2}\hbar$ in some arbitrary but definite direction; e.g., by a Stern-Gerlach device. Then, its spin state is again measured along a perpendicular direction; e.g., by another, differently oriented, Stern-Gerlach device. Two of the output ports, say the ones corresponding to positive angular momentum $+\frac{3}{2}\hbar$ and $+\frac{1}{2}\hbar$, are identified with the symbol ``$0$,'' the other two ports with the symbol ``$1$.'' In that way, a random sequence is obtained from quantum coin tosses which can be ensured to operate under the conditions of value indefiniteness in the sense of the Kochen-Specker theorem. Of course, this protocol can also be used to generate random sequences containing four symbols (one symbol per detector).
With respect to the use of beam splitters, the reader is kindly reminded of another issue related to the fact that beam splitters are {\em reversible} devices capable of only translating an incoming signal into an outgoing signal in a {\em one-to-one} manner. The ``nondestructive'' action of a beam splitter could also be demonstrated by ``reconstructing'' the original signal through a ``reversed'' identical beam splitter in a Mach-Zehnder interferometer~\cite{green-horn-zei}. In this sense, the signal leaving the output ports of a beam splitter is ``as good'' for cryptographic purposes as the one entering the device. This fact relegates considerations of the quality of quantum randomness to the quality of the source. Every care should thus be taken in preparing the source to assure that the state entering the input port (i) either is pure and could subsequently be used for measurements corresponding to conjugate bases, (ii) or is maximally mixed, resulting in a representation of its state in finite dimensions proportional to the unit matrix.
\subsection{Configurations with statistical value indefiniteness}
Protocols like the Ekert protocol \cite{ekert91} utilize two entangled two-state particles for a generation of a random key shared by two parties. The particular Einstein-Podolsky-Rosen configuration~\cite{epr} and the singlet Bell state communicated among the parties guarantee stronger-than-classical correlations of their sequences, resulting in a violation of Bell-type inequalities obeyed by classical probabilities.
Although criticized~\cite{PhysRevLett.68.557} on the grounds that the Ekert protocol in certain cryptanalytic aspects is equivalent to existing ones (see Ref.~\cite{benn-92b} for a reconciliation), it offers additional security in the light of quantum value indefiniteness, as it suggests to probe the nonclassical parts of quantum statistics. This can best be understood in terms of the impossibility to generate co-existing tables of all --- even the counterfactually possible --- measurement outcomes of the quantum observables used~\cite{peres222}. This, of course, can only happen for the four-dimensional Hilbert space configuration proposed by Ekert, and not for effectively two-dimensional ones of previous proposals. As a result, the Eckert protocol cannot be performed with chocolate balls. Formally, this is due to the nonexistence of two-valued states in four-dimensional Hilbert space.
Suppose one would nevertheless attempt to ``mimic'' the Ekert protocol with a classical ``singlet'' state which uses compositions of two balls of the form \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 0}}$}} \end{picture}---\unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 1}}$}} \end{picture} / \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 1}}$}} \end{picture}---\unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 0}}$}} \end{picture} / \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 0}}$}} \end{picture}---\unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4 ,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 1}}$}} \end{picture} / \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 1}}$}} \end{picture}---\unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 0}}$}} \end{picture}, with strictly different (alternatively strictly identical) particle types. The resulting probabilities and expectations would obey the classical Clauser-Horne-Shimony-Holt bounds~\cite{chsh}. This is due to the fact that generalized urn models have quasi-classical probability distributions which can be represented as convex combinations of the full set of separable two-valued states on their observables.
\subsection{Nonprobabilistic value indefiniteness}
In an attempt to fully utilize quantum value indefiniteness, we propose a generalization of the BB84 protocol on a propositional structure which does not allow any two-valued state. In principle, this could be any kind of finite configuration of observables in three- and higher-dimensional Hilbert space; in particular ones which have been proposed for a proof of the Kochen-Specker theorem. \begin{figure}
\caption{ Greechie orthogonality diagram of a ``short'' proof~\cite{cabello-96,cabello:210401} of the Kochen-Specker theorem in four dimensions containing 24 propositions in 24 tightly interlinked contexts~\cite{tkadlec-priv}. The graph cannot be colored by the two colors red (associated with truth) and green (associated with falsity) such that every context contains exactly one red and three green point. For the sake of a proof, consider just the six outer lines and the three outer ellipses. Then in a table containing the points of the contexts as columns and the enumeration of contexts as rows, every red point occurs in exactly {\em two} contexts, and there should be an {\em even} number of red points. On the other hand, there are nine contexts involved; thus by the rules it follows that there should be an {\em odd} number (nine) of red points in this table (exactly one per context). }
\label{2009-QvPR}
\end{figure}
\begin{figure}
\caption{ Subdiagrams of Figure~\ref{2009-QvPR} allowing (value definite) chocolate ball realizations. }
\label{2009-QvPRtria}
\end{figure}
For the sake of a concrete example, we shall consider the tightly interlinked collection of observables in four-dimensional Hilbert space presented by Cabello, Estebaranz and Garc{\'i}a-Alcaine~\cite{cabello-96,cabello:210401}, which is depicted in Figure~\ref{2009-QvPR}. Instead of two measurement bases of two-dimensional Hilbert space used in the BB84 protocol, nine such bases of four-dimensional Hilbert space, corresponding to the nine smooth (unbroken) orthogonal curves in Fig.~\ref{2009-QvPR} are used. In what follows, it is assumed that any kind of random decision has been prepared according to the protocol for generating random sequences sketched above.
\begin{enumerate} \item In the first step, ``Alice'' randomly picks an arbitrary basis from the nine available ones, and sends a random state to ``Bob.'' \item In the second step, Bob independently from Alice, picks another basis at random, and measures the particle received from Alice. \item In the third step, Alice and Bob compare their bases over a public channel, and keep only those events which were recorded either in a common basis, or in an observable interlinking two different bases. \item Both then exchange some of the remaining matching outcomes over a public channel to assure that nobody has attended their quantum channel. \item Bob and Alice encode the four outcomes by four or less different symbols. As a result, Bob and Alice share a common random key certified by quantum value indefiniteness. \end{enumerate}
The advantage of this protocol resides in the fact that is does not allow its realization by any partition of a set, or any kind of colored chocolate balls. Because if it did, any such coloring could be used to generate ``classical'' two-valued states, which in turn may be used towards a classical re-interpretation of the quantum observables; an option ruled out by the Kochen-Specker theorem.
Readers not totally convinced at this point might, for the sake of demonstration, consider a generalized urn model with nine colors, associated with the nine bases in Figure~\ref{2009-QvPR}. Suppose further that there is a uniform set of symbols, say $\{0,1,2,3\}$ for all four colors. If all varieties (permutations) contribute, the number of different types of balls should be $4^9$. Note, however, that every interlinked color must have {\em identical} (or at least unique ``partner'') symbols in the interlinking colors; a condition which cannot be satisfied ``globally'' for all the interlinks in Figure~\ref{2009-QvPR}.
A simplified version of the protocol, which is based on a subdiagram of Figure~\ref{2009-QvPR}, contains only three contexts, which are closely interlinked. The structure of observables is depicted in Figure~\ref{2009-QvPRtria}(a). The vectors represent observables in four-dimensional Hilbert space in their usual interpretation as projectors generating the one-dimensional subspaces spanned by them. In addition to this quantum mechanical representation, and in contrast to the Kochen-Specker configuration in Figure~\ref{2009-QvPR}, this global collection of observables still allows for value definiteness, as there are ``enough'' two valued states permitting the formation of a partition logic and thus a chocolate ball realization; e.g., $$ \begin{array}{c} \{ \{ \{1,2 \}, \{ 3,4,5,6,7 \}, \{ 8,9,10,11,12 \}, \{13,14 \} \}, \\ \{ \{1,4,5,9,10 \}, \{ 2,6,7,11,12 \}, \{ 3,8 \}, \{ 13,14 \} \}, \\ \{ \{ 1,2 \}, \{ 3,8 \}, \{ 4,6,9,11,13 \}, \{ 5,7,10,12,14 \} \} \}. \end{array} $$ The three partitions of the set $\{1,2,\ldots ,14\}$ have been obtained by indexing the atoms in terms of all the nonvanishing two-valued states on them~\cite{svozil-2001-eua,svozil-2008-ql}, as depicted in Figure~\ref{2009-qcho-f2vs}. They can be straightforwardly applied for a chocolate ball configuration with three colors (say green, red and blue) and four symbols (say 0, 1, 2, and 3). The 14 ball types corresponding to the 14 different two-valued measures are as follows: \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 0}}{\color{yellow} {\bf 0}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 1}}{\color{yellow} {\bf 0}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 2}}{\color{yellow} {\bf 1}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 0}}{\color{yellow} {\bf 2}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 0}}{\color{yellow} {\bf 3}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 1}}{\color{yellow} {\bf 2}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 1}}{\color{yellow} {\bf 3}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 2}}{\color{Turquoise1} {\bf 2}}{\color{yellow} {\bf 1}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 2}}{\color{Turquoise1} {\bf 0}}{\color{yellow} {\bf 2}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 2}}{\color{Turquoise1} {\bf 0}}{\color{yellow} {\bf 3}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 2}}{\color{Turquoise1} {\bf 1}}{\color{yellow} {\bf 2}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 2}}{\color{Turquoise1} {\bf 1}}{\color{yellow} {\bf 3}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 3}}{\color{Turquoise1} {\bf 3}}{\color{yellow} {\bf 2}}$}}\end{picture}, and \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 3}}{\color{Turquoise1} {\bf 3}}{\color{yellow} {\bf 3}}$}}\end{picture}.
Figure~\ref{2009-QvPRtria}(b) contains a three-dimensional subconfiguration with two complementary contexts interlinked in a single observable. It again has a value definite representation in terms of partitions of a set, and thus again a chocolate ball realization with three symbols in two colors; e.g., \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 0}}$}} \end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 1}}$}} \end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 2}}$}} \end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 2}}{\color{Turquoise1} {\bf 1}}$}} \end{picture}, and \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(8,8) \put(4,2){\circle*{8}} \put(4,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 2}}{\color{Turquoise1} {\bf 2}}$}} \end{picture}.
\begin{figure}
\caption{ Two-valued states interpretable as global truth functions of the observables depicted in Figure~\ref{2009-QvPRtria}(a). Encircled numbers count the states, smaller numbers label the observables. }
\label{2009-qcho-f2vs}
\end{figure}
\section{Noncommutative cryptography which cannot be realized quantum mechanically}
Quantum mechanics does not allow a ``triangular'' structure of observables similar to the one depicted in Fig.~\ref{2009-QvPRtria} with {\em three} instead of four atoms per block (context), since no geometric configuration of tripods exist in three-dimensional vector space which would satisfy this scheme. (For a different propositional structure not satisfiable by quantum mechanics, see Specker's programmatic article~\cite{specker-60} from 1960.) It contains six atoms $1,\ldots ,6$ in the blocks 1--2--3, 3--4--5, 5--6--1. In order to obtain a partition logic on which the chocolate ball model can be based, the four two-valued states are enumerated and depicted in Figure~\ref{2009-qcho-f2vs-2}. \begin{figure}
\caption{ Two-valued states on triangular propositional structure with three atoms per context or block. }
\label{2009-qcho-f2vs-2}
\end{figure}
The associated partition logic is given by $$ \begin{array}{c} \{ \{ \{1 \}, \{2 \}, \{ 3,4 \} \}, \\ \{ \{1,4 \}, \{ 2 \}, \{ 3 \} \}, \\ \{ \{ 1 \}, \{ 2,4 \}, \{ 3 \} \} \}. \end{array} $$ Every one of the three partitions of the set $\{1,\ldots ,4\}$ of ball types labelled by 1 through 4 corresponds to a color; and there are three symbols per colors. For the first (second/third) partition, the propositions associated with these protocols are: \begin{itemize} \item ``when seen through light of the first (second/third) color (e.g., pink/light blue/yellow), symbol ``0'' means ball type number 1 (2/3);'' \item ``when seen through light of the first (second/third) color (e.g., pink/light blue/yellow), symbol ``1'' means ball type number 3 or 4 (1 or 4/2 or 4);'' \item ``when seen through light of the first (second/third) color (e.g., pink/light blue/yellow), symbol ``2'' means ball type number 2 (3/1).'' \end{itemize} More explicitly, there are four ball types of the form \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 0}}{\color{Turquoise1} {\bf 1}}{\color{yellow} {\bf 2}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 2}}{\color{Turquoise1} {\bf 0}}{\color{yellow} {\bf 1}}$}}\end{picture}, \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 2}}{\color{yellow} {\bf 0}}$}}\end{picture}, and \unitlength 0.7mm \allinethickness{1pt}\begin{picture}(12,12)\put(6,2){\circle*{12}} \put(6,2){\makebox(0,0)[cc]{${\color{DeepPink1} {\bf 1}}{\color{Turquoise1} {\bf 1}}{\color{yellow} {\bf 1}}$}}\end{picture}. The resulting propositional structure is depicted in Fig.~\ref{2009-QvPRtria-2}. With respect to realizability, cryptographic protocols --- such as the one sketched above --- based on this structure are ``stranger than quantum mechanical'' ones. \begin{figure}
\caption{ Propositional structure allowing (value definite) chocolate ball realizations with three atoms per context or block which does not allow a quantum analog. }
\label{2009-QvPRtria-2}
\end{figure}
\section{Summary and discussion}
It has been argued that value indefiniteness should be used as a quantum resource against cryptanalytic attacks, as complementarity may not be a sufficient resource for the type of ``objective'' security envisaged by quantum cryptography. A necessary condition for this quantum resource is the presence of at least three mutually exclusive outcomes.
It may be objected that quantum complementarity suffices as resource against cryptanalytic attacks, and thus the original BB84 protocol needs not be amended. To this criticism I respond with a performance of the original BB84 protocols with chocolate balls~\cite{svozil-2005-ln1e}; or more formally, by stating that configurations with just two outcomes leave open the possibility of a quasi-classical explanation, as they cannot rule out the existence of sufficiently many two-valued states in order to construct homeomorphisms, i.e., structure-preserving maps between the quantum and classical observables. Thus, when it comes to fully ``harvesting'' the quantum, it appears prudent to utilize value indefiniteness, one of its most ``mind-boggling'' features encountered if one assumes the existence of nonoperational yet counterfactual observables.
$\;$\\ {\bf Acknowledgements} \\ The author gratefully acknowledges discussions with Cristian Calude and Josef Tkadlec, as well as the kind hospitality of the {\it Centre for Discrete Mathematics and Theoretical Computer Science (CDMTCS)} of the {\it Department of Computer Science at The University of Auckland.} This work was also supported by {\it The Department for International Relations} of the {\em Vienna University of Technology.} The pink--light blue--yellow coloring scheme is by Renate Bertlmann; communicated to the author by Reinhold Bertlmann.
\end{document} | arXiv |
Symplectic representation
In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (V, ω) which preserves the symplectic form ω. Here ω is a nondegenerate skew symmetric bilinear form
$\omega \colon V\times V\to \mathbb {F} $
where F is the field of scalars. A representation of a group G preserves ω if
$\omega (g\cdot v,g\cdot w)=\omega (v,w)$
for all g in G and v, w in V, whereas a representation of a Lie algebra g preserves ω if
$\omega (\xi \cdot v,w)+\omega (v,\xi \cdot w)=0$
for all ξ in g and v, w in V. Thus a representation of G or g is equivalently a group or Lie algebra homomorphism from G or g to the symplectic group Sp(V,ω) or its Lie algebra sp(V,ω)
If G is a compact group (for example, a finite group), and F is the field of complex numbers, then by introducing a compatible unitary structure (which exists by an averaging argument), one can show that any complex symplectic representation is a quaternionic representation. Quaternionic representations of finite or compact groups are often called symplectic representations, and may be identified using the Frobenius–Schur indicator.
References
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103..
| Wikipedia |
\begin{document} \title{The Gysin map is compatible with mixed Hodge Structures}
\section {Introduction} The language of Bivariant Theory was developed in \ci{fuma} and it turns out to be extremely useful in Intersection Theory and in Riemann-Roch-type questions (cfr. \ci{fulton}). The Chow-theoretic version associates a graded group $A^*(X \stackrel{f}{\to} Y)$ with a morphism $f:X \to Y$ of algebraic schemes.
\n There are a product structure, a proper push-forward and a pull-back.
\n In the case of the structural map $X \to Spec \,k$, we find the Chow groups $A^*(X \stackrel{}{\to} Spec \, k) \simeq A_{-*}(X)$, and in the case of the identity map the groups $A^*(X \stackrel{Id}{\to} X)$ are called the Chow cohomology groups.
\n When $X$ is nonsingular these latter groups agree with the Chow groups: $$ A^*(X \stackrel{Id}{\to} X) \simeq A_{\dim{X}-*}(X) $$ and the bivariant product structure agrees with the usual intersection product.
For varieties defined over the field of complex numbers, the topological counterpart, i.e. Topological Bivariant Theory, admits the characterization $$H^i(X \stackrel{f}{\to} Y) \simeq Hom_{D^b(Y)}(Rf_! \rat_X, \rat_Y[i]),$$ from which it follows immediately that $H^*(X \stackrel{}{\to} \{point\}) \simeq Hom (H^*_c(X), \rat)= H_{*}^{BM}(X)$, the Borel-Moore homology groups, and $H^i(X \stackrel{Id}{\to} X)= Hom_{D^b(Y)}( \rat_X, \rat_X[i])=H^i(X, \rat)$, the cohomology groups.
It is quite natural to expect that the topological bivariant groups can be endowed with natural mixed Hodge structures (MHS) and that the maps arising in the context of the bivariant formalism should be compatible with MHS.
\n It seems that this question can be satisfactorily dealt-with only after having developed the Hodge theory of maps, and in particular the Hodge-theoretic version of the Decomposition Theorem (cfr. \ci{decatmig2}).
In this note we shall be less ambitious and we shall deal with a very special yet useful case which does not seem to be available in the literature in the generality which is needed in some applications. See, for example, those contained in \ci{decatmig}, where morphisms between cohomology groups associated with correspondences between global finite group quotients of smooth, not necessarily complete, varieties are stated to be maps of MHS.
Consider a codimension $d$ regular embedding $h:Y \longrightarrow X$. There is a refined Gysin homomorphism $[h] \in H^{2d}(Y \stackrel {h}{\to} X) \simeq H^{2d}(X, X\setminus Y)$, cfr. \ci{fulton}, 19.2, which gives, for any map $X' \longrightarrow X$, the so-called refined Gysin maps $h^!: H_{*}^{BM}(X')\to H_{*-2d}^{BM}(Y \times _X X')$.
Theorem \ref{gyismhs} states the compatibility of the Gysin map $ h^!$ with the MHS involved and it is proved using the definition of the Gysin map via specialization to the normal cone (\ci{fulton}, \ci{verdier}).
We work with algebraic varieties and algebraic schemes defined over the field of complex numbers. We use cohomology etc. with rational coefficients.
The first-named author dedicates this paper to the memory of Meeyoung Kim.
\section{Remarks on mixed Hodge structures} \label{romhs} Given a morphism $f:Y \to X$ of algebraic schemes, the relative cohomology groups $H^*(X \hbox{mod}\,Y)$ are given a MHS which is functorial in the map $f,$ in the sense that given a commutative diagram of maps of algebraic schemes $$ \begin{array}{ccccc} Y' & \stackrel{i_1}\to & Y \\ \downarrow f' & & \downarrow f \\ X' & \stackrel{i_2}\to & X, \end{array} $$ the natural morphism $H^*(X \hbox{mod}\,Y) \to H^*(X' \hbox{mod} \, Y') $ is a morphism of Mixed Hodge structures (cfr. \ci{ho3}, Exemple 8.3.8; see also \ci{du}, $\S2$). The relative cohomology groups are the cohomology groups of the simplicial scheme $C(f).$
\n The diagram above gives rise to a map of simplicial schemes $C(f')\to C(f)$, hence the morphism of MHS.
Given an algebraic scheme $U,$ one has, for every open immersion $U \longrightarrow U'$ into a proper algebraic scheme, $H^l_c(U) \simeq H^l(U', U'\setminus U)$.
\n It follows that cohomology with compact supports admits a natural MHS which is functorial for open immersions and proper maps.
\n Dually, since $H^{BM}_l(U) \simeq H^l_c(U)^{\vee}$, the same is true for Borel-Moore homology.
\section{The deformation to the normal cone} \label{tdtnc} For more details on what follows, see \ci{fulton} and \ci{verdier}.
Let $h:Y \longrightarrow X$ be a closed embedding of algebraic schemes, and ${\cal I}_Y \subseteq {\cal O}_X$ be the corresponding sheaf of ideals.
\n The {\em normal cone to $Y$ in $X$} is defined to be the algebraic scheme $$ C_{Y}X \, :=\, Spec_{{\cal O}_Y} \, \bigoplus_{n\geq 0} {\cal I}_Y^n/{\cal I}_Y^{n+1} $$ There are natural maps $ Y \longrightarrow C_{Y}X \longrightarrow Y,$ where the second one is the (affine) cone-bundle projection to $Y$ and the first one is the closed embedding of the zero-section.
\n If the embedding of $Y$ in $X$ is regular of codimension $d$, then $C_Y X$ is naturally identified with the rank $d$ normal bundle $N_{Y, X}$ of $Y$ in $X.$
\n The exceptional divisor of the blowing-up $Bl_Y X \longrightarrow X$ of $X$ along $Y$ is the projectivisation $P(C_Y X).$
Let us recall how the deformation to the normal cone and the canonical maps which are associated with it are defined.
Let $M = Bl_{ Y \times \{0\} } X \times {\Bbb A}^1$ be the blowing-up of $X \times {\Bbb A}^1$ along $Y \times \{0\}.$
\n The exceptional divisor can be naturally identified with $P(C_YX \oplus {\cal O}_Y).$
\n The resulting map $\tilde{p}: M \longrightarrow {\Bbb A}^1$ is flat.
We have that:
\n -- the blowing-up $Bl_YX$ is naturally embedded in $\tilde{p}^{-1}(0) \subseteq M;$
\n --
$\tilde{p}^{-1} (0) = P(C_Y X \oplus {\cal O}_Y ) \cup Bl_Y X;$
\n -- $P(C_YX \oplus {\cal O}_Y ) \cap Bl_Y X $ can be viewed either as the exceptional divisor of the blowing-up $Bl_Y X,$ or as the divisor at infinity of $P(C_Y X \oplus {\cal O}_Y );$
\n -- $C_YX$ embeds in the exceptional divisor $P(C_YX \oplus {\cal O}_Y)$ as the complement of the divisor at infinity;
\n -- $Y$ embeds in $C_YX \subseteq P(C_YX \oplus {\cal O}_Y)$ as the zero section;
\n -- $Y \times {\Bbb A}^1$ embeds in $M,$ compatibly with the projection to ${\Bbb A}^1;$ in particular,
over ${\Bbb A}^1 \setminus \{0 \},$ it is just the product embedding into $X \times {\Bbb A}^1 \setminus \{0 \};$ $Y\times \{0 \}$ is embedded in $p^{-1}(0)$ as the zero-section of $C_Y X \subseteq P(C_YX \oplus O_Y).$
Let $$ M \setminus Bl_Y X \, =:\, M' \stackrel{p}\longrightarrow {\Bbb A}^1 $$ be the natural flat map. The map $p$ is not proper, even if $X$ is complete.
We say that the embedding $Y \subseteq X$ (i.e. the situation at $p^{-1}(t \neq 0)$) deforms to the embedding $Y \subseteq C_YX$ (i.e. the situation at $p^{-1}(0)$).
\n This construction is called the {\em deformation to the normal cone}.
There are specialization maps (\ci{verdier}) : $$ h^{?}: H^{BM}_l(X) \longrightarrow H^{BM}_l(C_Y X), \qquad {h^{?}}^{\vee}: H^l_c(C_Y X) \longrightarrow H^l_c(X) , $$ whose construction is recalled in the proof of \ref{gyismhs}.
In the case of a regular embedding the normal cone is in fact the normal bundle and the flat pull-back gives an isomorphism $H^{BM}_l(Y) \simeq H^{BM}_{l+2d} (C_Y X)$.
The Gysin map $h^!$ is, by definition, the composition $H^{BM}_l(X) \longrightarrow H^{BM}_l(C_Y X) \longrightarrow H^{BM}_{l-2d}(Y).$
\section{The main result} \label{tmr}
\begin{tm} \label{gyismhs} Let $h:Y \longrightarrow X$ and $C_Y X$ be as above. The natural maps $$ h^{?}: H^{BM}_l(X) \longrightarrow H^{BM}_l(C_Y X), \qquad {h^{?}}^{\vee}: H^l_c(C_Y X) \longrightarrow H^l_c(X) $$ are maps of MHS of type $(0,0).$
\n If $h$ is regular of pure codimension $d,$ then the Gysin map and its dual $$ h^{!} : H^{BM}_l(X) \longrightarrow H^{BM}_{l-2d}(Y), \qquad { h^! }^{\vee}: H^l_c(Y) \longrightarrow H^{l+2d}_c(X) $$ are maps of MHS of type $(-d,-d)$ and $(d,d),$ respectively. \end{tm}
\n {\em Proof.} We first consider a commutative diagram of maps of algebraic schemes $$ \begin{array}{ccccc} U & \stackrel{j}\longrightarrow & {\cal X} & \stackrel{i}\longleftarrow & B \\
& p \searrow & \downarrow \pi & \swarrow b & \\ & & S \end{array} $$ where $j$ is an open immersion, $B := {\cal X} \setminus U$
and $\pi$ is proper.
\n We have a distinguished triangle in the bounded derived category $D^b_{cc}({\cal X})$ of the category of constructible complexes of rational vector spaces on ${\cal X}:$ $$ Rj_! j^! \rat_{\cal X} (\simeq Rj_!\rat_{U})
\longrightarrow \rat_{\cal X} \longrightarrow Ri_* \rat_{B} \stackrel{+1}\longrightarrow , $$ to which we apply $R\pi_*\simeq R\pi_!$ and get $$ Rp_! \rat_{U} \longrightarrow R\pi_*\rat_{\cal X} \longrightarrow Rb_* \rat_{B} \stackrel{+1}\longrightarrow . $$ It follows that $$ {\Bbb H}^l(S,Rp_! \rat_{U}) \simeq H^l({\cal X}, B), \quad \forall l. $$
\n Let $s \in S$ be a point. By the remarks in $\S$\ref{romhs}, the map of pairs: $$ ( \pi^{-1}(s), b^{-1}(s) ) \longrightarrow ({\cal X}, B) $$ induces natural maps $H^l({\cal X}, B) \to H^l(\pi^{-1}(s), b^{-1}(s) )$ of MHS for every $l.$ Since $\pi$ is proper, $\pi^{-1}(s)$ is compact and the latter group can be identified with $H^l_c(p^{-1}(s)).$ This endows these latter groups with MHS.
\n To summarize, the natural maps $$ {\Bbb H}^l(S, Rp_! \rat_U) \to H_c^l(p^{-1}(s)) $$ are of MHS, for every $s \in S$ and for every $l.$
\n We apply what above to the following situation.
\n Let $Y' \subseteq X'$ be an algebraic compactification of $Y \subseteq X$, ${\cal X} := Bl_{Y'\times \{0 \} } (X' \times {\Bbb A}^1 )$, $B:= (X' \setminus X) \times ({\Bbb A}^1 \setminus \{0 \} ) \coprod Bl_{Y'}X' \times \{0 \} \coprod C_{Y' \setminus Y}(X' \setminus X) ,$ $U: = {\cal X} \setminus B.$ Let $\pi: {\cal X} \longrightarrow {\Bbb A}^1$ be the natural map and $p$ and $b$ the induced maps.
\n Note that $p^{-1}( \{ 0 \} ) = C_Y X.$
\n We have maps of MHS by choosing $s=0$ and $s =s_0 \neq 0$: $$ H_c^{l}(p^{-1}(0) ) \stackrel{a}\longleftarrow {\Bbb H}^{l}(Rp_! \rat_U)\simeq H^{l}€({\cal X}, B) \longrightarrow H_c(p^{-1}(s_0)). $$
\n CLAIM: $a$ is an isomorphism of MHS for every $l.$
\n {\em Proof.} It is enough to prove that $a$ is an isomorphism. Consider the fundamental system of neighborhoods of the point $\{0\} \in
{\Bbb A}^1$ given by disks $D_r= \{ z : \, |z| <r \}.$ Let ${\cal X}_r:= \pi^{-1}(D_r)$, $U_r: p^{-1}(D_r),$
$B_r=b^{-1}(D_r)$, $p_r:= p_{|U_r}.$ Since ${\cal X}$ is a product away from $\pi^{-1}( \{ 0 \} ),$ the homotopy axiom for relative cohomology ensures that the restriction maps ${\Bbb H}^l({\Bbb A}^1, Rp_! \rat_U) \simeq H^l({\cal X}, B) \longrightarrow H^l({\cal X}_r, B_r)\simeq {\Bbb H}^l(U_r, R{p_r}_! \rat_{U_r})$ are isomorphisms for every $r>0.$
\n The complex $Rp_! \rat_U$ is constructible with respect to the stratification $({\Bbb A}^1 \setminus \{0\}, \{ 0 \})$ of ${\Bbb A}^1$ and, by \ci{go-ma2}, $\S1.4$, this implies that the restriction map $H^l ({\cal X}, B)\simeq {\Bbb H}^l( {\Bbb A}^1, Rp_! \rat_U)) \stackrel{a}\longrightarrow H^l_c(p^{-1}(0))$
is an isomorphism.
\n The first assertion of the Theorem follows.
\n If the embedding $Y \subseteq X$ is regular of codimension $d$, then $C_Y X$ is the normal bundle of $Y$ in $X$ and we have an isomorphism of MHS $H^{BM}_l(Y) \simeq H^{BM}_{l+2d} (C_Y X)$ of type $(d,d)$ and the Gysin map $h^!$ being, by definition, the composition $H^{BM}_l(X) \longrightarrow H^{BM}_l(C_Y X) \longrightarrow H^{BM}_{l-2d}(Y),$ is therefore of MHS of type $(-d,-d).$ \blacksquare
\begin{rmk} \label{lci} {\rm After simple modifications to the statement and to the proof, Theorem \ref{gyismhs} holds when $h$ is a local complete intersection morphism. } \end{rmk}
The following corollary can be used when dealing with correspondences:
\begin{cor} \label{corrmhs} Let $X \stackrel{p}\longleftarrow \Gamma \stackrel{q}\longrightarrow Y$ be algebraic maps of algebraic varieties. Assume that the graph embedding $h: \Gamma \longrightarrow \Gamma \times X$ is regular and that $q$ is proper. Then the natural map $$ \Gamma_*: H^{BM}_{\bullet}(X) \longrightarrow H^{BM}_{ \bullet + 2(\dim{\Gamma} - \dim{X} )} (Y) $$ is of MHS of type $(\dim{\Gamma} - \dim{X},\dim{\Gamma} - \dim{X}).$ \end{cor} {\em Proof.} We have $\Gamma_*(x) = q_* ( h^! ( [\Gamma]\times x ) )$ and all operations involved are compatible with MHS. \blacksquare
\begin{rmk} \label{rgmq} {\rm The same statement above holds when $X$ is assumed to be smooth or a quotient $X=X'/G$ of a smooth variety by a finite group, in which case one modifies the statement by working with $X'$ and with the distinguished irreducible component of $\Gamma \times_X X'.$ We leave this task to the reader. } \end{rmk}
\section{Examples}
\label{ex} The paper \ci{decatmig} contains several examples of correspondences, stemming from maps between surfaces, from Hilbert schemes of points on surfaces, semi-small resolutions of singularities etc. In what follows, for the reader's convenience, we offer two of the applications of Theorem \ref{gyismhs} contained in \ci{decatmig}.
\subsection{Small resolutions} \label{sr} Let $f_{i}:X_{i} \longrightarrow Y,$ $i=1,\,2,$ be {\em small} resolutions of the singularities of an algebraic variety $Y.$ This means that the $X_{i}$ are nonsingular, the $f_{i}$ are proper and birational morphisms, and the not necessarily irreducible algebraic schemes $X_{i}\times_{Y} X_{i} $ have exactly one component of maximal dimension $\dim{Y}.$ This component is the unique component dominating the spaces $X_{i}$ under either projection. See \ci{decatmig} for more on this notion and more references. Note that the same will be true for $X_{1}\times_{Y}X_{2}.$
Let $D_{12} \subseteq X_{1}\times_{Y} X_{2}$ be the unique irreducible component of maximal dimension $\dim{Y}.$
Theorem \ref{gyismhs} and a simple calculation using
the calculus of correspondences
imply the following
\begin{pr}
\label{mhsisosm} The maps $$ {D_{ij}}_{*} \, : \, H^{BM}_{\bullet}(X_{i}) \longrightarrow H^{BM}_{\bullet}(X_{j}) $$ are isomorphisms of MHS with inverse ${D_{ji}}_{*}.$
\n In particular, the virtual Hodge-Deligne numbers of the varieties $X_{i},$ $i=1,\,2,$ coincide. \end{pr} {\em Proof.} See \ci{decatmig}, $\S$3. \blacksquare
\subsection{ Wreath products, rational double points and orbifolds} \label{wp} The combinatorial details of this example are rather lengthy. We omit them in favor of the end result. The interested reader can see \ci{decatmig}, $\S$7.3.
Let $Y'$ be a smooth algebraic surface and $G \subseteq SL_{2}(\comp)$ be a finite group acting on $Y'$ with only isolated fixed points.
Let $Y := Y'/G$ and $f: X \longrightarrow Y$ be its minimal resolution.
The semi-direct product $G_{n}$ (called the Wreath product) of $G^{n}$ with the symmetric group in $n$ letters $S_{n}$ acts on ${Y'}^{n}.$
The Hilbert scheme of $n-$points $X^{[n]}$ is a semi-small resolution of the singularities of ${Y'}^{n}/G_{n}.$
There is an explicit collection of correspondences that arises in this situation.
There is the notion of orbifold cohomology groups $H^{*}({Y'}^{n}/G_{n})_{orb}$ for the pair $({Y'}^{n}, G_{n}).$ These groups carry a natural MHS.
Theorem \ref{gyismhs} allows to prove, using the aforementioned collection of correspondences, the following
\begin{pr}
\label{orbimhs}
There is a canonical isomorphism of MHS
$$
H^{*}({Y}'^{n}/G_{n})_{orb} \, \simeq \, H^{*}(X^{[n]}).
$$
\end{pr}
\noindent Authors' addresses:
Mark Andrea A. de Cataldo, Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794, USA. \quad e-mail: {\em [email protected]}
Luca Migliorini, Dipartimento di Matematica, Universit\`a di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, ITALY. \quad e-mail: {\em [email protected]}
\end{document} | arXiv |
Adaptively partitioned block-based contrast enhancement and its application to low light-level video surveillance
Seungwon Lee1,
Nahyun Kim1 &
Joonki Paik1
This paper presents a dark region detection and enhancement method with low computational complexity for low-cost imaging devices. Conventional contrast enhancement methods generally have an oversaturation problem while brightness of the dark region increases. To solve this problem, the proposed method first divides an input image into dark object and bright background regions using adaptively partitioned blocks. Next, the contrast stretching is performed only in the dark region. The major advantage of the proposed method is the minimized block artifacts using optimally partitioned blocks using fuzzy logic and a refining step to accurately detect boundaries between two regions. Experimental results show that the proposed method can efficiently enhance the contrast of backlit images without the oversaturation problem. Because of low computational complexity, the proposed method can be applied to enhance very low light-level video sequences for video surveillance systems.
Recent advances in digital image processing systems enable users to easily acquire high quality images using compact, inexpensive digital cameras. However, a limited dynamic range is still a bottleneck of the camera technology (Debevec and Malik 1997). Because of the limited dynamic range, an image having both dark objects and bright background either loses object information or becomes over-saturated in the background region. An efficient image enhancement algorithm is required to enhance the contrast of the dark objects without over-saturation in the background.
Histogram equalization (HE) is a global contrast enhancement method for solving the unbalanced illumination problem in the image (Wang and Ye 2005). However, it tends to make the background saturated and amplifies the noise in the dark region of the image. For addressing this issue, several versions of improved HE algorithms have been proposed. The adaptive histogram equalization (AHE) method adaptively partitions the image into multiple sub-blocks for block-based local histogram equalization at the cost of blocking artifacts (Zimmerman et al. 1998). The bi-histogram equalization (BHE) method enhances the contrast of backlit images while preserving the average brightness. It is difficult to accurately separate background and object regions using a single threshold value to bisect the histogram (Kim 1997). The dualistic sub-image histogram equalization (DSIHE) method is similar to BHE except that the threshold value is selected using the median value of an image (Wan et al. 2003). As a result, DSIHE enhances the contrast of the images while preserving the mean brightness. However, the disordered histogram results in either over-saturation or under exposure. The recursive mean-separate histogram equalization (RMSHE) method performs iterative BHE for preserving the average brightness (Chen and Ramlli 1999). However, the effect of contrast enhancement decreases as the iteration continues. The gain controllable clipped histogram equalization (GC-CHE) method dynamically controls the clipping level of the histogram for appropriately re-distributing the dynamic range (Kim and Paik 2008).
For solving the problem of above mentioned global contrast enhancement methods, locally adaptive contrast enhancement methods have also been proposed. Kim et al. divided backlit and background regions using a set of optimal threshold values. Contrast enhancement is then performed only in the backlit region (Kim et al. 2013). However, blocking artifacts are generated in the boundary between the two regions. The retinex-based method can be considered as a locally adaptive contrast enhancement method that reduces the illumination dependency and stretches the dynamic range of only reflectance component using a Gaussian filter (Kim et al. 2011). However, if the size of the Gaussian filter is not appropriately selected, the processed image contains halo effect and color distortion.
In this context most conventional contrast enhancement methods have the problem of under- or over-saturation with color distortion. To overcome this problem, adaptively partitioned block-based dark region detection and enhancement is presented as shown in Fig. 1.
Block diagram of the proposed dark region detection and enhancement (top) and the corresponding output of each block (bottom)
As show in Fig. 1, the proposed method separates the image into the dark and background regions using adaptively partitioned blocks based on the optimal threshold value computed by fuzzy C-means clustering (FCM). More specifically, the proposed method partitions the input image into non-overlapped blocks of size \(64\times 64 \), and classifies them as dark, background, and ambiguous regions using the optimal threshold. The ambiguous blocks are further partitioned into four sub-blocks, which are then re-classified in the same manner. This partitioning process is repeated until the size of a block becomes \(4\times 4 \). Finally, the detected block-based dark region is refined using the guided filter for removing block artifacts in the enhanced image region (He et al. 2010). The filtered dark region is enhanced by contrast stretching, and the final output is obtained by fusing the enhance dark and input background regions.
This paper is organized as follows. "Adaptively partitioned block-based dark region detection and enhancement" section presents the automatic object segmentation algorithm, and "Experimental results" section presents experimental results, and "Conclusion" section concludes the paper.
Adaptively partitioned block-based dark region detection and enhancement
Because of the limited dynamic range of a digital camera, many consumers photographs are subject to backlit image degradation. The backlit image has a bi-modal histogram where one mode corresponds to the dark range and the other to the bright range in the background region.
Figure 2 compares histograms of a normally illuminated and backlit images. The normal image has evenly distributed intensity value as shown in Fig. 2a, while the backlit image has a two-mode histogram as shown in Fig. 2b. We can also observe that bright background region gives more visual information than the dark backlit region that has narrow dynamic range of low intensities.
Comparison of a normal and backlit images with their histograms: a a normally illuminated image and, b a backlit image
In order to selectively enhance the contrast with over-saturation problem, the dark backlit region is accurately detected using adaptively partitioned blocks. Contrast of the backlit region is then enhanced while preserving the brightness of the background region.
Adaptive backlit region detection
If the backlit region is segmented in the pixel level using a threshold of brightness, the low intensity in the background region is misclassified as the backlit region, and the high intensity in the backlit region is misclassified as the background region. This sub section presents the fast backlit region detection algorithm using adaptively partitioned blocks while minimizing the misclassification. Because the degree of brightness in the backlit region varies by image, an optimal threshold for detecting the backlit region has to be adaptively selected. Under assumption that the image can be divided into the background and backlit regions, the optimal threshold value is selected using a clustering method. Although the k-means clustering method is widely used for pattern classification because of its simplicity and robustness to noise, it is not suitable for region classification if an image contains ambiguous regions. To overcome the limitation of the k-means clustering algorithm, the proposed method uses the fuzzy C-mean (FCM) clustering method that estimates the distribution of the brightness in the image to adaptively select the optimal threshold value (Shen et al. 2005). In this work the number of clusters in the FCM algorithm is set to two for backlit and background regions. The optimal means of the two clusters are defined as
$$\begin{aligned} {J_{FCM}} = \sum \limits _{i = 1}^N {\sum \limits _{j = 1}^C {{{( {{u_{ij}}}) }^m}{{\left\| {{x_i} - {c_j}} \right\| }^2}} }, \end{aligned}$$
where \(x_{i}\) represents the brightness of the ith pixel in the image, \(c_{j}\) the mean of the jth cluster, N the total number of pixels in the image, C the number of clusters, m a weighting exponent on each fuzzy membership, and \(u_{ij}\) the degree of \(x_{i}\) contained in the ith cluster. In the experiment, \(C=2\), and \(m=2\) were used. To reduce the computational complexity, the input image is divided into non-overlapped \(64\times 64\) blocks. \(x_{i}\) is then used as the mean of the brightness of each non-overlapped block. The optimal thresholds \(c_{1}\) and \(c_{2}\) for dark and background regions, respectively, can be selected by minimizing (1).
Figure 3 shows the optimal threshold decision process using FCM. Figure 3a shows an input backlit image and its histogram. The intensity distribution is concentrated in both dark and bright ranges. The histogram of the input image is segmentation by the optimal thresholds \(c_1 \) and \(c_2 \) as shown in Fig. 3b. If an intensity value is less than \(c_1\), the corresponding pixel is classified as the dark backlit region. On the other hand, if an intensity value is larger than \(c_2 \), the corresponding pixel is classified as the bright background region. Figure 3c shows the result of the histogram segmentation of Fig. 3a using FCM.
The optimal thresholds decision using FCM: a an input image and its histogram, b concept of partitioning the three regions using two optimal thresholds \(c_1\) and \(c_2\), c result of histogram (red dark region, green ambiguous region, blue bright region)
The proposed backlit region detection partitions the input image into non-overlapped blocks of size \(64\times 64\) and classifies each block as one of dark, background, and bright regions using the optimal thresholds. The classification of the blocks is determined as
$$ L_{{k,l}} = \left\{ {\begin{array}{*{20}ll} {2,} & {b_{{\max}} ^{{{{k,l}} }} \le c_{1} \;or,\;\frac{{D_{{k,l}} }}{{N_{k} }} > 0.75} \\ {1,} & {c_{2} \le b_{{\min}} ^{{{{k,l}} }} } \\ {0,} & {otherwise} \\ \end{array} } \right., $$
where k represents the step of the partitioning blocks, and l the block number in the kth step. \(L=2\) corresponds to a dark block, 1 to a bright block, and 0 to an ambiguous block. \(b_{max}\) and \(b_{min},\) respectively represent the maximum and minimum brightness values in the corresponding block. \(N_{k}\) represents the total number of pixels in the block at the kth step. \(D_{k,l}\) represents the number of pixels smaller than \(c_{1}\) in the lth block, and \(\frac{{{D_{k,l}}}}{{{N_k}}}\) the ratio of the dark pixels in the block, which the misclassification caused by illumination change, noise, and small bright colors in the dark region. The classified ambiguous blocks are further partitioned into four sub-blocks, which are then re-classified in the same manner. The hierarchical partitioning repeats until the block size becomes \(4\times 4\).
The guided filter is used to refine the boundary of the dark region while preserving the edge of the original image (Paris et al. 2008). The guidance image I is used to filter a guided image L that has block-wise partitioned dark and background regions. Let \(I_{x}\) and \(L_{x}\) be the \(1\times 3\) color vector of the guidance image at pixel x and the corresponding label, and \(w_{k}\) be the kernel window centered at x, then the guided filter is formulated as
$$\begin{aligned} {G_x} = \mathrm{{a}}_x^\mathrm{{T}}{\mathrm{{I}}_x} + {b_x}, \end{aligned}$$
where \(a_{x}\) is a \(3\times 1\) coefficient vector, and \(b_x\) is a scalar defined as
$$\begin{aligned} {\mathrm{{a}}_x} = {\left(\sum \limits _x + \varepsilon U\right)^{ - 1}}\left(\frac{1}{{\left| \omega \right| }}\sum \limits _{i \in {\omega _x}} {{\mathrm{{I}}_i}{G_i} - {\mu _x}{{\bar{G}}_x}}\right), \end{aligned}$$
$$\begin{aligned} {b_x} = {\bar{G}_x} - \mathrm{{a}}_x^\mathrm{{T}}{\mu _x}, \end{aligned}$$
where \(\mu _{x}\) represents the mean of \(I_x\) in \(\omega _{x}\), \(\left| {{\omega _x}} \right| \) the number of pixels in \(\omega _{x}\), \({\bar{G}_x} = \frac{1}{{\left| \omega \right| }}\sum \nolimits _{i \in {\omega _x}} {{G_i}}\) the mean of G in \(\omega _{x}\). \(\sum \nolimits _x\) the \(3\times 3\) covariance matrix of \(I_x\) in \(\omega _{x}\), and U the \(3\times 3\) identity matrix. The degree of smoothing of the guided filter is adjusted by parameter \(\varepsilon \). The larger \(\varepsilon \) is, the smoother the filtered image becomes.
The detected dark region using the adaptively partitioned blocks is shown in Fig. 4. The original image has obvious contrast of the brightness between the backlit and background regions as shown Fig. 4a. Figure 4b and c, respectively show the first and last steps of hierarchical block partitioning. In case of k = 5, the detected backlit region looks almost the same to the actual backlit region, but there still remains blocking artifacts and ambiguous regions at the boundary between backlit and background regions. The image enhancement with the this coarsely segmented region occurs unnatural boundary and halo effects. In order to reduce the blocking artifacts shown in Fig. 3c, the guided filter is used to generate a continuous-valued weighting map for the ambiguous region. Figure 3d shows the result of the guided filter applied to the fifth step of partitioned blocks to generate a naturally looking dark region.
Results of the proposed adaptively partitioned block-based dark region detection method. (Black ambiguous regions, white dark region, and gray background region): a input image, b \(k=1\,({n=64}) \), c \(k=5\,( {n=4}) \), and d the finally refined dark region
Contrast enhancement of the dark region
If a global contrast stretching method is used to enhance objects in the dark region, the bright background is over-saturated. To solve this problem, contrast stretching is performed only for the detected dark region. The traditional contrast stretching method changes the brightness of the entire image using the minimum and maximum intensity values as
$$\begin{aligned} {\hat{I}_x} = \frac{{{I_x} - {I_{\min }}}}{{{I_{\max }} - {I_{\min }}}} \times 255, \end{aligned}$$
where \(I_{min}\) and \(I_{max}\) respectively represent the minimum and maximum intensity value of the image (Gonzalez and Woods 2008). For this reason, the bright background region is over-saturated by increasing the intensity of the dark region. To avoid this problem, the contrast stretching ends-in search (CSES) method has been proposed in Srinivasan and Balram (2006), which uses user-selected thresholds \(I_{min}\) and \(I_{max}\). Although CSES can efficiently enhance the contrast for the desired region, the background region is still over-saturated. On the other hand, the proposed method adaptively selects the thresholds, and performs contrast stretching in the refined dark region using a guided filter.
The proposed contrast stretching method can be formulated as
$$\begin{aligned} {\hat{I}_x} = \frac{{{I_x} - {I_{\min }}}}{{{c_1} - {I_{\min }}}} \times {c_2}, \end{aligned}$$
where \(c_{1}\) and \(c_{2}\) represent the adaptively chosen minimum and maximum thresholds, respectively. Since a backlit image consists of both dark and bright regions, the proposed method considers only low brightness pixels in the dark region. \(c_{1}\) and \(c_{2}\) determine if the corresponding pixel falls into either dark or background region. Simple multiplication of an appropriate constant and a low intensity value in the background region results in unnatural boundary between dark and background regions. On the other hand, multiplication by \(c_{2}\) prevents dark pixels from becoming excessively bright, and thus reduces unnatural boundary artifacts. The efficient contrast stretching is performed using adaptive thresholds estimated by FCM. The finally enhanced image is created by the adaptive combination of contrast enhanced and the original images using the refined dark region as
$${E_x} = {G_x} \times \hat{I_x} + (1 - {G_x}) \times {I_x},$$
where \({G_x}\) represents the refined dark region, \({\hat{I}_x}\) the result of contrast stretching, and \({I_x}\) the input image. The finally enhanced image is generated by the combination of the enhanced image by (7) and the original image using the ratio of \(G_{x}\), which makes the boundary smooth.
Figure 5 shows results of the enhancement of the proposed dark region detection method using a guided filter. The boundary between the dark and background regions is blocky as shown in 5a. The boundary of the object in the backlit region using the guided filter is well-refined as shown in Fig. 5b. The result of the enhancement of the backlit image smoothly changes the brightness at the boundary between the dark and background regions.
Results of the proposed adaptively partitioned block-based dark region detection method block-partitioning results of a the first step [\(k=1\,( {n=64}) \)] and b the fifth step [\(k=5\,( {n=4}) \)]
This section presents experimental results of the proposed adaptively partitioned block-based contrast enhancement method. The proposed enhancement method first transforms the input color image into the hue-saturation-value (HSV) color space, and performs contrast enhancement for only V channel, which contains brightness information. To evaluate the effectiveness of the proposed method, this paper compared the proposed method with gain-controllable clipped histogram equalization (GC-CHE) (Kim and Paik 2008) and multi scale Retinex with color restoration (MSRCR) method (Jobson et al. 1997). The MSRCR method restores the color contrast by applying the conventional Retinex method to each RGB channel.
Figure 6 compares the performance of the proposed method with conventional contrast enhancement methods. The original image as shown in Fig. 6a lost the information of the objects in the dark backlit region. The result of GC-CHE overcame the over-saturation problem in the background region such as the sky as shown in Fig. 6b. However, it cannot significantly enhance the backlit region. Figure 6c is the result of MSRCR, which enhances the detail of objects in the backlit region better than the other two methods. However, MSRCR still exhibits color distortion and over-saturation. Figure 6e shows the enhancement result of the proposed method. The detail of objects were better restored than existing methods. Since it performs the enhancement only in the detected dark region as shown in Fig. 6d.
Results of four different contrast enhancement methods: a input image, b GC-CHE, c MSRCR, d alpha map of guided filter based backlit region, and e the proposed method
A low light video containing a vehicle head lamp or streetlamps has a bi-modal histogram similar to the backlit image by the sum. Therefore, the proposed method can also be used to enhance such low light video sequences.
Figure 7 shows the result of the proposed enhancement method for a low light image with the highlighted region in the center. Figure 7a is a low light image acquired by a video camera. The background region becomes darker because of vehicle lamps makes the bi-modal histogram like the sun in a backlit image. Figure 7b shows the detected highlighted region, where the block area which represents the bright lamps. Figure 7c shows the result of contrast enhancement using detected highlighted region shown in Fig. 7b. The detail of the objects in dark regions is better restored than the input image without over-saturation or color distortion in the highlighted region.
Result of the proposed contrast enhancement method for a low light video: a input image frame, b detected highlighted region, and c result of the proposed enhancement method for the low light image
Figure 8 compares the performance of the proposed method with the HE and MSRCR methods. The enhanced image using HE can restore the detail of objects in the backlit region at the cost of over-saturation in the highlighted region as shown in Fig. 8b. On the other hand, MSRCR enhances the detail of the objects in the backlit region better than the HE at the cost insufficient enhancement in the backlit region as shown in Fig. 8c. As shown in Fig. 8d, the proposed method can enhance the contrast in both dark and bright regions without only color distortion problems.
Comparison of various contrast enhancement methods for a low light video. Four frames of a an input low light video, b enhance results using HE, c MSRCR, and d the proposed method
For objective evaluation of the performance of contrast enhancement methods, average entropy (AE) is computed as
$$\begin{aligned} {AE} = \frac{1}{N}\sum \limits _{k = 0}^{255} {P_{out}(X_k)} \times {log_{2} P_{out}(X_k)}, \end{aligned}$$
where \(P_{out}(X_k)\) is the normalized probability of the kth gray level, and N the number of pixels in the image.
Table 1 Quantitative evaluating using AE
Table 1 shows AE value of various contrast enhancement methods on the set of three test images. The higher AE value indicates that more detail of the image is restored by the enhancement method. According to Table 1, the proposed method produces higher AE value than any other conventional methods.
The gain-controlled clipped histogram equalization (GC-CHE) method (Kim and Paik 2008) is compared with the proposed method since it is known as the best histogram modification approach in enhancing the contrast of digital images. Although various improved or modified versions were proposed in the literature, the original work of Kim and Paik (2008) is the best candidate of the performance comparison without significantly increasing the computational load. The multi-scale retinex method (Jobson et al. 1997) is also known as the first work or retinex theory-based contrast enhancement method. Although various different versions of retinex-based methods were proposed in the literature, the original work of Jobson et al. (1997) is the most appropriate for performance comparison with similar amount of computational load.
In order to demonstrate the performance of the proposed method, we first used subjective comparison as shown in Fig. 8. The input low-light video frames shown in Fig. 8a is suitable to evaluate the performance of contrast enhancement since it contains both dark and saturated regions. In addition to subjective comparison, we evaluated average entropy to represent how evenly the brightness is spread in the processed image.
This paper has presented a method to enhance the contrast of two-mode brightness image. Conventional contrast enhancement methods have over-saturation and color distortion problems. To solve these problems, the proposed method divides the image into dark and background regions using adaptively partitioned blocks by two optimal threshold values computed by fuzzy C-means clustering in the V channel of the HSV color space. The proposed contrast stretching process is performed only in the detected dark region. The major advantage of the proposed method is the minimized block artifacts due to adaptively partitioning the image according to the optimal threshold and the refining step to detect the dark regions. The proposed method automatically segments backlit region and the background region. It does not need manual seed region selection for segmentation and has low segmentation complexity than heavy segmentation methods such as graph cut-based method. Experimental results showed that the proposed method can better enhance the contrast than existing methods in the sense of both minimizing over-saturation in the bright background region and preserving details in the dark region.
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SL initiated the research, and developed major algorithms, NK performed experiment, and JP wrote the paper. All authors read and approved the final manuscript.
ICT R&D program of MSIP/IITP. (14-824-09-002, Development of global multi-target tracking and event prediction techniques based on real-time large-scale video analysis), and the Technology Innovation Program (Development of Smart Video/Audio Surveillance SoC and Core Component for Onsite Decision Security System) under Grant 10047788.
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interests.
Chung-Ang University, 221 Heukseok-Dong, Dongjak-Gu, Seoul, 156-756, Korea
Seungwon Lee, Nahyun Kim & Joonki Paik
Seungwon Lee
Nahyun Kim
Joonki Paik
Correspondence to Joonki Paik.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Lee, S., Kim, N. & Paik, J. Adaptively partitioned block-based contrast enhancement and its application to low light-level video surveillance. SpringerPlus 4, 431 (2015). https://doi.org/10.1186/s40064-015-1226-x
Backlighting compensation
Guided filter | CommonCrawl |
\begin{document}
\title{Pro-groups and generalizations of a theorem of Bing}
\thanks{2010 {\it Mathematics Subject Classification}. Primary 52C23, 57R05, 54F15, 37B45; Secondary 53C12, 57N55 }
\author{Alex Clark} \address{Alex Clark, School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK} \email{[email protected]}
\author{Steven Hurder} \address{Steven Hurder, Department of Mathematics, University of Illinois at Chicago, 322 SEO (m/c 249), 851 S. Morgan Street, Chicago, IL 60607-7045} \email{[email protected]}
\author{Olga Lukina} \address{Olga Lukina, Department of Mathematics, University of Illinois at Chicago, 322 SEO (m/c 249), 851 S. Morgan Street, Chicago, IL 60607-7045} \email{[email protected]}
\thanks{Version date: November 1, 2018 }
\date{}
\begin{abstract}
A matchbox manifold is a generalized lamination, which is a continuum whose path components define the leaves of a foliation of the space. A matchbox manifold is $M$-like if it has the shape of a fixed topological space $M$. When $M$ is a closed manifold, in a previous work, the authors have shown that if ${\mathfrak{M}}$ is a matchbox manifold which is $M$-like, then it is homeomorphic to a weak solenoid. In this work, we associate to a weak solenoid a pro-group, whose pro-isomorphism class is an invariant of the homeomorphism class of ${\mathfrak{M}}$. We then show that an $M$-like matchbox manifold is homeomorphic to a weak solenoid whose base manifold has fundamental group which is non co-Hopfian; that is, it admits a non-trivial self-embedding of finite index. We include a collection of examples illustrating this conclusion. \end{abstract}
\maketitle
\section{Introduction} \label{sec-intro}
A \emph{continuum} is a compact \emph{connected} metric space. From the very beginnings of the subject of Topology in the first half of the twentieth century, examples of continua with unusual properties were a source of motivation for the development of the subject. For example, the Warsaw circle, the Sierpinsky carpet \cite{Sierpinski1916}, and the 1-dimensional solenoids introduced by Vietoris \cite{Vietoris1927} and van Danzig \cite{vanDantzig1930} are all well-known examples of ``non-trivial'' continua. Continua frequently arise as invariant sets for dynamical systems, and moreover as quoted from Barge and Kennedy \cite{BargeKennedy1990}: \begin{quote} One can argue quite convincingly that continuum theory first arose from problems in dynamics even before there was a definition of a topological space. \end{quote}
The study of the invariant continua for a dynamical system is a basic technique for the analysis of its global dynamics. There are many established approaches for the study of continua, as discussed in the book by Sam Nadler \cite{Nadler1992}. One of these is the following fundamental result of Hans Freudenthal:
\begin{thm}[Freudenthal \cite{Freudenthal1936}] \label{thm-Freudenthal} Let ${\mathfrak{M}}$ be a continuum, then there exists a sequence of maps \begin{equation}\label{eq-presentationFreud} {\mathcal P} = \{\, f_{\ell+1} \colon X_{\ell+1} \to X_\ell \mid \ell \geq 0\} \end{equation} where for each $\ell \geq 0$, $X_{\ell}$ is a finite connected polyhedron and each map $f_{\ell +1}$ is a continuous surjection, such that ${\mathfrak{M}}$ is homeomorphic to an inverse limit, \begin{equation}\label{eq-presentation-top} \Phi_{{\mathcal P}} \colon {\mathfrak{M}} ~ \cong ~ {\mathcal S}_{{\mathcal P}} \equiv \varprojlim \,\{\, f_{\ell+1} \colon X_{\ell+1} \to X_\ell \mid \ell \geq 0\} ~ , \end{equation}
The maps $f_{\ell +1}$ in \eqref{eq-presentationFreud} are called the \emph{bonding maps} of the presentation ${\mathcal P}$. \end{thm}
The inverse limit is considered as a subset of the Cartesian product of the spaces $\{X_{\ell} \mid \ell \geq 0\}$ with the Tychonoff topology, and a point is represented by a sequence $x = (x_{\ell})_{\ell \geq 0}$ where $x_{\ell} \in X_{\ell}$ and $f_{\ell +1}(x_{\ell+1}) = x_{\ell}$ for $\ell \geq 0$. A presentation ${\mathcal P}$ is said to be \emph{trivial} if all of the maps $f_{\ell +1}$ are homeomorphisms, and thus ${\mathfrak{M}}$ is homeomorphic to a finite connected polyhedron.
Observe that given a presentation ${\mathcal P}$ for a continuum ${\mathfrak{M}}$ as in \eqref{eq-presentation-top}, for all $\ell \geq 0$, there is a continuous surjection $\Pi_{\ell} \colon {\mathfrak{M}} \to X_{\ell}$ given by $\Pi_{\ell}((x_{\ell})_{\ell \geq 0}) = x_{\ell}$. It then follows that for any ${\epsilon} > 0$, there exists some $\ell_{{\epsilon}}$ such that $\Pi_{\ell_{\epsilon}}$ is an ${\epsilon}$-mapping; that is, for all $y \in X_{\ell_{\epsilon}}$ the inverse image set $\Pi_{\ell_{\epsilon}}^{-1}(y) \subset {\mathfrak{M}}$ has diameter less than ${\epsilon}$.
The notion of an ${\epsilon}$-map was introduced in 1928 by Alexandroff \cite{Alexandroff1928}: \begin{defn} \label{def-Ylike}
Let ${\mathfrak{M}}$ be a continuum, $Y$ a topological space and ${\epsilon} > 0$ a constant. A map $f \colon {\mathfrak{M}} \to Y$ is said to be an ${\epsilon}$-map if $f$ is a continuous surjection, and for each point $y\in Y$, the inverse image $f^{-1}(y)$ has diameter less than ${\epsilon}$. A continuum ${\mathfrak{M}}$ is said to be \emph{$Y$--like}, for some topological space $Y$, if for every ${\epsilon}>0$, there is an ${\epsilon}$-map $f_{\epsilon} \colon {\mathfrak{M}} \to Y$.
More generally, let ${\mathcal Y}$ be a collection of compact topological spaces, then ${\mathfrak{M}}$ is ${\mathcal Y}$-like if for all ${\epsilon} > 0$, there exists $Y_{\epsilon} \in {\mathcal Y}$ and an ${\epsilon}$-map $f_{\epsilon} \colon {\mathfrak{M}} \to Y_{\epsilon}$.
\end{defn}
In this work, we consider the following special case of the above definition: \begin{defn} \label{def-Mlike}
A continuum ${\mathfrak{M}}$ is \emph{manifold-like}, if given any ${\epsilon} > 0$, there exists a closed manifold $M_{{\epsilon}}$ and an ${\epsilon}$-map
$f \colon {\mathfrak{M}} \to M_{{\epsilon}}$. For a closed manifold $M$, a continuum ${\mathfrak{M}}$ is \emph{$M$-like}, if given any ${\epsilon} > 0$, there exists an ${\epsilon}$-map
$f \colon {\mathfrak{M}} \to M$.
\end{defn}
The following is then a natural problem to investigate:
\begin{prob}\label{prob-Mlike} For a closed $n$-manifold $M$, characterize the continua which are $M$-like.
\end{prob}
Problem~\ref{prob-Mlike} is too broad to obtain definitive results, as there exist too many pathological constructions of $M$-like spaces. The simplest case, where $M = [0,1]$ is an interval, has been extensively studied for many special classes of bonding maps, such as the ``tent maps'', where many results are known for the inverse limit spaces. However, without some sort of ``regularity'' restriction, even the continua modeled on bonding maps to $[0,1]$ are too wild to hope to classify. Thus, one imposes additional assumptions on the continua considered, such as the following.
Recall that a topological space ${\mathfrak{M}}$ is \emph{homogeneous} if for every $x, y \in {\mathfrak{M}}$, there exists a homeomorphism $h \colon {\mathfrak{M}} \to {\mathfrak{M}}$ such that $h(x) = y$. A continuum $X$ is \emph{circle-like} if it is $M$-like, where $M = {\mathbb S}^1$ is the circle. The following is a well-known result of R.H. Bing, which has inspired many generalizations: \begin{thm} [Bing \cite{Bing1960}] \label{thm-bing} Let ${\mathfrak{M}}$ be a homogeneous, circle-like continuum that contains an arc. Then either ${\mathfrak{M}}$ is homeomorphic to ${\mathbb S}^1$, or to an inverse limit of proper finite coverings of ${\mathbb S}^1$. \end{thm}
The hypothesis in Theorem~\ref{thm-bing} that ${\mathfrak{M}}$ is homogeneous and contains an arc implies that for every point $x \in {\mathfrak{M}}$, there is an arc containing $x$. One generalization of this hypothesis is suggested by the properties of continua arising from the investigation of the dynamical properties of smooth foliations of compact manifolds, for example as discussed in \cite{Hurder2014}. In this case, the invariant continua have the structure of foliated spaces, in the sense of \cite{MS2006}. That is, for each $x \in {\mathfrak{M}}$, there is a neighborhood $x \in U_x \subset {\mathfrak{M}}$ where $U_x$ is homeomorphic to a product of an open subset of ${\mathbb R}^n$, for some $n \geq 1$, with a closed subset ${\mathfrak{T}}_x$ of some Polish space. If the transversal spaces ${\mathfrak{T}}_x$ are \emph{totally disconnected} for all $x \in {\mathfrak{M}}$, and are not singleton spaces, then we say that ${\mathfrak{M}}$ is a \emph{matchbox manifold}, and ${\mathfrak{M}}$ then admits a decomposition ${\mathcal F}_{{\mathfrak{M}}}$ into path-connected components of constant dimension $n$ which are the leaves of ${\mathcal F}_{{\mathfrak{M}}}$. A matchbox manifold with $2$-dimensional leaves is a lamination by surfaces in the sense of Ghys \cite{Ghys1999} and Lyubich and Minsky \cite{LM1997}, while Sullivan called them ``solenoidal spaces'' in \cite{Sullivan2014,Verjovsky2014}. The terminology ``matchbox manifold'' follows the usage introduced in \cite{AM1988,AO1991,AO1995}, and as used in the authors' works \cite{ClarkHurder2011,ClarkHurder2013,CHL2014,CHL2018a,CHL2018b,DHL2016a,DHL2016b,DHL2016c,HL2017,HL2018}
which study this class of continua.
In the authors' previous work \cite{CHL2018b} we have shown:
\begin{thm}\cite[Corollary~1.6]{CHL2018b} \label{thm-weak} Let ${\mathfrak{M}}$ be a manifold-like matchbox manifold ${\mathfrak{M}}$. Then there exists a presentation ${\mathcal P} = \{ p_{\ell +1} \colon M_{\ell +1} \to M_{\ell} \mid \ell \geq 0 \}$,
where each bonding map $p_{\ell +1}$ is a proper covering map, and a homeomorphism
\begin{equation}\label{eq-ws} \Phi_{{\mathcal P}} \colon {\mathfrak{M}} ~ \cong ~ {\mathcal S}_{{\mathcal P}} \equiv \varprojlim\, \{ p_{\ell +1} \colon M_{\ell +1} \to M_{\ell} \mid \ell \geq 0 \} ~. \end{equation} \end{thm} A continuum given by the inverse limit of a sequence of proper covering maps is called a \emph{weak solenoid}, following the usage in the the works of Rogers and Tollefson \cite{RogersJT1970,RT1971a,RT1971b,RT1972a}. The properties of these spaces are discussed further in Section~\ref{sec-solenoids}.
Note that a homeomorphism $\Phi_{{\mathcal P}} \colon {\mathfrak{M}} \to {\mathcal S}_{{\mathcal P}}$ preserves path components, hence maps the leaves of the foliation ${\mathcal F}_{{\mathfrak{M}}}$ homeomorphically onto the leaves of the foliation ${\mathcal F}_{\cP}$ of ${\mathcal S}_{{\mathcal P}}$.
Mardesic and Segal discuss pointed shape expansions in \cite{MardesicSegal1982}, and show there is well-defined notion of pro-homotopy groups {\it pro}-$\pi_k({\mathfrak{M}}, z)$ of a space ${\mathfrak{M}}$ using pointed shape exapnsions, though it may depend on the choice of basepoint $z \in {\mathfrak{M}}$. For example, see Example~4, \cite[Chapter II, Section 3.3]{MardesicSegal1982}. One of the main technical points of this work, is that the pro-homotopy groups are well-defined for a weak solenoid, independent of the choice of basepoint $z \in {\mathfrak{M}}$.
Here is our first result, which is a general property of weak solenoids, and can be considered a formal version of the ideas of Fokkink and Oversteegen in \cite{FO2002}.
\begin{thm}\label{thm-MMprogroups} Let ${\mathfrak{M}}$ be an $M$-like matchbox manifold, then the pro-group {\it pro}-$\pi_1({\mathfrak{M}}, z)$ is independent of the choice of basepoint $z \in {\mathfrak{M}}$ up to isomorphism, and depends only on the homeomorphism class of ${\mathfrak{M}}$.
\end{thm} Pro-groups are discussed in Section~\ref{sec-progroups}, then the definition of {\it pro}-$\pi_1({\mathfrak{M}}, z)$ is given in Section~\ref{sec-progroupsWS} and we show that it is well-defined. Then in Section~\ref{sec-progroupsMlike}, we calculate the pro-group associated to an $M$-like matchbox manifold, which leads to the following generalization of Theorem~\ref{thm-bing} by Bing:
\begin{thm}\label{thm-main1}
Let ${\mathfrak{M}}$ be an $n$-dimensional matchbox manifold which is $M$-like, for a closed manifold $M$.
Then ${\mathfrak{M}}$ admits a presentation
$\displaystyle {\mathcal P} = \{\, q_{\ell+1} \colon N_{\ell+1} \to N_{\ell} \mid \ell \geq 0\}$,
where each $N_{\ell}$ is a closed $n$-manifold whose fundamental group is isomorphic to that of $M$, and each $q_{\ell+1} \colon N_{\ell +1} \to N_{\ell}$ is a proper covering map for $\ell \geq 0$, such that there is a homeomorphism
\begin{equation}\label{eq-scale1} \Phi_{{\mathcal P}} \colon {\mathfrak{M}} ~ \cong ~ {\mathcal S}_{{\mathcal P}} \equiv \varprojlim \,\{\, q_{\ell+1} \colon N_{\ell +1} \to N_{\ell} \mid \ell \geq 0\} ~ . \end{equation}
\end{thm}
The conclusion of Theorem~\ref{thm-main1} can be strengthened if we impose additional hypotheses on the target space $M$. Recall that a finite $CW$-complex $Y$ is \emph{aspherical} if it is connected and its universal covering space is contractible. The \emph{Borel Conjecture} is that if $Y_1$ and $Y_2$ are homotopy equivalent, aspherical closed manifolds, then a homotopy equivalence between $Y_1$ and $Y_2$ is homotopic to a homeomorphism between $Y_1$ and $Y_2$. The Borel Conjecture has been proven for many classes of aspherical manifolds: \begin{enumerate} \item the torus ${\mathbb T}^n$ for all $n \geq 1$, \item all infra-nilmanifolds, \item closed Riemannian manifolds $Y$ with negative sectional curvatures, \item closed Riemannian manifolds $Y$ of dimension $n \ne 3,4$ with non-positive sectional curvatures. \end{enumerate}
The above list is not exhaustive, and only cases (1) and (2) are used in this paper. The history and status of the Borel Conjecture is discussed, for example, in the surveys of Davis \cite{Davis2012} and L\"{u}ck \cite{Luck2012}.
A manifold for which all finite coverings satisfy the Borel Conjecture was called \emph{strongly Borel} in the authors work \cite{CHL2018a}. Note that the class of ``D-like'' spaces introduced in \cite[Section~5]{RT1972b} consists of manifolds which are strongly Borel.
Here is our second main result, which is a further generalization of Theorem~\ref{thm-bing} by Bing:
\begin{thm}\label{thm-main2}
Let $M$ be a closed aspherical $n$-manifold $M$ which satisfies the Borel Conjecture. If ${\mathfrak{M}}$ is a matchbox manifold which is $M$-like,
then ${\mathfrak{M}}$ admits a presentation
$\displaystyle {\mathcal P} = \{\, q_{\ell+1} \colon M \to M \mid \ell \geq 0\}$, where each $q_{\ell+1} \colon M \to M$ is a proper covering map for $\ell \geq 0$, such that there is a homeomorphism
\begin{equation}\label{eq-scale2} \Phi_{{\mathcal P}} \colon {\mathfrak{M}} ~ \cong ~ \varprojlim \,\{\, q_{\ell+1} \colon M \to M \mid \ell \geq 0\} ~ . \end{equation}
\end{thm}
As an example, note that the $n$-torus ${\mathbb T}^n$ satisfies the hypotheses of Theorem~\ref{thm-main2}, so we conclude:
\begin{cor}\label{cor-toral} Let ${\mathfrak{M}}$ be a matchbox manifold which is ${\mathbb T}^n$-like, then it admits a presentation as in \eqref{eq-scale2} for $M = {\mathbb T}^n$.
\end{cor}
Recall that a group $H$ is said to be \emph{(finitely) non co-Hopfian}, or just non co-Hopfian in this work, if there is an injective homomorphism $\iota \colon H \to H$ whose image has finite index, but is not onto. That is, $H$ contains a subgroup of finite index isomorphic to itself. The class of non co-Hopfian groups has been extensively studied, as discussed further in Section~\ref{sec-examples}, providing many examples. The general philosophy is that the group $H$ should be ``essentially'' nilpotent. A complete solution of Problem~\ref{prob-Mlike} in the case of matchbox manifolds then requires a classification of the finitely-generated, non co-Hopfian groups. The works of van~Limbeek \cite{vanLimbeek2018,vanLimbeek2017} give some indications of the available techniques for the study of this problem, as well as some of the difficulties it presents.
The proofs of Theorems~\ref{thm-main1} and \ref{thm-main2} are given in Section~\ref{sec-proofs}. Finally, Section~\ref{sec-examples} gives collections of examples of weak solenoids and non co-Hopfian groups illustrating the conclusions of theorems, and showing that stronger results are not possible without further assumptions.
\section{Weak solenoids}\label{sec-solenoids}
A \emph{weak solenoid} ${\mathcal S}_{\mathcal P}$ of dimension $n \geq 1$, is the inverse limit space of a sequence
\begin{eqnarray} {\mathcal S}_{{\mathcal P}} & \equiv & \varprojlim ~ \{ p_{\ell +1} \colon M_{\ell +1} \to M_{\ell}\} \label{eq-presentationinvlim} \\ & = & \{(x_0, x_1, \ldots ) \in {\mathcal S}_{{\mathcal P}} \mid p_{\ell +1 }(x_{\ell + 1}) = x_{\ell} ~ {\rm for ~ all} ~ \ell \geq 0 ~\} ~ \subset \prod_{\ell \geq 0} ~ M_{\ell} ~ \nonumber \end{eqnarray}
where each $M_\ell$ is a compact connected $n$-dimensional manifold without boundary, and each $p_{\ell +1 }$ is a covering map of finite degree greater than one. The set ${\mathcal S}_{{\mathcal P}}$ is given the relative topology, induced from the product (Tychonoff) topology, so that ${\mathcal S}_{{\mathcal P}}$ is itself compact and connected. The collection \begin{equation}\label{eq-presentation} {\mathcal P} = \{p_{\ell +1} \colon M_{\ell +1} \to M_{\ell} \mid \ell \geq 0\}, \end{equation}
is called a \emph{presentation} for ${\mathcal S}_{\mathcal P}$.
For example, a \emph{Vietoris solenoid} \cite{vanDantzig1930,Vietoris1927} is a $1$-dimensional solenoid, where each $M_{\ell}$ is a circle, as arises in the conclusion of Theorem~\ref{thm-bing}.
Let ${\mathcal S}_{\mathcal P}$ be a weak solenoid with presentation ${\mathcal P}$ as in \eqref{eq-presentation}. For each $\ell \geq 1$, the composition \begin{equation}\label{eq-coverings} p^0_{\ell} = p_{1} \circ \cdots \circ p_{\ell -1} \circ p_{\ell} \colon M_{\ell} \to M_0 ~ \end{equation} is a finite-to-one covering map of the base manifold $M_0$.
For each $\ell \geq 0$, projection onto the $\ell$-th factor in the product $\displaystyle \prod_{\ell \geq 0} ~ M_{\ell}$ in \eqref{eq-presentationinvlim} yields a surjective map denoted by $\Pi_{\ell}^{{\mathcal P}} \colon {\mathcal S}_{{\mathcal P}} \to M_{\ell}$. Then for $\ell \geq 1$, we have $ \Pi_0^{{\mathcal P}} = p^0_{\ell} \circ \Pi_{\ell}^{{\mathcal P}} \colon {\mathcal S}_{{\mathcal P}} \to M_0$.
McCord showed in \cite{McCord1965} that for each $\ell > 0$, the map $\Pi_{\ell}^{{\mathcal P}} \colon {\mathcal S}_{{\mathcal P}} \to M_{\ell}$ is a fibration over $M_{\ell}$. For a choice of a basepoint $x_0 \in M_0$, the assumption that the fibers of each map $p_{\ell+1}$ have cardinality at least $2$ implies the fiber ${\mathfrak{X}}_0 = (\Pi_0^{{\mathcal P}})^{-1}(x_0)$ is a Cantor set. McCord also observed that ${\mathcal S}_{\mathcal P}$ has a local product structure, which implies that the path-connected components of ${\mathcal S}_{\mathcal P}$ define a foliation denoted by ${\mathcal F}_{\cP}$. That is,
$n$-dimensional weak solenoids are examples of $n$-dimensional matchbox manifolds. The properties of weak solenoids have been studied in the works by Schori \cite{Schori1966}, Rogers and Tollefson \cite{RogersJT1970,RT1971a,RT1971b,RT1972a}, Fokkink and Oversteegen in \cite{FO2002}, and by the authors.
Given a basepoint $x \in {\mathcal S}_{{\mathcal P}}$ for each $\ell \geq 0$ we then obtain basepoints $x_{\ell} = \Pi_{\ell}(x) \in M_{\ell}$. Denote by $G_0 = \pi_1(M_{0}, x_{0})$ the fundamental group of $M_0$ with basepoint $x_0$, and for $\ell > 0$ let \begin{equation}\label{eq-imahes} G_{\ell} = {\rm image}\left\{ p^0_{\ell} \colon \pi_1(M_{\ell}, x_{\ell})\longrightarrow G_{0}\right\} \end{equation}
where by a small abuse of notation, we let $p^0_{\ell}$ also denote the induced map on fundamental groups, and also suppress the dependence on the choice of basepoint $x$ in the notation $G_{\ell}$.
Note that each $G_{\ell}$ is a subgroup of finite index in $G_0$.
In this way, associated to the presentation ${\mathcal P}$ and basepoint $x \in {\mathfrak{X}}_0$, we obtain a descending chain of subgroups of finite index
\begin{equation}\label{eq-descendingchain} {\mathcal G}_x \equiv \{ G_{0} \supset G_{1} \supset G_{2} \supset \cdots \supset G_{\ell} \supset \cdots \} ~ . \end{equation}
Each quotient $X_{\ell} = G_{0}/G_{\ell}$ is a finite set equipped with a left $G_0$-action, and there are surjections $X_{\ell +1} \to X_{\ell}$ which commute with the action of $G_0$. The inverse limit, \begin{equation}\label{eq-Galoisfiber} X({\mathcal G}) = \varprojlim ~ \{ p_{\ell +1} \colon X_{\ell +1} \to X_{\ell} \mid \ell \geq 0\} = \{(eG_0, g_1G_1, \ldots) \mid g_{\ell} G_{\ell} = g_{\ell+1}G_{\ell}\} ~ \subset \prod_{\ell \geq 0} ~ X_{\ell} \end{equation} is then a totally-disconnected, compact, perfect set for the topology induced from the Tychonoff topology on the product, so is a Cantor set. Note that there is a canonical basepoint $(eG_{\ell}) \in X({\mathcal G})$ where $e \in G_0$ is the identity element of the group. The fundamental group $G_0$ acts on the left on $X({\mathcal G})$ via the coordinate-wise multiplication on the product in \eqref{eq-Galoisfiber}. The left action of $G_0$ on each quotient space $X_{\ell}$ is transitive, so each orbit of $G_0$ acting on $X({\mathcal G})$ is dense. We denote this minimal Cantor action by $\Phi_0 \colon G_0 \times X({\mathcal G}) \to X({\mathcal G})$, or by $(X({\mathcal G}) , G_0 , \Phi_0)$.
The space $X({\mathcal G})$ admits a natural identification with the fiber ${\mathfrak{X}}_0 \equiv \Pi_0^{-1}(x_0)$. The Cantor action $(X({\mathcal G}) , G_0 , \Phi_0)$ is then conjugate with the monodromy action of $G_0$ on ${\mathfrak{X}}_0$ induced by the leaves of the foliation ${\mathcal F}_{\cP}$. See \cite{DHL2016c} for more details on this identification, and also the dependence on the choice of the basepoint $x \in {\mathfrak{X}}_0$.
A group chain ${\mathcal G}$ as in \eqref{eq-descendingchain} determines a presentation ${\mathcal P}$ with pointed base manifold $(M_0, x_0)$, and thus a weak solenoid ${\mathcal S}_{{\mathcal P}}$. We next consider two equivalence relations on group chains, which were introduced by Rogers and Tollefson in \cite{RT1971b} and
Fokkink and Oversteegen in \cite{FO2002}.
Let $\mathfrak{G}$ denote the collection of all subgroup chains in $G_0$. The first notion of equivalence for elements of $\mathfrak{G}$ corresponds to the standard notion of intertwined chains: \begin{defn} (Rogers-Tollefson, \cite{RT1971b}) \label{defn-greq} Group chains ${\mathcal G} = \{G_{\ell}\}_{\ell \geq 0}$ and ${\mathcal H} = \{H_{\ell}\}_{\ell \geq 0}$ are \emph{equivalent} if $G_0=H_0$ and there is a group chain ${\mathcal K} = \{K_{\ell}\}_{\ell \geq 0}$ with $K_0 = G_0$, and infinite subsequences $\{G_{\ell_k}\}_{k \geq 0}$ and $\{H_{j_k}\}_{k \geq 0}$ such that $K_{2k} = G_{\ell_k}$ and $K_{2k+1} = H_{j_k}$ for $k \geq 0$. \end{defn}
The second notion of equivalence is more subtle, as it corresponds to conjugating a group chain by an element of the profinite core group $C_{\infty}$ as discussed in \cite{DHL2016a}. \begin{defn} (Fokkink-Oversteegen, \cite{FO2002}) \label{defn-conjequiv} Group chains ${\mathcal G} = \{G_{\ell}\}_{\ell \geq 0}$ and ${\mathcal H} = \{H_{\ell}\}_{\ell \geq 0}$ in $\mathfrak{G}$ are \emph{conjugate equivalent} if $G_0=H_0$, and: \begin{enumerate} \item there exists a sequence $(g_{\ell})_{\ell \geq 0}$ where each $g_{\ell} \in G_0$; \item the compatibility condition $g_{\ell}G_{\ell} = g_{\ell +1} G_{\ell}$ for all $\ell\geq 0$ is satisfied; \item the group chains ${\mathcal G}^{(g_{\ell})} = \{g_{\ell} G_{\ell} g_{\ell}^{-1}\}_{\ell \geq 0}$ and ${\mathcal H} = \{H_{\ell}\}_{\ell \geq 0}$ are equivalent. \end{enumerate} \end{defn}
The dynamical meaning of the equivalences in Definitions~\ref{defn-greq} and \ref{defn-conjequiv} is given by the following theorem, which follows from results in \cite{FO2002}; see also \cite{DHL2016a}.
\begin{thm}\label{equiv-rel-11} Let ${\mathcal G} = \{G_{\ell}\}_{\ell \geq 0}$ and ${\mathcal H} = \{H_{\ell}\}_{\ell \geq 0}$ be group chains with $H_0 = G_0$, and let \begin{eqnarray*} X({\mathcal G}) & = & \varprojlim \, \{G_0/G_{\ell+1} \to G_0/G_{\ell}\} \ , \\ X({\mathcal H}) & = & \varprojlim \, \{H_0/H_{\ell+1} \to H_0/H_{\ell}\} \ . \end{eqnarray*} Then: \begin{enumerate} \item \label{er-item1} The group chains ${\mathcal G}$ and ${\mathcal H}$ are \emph{equivalent} if and only if there exists a homeomorphism $\tau \colon X({\mathcal G}) \to X({\mathcal H})$ equivariant with respect to the $G_0$-actions on $X({\mathcal G})$ and $X({\mathcal H})$, and such that $\phi(eG_{\ell}) = (eH_{\ell})$. \item The group chains ${\mathcal G}$ and ${\mathcal H}$ are \emph{conjugate equivalent} if and only if there exists a homeomorphism $\tau \colon X({\mathcal G}) \to X({\mathcal H})$ equivariant with respect to the $G_0$-actions on $X({\mathcal G})$ and $X({\mathcal H})$. \end{enumerate} \end{thm}
That is, an equivalence of two group chains corresponds to the existence of a \emph{basepoint-preserving} equivariant homeomorphism between their inverse limit systems, while a conjugate equivalence of two group chains corresponds to the existence of a equivariant conjugacy between their inverse limit systems, which need not preserve the basepoint.
\begin{remark}\label{rmk-kernels} {\rm The \emph{kernel} of a group chain ${\mathcal G} = \{G_{\ell}\}_{\ell \geq 0}$ is the subgroup of $G_0$ given by \begin{equation}\label{eq-kernel} K({\mathcal G}) = \bigcap_{\ell \geq 0} ~ G_{\ell} \ . \end{equation} It follows immediately from the definitions that if the group chains ${\mathcal G} = \{G_{\ell}\}_{\ell \geq 0}$ and ${\mathcal H} = \{H_{\ell}\}_{\ell \geq 0}$ with $G_0=H_0$ are equivalent, then $K({\mathcal G}) = K({\mathcal H}) \subset G_0$. If the chains ${\mathcal G}$ and ${\mathcal H}$ are only conjugate equivalent, then the kernels need not be equal. Thus, the kernel $K({\mathcal G})$ of a group chain ${\mathcal G}$ is an invariant of equivalence, but is not necessarily invariant by conjugate equivalence.
The kernel $K({\mathcal G})$ has an interpretation in terms of the topology of the leaves of the foliation ${\mathcal F}_{\cP}$ of the weak solenoid ${\mathcal S}_{{\mathcal P}}$. Choose a basepoint $x \in \Pi_0^{-1}(x_0)$ and let $L_x \subset {\mathcal S}_{{\mathcal P}}$ be the leaf containing $x$. The restriction of the bundle projection $\Pi_0|_{L_x} \colon L_x \to M_0$ is a covering map. Let $\widetilde{M_0}$ be the universal cover of $M_0$. Then by standard arguments of covering space theory (see also McCord \cite{McCord1965}) there is a homeomorphism $\displaystyle \widetilde{M_0}/ K({\mathcal G}) \to L_x$.
In particular, if $L_x$ is simply connected, then $K({\mathcal G}) = \{e\}$. }
\end{remark}
A standard fact about weak solenoids, is that their homeomorphism type remains unchanged if a finite number of the initial maps in a presentation ${\mathcal P}$ as in \eqref{eq-presentation} are deleted. See \cite{CHL2018a,DHL2016c,HL2018} for discussions of this property and its significance. This fact suggests a third type of equivalence relation on group chains. For a group chain ${\mathcal G}= \{G_{\ell}\}_{\ell \geq 0}$ and $k \geq 0$, define the ``truncated group chain'' ${\mathcal G}^{(k)}$ by setting where $G^{(k)}_{\ell} = G_{\ell + k}$, then
\begin{equation}\label{eq-truncatedchain1} {\mathcal G}^{(k)} \colon G^{(k)}_{0} \supset G^{(k)}_{1} \supset G^{(k)}_{2} \supset \cdots \supset G^{(k)}_{\ell} \supset \cdots \, . \end{equation}
\begin{defn} \label{defn-return} Group chains ${\mathcal G}= \{G_{\ell}\}_{\ell \geq 0}$ and ${\mathcal H} = \{H_{\ell}\}_{\ell \geq 0}$ are \emph{return equivalent} if there exists integers $k, m \geq 0$ such that the truncated group chains ${\mathcal G}^{(k)}$ and ${\mathcal H}^{(m)}$ are equivalent in the sense of Definition~\ref{defn-greq}. We say they are \emph{conjugate return equivalent} if there exists integers $k, m \geq 0$ such that the truncated group chains ${\mathcal G}^{(k)}$ and ${\mathcal H}^{(m)}$ are equivalent in the sense of Definition~\ref{defn-conjequiv}. \end{defn}
Recall that for the Tychonoff topology on $X({\mathcal G})$, we have a descending chain $\{U_k({\mathcal G}) \mid k =0,1,2, \ldots\}$ of clopen neighborhoods of the basepoint $(eG_{\ell})$, where \begin{equation}\label{eq-truncatedchain2} U_k({\mathcal G}) = \varprojlim \, ~ \{ p_{\ell +1} \colon X_{\ell +1} \to X_{\ell} \mid \ell \geq k\} \end{equation} The spaces $U_k({\mathcal G})$ and $X({\mathcal G}^{(k)})$ are canonically isomorphic, and the action of $G_k$ on $X({\mathcal G}^{(k)})$ is conjugate to the restricted action of $G_k$ on $U_k({\mathcal G})$. Thus, in the terminology of \cite{HL2018} we have: \begin{prop}\label{prop-re} Given group chains ${\mathcal G}= \{G_{\ell}\}_{\ell \geq 0}$ and ${\mathcal H} = \{H_{\ell}\}_{\ell \geq 0}$, then: \begin{enumerate} \item ${\mathcal G}$ and ${\mathcal H}$ are return equivalent if an only if the minimal Cantor actions $(X({\mathcal G}) , G_0 , \Phi_0)$ and $(X({\mathcal H}) , H_0 , \Psi_0)$ are return equivalent by a basepoint preserving local homeomorphism. \item ${\mathcal G}$ and ${\mathcal H}$ are conjugate return equivalent if and only if the minimal Cantor actions $(X({\mathcal G}) , G_0 , \Phi_0)$ and $(X({\mathcal H}) , H_0 , \Psi_0)$ are return equivalent. \end{enumerate}
\end{prop}
Finally, we recall a basic result from \cite{CHL2018a} that was mentioned above as the motivation for introducing the equivalence relation in Definition~\ref{defn-return}: \begin{thm}\label{thm-re} Let ${\mathcal S}_{{\mathcal P}}$ and ${\mathcal S}_{{\mathcal P}'}$ be weak solenoids associated to presentations ${\mathcal P}$ and ${\mathcal P}'$ with a common base manifolds, $M_0 = M_0'$. Let ${\mathcal G}$ be the group chain associated to ${\mathcal P}$, and ${\mathcal G}'$ be the group chain associated to ${\mathcal P}'$. If ${\mathcal S}_{{\mathcal P}}$ and ${\mathcal S}_{{\mathcal P}'}$ are homeomorphic as continua, then the group chains ${\mathcal G}$ and ${\mathcal G}'$ are conjugate return equivalent. \end{thm} There is a converse to this result, which requires additional assumptions on the base manifold $M_0$ and the geometry of the foliations, as given in \cite[Theorem 1.5]{CHL2018a}. The conclusion of Theorem~\ref{thm-re} is one of the motivations for the study of return equivalence of group chains, with the goal of obtaining computable invariants. The \emph{asymptotic discriminant}, introduced in \cite{HL2017} and further studied in \cite{HL2018}, is an invariant of conjugate return equivalence. The \emph{growth of orbits} as discussed in \cite{DHL2016b} is another such invariant.
\section{Induced maps between continua}\label{sec-induced}
We recall some results from the literature, concerning the properties of induced maps between continua as defined by maps between their presentations as in \eqref{eq-presentation-top}. The notion of equivalence is a fundamental aspect of the theory of shape equivalence, as developed for example in \cite{MardesicSegal1982}. In order to apply these results to the group chains associated to presentations, care must be taken with respect to basepoints. For this reason, we take our definitions from the original works by Bousfield and Kan \cite{BousfieldKan1972}, Fort and McCord \cite{FM1966}, Mioduszewski \cite{Mioduszewski1963}, and Rogers and Tollefson \cite{RT1971a,RT1971b}. These results are then applied for the presentations associated to a matchbox manifold which is $M$-like.
Assume there are given presentations
\begin{eqnarray} {\mathcal P} & = & \{\, f_{\ell+1} \colon X_{\ell+1} \to X_\ell \mid \ell \geq 0\} \\ {\mathcal Q} & = & \{\, g_{\ell+1} \colon Y_{\ell+1} \to Y_\ell \mid \ell \geq 0\} ~ , \end{eqnarray} where for all $\ell \geq 0$, the spaces $X_{\ell}$ and $Y_{\ell}$ are compact connected polyhedra, and the maps $f_{\ell+1}$ and $g_{\ell+1}$ are continuous surjections. Let ${\mathcal S}_{{\mathcal P}}$ be the inverse limit defined by ${\mathcal P}$, and ${\mathcal S}_{{\mathcal Q}}$ the inverse limit of ${\mathcal Q}$. A \emph{pro-map} from ${\mathcal P}$ to ${\mathcal Q}$ is a collection $\widehat{\sigma} \equiv \{ \sigma_{\ell} \colon X_{m_{\ell}} \to Y_{\ell} \mid \ell \geq 0\}$, where \begin{enumerate} \item $0 \leq m_0 <m_1 < m_2 < \cdots$ is an increasing sequence, \item $\sigma_{\ell} \colon X_{m_{\ell}} \to Y_{\ell}$ are continuous maps for $\ell \geq 0$, \end{enumerate} and moreover, for each $\ell > 0$ and $f^{m_{\ell+1}}_{m_{\ell}} = f_{m_{\ell} +1} \circ \cdots \circ f_{m_{\ell+1}}$, the following diagram commutes \begin{align}\label{eq-commutingdiagram} \xymatrix{
X_{m_{\ell}}\ar[d]^{\sigma_{\ell}} & \hspace{2mm} X_{m_{\ell+1}} \ar[d]^{\sigma_{\ell+1} ~ .} \ar[l]_{f^{m_{\ell+1}}_{m_{\ell}}}\\ Y_{\ell} & \hspace{2mm} Y_{\ell+1} \ar[l]^{g_{\ell+1}} } \end{align} A pro-map $\widehat{\sigma}$ determines a continuous map $S(\widehat{\sigma}) \colon {\mathcal S}_{{\mathcal P}} \to {\mathcal S}_{{\mathcal Q}}$ called the \emph{induced map} for $\widehat{\sigma}$.
Recall that $\Pi_{\ell}^{{\mathcal P}} \colon {\mathcal S}_{{\mathcal P}} \to X_{\ell}$ and $\Pi_{\ell}^{{\mathcal Q}} \colon {\mathcal S}_{{\mathcal Q}} \to Y_{\ell}$ denote the projections on the factors of these spaces. Then the defining property for an induced map is that for all $\ell > 0$, we have \begin{equation}\label{eq-commutingsquare} \Pi_{\ell}^{{\mathcal Q}} \circ S(\widehat{\sigma})= \sigma_{\ell} \circ \Pi_{m_{\ell}}^{{\mathcal P}} \colon {\mathcal S}_{{\mathcal P}} \to Y_{\ell} ~ . \end{equation} A fundamental question is when is there an induced map $S(\widehat{\sigma})$ which equals $h$, or is approximately equal to $h$ in an appropriate sense? This problem has been analyzed in the works \cite{Mioduszewski1963,FM1966,Kaul1967,RogersJT1970,RT1971a,RT1972b,Dydak1975,Dydak1980}, and we recall their conclusions as needed for the study of $M$-like spaces.
The first result was obtained by Mioduszewski in \cite{Mioduszewski1963}, where he introduced a generalization of the notion of a map between resolutions, where the hypothesis that the diagram \eqref{eq-commutingdiagram} commutes is replaced by a condition that it \emph{almost commutes}. To be more precise, first assume that each space $Y_{\ell}$ has a metric $d_{Y_{\ell}}$ defining its topology, for $\ell \geq 0$. Let $D(Y_{\ell}) > 0$ be the diameter of $Y_{\ell}$ for this metric, then and give the inverse limit ${\mathcal S}_{{\mathcal Q}}$ the metric \begin{equation}\label{eq-hamming} d_{{\mathcal S}_{{\mathcal Q}}}((y_{\ell})_{\ell \geq 0}) , (y'_{\ell})_{\ell \geq 0})) = \sum_{\ell \geq 0} ~ \frac{1}{2^{-\ell} D(Y_{\ell}) } \cdot d_{Y_{\ell}}(y_{\ell} , y'_{\ell}) ~ . \end{equation} Assume that there is given a descending sequence $\widehat{\epsilon} \equiv \{{\epsilon}_{0} > {\epsilon}_1 > {\epsilon}_2 > \cdots\}$ with $\displaystyle \lim_{\ell \to \infty} \, {\epsilon}_{\ell} = 0$. Then a sequence of maps $\widehat{\sigma}$ as above is said to be an $\widehat{{\epsilon}}$-pro-map if, in place of the assumption that the diagrams \eqref{eq-commutingdiagram} are commutative, we assume that for all $\ell > 0$, \begin{equation}\label{eq-epromap} d_{Y_{\ell}}(g_{\ell+1} \circ \sigma_{\ell +1}(x), \sigma_{\ell} \circ f^{m_{\ell+1}}_{m_{\ell}}(x)) < {\epsilon}_{\ell} \quad {\rm for ~ all} ~ x \in X_{m_{\ell +1}} ~ . \end{equation}
By \cite[Theorem 2']{Mioduszewski1963}, an $\widehat{{\epsilon}}$-pro-map $\widehat{\sigma}$ determines a continuous map $S(\widehat{\sigma}) \colon {\mathcal S}_{{\mathcal P}} \to {\mathcal S}_{{\mathcal Q}}$. Moreover, by the results of \cite[Section 3]{Mioduszewski1963}, if there is given a homeomorphism $\Phi \colon {\mathcal S}_{{\mathcal P}} \to {\mathcal S}_{{\mathcal Q}}$ then there exists $\widehat{\epsilon}$ as above, and an $\widehat{{\epsilon}}$-pro-map $\widehat{\sigma}$ such that the induced map $S(\widehat{\sigma}) \colon {\mathcal S}_{{\mathcal P}} \to {\mathcal S}_{{\mathcal Q}}$ is a homeomorphism. A natural problem is to then ask when the induced map $S(\widehat{\sigma})$ is equal to the map $\Phi$. Fort and McCord gave a partial answer to this question, in the case where the target space ${\mathcal S}_{{\mathcal Q}}$ is a weak solenoid: \begin{thm} \cite[Theorem 1]{FM1966} \label{thm-FMc} Let ${\mathcal P}$ be a presentation whose spaces $X_{\ell}$ are compact connected polyhedra, and let ${\mathcal Q}$ be a presentation consisting of covering maps. Assume that all spaces have metrics as above. Then for any map $\Phi \colon {\mathcal S}_{{\mathcal P}} \to {\mathcal S}_{{\mathcal Q}}$ and ${\epsilon} > 0$, there exists a descending sequence $\widehat{\epsilon}$ as above, and an $\widehat{{\epsilon}}$-pro-map $\widehat{\sigma}$ such that the induced map $S(\widehat{\sigma}) \colon {\mathcal S}_{{\mathcal P}} \to {\mathcal S}_{{\mathcal Q}}$ is ${\epsilon}$-homotopic to $\Phi$. In particular, the maps $\Phi$ and $S(\widehat{\sigma})$ are ${\epsilon}$-close. \end{thm}
In the proof of this result, the fact that each bonding map $q_{\ell}$ is a covering is used to lift a homotopy from the approximation at level $\ell$ to one at level $\ell+1$ (see the proof of \cite[Lemma 5]{FM1966}). Rogers and Tollefson gave a further refinement of Theorem~\ref{thm-FMc} in \cite{RT1971a}, characterizing the maps for which one can obtain $h = S(\widehat{\sigma})$ in terms of whether $h$ is locally fiber preserving, and in \cite{RT1972b} they applied their method to the case where both presentations consists of covering maps.
Next, let ${\mathfrak{M}}$ be a matchbox manifold which is $M$-like, where $M$ is a closed manifold. Then by Theorem~\ref{thm-weak},
there exists a presentation ${\mathcal P} = \{ p_{\ell +1} \colon M_{\ell +1} \to M_{\ell} \mid \ell \geq 0 \}$
where each bonding map $p_{\ell +1}$ is a proper covering map, and a homeomorphism
\begin{equation}\label{eq-ws2} \Phi_{{\mathcal P}} \colon {\mathfrak{M}} ~ \cong ~ {\mathcal S}_{{\mathcal P}} \equiv \varprojlim\, \{ p_{\ell +1} \colon M_{\ell +1} \to M_{\ell} \mid \ell \geq 0 \} ~. \end{equation} The choice of such a presentation is not unique. The result of Rogers and Tollefson \cite[Theorem 4.6]{RT1972b} relates two such presentations. \begin{thm} \label{thm-RTcoverings} Suppose there are two presentations as weak solenoids
\begin{eqnarray} {\mathcal P} & = & \{ p_{\ell +1} \colon M_{\ell +1} \to M_{\ell} \mid \ell \geq 0 \} \\ {\mathcal Q} & = & \{ q_{\ell +1} \colon N_{\ell +1} \to N_{\ell} \mid \ell \geq 0 \} ~ , \end{eqnarray}
a homeomorphism $\Phi \colon {\mathcal S}_{{\mathcal P}}\to {\mathcal S}_{{\mathcal Q}}$ and ${\epsilon} > 0$. Then there exists a decreasing sequence $\widehat{\epsilon}$ with ${\epsilon}_0 \leq {\epsilon}$, \begin{enumerate} \item an intertwined increasing sequence $0 \leq j_0 < i_0 <j_1 <i_1 < j_2 < i_2 < \cdots$, \item covering maps $\lambda_{\ell} \colon M_{i_{\ell +1}} \to N_{j_{\ell}}$ for $\ell \geq 0$, \item covering maps $\mu_{\ell} \colon N_{j_{\ell}} \to M_{i_{\ell}}$ for $\ell \geq 0$ ~ , \end{enumerate} such that the following diagram $\widehat{\epsilon}$-commutes:
\begin{align} \label{eq-commutativediagram}
\xymatrix{
M_{i_{0}} & \ar[l] M_{i_1} \ar[ld]_{\lambda_0} & \ar[l] M_{j_1} & \ar[l] M_{i_2} \ar[ld]_{\lambda_1} & \ar[l] M_{j_2} \cdots \\ N_{j_{0}} & \ar[l] N_{i_1} & \ar[l] N_{j_1} \ar[lu]_{\mu_1} & \ar[l] N_{i_2} & \ar[l] N_{j_2} \ar[lu]_{\mu_2} \cdots } \end{align} Moreover, for $\widehat{\lambda} = \{\lambda_{\ell} \colon M_{i_{\ell +1}} \to N_{j_{\ell}} \mid \ell \geq 0\}$, the induced map $S(\widehat{\lambda})$ is ${\epsilon}$-homotopic to $\Phi$, and similarly the induced map $S(\widehat{\mu})$ is ${\epsilon}$-homotopic to $\Phi^{-1}$. \end{thm}
We next recall an observation from Fokkink and Oversteegen in \cite{FO2002}, based on \cite[Theorem 2]{RT1971a}. Let ${\mathcal P}$ and ${\mathcal Q}$ be presentations for weak solenoids and $\Phi \colon {\mathcal S}_{{\mathcal P}} \to {\mathcal S}_{{\mathcal Q}}$ a homeomorphism, as in Theorem~\ref{thm-RTcoverings}.
Choose a basepoint $x \in {\mathcal S}_{{\mathcal P}}$ and set $y = \Phi(x) \in {\mathcal S}_{{\mathcal Q}}$.
For $\ell \geq 0$, let $\Pi^{{\mathcal P}}_{\ell} \colon {\mathcal S}_{{\mathcal P}} \to M_{\ell}$ be the projection map onto the factor space $M_{\ell}$, and set $x_{\ell} = \Pi^{{\mathcal P}}_{\ell}(x) \in M_{\ell}$. Then $X({\mathcal P})_{\ell} = (\Pi^{{\mathcal P}}_{\ell})^{-1}(x_{\ell}) \subset {\mathcal S}_{{\mathcal P}}$. Similarly, let $\Pi^{{\mathcal Q}}_{\ell} \colon {\mathcal S}_{{\mathcal Q}} \to N_{\ell}$ be the projection map onto the factor space $N_{\ell}$, and set $y_{\ell} = \Pi^{{\mathcal Q}}_{\ell}(y) \in N_{\ell}$. Then $X({\mathcal Q})_{\ell} = (\Pi^{{\mathcal Q}}_{\ell})^{-1}(y_{\ell}) \subset {\mathcal S}_{{\mathcal Q}}$.
\begin{remark}\cite[Lemma 21]{FO2002} \label{rmk-FO} {\rm In the conclusion of Theorem~\ref{thm-RTcoverings}, for $j_0 \geq 0$ and $i_1 > j_0$ sufficiently large, we can choose $\widehat{\lambda}$ so that the induced map satisfies $S(\widehat{\lambda})(X({\mathcal P})_{i_1}) = X({\mathcal Q})_{j_0}$. } \end{remark}
Fokkink and Oversteegen use this remark in \cite{FO2002} to show that, in essence, associated to a weak solenoid ${\mathfrak{M}}$, there are well-defined pro-fundamental groups. This shown in Theorem~\ref{thm-solprogroups} below.
Next, we recall the following sharper version of Theorem~\ref{thm-Freudenthal}, due to Marde\v{s}i\'{c} and Segal:
\begin{thm} \cite[Theorem~1*]{MardesicSegal1963} \label{thm-MardSeg} Let ${\mathfrak{M}}$ be a continuum which is $M$-like for some closed connected manifold $M$. Then there exists a presentation ${\mathcal Q} = \{ q_{\ell +1} \colon M \to M \mid \ell \geq 0 \}$, where each map $q_{\ell +1}$ is a continuous surjection, and a homeomorphism $\Phi_{{\mathcal Q}} \colon {\mathfrak{M}} \cong {\mathcal S}_{{\mathcal Q}}$. \end{thm}
For an $M$-like matchbox manifold ${\mathfrak{M}}$, there is a presentation ${\mathcal P}$ as a weak solenoid and a homeomorphism $\Phi_{{\mathcal P}} \colon {\mathfrak{M}} \to {\mathcal S}_{{\mathcal P}}$, as in \eqref{eq-ws2}, and a presentation ${\mathcal Q}$ and a homeomorphism $\Phi_{{\mathcal Q}} \colon {\mathfrak{M}} \to {\mathcal S}_{{\mathcal Q}}$, as in Theorem~\ref{thm-MardSeg}. Set $\Phi = {\mathcal S}_{{\mathcal Q}} \circ {\mathcal S}_{{\mathcal P}}^{-1} \colon {\mathcal S}_{{\mathcal P}} \to {\mathcal S}_{{\mathcal Q}}$. Combining the results of this section, we then have: \begin{thm} \label{thmRT} Suppose there are presentations
\begin{eqnarray} {\mathcal P} & = & \{ p_{\ell +1} \colon M_{\ell +1} \to M_{\ell} \mid \ell \geq 0 \} \label{eq-presentationWS}\\ {\mathcal Q} & = & \{ q_{\ell +1} \colon M \to M \mid \ell \geq 0 \} \label{eq-presentationML} \end{eqnarray} where for $\ell \geq 0$, each $p_{\ell +1}$ is a covering map, each $q_{\ell +1}$ is a continuous surjection, and there is a homeomorphism $\Phi \colon {\mathcal S}_{{\mathcal P}} \to {\mathcal S}_{{\mathcal Q}}$. Assume that the presentations have metrics as in \eqref{eq-hamming}. Then for ${\epsilon} > 0$ there exists a descending sequence $\widehat{\epsilon}$ with $0 < {\epsilon}_0 < {\epsilon}$ and: \begin{enumerate} \item intertwined increasing sequence $0 \leq j_0 < i_0 <j_1 <i_1 < j_2 < i_2 < \cdots$, \item continuous surjections $\lambda_{\ell} \colon M_{i_{\ell +1}} \to M$ for $\ell \geq 0$, \item continuous surjections $\mu_{\ell} \colon M \to M_{i_{\ell}}$ for $\ell \geq 0$, \end{enumerate} such that the following diagram $\widehat{\epsilon}$-commutes:
\begin{align} \label{eq-commutativediagram2}
\xymatrix{
M_{i_{0}} & \ar[l] M_{i_1} \ar[ld]_{\lambda_0} & \ar[l] M_{j_1} & \ar[l] M_{i_2} \ar[ld]_{\lambda_1} & \ar[l] M_{j_2} \cdots \\ M & \ar[l] M & \ar[l] M \ar[lu]_{\mu_1} & \ar[l] M & \ar[l] M \ar[lu]_{\mu_2} \cdots } \end{align}
Moreover, for $\widehat{\lambda} = \{\lambda_{\ell} \colon M_{i_{\ell +1}} \to M \mid \ell \geq 0\}$, the induced map $S(\widehat{\lambda}) \colon {\mathcal S}_{{\mathcal P}} \to {\mathcal S}_{{\mathcal Q}}$ is ${\epsilon}$-close to $\Phi$.
For $\widehat{\mu} = \{\mu_{\ell} \colon M \to M_{i_{\ell}} \mid \ell \geq 0\}$, the induced map $S(\widehat{\mu}) \colon {\mathcal S}_{{\mathcal Q}} \to {\mathcal S}_{{\mathcal P}}$ is ${\epsilon}$-homotopic to $\Phi^{-1}$, and in particular the maps $\Phi^{-1}$ and $S(\widehat{\mu})$ are ${\epsilon}$-close. \end{thm}
\begin{remark}\label{rmk-compare} {\rm In Section~\ref{sec-progroupsWS}, we compare in Theorem~\ref{thm-solprogroups} the pro-groups obtained from the fundamental groups for the spaces in the top and bottom rows of \eqref{eq-commutativediagram} in Theorem~\ref{thm-RTcoverings}, and the fact that we are given that the squares formed by the maps in $\widehat{\lambda}$ and $\widehat{\mu}$ are $\widehat{{\epsilon}}$-homotopic, and that the maps in $\widehat{\lambda}$ and $\widehat{\mu}$ are coverings, suffices to imply that these pro-groups are isomorphic.
Then in Section~\ref{sec-progroupsMlike}, we compare in Theorem~\ref{thm-proM} the pro-groups obtained from the fundamental groups for the spaces in the top and bottom rows of \eqref{eq-commutativediagram2} in Theorem~\ref{thmRT}. However, in this case, it is no longer given that the maps in $\widehat{\lambda}$ and $\widehat{\mu}$ are coverings. We note that the proof of Theorem~\ref{thmRT} by Marde\v{s}i\'{c} and Segal shows that there are simplicial spaces $N_{\ell}$, with each $N_{\ell}$ homeomorphic to $M$, such that the maps $q_{\ell +1} \colon M \to M $ for $\ell \geq 0$ are induced by simplicial maps $q'_{\ell +1} \colon N_{\ell+1} \to N_{\ell}$. Thus, we can use the results of James W. Rogers in \cite{RogersJW1973} to conclude that the squares formed by the maps in $\widehat{\lambda}$ and $\widehat{\mu}$ are in fact $\widehat{{\epsilon}}$-homotopic. However, it is not given that the maps in $\widehat{\lambda}$ and $\widehat{\mu}$ are locally onto, as they need not be covering maps, so that one does not have the injectivity property on fundamental groups that covering maps enjoy. Thus, an additional argument is needed to compare the pro-groups obtained from the fundamental groups for the spaces in the top and bottom rows of \eqref{eq-commutativediagram2}. This additional fact is given by Proposition~\ref{prop-leafonto}, which shows the local onto property for maps restricted to leaves, which is then used to establish an ${\epsilon}$-homotopy lifting property that is sufficient to compare these pro-groups by the path-lifting property. } \end{remark}
\section{Pro-groups and group chains}\label{sec-progroups}
In this section, we recall the general definition of a pro-group, and equivalence between pro-groups, following the development in \cite[Chapter II, Section 2]{MardesicSegal1982}. We then compare this notion with the group chains introduced in the previous section.
\begin{defn}\label{defn-progroup}
A pro-group is a collection
$${\mathcal G}(G_{\lambda}, p^{\lambda}_{\lambda'} , \Lambda) = \{ p^{\lambda}_{\lambda'} \colon G_{\lambda'} \to G_{\lambda} \mid \lambda \leq \lambda' , ~ \lambda, \lambda' \in \Lambda \}$$
where the $G_{\lambda}$ are groups, $p^{\lambda}_{\lambda'} $ are group homomorphisms, and $\Lambda$ is a directed index set.
\end{defn}
A \emph{morphism} $(f_{\gamma}, \phi)$ between pro-groups ${\mathcal G}(G_{\lambda}, p^{\lambda}_{\lambda'} , \Lambda)$ and ${\mathcal G}(H_{\lambda}, q^{\gamma}_{\gamma'} , \Gamma) $ is given by an order-preserving map $\phi \colon \Gamma \to \Lambda$ and homomorphisms $f_{\gamma} \colon G_{\phi(\gamma)} \to H_{\gamma}$
such that, whenever $\gamma < \gamma'$, then there is $\lambda \in \Lambda$, $\lambda \geq \phi(\gamma)$ and $\lambda \geq \phi(\gamma')$, for which
$f_{\gamma} \circ p^{\phi(\gamma)}_{\lambda} = q^{\gamma}_{\gamma'} \circ f_{\gamma'} \circ p^{\phi(\gamma')}_{\lambda}$. That is, the following commutes:
\begin{align} \label{eq-promorphism}
\xymatrix{
G_{\phi(\gamma)} \ar[d]_{f_{\gamma}} & & \ar[ll]_{p^{\phi(\gamma)}_{\lambda}} G_{\lambda} \ar[rr]^{p^{\phi(\gamma')}_{\lambda}} & & G_{\phi(\gamma')} \ar[d]^{f_{\gamma'}} \\ H_{\gamma} & & & & \ar[llll]_{q^{\gamma}_{\gamma'}} H_{\gamma'} } \end{align} The diagram \eqref{eq-promorphism} is called an \emph{equalizer} between the maps $f_{\gamma}$ and $f_{\gamma'}$.
The composition of morphisms of pro-groups is again a morphism, and the operation is associative. The details of the proofs of these statements, along with the notion of equivalence between morphisms, can be found in \cite[Chapter I, Section 1.1]{MardesicSegal1982}. Also, the notion of monomorphism, epimorphism and isomorphism between pro-groups is defined there. We recall a reformulation of these notions using the results in \cite[Chapter I, Sections 1.1, 1.2]{MardesicSegal1982}.
Given a morphism $(f_{\gamma}, \phi)$, a pair $(\lambda, \gamma) \in \Lambda \times \Gamma$ is said to be \emph{admissible} if $\lambda \geq \phi(\gamma)$. For an admissible pair $(\lambda, \gamma)$ define the homomorphism \begin{equation}\label{eq-compos} f^{\gamma}_{\lambda} = f_{\gamma} \circ p^{\phi(\gamma)}_{\lambda} ~ . \end{equation}
\begin{defn}\label{defn-promono}
\cite[Chapter II, Section 2.1, Theorem 2]{MardesicSegal1982}
A morphism $(f_{\gamma}, \phi)$ between pro-groups ${\mathcal G}(G_{\lambda}, p^{\lambda}_{\lambda'} , \Lambda)$ and ${\mathcal G}(H_{\lambda}, q^{\gamma}_{\gamma'} , \Gamma) $ is a \emph{monomorphism} if for every admissible pair $(\lambda, \gamma)$ for $(f_{\gamma}, \phi)$, there is an admissible pair $(\lambda', \gamma')$ for $(f_{\gamma}, \phi)$ with $\lambda' \geq \lambda$ and $\gamma' \geq \gamma$, such that \begin{equation}\label{eq-promono} {\rm Ker}\{f^{\gamma'}_{\lambda'} \colon G_{\lambda'} \to H_{\gamma'} \} \subset {\rm Ker}\{p^{\lambda}_{\lambda'} \colon G_{\lambda'} \to G_{\lambda}\} ~ . \end{equation} \end{defn}
\begin{defn}\label{defn-proepi}
\cite[Chapter II, Section 2.1, Theorem 4]{MardesicSegal1982}
A morphism $(f_{\gamma}, \phi)$ between pro-groups ${\mathcal G}(G_{\lambda}, p^{\lambda}_{\lambda'} , \Lambda)$ and ${\mathcal G}(H_{\lambda}, q^{\gamma}_{\gamma'} , \Gamma) $ is an \emph{epimorphism} if for every admissible pair $(\lambda, \gamma)$ for $(f_{\gamma}, \phi)$, there is an admissible pair $(\lambda', \gamma')$ for $(f_{\gamma}, \phi)$ with $\lambda' \geq \lambda$ and $\gamma' \geq \gamma$, such that \begin{equation}\label{eq-proepi} {\rm Image}\{q^{\gamma}_{\gamma'} \colon H_{\gamma'} \to H_{\gamma} \} \subset {\rm Image}\{f^{\gamma}_{\lambda} \colon G_{\lambda} \to H_{\gamma}\} ~ . \end{equation} \end{defn}
The following characterization of isomorphism of pro-groups is then equivalent to Morita's Lemma:
\begin{defn}\label{defn-proiso}
\cite[Chapter II, Section 2.2, Theorem 6]{MardesicSegal1982}
A morphism $(f_{\gamma}, \phi)$ between pro-groups ${\mathcal G}(G_{\lambda}, p_{\lambda, \lambda'} , \Lambda)$ and ${\mathcal G}(H_{\lambda}, q_{\gamma, \gamma'} , \Gamma) $ is an \emph{isomorphism} if and only if $(f_{\gamma}, \phi)$ is a monomorphism and an epimorphism.
\end{defn}
For our purposes, the ordered sets $\Lambda$ and $\Gamma$ above will always be assumed to be a subset of the non-negative integers ${\mathbb N}$ with the natural order.
Finally, we compare the notions of group chains and the pro-groups they determine.
Let ${\mathcal G} = \{G_{\ell}\}$ and ${\mathcal H} = \{H_{\ell}\}$ be two group chains, which determine pro-groups
\begin{eqnarray*} {\mathcal G}(G_{\ell}, p^{\ell}_{\ell'}) & = & \{ p^{\ell}_{\ell'} \colon G_{\ell'} \to G_{\ell} \mid \ell' \geq \ell \geq 0, ~ \ell, \ell' \in {\mathbb N} \} \\ {\mathcal G}(H_{\ell}, q^{\ell}_{\ell'}) & = & \{ q^{\ell}_{\ell'} \colon H_{\ell'} \to H_{\ell} \mid \ell' \geq \ell \geq 0, ~ \ell, \ell' \in {\mathbb N} \} ~ . \end{eqnarray*} where the maps $p^{\ell}_{\ell'}$ and $q^{\ell}_{\ell'}$ are given by the inclusions of subgroups.
Suppose the group chains ${\mathcal G}$ and ${\mathcal H}$ are equivalent in the sense of Definition~\ref{defn-greq}. Then there are infinite increasing sequences $\{\ell_k\}$ and $\{j_k\}$, and inclusion maps $$\lambda_k \colon K_{2k+2} = G_{\ell_{k+1}} \subset K_{2k+1} = H_{j_k} \quad {\rm and} \quad \mu_k \colon K_{2k+1} = H_{j_k} \subset K_{2k} = G_{\ell_k} ~ .$$ Introduce the group chains ${\mathcal A} = \{A_k \equiv G_{\ell_k} \mid k \geq 1\}$ and ${\mathcal B} = \{B_k \equiv H_{j_k} \mid k \geq 1\}$. Then we obtain a pro-group ${\mathcal G}(A_k, \iota^k_{k'})$, where $\iota^k_{k'} \colon A_{k'} \subset A_k$ is the inclusion map. It is immediate that ${\mathcal G}(G_{\ell}, p^{\ell}_{\ell'})$ and ${\mathcal G}(A_k, \iota^k_{k'})$ are isomorphic pro-groups. Likewise, form the pro-group ${\mathcal G}(B_k, \xi^k_{k'})$ where $\xi^k_{k'} \colon B_{k'} \subset B_k$ is the inclusion map. It is immediate that ${\mathcal G}(H_{\ell}, q^{\ell}_{\ell'})$ and ${\mathcal G}(B_k, \xi^k_{k'})$ are isomorphic pro-groups.
For $k \geq 1$, set $\phi(k) = k+1$, and let $f_{k} = \lambda_k \colon A_{k+1} \to B_k$. Then the map of pro-groups $(f_k, \phi)$ is a monomorphism, as each $\lambda_k$ is a monomorphism. This map is also an epimorphism, which follows using the maps $\{\mu_k\}$. It follows that the pro-groups ${\mathcal G}(G_{\ell}, p^{\ell}_{\ell'}) $ and ${\mathcal G}(H_{\ell}, q^{\ell}_{\ell'})$ are isomorphic.
If the group chains ${\mathcal G}$ and ${\mathcal H}$ are equivalent in the sense of Definition~\ref{defn-conjequiv}, then there exists
a sequence $(g_{\ell})_{\ell \geq 0}$ where each $g_{\ell} \in G_0$ and $g_{\ell}G_{\ell} = g_{\ell +1} G_{\ell}$ for all $\ell\geq 0$, so that the groups chains ${\mathcal G}^{(g_{\ell})} = \{g_{\ell} G_{\ell} g_{\ell}^{-1}\}_{\ell \geq 0}$ and ${\mathcal H}$ are equivalent. Define \begin{equation}\label{eq-chainstopro} \lambda_k(g) = g_{\ell_{k+1}} g g_{\ell_{k+1}}^{-1} ~ {\rm for}~ g \in G_{\ell_{k+1}} \quad , \quad
\mu_k(h) = g_{j_k}^{-1} h g_{j_k} ~ {\rm for} ~ h \in H_{j_k} ~ , \end{equation}
then as before, this yields an isomorphism between the pro-groups ${\mathcal G}(G_{\ell}, p^{\ell}_{\ell'}) $ and ${\mathcal G}(H_{\ell}, q^{\ell}_{\ell'})$.
Next, suppose that we have an isomorphism $(f_{\gamma}, \phi)$ between pro-groups ${\mathcal G}(G_{\ell}, p^{\ell}_{\ell'})$ and ${\mathcal G}(H_{\ell}, q^{\ell}_{\ell'})$. Then we have monomorphisms $f_{\ell} \colon G_{\phi(\ell)} \to H_{\ell}$.
Define $H'_{\ell} = f_{\ell}(G_{\phi(\ell)})$, then by the commutative properties in diagram \eqref{eq-promorphism} and the isomorphism conditions \eqref{eq-promono} and \eqref{eq-proepi}, we obtain a group chain
\begin{equation} H_0 \supset H_1 \supset H'_1 \supset H_2 \supset H'_2 \supset \cdots ~ . \end{equation} It follows that the group chains ${\mathcal H} = \{H_{\ell}\}$ and ${\mathcal H}' = \{H'_{\ell}\}$ are equivalent in the sense of Definition~\ref{defn-greq}.
Now suppose that the pro-groups are obtained from group chains ${\mathcal G}$ and ${\mathcal H}$ as above. For simplicity, assume that $G_0 = H_0$. Assume that the pro-groups are isomorphic, then by the above, we have that ${\mathcal H}$ and ${\mathcal H}'$ are equivalent group chains. However, this does not imply that the group chains ${\mathcal G}$ and ${\mathcal H}$ in $G_0$ are conjugate equivalent, as the conditions in Definition~\ref{defn-conjequiv} require that each monomorphism $f_{\ell}$ be implemented by conjugation by some element $g_{\ell} \in G_0$. In other words, given two isomorphic subgroups of a group, they need not be in the same conjugacy class. If we consider the weaker notion of return equivalence in Definition~\ref{prop-re}, then the requirement becomes that there exists some $\ell_0 > 0$ such that the maps $f_{\ell}$ are implemented by a conjugacy for $\ell \geq \ell_0$ and this need not hold in general either. Thus, the notion of isomorphism between the pro-groups defined by group chains ${\mathcal G}$ and ${\mathcal H}$ may be weaker than the notion of return equivalence between the group chains.
\begin{quest}
Is there a natural algebraic condition on a group $G$, which implies that two group chains ${\mathcal G}$ and ${\mathcal H}$ in $G$ which are isomorphic as pro-groups must be return equivalent?
\end{quest}
\section{Pro-fundamental groups of weak solenoids}\label{sec-progroupsWS}
A presentation for a continuum ${\mathfrak{M}}$ can be viewed as a particular type of shape expansion for ${\mathfrak{M}}$, and so defines its pro-homotopy type, in the sense of Marde\v{s}i\'{c} and Segal \cite{MardesicSegal1982}, or in the foundational work by Bousfield and Kan \cite{BousfieldKan1972}.
One can apply various functors to this shape expansion to obtain invariants of ${\mathfrak{M}}$. Marde\v{s}i\'{c} and Segal show in \cite[Chapter II, Section 3.1]{MardesicSegal1982} that the pro-homology groups {\it pro}-$H_k({\mathfrak{M}})$ for $k > 0$ are well-defined, and show in \cite[Chapter II, Section 3.3]{MardesicSegal1982} that the pro-homotopy groups {\it pro}-$\pi_k({\mathfrak{M}}, z)$ for $k > 1$ are well-defined.
The definition of the pro-fundamental group {\it pro}-$\pi_1({\mathfrak{M}}, z)$ for a weak solenoid requires extra care, due to basepoint issues. If one assumes that there is a shape expansion which preserves basepoints, and that maps between expansions also preserves basepoints, then the results of \cite[Chapter II, Section 3.3]{MardesicSegal1982} show that the pro-fundamental group is well-defined. However, the maps between presentations in Section~\ref{sec-induced} above are not assumed to preserve basepoints. The work of Fox in
\cite{Fox1972} introduces the notion of \emph{tropes} to circumvent these difficulties, and define covering spaces for shape approximations. This topic is addressed further in the work of
Marde\v{s}i\'{c} and Matijevi\'{c} \cite{MardesicMatijevic2001}, and the work of
Eda, Mandi\'{c} and Matijevi\'{c} in \cite{EMM2005} illustrates some of the subtleties of the issues involved. In any case, this approach does not suffice for the study of the equivalence of pro-fundamental groups for the inverse limits spaces we consider in this work. Instead, we develop an approach based on the fact that the shape expansions we work with are derived from a matchbox manifold, so that we can use the geometry of the leaves of the foliation of this space to show the pro-groups are well-defined.
Now assume that there is a solenoidal presentation ${\mathcal P}$ and a homeomorphism $\Phi_{{\mathcal P}} \colon {\mathfrak{M}} \to {\mathcal S}_{{\mathcal P}}$. Choose a basepoint $z \in {\mathfrak{M}}$, then set $x = \Phi_{{\mathcal P}}(z)$. Let $\Pi^{{\mathcal P}}_{\ell} \colon {\mathcal S}_{{\mathcal P}} \to M_{\ell}$ be the projection map onto the factor space $M_{\ell}$, set $x_{\ell} = \Pi^{{\mathcal P}}_{\ell}(x) \in M_{\ell}$ and let $X({\mathcal P})_{\ell} = (\Pi^{{\mathcal P}}_{\ell})^{-1}(x_{\ell}) \subset {\mathcal S}_{{\mathcal P}}$. Define the pro-group \begin{equation}\label{eq-weakprogroupdef} {\mathcal G}({\mathcal P},x) = \{p_{\ell +1} \colon \pi_1(M_{\ell +1} , x_{\ell+1}) \to \pi_1(M_{\ell} , x_{\ell}) \mid \ell \geq 0 \} \end{equation} where by a small abuse of notation, we let $p_{\ell +1}$ also denote the induced map on fundamental groups.
We then have the following result, which can be considered a formal restatement of results in \cite{FO2002}:
\begin{thm}\label{thm-solprogroups} Let ${\mathfrak{M}}$ be an $M$-like matchbox manifold, and let ${\mathcal P}$ be a presentation for which there is a homeomorphism $\Phi_{{\mathcal P}} \colon {\mathfrak{M}} \to {\mathcal S}_{{\mathcal P}}$. Choose $z \in {\mathfrak{M}}$ and set $x = \Phi_{{\mathcal P}}(z)$. Then the isomorphism class of the pro-group {\it pro}-$\pi_1({\mathfrak{M}},z)$ derived from ${\mathcal G}({\mathcal P},x)$ is independent of the choice of presentation ${\mathcal P}$ and basepoint $x$, and thus is invariant of the homeomorphism class of ${\mathfrak{M}}$.
\end{thm}
\proof
Suppose that ${\mathcal Q}$ is another presentation for ${\mathfrak{M}}$ as a weak solenoid, then we are given the hypotheses and conclusions of Theorem~\ref{thm-RTcoverings}. We use the notation there.
Choose a basepoint $z \in {\mathfrak{M}}$, let $x = \Phi_{{\mathcal P}}(z)$ and $y = \Phi_{{\mathcal Q}}(z)$. Set $\Phi = \Phi_{{\mathcal Q}}^{-1} \circ \Phi_{{\mathcal P}}$.
For the pro-morphisms $\widehat{\lambda}$ and $\widehat{\mu}$ obtained from the maps in diagram \eqref{eq-commutativediagram} which $\widehat{{\epsilon}}$-commutes, the induced map $S(\widehat{\lambda})$ is ${\epsilon}$-homotopic to $\Phi$, and similarly the induced map $S(\widehat{\mu})$ is ${\epsilon}$-homotopic to $\Phi^{-1}$.
For $\ell \geq 0$ set $x_{\ell} = \Pi^{{\mathcal P}}_{\ell}(x) \in M_{\ell}$ and $y_{\ell} = \Pi^{{\mathcal Q}}_{\ell}(y) \in N_{\ell}$, then we have: \begin{equation}\label{eq-ehomotopic} d_{N_{j_{\ell}}}( \lambda_{\ell}(x_{i_{\ell +1}}) , y_{j_{\ell}}) \leq {\epsilon}_{\ell} \quad {\rm and} \quad d_{M_{i_{\ell}}}( \mu_{\ell}(y_{j_{\ell}}) , x_{i_{\ell}}) \leq {{\epsilon}_{\ell}} ~ . \end{equation} The ${\epsilon}$-homotopy from $S(\widehat{\lambda})$ to $\Phi$ defines a path from the basepoint $\lambda_{\ell}(x_{i_{\ell +1}})$ to the basepoint $y_{j_{\ell}}$, and similarly from the basepoint $\mu_{\ell}(y_{j_{\ell}})$ to the basepoint $x_{i_{\ell}}$. Moreover, this homotopy induces a homotopy of paths at these basepoints, so we obtain from \eqref{eq-commutativediagram} a commutative diagram:
\begin{align} \label{eq-commutativediagrampi1}
\xymatrix{
& \pi_1(M_{i_1} , x_{i_1}) \ar[ld]_{\lambda_0} & \ar[l] \pi_1(M_{j_1} , x_{j_1}) & \ar[l] \pi_1(M_{i_2} , x_{i_2}) \ar[ld]_{\lambda_1} & \ar[l] \pi_1(M_{j_2} , x_{j_2}) \cdots \\ \pi_1(N_{j_0} , y_{j_0}) & \ar[l] \pi_1(N_{i_1} , y_{i_1}) & \ar[l] \pi_1(N_{j_1} , y_{j_1}) \ar[lu]_{\mu_1} & \ar[l] \pi_1(N_{i_2} , y_{i_2}) & \ar[l] \pi_1(N_{j_2} , y_{j_2}) \ar[lu]_{\mu_2} \cdots } \end{align} where we again abuse notation, by denoting a map and its induced map on fundamental groups the same. All maps in \eqref{eq-commutativediagrampi1} are monomorphisms, as they are induced from covering maps.
We check that the conditions of Definitions~\ref{defn-promono} and \ref{defn-proepi} are satisfied for $G_i = \pi_1(M_i,x_i)$ and $H_j = \pi_1(N_j,y_j)$. We first translate the conditions \eqref{eq-promono} and the conditions \eqref{eq-proepi} into the nomenclature of diagram \eqref{eq-commutativediagrampi1}. The corresponding indexing sets are $\Lambda = \{i_1 , i_2, \ldots \}$ and $\Gamma = \{j_1 , j_2, j_3, \ldots \}$. Recall that these indices are chosen so that $ j_{\ell} < i_{\ell} <j_{\ell + 1} <i_{\ell + 1}$ for all $\ell \geq 0$.
The maps $f_{\lambda}$ correspond to the maps $\lambda_{\ell} = G_{i_{\ell+1}} \to H_{j_{\ell}}$ so we have $\phi(j_{\ell}) = i_{\ell+1}$.
The admissible condition $\lambda \geq \phi(\gamma)$ then becomes, for $\lambda = k$ and $\gamma = \ell$, that $i_{k} \geq i_{\ell +1}$ so by our choice of the sequences, that $k \geq \ell+1$.
The maps $f_{\gamma} \colon G_{\phi(\gamma)} \to H_{\gamma}$ correspond to the maps $\lambda_{\ell} \colon G_{i_{\ell +1}} \to H_{j_{\ell}}$, and so the maps $f^{\gamma}_{\lambda} = f_{\gamma} \circ p^{\phi(\gamma)}_{\lambda}$ defined in \eqref{eq-compos} correspond to the maps $f^{\ell}_{k} = \lambda_{\ell} \circ p^{\phi(j_\ell)}_{k}$.
Then the monomorphism condition \eqref{eq-promono} becomes \begin{equation}\label{eq-monoaltnotation} {\rm Ker}\{\lambda_{\ell'} \circ p^{\phi(j_{\ell'})}_{k'} \colon G_{k'} \to H_{\ell'} \} \subset {\rm Ker}\{p^{k}_{k'} \colon G_{k'} \to G_{k}\} \end{equation}
and the epimorphism condition \eqref{eq-proepi} becomes \begin{equation}\label{eq-epialtnotation} {\rm Image}\{q^{\ell}_{\ell'} \colon H_{\ell'} \to H_{\ell} \} \subset {\rm Image}\{\lambda_{\ell} \circ p^{\phi(j_{\ell})}_{k} \colon G_{k} \to H_{\ell}\} ~ . \end{equation} The induced maps $\lambda_{\ell}$ and $p_{\ell}$ are monomorphisms, as they are induced by covering maps, so condition \eqref{eq-monoaltnotation} is satisfied.
To show condition \eqref{eq-epialtnotation}, we use the commutativity of the diagram \eqref{eq-commutativediagrampi1} which implies that $q^{\ell}_{\ell'} = \lambda_{\ell} \circ p^{i_{\ell +1}}_{i_{\ell'}} \circ \mu_{\ell'}$ so that \eqref{eq-epialtnotation} is satisfied for $k = \ell' \geq \ell+1$.
Thus, the pro-morphisms $\widehat{\lambda}$ and $\widehat{\mu}$ induce isomorphisms between the pro-groups defined by the top and bottom rows of \eqref{eq-commutativediagrampi1}. This shows that the pro-group ${\mathcal G}({\mathcal P},x)$ defined in \eqref{eq-weakprogroupdef} is independent of the choice of presentation ${\mathcal P}$.
It remains to show that for two choices $z,z' \in {\mathfrak{M}}$, the pro-groups ${\mathcal G}({\mathcal P},x)$ and ${\mathcal G}({\mathcal P},x')$ are isomorphic for $x = \Phi_{{\mathcal P}}(z)$ and $x' = \Phi_{{\mathcal P}}(z')$. By Remark~\ref{rmk-FO}, we can assume that $x' \in {\mathfrak{X}}_0$. Then by Theorem~\ref{thm-re}, the group chains ${\mathcal G}_x$ and ${\mathcal G}_{x'}$ are conjugate return equivalent in the sense of Definition~\ref{defn-return}.
Then as in the discussion at the end of Section~\ref{sec-progroups}, observe that the conjugation maps in Definition~\ref{defn-conjequiv} induce isomorphisms between the subgroups $G_{\ell}$ and $H_{\ell}$ for $\ell > 0$, and the notion of equivalence of group chains in Definition~\ref{defn-greq} induces an isomorphism of the pro-groups ${\mathcal G}({\mathcal P},x)$ and ${\mathcal G}({\mathcal P},x')$ they determine, as was to be shown. \endproof
\begin{remark}
{\rm
We note a curious aspect of the conclusion of Theorem~\ref{thm-solprogroups}.
The isomorphism class {\it pro}-$\pi_1({\mathfrak{M}},z)$ is independent of the choice of basepoint used to define the group chain ${\mathcal G}^x$, for $x = \Phi_{{\mathcal P}}(z)$, while by
Remark~\ref{rmk-kernels} the kernel of the associated group chain $K({\mathcal G}^x)$ may depend on the choice of $x$.
Thus, there is some loss of information when we pass from a group chain ${\mathcal G}^x$ to the pro-group it determines. The reason is simply that the group chain ${\mathcal G}^x$ contains the information on how the groups are embedded in $G_0$, while the pro-group {\it pro}-$\pi_1({\mathfrak{M}},x)$ does not.
}
\end{remark}
\section{Pro-fundamental groups of M-like spaces}\label{sec-progroupsMlike}
In this section we prove the following, which can be considered the main result of this work. \begin{thm}\label{thm-proM}
Let ${\mathfrak{M}}$ be a matchbox manifold which is $M$-like. Then {\it pro}-$\pi_1({\mathfrak{M}}, z)$ is pro-isomorphic to the pro-group defined by maps $\{g_{\ell} \colon G \to G \mid \ell \geq 0\}$, where $G= \pi_1(M, y)$.
\end{thm}
\proof The idea is to use Theorem~\ref{thm-MardSeg} to obtain a diagram analogous to \eqref{eq-commutativediagrampi1}, and then follow the outline of the proof of Theorem~\ref{thm-solprogroups}. This requires that we first choose basepoints that ``almost commute'', so the pro-morphisms $\widehat{\lambda}$ and $\widehat{\mu}$ are well-defined. The main issue then is to show the maps satisfy the conditions \eqref{eq-monoaltnotation} and \eqref{eq-epialtnotation}. It follows from Theorem~\ref{thmRT} that the maps in $\widehat{\mu}$ induce commutative squares in the diagram \eqref{eq-commutativediagram2}. Unfortunately, the maps $\widehat{\lambda}$ in the diagram \eqref{eq-commutativediagram2} are only known to $\widehat{\epsilon}$-commute, and neither collection of maps $\widehat{\lambda}$ and $\widehat{\mu}$ are known to be coverings. Thus, another approach is required, and we use a technique that is analogous to the approach of Rogers and Tollefson in \cite[Section 3]{RT1972b}.
We first recall some basic results about the leafwise geometry of foliated spaces and matchbox manifolds, as discussed in detail for example in the works \cite{CandelConlon2000,ClarkHurder2013,CHL2018b}.
\begin{thm}\label{thm-riemannian} Let ${\mathfrak{M}}$ be a smooth matchbox manifold with foliation ${\mathcal F}_{{\mathfrak{M}}}$. Then there exists a leafwise Riemannian metric for ${\mathcal F}_{{\mathfrak{M}}}$, such that for each $x \in {\mathfrak{M}}$, the leaf $L_x$ inherits the structure of a complete Riemannian manifold with bounded geometry, and the Riemannian metric and its covariant derivatives depend continuously on $x$ . \end{thm} Bounded geometry implies, for example, that for each $x \in {\mathfrak{M}}$, there is a leafwise exponential map $\exp^{{\mathcal F}}_x \colon T_x{\mathcal F}_{{\mathfrak{M}}} \to L_x$ which is a surjection, and the composition with the inclusion map, $\exp^{{\mathcal F}}_x \colon T_x{\mathcal F}_{{\mathfrak{M}}} \to L_x \subset {\mathfrak{M}}$, depends continuously on $x$ in the compact-open topology on maps.
Each leaf $L \subset {\mathfrak{M}}$ has a complete path-length metric, induced from the leafwise Riemannian metric:
$$d_{{\mathcal F}}(x,y) = \inf \left\{\| \gamma\| \mid \gamma \colon [0,1] \to L ~{\rm is ~ piecewise ~~ C^1}~, ~ \gamma(0) = x ~, ~ \gamma(1) = y ~, ~ \gamma(t) \in L \quad \forall ~ 0 \leq t \leq 1\right\}$$
where $\| \gamma \|$ denotes the path-length of the piecewise $C^1$-curve $\gamma(t)$. If $x,y \in {\mathfrak{M}}$ are not on the same leaf, then set $d_{{\mathcal F}}(x,y) = \infty$.
For each $x \in {\mathfrak{M}}$ and $r > 0$, let $D_{{\mathcal F}}(x, r) = \{y \in L_x \mid d_{{\mathcal F}}(x,y) \leq r\}$.
For each $x \in {\mathfrak{M}}$, the {Gauss Lemma} implies that there exists $\lambda_x > 0$ such that $D_{{\mathcal F}}(x, \lambda_x)$ is a \emph{strongly convex} subset for the metric $d_{{\mathcal F}}$. That is, for any pair of points $y,y' \in D_{{\mathcal F}}(x, \lambda_x)$ there is a unique shortest geodesic segment in $L_x$ joining $y$ and $y'$ and contained in $D_{{\mathcal F}}(x, \lambda_x)$. This standard concept of Riemannian geometry is discussed in detail in \cite{BC1964}, and in \cite[Chapter 3, Proposition 4.2]{doCarmo1992}. Then for all $0 < \lambda < \lambda_x$ the disk $D_{{\mathcal F}}(x, \lambda)$ is also strongly convex. Then we have: \begin{lemma}\label{lem-stronglyconvex} There exists ${\lambda_{\mathcal F}} > 0$ such that for all $x \in {\mathfrak{M}}$, $D_{{\mathcal F}}(x, {\lambda_{\mathcal F}})$ is strongly convex. \end{lemma}
Choose a presentation ${\mathcal Q} = \{ q_{\ell +1} \colon M \to M \mid \ell \geq 0 \}$ as in Theorem~\ref{thm-MardSeg}. The manifold $M$ is assumed to be smooth and without boundary, and choose a Riemannian metric $d_M$ for $M$. Let $\lambda_M > 0$ be such that for $u \in M$, the geodesic ball $B_M(u, \lambda_M) \subset M$ is strongly convex.
Give ${\mathcal S}_{{\mathcal Q}}$ the metric \eqref{eq-hamming}, where each factor space $Y_{\ell} = M$ is given the metric $d_M$.
Choose a presentation ${\mathcal P} = \{ p_{\ell +1} \colon M_{\ell +1} \to M_{\ell} \mid \ell \geq 0 \}$ for which there is a homeomorphism $\Phi_{{\mathcal P}} \colon {\mathfrak{M}} \cong {\mathcal S}_{{\mathcal P}}$,
where each $M_\ell$ is a compact connected $n$-dimensional manifold without boundary, and each $p_{\ell +1 }$ is a covering map of finite degree greater than one.
Choose a Riemannian metric on $M_0$ such that every geodesic ball of radius at most ${\epsilon}'' > 0$ is strongly convex, then for $\ell > 0$, let $M_{\ell}$ have the Riemannian metric induced by the covering map $p^0_{\ell} \colon M_{\ell} \to M_0$ in \eqref{eq-presentation}. Give ${\mathcal S}_{{\mathcal P}}$ the metric as in \eqref{eq-hamming}.
It then follows that for each $b \in M_0$ and closed ball $B_{M_0}(b, {\epsilon}''/2) \subset M_0$, the inverse image $\Pi_0^{-1}(B_{M_0}(b, {\epsilon}''/2)) \subset {\mathcal S}_{{\mathcal P}}$ is a chart for the foliation ${\mathcal F}_{\cP}$ on the inverse limit ${\mathcal S}_{{\mathcal P}}$. The proof by McCord in \cite{McCord1965} that weak solenoids are foliated spaces used just such charts. Also, note that each leaf $L \subset {\mathcal S}_{{\mathcal P}}$ is a covering of $M_0$. The lift of the Riemannian metric on $M_0$ to $L$ defines a path length metric on each leaf $L$ of ${\mathcal F}_{\cP}$.
Moreover, the homeomorphism $\Phi_{{\mathcal P}}$ preserves the path components of ${\mathfrak{M}}$ and ${\mathcal S}_{{\mathcal P}}$ so for each leaf $L \subset {\mathfrak{M}}$ the restriction $\Phi_{{\mathcal P}} | L$ is a homeomorphism onto a leaf of ${\mathcal F}_{\cP}$. Moreover, the compactness of ${\mathfrak{M}}$ implies that each such restriction is uniformly continuous for the leafwise metrics on ${\mathcal F}_{{\mathfrak{M}}}$ and ${\mathcal F}_{\cP}$. Thus, we can choose ${\epsilon}'' > 0$ sufficiently small so that for each $w \in {\mathcal S}_{{\mathcal P}}$ the leafwise ball $B_{{\mathcal S}_{{\mathcal P}}}(w, {\epsilon}'') \subset L_z$ is contained in the image $\Phi(B_{{\mathfrak{M}}}(u, {\lambda_{\mathcal F}}))$, where $u = \Phi_{{\mathcal P}}^{-1}(w)$.
By Theorem~\ref{thmRT}, for ${\epsilon} = \min \{{\epsilon}'/4, {\epsilon}''/4\}$, there exists a sequence $\widehat{\epsilon} = \{{\epsilon} \geq {\epsilon}_0 > {\epsilon}_1 > \cdots\}$ descending to zero, and \begin{enumerate} \item an intertwined increasing sequence $0 \leq j_0 < i_0 <j_1 <i_1 < j_2 < i_2 < \cdots$, \item continuous surjections $\widehat{\lambda} \equiv \{\lambda_{\ell} \colon M_{i_{\ell +1}} \to M \mid \ell \geq 0\}$, \item continuous surjections $\widehat{\mu} \equiv \{\mu_{\ell} \colon M \to M_{i_{\ell}} \mid \ell \geq 0\}$, \item a homeomorphism $\Phi_{{\mathcal Q}} \colon {\mathfrak{M}} \to {\mathcal S}_{{\mathcal Q}}$, \end{enumerate}
so that the diagram \eqref{eq-commutativediagram3} below $\widehat{\epsilon}$-commutes, where for the convenience of the arguments to follow we use $N_{\ell} = M$ so that the indexing is indicated:
\begin{align} \label{eq-commutativediagram3}
\xymatrix{
M_{i_{0}} & \ar[l] M_{i_1} \ar[ld]_{\lambda_0} & \ar[l] M_{j_1} & \ar[l] M_{i_2} \ar[ld]_{\lambda_1} & \ar[l] M_{j_2} \cdots \\ N_{j_{0}} & \ar[l] N_{i_1} & \ar[l] N_{j_1} \ar[lu]_{\mu_1} & \ar[l] N_{i_2} & \ar[l] N_{j_2} \ar[lu]_{\mu_2} \cdots } \end{align}
This means that for each $\ell > 0$, we have
\begin{eqnarray} d_M( \lambda_{\ell} \circ \mu_{\ell +1}(y) , q^{j_{\ell}}_{j_{\ell +1}}(y)) & \leq & {\epsilon}_{\ell} \quad {\rm for ~ all} ~ y \in N_{j_{\ell +1}} = M \label{eq-approx1ell}\\
d_{M_{i_{\ell}}} ( \mu_{\ell} \circ \lambda_{\ell}(x) , p^{i_{\ell}}_{i_{\ell+1}}(x)) & \leq & {\epsilon}_{\ell} \quad {\rm for ~ all} ~ y \in M_{i_{\ell+1}} ~ . \label{eq-approx2ell} \end{eqnarray}
Choose $z \in {\mathfrak{M}}$, let $x = \Phi_{{\mathcal P}}(z)$ and $y = \Phi_{{\mathcal Q}}(z)$. Set $x_{\ell} = \Pi^{{\mathcal P}}_{\ell}(x) \in M_{\ell}$ and $y_{\ell} = \Pi^{{\mathcal Q}}_{\ell}(y) \in M$.
By \eqref{eq-approx1ell} we have that $\mu_{\ell}(y_{j_{\ell}}) \in B_{M_{i_{\ell}}}(x_{i_{\ell}}, {\epsilon}_{i_{\ell}})$ which is a contractible disk. Thus, we obtain a well-defined map $\mu_{\ell} \colon \pi_1(M, y_{j_{\ell}}) \to \pi_1(M_{i_{\ell}} , x_{i_{\ell}})$ where we again use the same notation for the map on spaces and the induced map on homotopy groups. Similarly, \eqref{eq-approx2ell} implies that there well-defined map $\lambda_{\ell} \colon \pi_1(M_{i_{\ell+1}}, x_{i_{\ell +1}}) \to \pi_1(M , y_{i_{\ell}})$. Thus we obtain the following diagram: \begin{align} \label{eq-commutativediagrampi12}
\xymatrix{
& \pi_1(M_{i_1} , x_{i_1}) \ar[ld]_{\lambda_0} & \ar[l] \pi_1(M_{j_1} , x_{j_1}) & \ar[l] \pi_1(M_{i_2} , x_{i_2}) \ar[ld]_{\lambda_1} & \ar[l] \pi_1(M_{j_2} , x_{j_2}) \cdots \\ \pi_1(M , y_{j_0}) & \ar[l] \pi_1(M , y_{i_1}) & \ar[l] \pi_1(M , y_{j_1}) \ar[lu]_{\mu_1} & \ar[l] \pi_1(M , y_{i_2}) & \ar[l] \pi_1(M , y_{j_2}) \ar[lu]_{\mu_2} \cdots } \end{align} The maps in the top row of \eqref{eq-commutativediagrampi12} are all monomorphisms as they are induced by covering maps, and the squares formed by the maps in $\widehat{\mu}$ are commutative as the induced map $S(\widehat{\mu}) \colon {\mathcal S}_{{\mathcal Q}} \to {\mathcal S}_{{\mathcal P}}$ is ${\epsilon}$-homotopic to $\Phi^{-1}$. We claim that the squares formed by the maps in $\widehat{\lambda}$ on fundamental groups in \eqref{eq-commutativediagrampi12} are commutative, and then show that the properties \eqref{eq-monoaltnotation} and \eqref{eq-epialtnotation} are satisfied.
The idea of the proof of these claims is as follows. Let $L \subset {\mathfrak{M}}$ be a leaf of ${\mathcal F}_{{\mathfrak{M}}}$, endowed with the topology defined by the path-length metric (which is called the model $\widehat{L}$ for $L$ in \cite[Section 3]{RT1972b}). Recall that $\Phi_{{\mathcal P}}(L)$ is a leaf of ${\mathcal F}_{\cP}$ as the homeomorphism $\Phi_{{\mathcal P}}$ preserves path components, and then
the restriction $\Pi^{{\mathcal P}}_{\ell} | \Phi_{{\mathcal P}}(L)$ is a covering map by construction, for each $\ell > 0$. Thus, the restriction $\Pi^{{\mathcal P}}_{\ell} \circ \Phi_{{\mathcal P}} \colon L \to M_{\ell}$ is a covering map. However, the composition $\Pi^{{\mathcal Q}}_{\ell} \circ \Phi_{{\mathcal Q}} \colon L \to M$ need not be a covering map. What we claim is that this restriction is an approximate covering map, in that it satisfies an approximate lifting property, to be defined precisely below.
There have been many works studying the properties of ${\epsilon}$-maps between compact manifolds, starting with those of Eilenberg \cite{Eilenberg1938} and Ganea \cite{Ganea1959}, and more recently by
Chapman and Ferry \cite{ChapmanFerry1979}, and Ferry \cite{Ferry1979}. The idea in the following, is to adapt some of the more elementary ideas from these works to the restrictions $\Pi^{{\mathcal Q}}_{\ell} \circ \Phi_{{\mathcal Q}} | L$. There is an immediate difficulty, in that
an ${\epsilon}$-map $f_{{\epsilon}} \colon {\mathfrak{M}} \to M$ is assumed to be onto, but it is not given that the restriction $\Pi^{{\mathcal Q}}_{\ell} \circ \Phi_{{\mathcal Q}} | L$ is locally onto. However, this property is proved in Proposition~\ref{Aprop-coverage} below. Recall that ${\lambda_{\mathcal F}} > 0$ is chosen so that for all $x \in {\mathfrak{M}}$, the leafwise closed disk $D_{{\mathcal F}}(x, {\lambda_{\mathcal F}})$ is strongly convex. Also, recall that $\lambda_M > 0$ was chosen so that for every $u \in M$, the closed disk $D_M(u,\lambda_M) \subset M$ is strictly convex.
\begin{prop} \cite[Proposition A.1]{CHL2018b} \label{Aprop-coverage} Let ${\mathfrak{M}}$ be a matchbox manifold with leafwise Riemannian metric on ${\mathcal F}_{{\mathfrak{M}}}$ and let $M$ be a closed Riemannian manifold. Then there exists $0 < {\epsilon_{{\mathcal F}}} \leq {\lambda_{\mathcal F}}/4$ such that, if $f \colon {\mathfrak{M}} \to M$ is an ${\epsilon_{{\mathcal F}}}$-map, then there exists $\delta_1 > 0$ such that for $x_0 \in {\mathfrak{M}}$ with $w_0 = f(x_0)$, \begin{equation}\label{eq-onto} D_M(w_0,\delta_1) \subset f(D_{{\mathcal F}}(x_0, {\lambda_{\mathcal F}}/2 )) ~ . \end{equation} \end{prop}
Thus, \eqref{eq-onto} implies that every sufficiently small disk in $M$ is contained in the image of a strictly convex disk in some leaf of ${\mathcal F}_{{\mathfrak{M}}}$.
\begin{prop}\label{prop-leafonto} There exists $\ell_2 > 0$ such that for $\ell \geq \ell_2$ the map
$\Pi^Q_{\ell} \circ \Phi_{{\mathcal Q}} \colon {\mathfrak{M}} \to M$ satisfies an approximate homotopy lifting property. That is, given an integer $k \geq 1$, constant $0 < \delta_2 < \lambda_M$ and $x \in {\mathfrak{M}}$, let $L$ be the leaf through $x$, and given a continuous map $\sigma \colon [0,1]^k \to M$ with $\sigma(\vec{0}) = y_{\ell} = \Pi^Q_{\ell} \circ \Phi_{{\mathcal Q}}(x)$, then there exists a continuous map $\widehat{\sigma} \colon [0,1]^k \to L$ with $\widehat{\sigma}(\vec{0}) = x$, for which \begin{equation} d_{M}(\Pi^Q_{\ell} \circ \Phi_{{\mathcal Q}} \circ \widehat{\sigma}(\vec{v}) , \sigma(\vec{v})) \leq \delta_2 \quad {\rm for ~ all} ~ \vec{v} \in [0,1]^k~ . \end{equation} Moreover, the maps $\sigma$ and $\Pi^Q_{\ell} \circ \Phi_{{\mathcal Q}} \circ \widehat{\sigma}$ are $\delta_2$-homotopic. \end{prop} \proof Choose $0 < {\epsilon}_2 < {\lambda_{\mathcal F}}/4$, then there exists $\ell_2 > 0$ sufficiently large so that $\ell \geq \ell_2$ implies that the composition $\Pi^Q_{\ell} \circ \Phi_{{\mathcal Q}}$ is an ${\epsilon}_2$-map. Fix a choice of $\ell \geq \ell_2$.
Let $\delta_2 \leq \lambda_M$ be as in Proposition~\ref{Aprop-coverage}, so that for all $x_0 \in {\mathfrak{M}}$ with $w_0 = f(x_0)$, then \begin{equation} D_M(w_0,\delta_2) \subset \Pi^Q_{\ell} \circ \Phi_{{\mathcal Q}}(D_{{\mathcal F}}(x_0, {\epsilon}_2 )) ~ . \end{equation} Let $B(\vec{v}, \delta_3)$ denote the ball of radius $\delta_3$ in ${\mathbb R}^k$ for the Euclidean metric on ${\mathbb R}^k$. Then choose $\delta_3 > 0$ such that for any $\vec{v} \in [0,1]^k$, we have $\sigma(B(\vec{v}, \delta_3) \cap [0,1]^k) \subset B_M(\sigma(\vec{v}), \delta_2)$.
Choose $n_3 > 0$ so that the ``small cube''' $[0,1/n_3]^k$ has diameter less than $\delta_3$.
Starting with the given value $\sigma(0) = y_{\ell} \in M$ we use the above choices to recursively construct the lifting $\widehat{\sigma}$, small cube by small cube for a cubical decomposition of $[0,1]^k$. This is a standard construction, usually proved for the assumption that the target space $M$ is locally contractible, with the only nuance being the estimates on the control of the maps. We omit the details. \endproof
We now return to the proof of Theorem~\ref{thm-proM}.
We are given a solenoidal presentation ${\mathcal P}$ and a homeomorphism $\Phi_{{\mathcal P}} \colon {\mathfrak{M}} \to {\mathcal S}_{{\mathcal P}}$, and a presentation ${\mathcal Q} = \{ q_{\ell +1} \colon M \to M \mid \ell \geq 0 \}$, where each map $q_{\ell +1}$ is a continuous surjection, and a homeomorphism $\Phi_{{\mathcal Q}} \colon {\mathfrak{M}} \cong {\mathcal S}_{{\mathcal Q}}$.
Let ${\lambda_{\mathcal F}} > 0$ as in Lemma~\ref{lem-stronglyconvex}, chosen so that for all $x \in {\mathfrak{M}}$, $D_{{\mathcal F}}(x, {\lambda_{\mathcal F}})$ is strongly convex.
Let ${\epsilon}_{{\mathcal P}} > 0$ be such that for all $\xi \in {\mathfrak{M}}$ then $B_{{\mathcal S}_{{\mathcal P}}}(\Phi_{{\mathcal P}}(\xi) , {\epsilon}_{{\mathcal P}}) \subset \Phi_{{\mathcal P}}(B_{{\mathfrak{M}}}(\xi, {\lambda_{\mathcal F}}/4))$.
Let ${\epsilon}_{{\mathcal Q}} > 0$ be such that for all $\xi \in {\mathfrak{M}}$ then $B_{{\mathcal S}_{{\mathcal Q}}}(\Phi_{{\mathcal Q}}(\xi) , {\epsilon}_{{\mathcal Q}}) \subset \Phi_{{\mathcal Q}}(B_{{\mathfrak{M}}}(\xi, {\lambda_{\mathcal F}}/4))$.
Set ${\epsilon}_{\Phi} \equiv \min \{{\epsilon}_{{\mathcal P}}, {\epsilon}_{{\mathcal Q}}, \lambda_M/4\}$, and let
$\widehat{\epsilon} \equiv \{{\epsilon}_{0} > {\epsilon}_1 > {\epsilon}_2 > \cdots\}$ with $\displaystyle \lim_{\ell \to \infty} \, {\epsilon}_{\ell} = 0$
and ${\epsilon}_0 < {\epsilon}_{\Phi}$.
We are given the diagram \eqref{eq-commutativediagram2}, and the homeomorphism $\Phi = {\mathcal S}_{{\mathcal Q}} \circ {\mathcal S}_{{\mathcal P}}^{-1} \colon {\mathcal S}_{{\mathcal P}} \to {\mathcal S}_{{\mathcal Q}}$, for which
\begin{enumerate} \item the maps $\widehat{\lambda} = \{\lambda_{\ell} \colon M_{i_{\ell+1}} \to M \mid \ell \geq 0\}$ are continuous surjections which $\widehat{\epsilon}$-commute; \item the maps $\widehat{\mu} = \{\mu_{\ell} \colon M \to M_{i_{\ell}} \mid \ell \geq 0\}$ are covering maps which $\widehat{\epsilon}$-homotopy commute; \item $S(\widehat{\mu}) \colon {\mathcal S}_{{\mathcal Q}} \to {\mathcal S}_{{\mathcal P}}$ is ${\epsilon}_{\Phi}$-homotopic to $\Phi^{-1} = \Phi = {\mathcal S}_{{\mathcal P}}\circ {\mathcal S}_{{\mathcal Q}}^{-1}$. \end{enumerate}
Choose a basepoint $z \in {\mathfrak{M}}$, then set $x = \Phi_{{\mathcal P}}(z)$ and $y = \Phi_{{\mathcal Q}}(z)$.
We show that the squares in \eqref{eq-commutativediagrampi123} below are commutative, and that the properties \eqref{eq-monoaltnotation} and \eqref{eq-epialtnotation} are satisfied. \begin{align} \label{eq-commutativediagrampi123}
\xymatrix{
\cdots & \ar[l] \pi_1(M_{i_{\ell+1}} , x_{i_{\ell+1}}) \ar[ld]_{\lambda_{\ell}} & \ar[l] \pi_1(M_{j_{\ell+1}} , x_{j_{\ell+1}}) & \ar[l] \pi_1(M_{i_{\ell+2}} , x_{i_{\ell +2}}) \ar[ld]_{\lambda_{\ell+1}} & \ar[l] \cdots \\ \pi_1(M , y_{j_{\ell}}) & \ar[l] \pi_1(M , y_{i_{\ell+1}}) & \ar[l] \pi_1(M , y_{j_{\ell+1}}) \ar[lu]_{\mu_{\ell+1}} & \ar[l] \pi_1(M , y_{i_{\ell+2}}) & \ar[l] \cdots } \end{align}
As noted before, that the maps $\widehat{\mu}$ induce commutative squares in the diagram \eqref{eq-commutativediagrampi123}.
We first show that the maps $\widehat{\lambda}$ induce commutative squares in the diagram \eqref{eq-commutativediagrampi123}.
Let $\alpha \in \pi_1(M_{i_{\ell+2}} , x_{i_{\ell+2}})$ be represented by a closed smooth curve
$$\gamma_{\alpha} \colon [0,1] \to M_{i_{\ell+2}} \quad , \quad \gamma_{\alpha}(0) = \gamma_{\alpha}(1) = x_{i_{\ell+2}} ~ .$$
Then $\lambda_{\ell} \circ p_{i_{\ell +2}}^{i_{\ell+1}}(\alpha) \in \pi_1(M , y_{j_{\ell}}) $
is represented by the closed curve
$\lambda_{\ell} \circ p_{i_{\ell +2}}^{i_{\ell+1}}\circ \gamma_{\alpha} \colon [0,1] \to M$,
and $q_{j_{\ell}}^{j_{\ell+1}} \circ \lambda_{\ell+1}(\alpha)$ is represented by the closed curve
$q_{j_{\ell}}^{j_{\ell+1}} \circ \lambda_{\ell+1} \circ \gamma_{\alpha} \colon [0,1] \to M$. We claim these two curves are homotopic.
Let $\widetilde{\gamma_{\alpha}} \colon [0,1] \to L$ with $\widetilde{\gamma_{\alpha}}(0) = z$ be the lift of $\gamma_{\alpha}$ via the covering map $\Pi_{i_{\ell +2}}^{{\mathcal P}} \circ \Phi_{{\mathcal P}} \colon L \to M_{i_{\ell +2}}$.
Then $\Pi_{j_{\ell +1}}^{{\mathcal Q}} \circ \Phi_{{\mathcal Q}} \circ \widetilde{\gamma_{\alpha}} \colon [0,1] \to M$. It is given that the maps $\widehat{\lambda}$ all $\widehat{\epsilon}$-commute, which by the definition \eqref{eq-epromap} implies that
\begin{equation} d_M( \lambda_{\ell+1} \circ \Pi_{i_{\ell +2}}^{{\mathcal P}} \circ \Phi_{{\mathcal P}} \circ \widetilde{\gamma_{\alpha}}(t) ~ , ~ \Pi_{j_{\ell +1}}^{{\mathcal Q}} \circ \Phi_{{\mathcal Q}} \circ \widetilde{\gamma_{\alpha}}(t)) ~ \leq ~ {\epsilon}_{j_{\ell+1}} ~ 0 \leq t \leq 1 ~ . \end{equation} As ${\epsilon}_{j_{\ell+1}} \leq \lambda_M/4$ by the choice of ${\epsilon}_{\Phi}$ for each $0 \leq t \leq 1$, there is a geodesic segment between $\lambda_{\ell+1} \circ \Pi_{i_{\ell +2}}^{{\mathcal P}} \circ \Phi_{{\mathcal P}} \circ \widetilde{\gamma_{\alpha}}(t)$ and $\Pi_{j_{\ell +1}}^{{\mathcal Q}} \circ \Phi_{{\mathcal Q}} \circ \widetilde{\gamma_{\alpha}}(t)$. Thus, if we form a closed curve in $M$ by joining the endpoints of $\Pi_{j_{\ell +1}}^{{\mathcal Q}} \circ \Phi_{{\mathcal Q}} \circ \widetilde{\gamma_{\alpha}}$ to $y_{j_{\ell +1}}$ by a geodesic segments in $B_M(y_{j_{\ell +1}}, \lambda_M/4)$, we obtain a curve which represents $\lambda_{\ell+1}(\alpha)$. A similar argument shows that
\begin{equation}\label{eq-composingequality}
\Pi_{j_{\ell }}^{{\mathcal Q}} \circ \Phi_{{\mathcal Q}} \circ \widetilde{\gamma_{\alpha}} = q_{j_{\ell+1}}^{j_{\ell}} \circ \Pi_{j_{\ell +1}}^{{\mathcal Q}} \circ \Phi_{{\mathcal Q}} \circ \widetilde{\gamma_{\alpha}} \colon [0,1] \to M \end{equation} represents the class $\lambda_{\ell} \circ p_{i_{\ell +2}}^{i_{\ell+1}}(\alpha) \in \pi_1(M , y_{j_{\ell}})$, while the equality \eqref{eq-composingequality} implies that it also represents the class $q_{j_{\ell}}^{j_{\ell+1}} \circ \lambda_{\ell+1}(\alpha)$. Thus, they are equal as was to be shown.
We next show that the induced maps $\lambda_{\ell}$ and $\mu_{\ell}$ are monomorphisms. Let $\alpha \in \pi_1(M_{i_{\ell+1}} , x_{i_{\ell+1}})$ and suppose that $\lambda_{\ell}(\alpha) \in \pi_1(M , y_{j_{\ell}})$ is trivial. We claim this implies $\alpha$ is also trivial. As above, let $\alpha$ be represented by a closed smooth curve
$$\gamma_{\alpha} \colon [0,1] \to M_{i_{\ell+1}} \quad , \quad \gamma_{\alpha}(0) = \gamma_{\alpha}(1) = x_{i_{\ell+1}} ~ ,$$ and let $\widetilde{\gamma_{\alpha}} \colon [0,1] \to L$ with $\widetilde{\gamma_{\alpha}}(0) = z$ be the lift of $\gamma_{\alpha}$ via the covering map $\Pi_{i_{\ell +1}}^{{\mathcal P}} \circ \Phi_{{\mathcal P}} \colon L \to M_{i_{\ell +1}}$. Then $\Pi_{j_{\ell}}^{{\mathcal Q}} \circ \Phi_{{\mathcal Q}} \circ \widetilde{\gamma_{\alpha}} \colon [0,1] \to M$ represents $\lambda_{\ell}(\alpha)$, where as before we obtain a closed curve by adjoining short geodesic segments to the endpoints. The assumption $\lambda_{\ell}(\alpha) $ is trivial class implies that $\lambda_{\ell}\circ \alpha \colon [0,1] \to M$ extends to a homotopy to the constant map, which is given by a map $H(t,s) \colon [0,1]^2 \to M$ with $H(t,0) = \lambda_{\ell}\circ \alpha(t)$ and $H(t,1) = H(0,s) = H(1,s) = y_{j_{\ell}}$ for all $0\leq t,s \leq 1$. Then by Proposition~\ref{prop-leafonto} there is an approximate lift $\widehat{H} \colon [0,1]^2 \to L$ such that $\Pi_{j_{\ell}}^{{\mathcal Q}} \circ \Phi_{{\mathcal Q}} \circ \widehat{H}$ is ${\epsilon}_{j_{\ell}}$ close to $H$, and hence is ${\epsilon}_{j_{\ell}}$-homotopic. Then the composition $\Pi_{i_{\ell+1}}^{{\mathcal P}} \circ \Phi_{{\mathcal P}} \circ \widehat{H} \colon [0,1]^2 \to M_{i_{\ell +1}}$ defines a homotopy of $\alpha$ to an almost constant map, hence $\alpha$ is the trivial class, as was to be shown.
The claim that the maps $\mu_{\ell}$ induce injective maps on fundamental groups follows similarly.
The claim that the properties \eqref{eq-monoaltnotation} and \eqref{eq-epialtnotation} are satisfied now follows in exactly the same way as in the proof of Theorem~\ref{thm-solprogroups}. This completes the proof of Theorem~\ref{thm-proM}.
\endproof
As remarked previously, Marde\v{s}i\'{c} and Segal show in \cite[Chapter II, Section 3.3]{MardesicSegal1982} that the pro-homotopy groups {\it pro}-$\pi_k({\mathfrak{M}},z)$ are well-defined for pointed shape expansions. The key technical fact used in this section, Proposition~\ref{prop-leafonto}, applies equally well for the study of the higher pro-homotopy groups {\it pro}-$\pi_k({\mathfrak{M}}, z)$ of a weak solenoid, for $k > 1$. Instead of considering lifts of paths to the leaf $L$, one considers lifts of cubes. Then essentially the same arguments as above yield a proof of the following result:
\begin{thm}\label{thm-proM2}
Let ${\mathfrak{M}}$ be a matchbox manifold which is $M$-like. For $k > 1$, {\it pro}-$\pi_k({\mathfrak{M}}, z)$ is isomorphic to the pro-group defined by maps $\{g_{\ell} \colon \pi_k(M, y) \to \pi_k(M, y) \mid \ell \geq 0\}$. In particular,
{\it pro}-$\pi_k({\mathfrak{M}}, z)$ is independent of the choice of basepoint $z \in {\mathfrak{M}}$.
\end{thm}
\section{Proofs of main theorems}\label{sec-proofs}
In this section, we show how the conclusions and proof of Theorem~\ref{thm-proM} imply Theorems~\ref{thm-main1} and \ref{thm-main2} from the introduction. We assume that ${\mathfrak{M}}$ be an $n$-dimensional matchbox manifold which is $M$-like, for a closed manifold $M$. By Theorem~\ref{thm-weak}, there exists a presentation $\displaystyle {\mathcal P} = \{\, f_{\ell+1} \colon M_{\ell+1} \to M_{\ell} \mid \ell \geq 0\}$ as a weak solenoid, and a homeomorphism $\Phi_{{\mathcal P}} \colon {\mathfrak{M}} \to {\mathcal S}_{{\mathcal P}}$.
By Theorem~\ref{thm-MardSeg}, there exists a presentation ${\mathcal Q} = \{ q_{\ell +1} \colon M \to M \mid \ell \geq 0 \}$, where all the maps $q_{\ell +1}$ are continuous surjections, and a homeomorphism $\Phi_{{\mathcal Q}} \colon {\mathfrak{M}} \to {\mathcal S}_{{\mathcal Q}}$. Choose a basepoint $z \in {\mathfrak{M}}$ and set $x = \Phi_{{\mathcal P}}(z)$ and $y = \Phi_{{\mathcal Q}}(z)$.
By the proof of Theorem~\ref{thm-proM}, we can assume that all of the squares in the diagram \eqref{eq-commutativediagrampi12} are commuting.
Define the sub-presentation ${\mathcal P}'$ of ${\mathcal P}$ given by the maps
\begin{equation} {\mathcal P}' = \{p_{i_{\ell+1}}^{i_{\ell}} \colon M_{i_{\ell+1}} \to M_{i_{\ell}} \mid \ell \geq 0\} ~ . \end{equation} Then by the basic properties of inverse limits, there is a canonical homeomorphism $\Phi' \colon {\mathcal S}_{{\mathcal P}'} \to {\mathcal S}_{{\mathcal P}}$.
From the proof of Theorem~\ref{thm-proM}, the maps $\lambda_{\ell}$ and $\mu_{\ell}$ in the diagram \eqref{eq-commutativediagrampi12} are monomorphisms, for all $\ell \geq 1$. Thus, we have the extended commutative diagram of group monomorphisms:
\begin{align} \label{eq-bigcommutativediagram}
\xymatrix{
\cdots \quad \quad & \ar[l]_{p_{i_{\ell}}^{i_{\ell-1}}} \pi_1(M_{i_{\ell}} , x_{i_{\ell}}) & & \ar[ll]_{p_{i_{\ell+1}}^{i_{\ell}}} \pi_1(M_{i_{\ell+1}} , x_{i_{\ell+1}}) & \ar[l]_{p_{i_{\ell+2}}^{i_{\ell+1}}} \quad \cdots \\ \cdots \pi_1(M , y_{j_{\ell-1}}) \ar[d]_{\cong} & \ar[l]_{\lambda_{\ell}} \pi_1(M_{i_{\ell}} , x_{i_{\ell}}) \ar[u]_{\cong} & \ar[l]_{\mu_{\ell}} \pi_1(M , y_{j_{\ell}}) \ar[d]_{\cong} & \ar[l]_{\lambda_{\ell+1}} \pi_1(M_{i_{\ell+1}} , x_{i_{\ell+1}}) \ar[u]_{\cong} & \ar[l]_{\mu_{\ell+1}} \ar[d]_{\cong}\pi_1(M , y_{j_{\ell+1}}) \cdots \\ \cdots \pi_1(M , y_{j_{\ell-1}}) & & \ar[ll]_{\lambda_{\ell}\circ \mu_{\ell}} \pi_1(M , y_{j_{\ell}}) & & \ar[ll]_{\lambda_{\ell+1}\circ \mu_{\ell+1}} \pi_1(M , y_{j_{\ell+1}}) \cdots } \end{align} Recall that in \eqref{eq-imahes} we defined $G_{\ell} = {\rm image}\left\{ p^0_{\ell} \colon \pi_1(M_{\ell}, x_{\ell})\longrightarrow G_{0}\right\}$, and the group chain ${\mathcal G}$ in \eqref{eq-descendingchain} associated to the presentation ${\mathcal P}$ and basepoint $x_0 \in M_0$. Define the group chain \begin{equation} {\mathcal H} = H_{\ell} = {\rm image}\left\{ p^0_{\ell} \circ \mu_{\ell} \colon \pi_1(M , y_{j_{\ell}}) \longrightarrow G_{0}\right\} ~, \end{equation} then \eqref{eq-bigcommutativediagram} implies we then obtain a group chain \begin{equation}\label{eq-interlacedGH} G_0 \supset G_{i_1} \supset H_{j_1} \supset G_{i_2} \supset \cdots \supset G_{i_{\ell}} \supset H_{j_{\ell}} \supset G_{i_{\ell+1}} \supset H_{j_{\ell +1}} \supset \cdots ~ . \end{equation} Set $H = \pi_1(M, y_0)$. As $M$ is connected, for all $\ell > 0$ there is an isomorphism $H_{\ell} \cong H$.
Now define $N_{\ell}$ to be the covering space of $M_0$ associated to the subgroup $H_{\ell}$. The chain of subgroups in \eqref{eq-interlacedGH} then yields a presentation ${\mathcal P}' = \{q_{\ell+1} \colon N_{\ell+1} \to N_{\ell} \mid \ell \geq 1\}$. By Theorem~\ref{thm-re}, the spaces ${\mathcal S}_{{\mathcal P}}$ and ${\mathcal S}_{{\mathcal P}'}$ are homeomorphic. This completes the proof of Theorem~\ref{thm-main1}.
For the proof of Theorem~\ref{thm-main2}, recall that $M$ aspherical means that $\pi_k(M, y_0) = \{0\}$ is the trivial group for all $k > 1$ and choice of basepoint $y_0$.
The key technical fact used in the proof of Theorem~\ref{thm-proM}, the approximate homotopy lifting property in Proposition~\ref{prop-leafonto}, applies equally well for the study of the maps between higher homotopy groups in diagram \eqref{eq-commutativediagram3}. Instead of considering lifts of paths to the leaf $L \subset {\mathfrak{M}}$, one considers lifts of cubes. We require only the conclusion that the following diagram commutes, for each $k > 1$ and $\ell \geq 1$:
\begin{align} \label{eq-commutativediagrampi1234}
\xymatrix{
\pi_k(M_{i_{\ell}} , x_{i_{\ell}}) & & \ar[ll]_{p_{i_{\ell+1}}^{i_{\ell}}} \pi_k(M_{i_{\ell+1}} , x_{i_1}) \ar[ld]_{\lambda_{\ell+1}} \\
& \pi_k(M , y_{j_{\ell}}) \ar[lu]_{\mu_{\ell}} & } \end{align} The covering map $p_{\ell+1} \colon M_{\ell+1} \to M_{\ell}$ induces an isomorphism on homotopy groups for $k > 1$, hence the horizontal map $p_{i_{\ell+1}}^{i_{\ell}}$ in \eqref{eq-commutativediagrampi1234} is an isomorphism. Thus, $ \pi_k(M_{i_{\ell}} , x_{i_{\ell}}) = \{0\}$ is trivial for all $\ell \geq 1$, and also $\pi_k(N_{\ell}, n_{\ell}) = \{0\}$ is trivial for the covering space $N_{\ell} \to M_{i_{\ell}}$ defined above, where $n_{\ell} \in N_{\ell}$ is some lift of $x_{i_{\ell}}$. Hence, each $N_{\ell}$ is an aspherical space. Thus, the lift of $\mu_{\ell}$ to a map $h_{\ell} \colon M \to N_{\ell}$ induces isomorphisms on all homotopy groups, so is a homotopy equivalence. By the assumption that $M$ satisfies the Borel Conjecture, it follows that $h_{\ell}$ is homotopic to a homeomorphism $f_{\ell}$ for each $\ell \geq 1$. Using the homeomorphisms $\{f_{\ell}\}$ we obtain a presentation ${\mathcal P}'' = \{f_{\ell}^{-1} \circ q_{\ell+1} \circ f_{\ell+1} \colon M \to M \mid \ell \geq 1\}$ for which ${\mathcal S}_{{\mathcal P}''}$ is homeomorphic to ${\mathcal S}_{{\mathcal P}'}$ and hence to ${\mathcal S}_{{\mathcal P}}$, which shows the claim of Theorem~\ref{thm-main2}.
\section{Examples}\label{sec-examples}
In this section, we give a collection of examples, some from the literature and some novel, which illustrate the conclusions of Theorems~\ref{thm-main1} and \ref{thm-main2}. These examples are based on the basic observation of this work, that the $M$-like hypothesis on a solenoid is a shape version of the non co-Hopfian property for manifolds.
Recall that a group $G$ is \emph{co-Hopfian} if there does not exist an embedding of $G$ to a proper subgroup of itself with finite index, and \emph{non co-Hopfian} otherwise. (In some literature, a group is said to be co-Hopfian if every embedding of $G$ to a proper subgroup of itself must be an isomorphism, and they refer to our definition as the \emph{finitely co-Hopfian property}. By non co-Hopfian, we will always mean finitely non co-Hopfian.) The co-Hopfian concept for groups was first studied by Baer in \cite{Baer1944}, where they are referred to as ``S-groups''.
A closed connected manifold $M$ is co-Hopfian if every finite covering map $\pi \colon M \to M$ is a homeomorphism, and non co-Hopfian otherwise. Given a proper self-covering map $\pi \colon M \to M$, choose a basepoint $x_1 \in M$ and set $x_0 = \pi(x_1)$. Then $G_0 = \pi_1(M, x_0)$ is a non co-Hopfian group.
Given a non co-Hopfian manifold $M$ with proper self-covering map $\pi \colon M \to M$, we obtain a presentation ${\mathcal P} = \{p_{\ell+1} \colon M_{\ell+1} \to M_{\ell} \mid \ell \geq 0\}$, where each $M_{\ell} = M$ and $p_{\ell+1} = \pi$. As remarked in Section~\ref{sec-intro}, the associated weak solenoid ${\mathcal S}_{{\mathcal P}}$ is always $M$-like.
Without additional hypotheses on $M$, the property that $G_0 = \pi_1(M, x_0)$ is a non co-Hopfian group does not imply that $M$ is a non co-Hopfian manifold. Here is a simple construction to show that.
Let $B$ be a non co-Hopfian manifold of dimension $m \geq 4$.
Let $M_0$ be the result of attaching the product space ${\mathbb S}^2 \times {\mathbb S}^{m-2}$ to $B$ along the boundary of a disk in each manifold. Then the fundamental group of $M_0$ equals that of $B$. Note that the second homotopy group $\pi_2(M_0, x_0)$ contains a free abelian summand corresponding to the attached copy of ${\mathbb S}^2$. For a proper covering $\pi \colon M_1 \to M_0$, the rank of $\pi_2(M_1, x_1)$ is then greater than that of $M_0$, hence they cannot be homeomorphic.
This construction illustrates the requirement that the base manifold $M_0$ be aspherical in Theorem~\ref{thm-main2}. The additional requirement that the coverings of $M_0$ also satisfy the Borel Conjecture is made so that all coverings of $M_0$ are then homeomorphic to $M_0$. Thus, a closed aspherical manifold $M$ which satisfies the strongly Borel property \cite[Definition~1.4]{CHL2018a} is co-Hopfian if and only if its fundamental group is co-Hopfian.
The non co-Hopfian property was also used in the work of Bi\'s, Hurder and Shive \cite[Section 3]{BHS2006} to construct many new classes of foliations which generalize the original construction of Hirsch in \cite{Hirsch1975} that was based on the construction of the Smale solenoid by self-embeddings of a solid torus.
We next recall some examples of weak solenoids obtained by explicit constructions of non co-Hopfian closed manifolds, and then discuss some examples obtained from more abstract methods.
\subsection{Abelian group chains}\label{subsec-abelian}
Let $G_0 ={\mathbb Z}^k$ for $k \geq 1$ be the free abelian group of rank $k$, with basis vectors $\{\vec{e}_1, \ldots , \vec{e}_k\}$. Choose any collection of vectors $\{\vec{v}_1, \ldots, \vec{v}_k\} \subset {\mathbb Z}^k$ which are linearly independent over ${\mathbb Q}$. Then define $\phi \colon {\mathbb Z}^k \to {\mathbb Z}^k$ by setting $\phi(\vec{e}_i) = \vec{v}_i$ for $1 \leq i \leq k$. The map $\phi$ is an endomorphism of ${\mathbb Z}^k$, and for most choices we obtain a proper embedding. We obtain a group chain by setting ${\mathcal G}_{\phi} = \{G_{\ell} \equiv \phi^{\ell}(G_0) \mid \ell \geq 0\}$.
We illustrate this with the simplest example, where $k=1$. Choose distinct primes $p, q \geq 2$. Define $G_{\ell} = \{n \cdot p^{\ell} \mid n \in {\mathbb Z}\}$ and $H_{\ell} = \{n \cdot q^{\ell} \mid n \in {\mathbb Z}\}$, so we obtain group chains defined by the self-embeddings, ${\mathcal P} = \{G_{\ell} \mid \ell \geq 1\}$ and ${\mathcal Q} = \{H_{\ell} \mid \ell \geq 1\}$. Criteria for the equivalence of group chains in ${\mathbb Z}$ were given by Bing \cite{Bing1960}, McCord \cite{McCord1965}, Aarts and Fokkink \cite{AartsFokkink1991}, as well as in \cite[Section 5]{CHL2018a}. In particular, ${\mathcal P}$ and ${\mathcal Q}$
are not return equivalent, so we obtain the well-known fact that the solenoids ${\mathcal S}_{{\mathcal P}}$ and ${\mathcal S}_{{\mathcal Q}}$ are not homeomorphic.
More generally, every group chain ${\mathcal G}$ in ${\mathbb Z}$ defines a scale in the sense of \cite{NekkyPete2011}, see also the discussion in Section \ref{subsec-scale}. On the other hand, a monomorphism $\phi \colon {\mathbb Z} \to {\mathbb Z}$ is determined by the integer $m = \phi(1)$, which has a finite number of prime divisors, where $m = q_1 \cdots q_k$ with each $q_i$ prime though possibly not distinct. As an example, let $\{p_1 < p_2 < \cdots\}$ be an infinite collection of increasing primes. Define $K_{\ell} = \{n \cdot p_1 p_2 \cdots p_{\ell} \mid n \in {\mathbb Z}\} \cong {\mathbb Z}$ and let ${\mathcal K} = \{K_{\ell} \mid \ell \geq 0\}$ be the corresponding group chain. Then the group chain ${\mathcal K}$ is not return equivalent to any group chain defined by a monomorphism $\phi: {\mathbb Z} \to {\mathbb Z}$, and thus the weak solenoid ${\mathcal S}_{{\mathcal K}}$ is not homeomorphic to ${\mathcal S}({\mathcal G}_{\phi})$.
As a consequence, we see that there are uncountably many circle-like weak solenoids which are distinct up to homeomorphism, and not homeomorphic to the solenoid obtained from a covering map $\pi \colon {\mathbb S}^1 \to {\mathbb S}^1$.
Thus, the conclusion of Theorem~\ref{thm-main2} cannot be strengthened in general.
\subsection{Coverings of the Klein bottle}\label{subsec-RT}
This example is a generalization to arbitrary integer $d>1$ of the example due to Rogers and Tollefson \cite{RT1972a}, see also Fokkink and Oversteegen \cite{FO2002}. Consider a map of the plane, given by a translation by $\frac{1}{2}$ in the first component, and by reflection in the second component, i.e.
$$r \times i \colon {\mathbb R}^2 \to {\mathbb R}^2 ~ {\rm where} ~ (x,y) \mapsto (x+\frac{1}{2},-y).$$ This map commutes with translations by the elements in the integer lattice ${\mathbb Z}^2 \subset {\mathbb R}^2$, and so induces the map $r \times i \colon {\mathbb T}^2 = {\mathbb R}^2/{\mathbb Z}^2 \to {\mathbb T}^2$ of the torus. This map is an involution, and the quotient space $K = {\mathbb T}^2/(x,y) \sim r\times i(x,y)$ is homeomorphic to the Klein bottle.
Let $L \colon {\mathbb T}^2 \to {\mathbb T}^2$ be the $d$-fold covering map given by $L(x,y) = (x,dy)$, and form the inverse limit ${\displaystyle {\mathbb T}_\infty = \lim_{\longleftarrow} \{L \colon {\mathbb T}^2 \to {\mathbb T}^2 \}}$, which is a solenoid with $2$-dimensional leaves. Let $x_0 = (0,0) \in M_0 = {\mathbb T}^2$. The fundamental group $G_0 = {\mathbb Z}^2$ is abelian, so for any $x,y \in {\mathfrak{X}}_0$ the kernels $K({\mathcal G}^x) = K({\mathcal G}^y) \cong {\mathbb Z}$, and every leaf is homeomorphic to an open two-ended cylinder.
The involution $r \times i$ is compatible with the covering maps $L$, and so it induces an involution $(r \times i)_\infty \colon {\mathbb T}_\infty \to {\mathbb T}_\infty$, which is seen to have a single fixed point $(0,0, \ldots) \in {\mathbb T}_\infty$, and permutes the other path-connected components. Let $p \colon K \to K$ be the $d$-fold covering of the Klein bottle by itself, given by $p(x,y) = (x,dy)$, and consider the inverse limit space ${\displaystyle K_\infty = \lim_{\longleftarrow} \{p \colon K \to K \}}$. Note that taking the quotient by the involution $r \times i$ is compatible with the covering maps $L$ and $p$; that is, $p \circ (r \times i ) = L$, and so induces the map $i_\infty \colon {\mathbb T}_\infty \to K_\infty$ of the inverse limit spaces. Under this map, the path-connected component of the fixed point $(0,0, \ldots)$ is identified so as to become a non-orientable one-ended cylinder. The image of any other path-connected component is an orientable $2$-ended cylinder, so the space $ K_\infty$ is not homogeneous.
Let $x =(x_{\ell})\in K_\infty$ for $x_{\ell} \in K$. Then $G_0 = \pi_1(K,x_0) = \langle a,b \mid bab^{-1} = a^{-1} \rangle$. The group chain associated to this solenoid is given by the subgroups $G_{\ell} = \langle a^{d^{\ell}}, b \rangle \subset G_0$. For the group chain ${\mathcal G}^x$, we have that the kernel $K({\mathcal G}^x) = \langle b \rangle $. If a point $y \in K_\infty$ is in a leaf which is orientable, then the kernel of the group chain ${\mathcal G}^y = \langle b^2 \rangle$. If $y \in K_\infty$ is in the non-orientable leaf, then the kernel $K({\mathcal G}^y)$ is conjugate to $K({\mathcal G}^x)$.
It follows that the effective action of $G_0$ on the fiber of the solenoid is that of the dihedral group $\langle a,b \mid bab^{-1} = a^{-1} , b^2 = 1 \rangle$. This action is that of the iterated monodromy group associated to a Chebyshev polynomial of degree $d$, as discussed in \cite{Lukina2018b}.
\subsection{Heisenberg group chains}\label{subsec-heisenberg}
Let ${\mathcal H}$ be the $3$-dimensional Heisenberg Lie group, and let $G_0 = {\mathbb H}$ be the discrete Heisenberg group. These groups are presented as the upper-triangular subgroup of the $3 \times 3$ matrices:
\begin{equation} {\mathcal H} \equiv \left( \begin{array}{ccc} 1 & x & z\\ 0 & 1 & y\\ 0 & 0 & 1\end{array}\right) ~ , ~ x,y,z \in {\mathbb R} \quad , \quad {\mathbb H} \equiv \left( \begin{array}{ccc} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\end{array}\right) ~ , ~ a,b,c \in {\mathbb Z} \end{equation} The group ${\mathcal H}$ can also be presented in the form $\{{\mathbb R}^3, *\}$, with the group operation $*$ given by $(x,y,z)*(x',y',z')=(x+x',y+y',z+z'+xy')$. The group ${\mathbb H}$ is the simplest example of a torsion-free nilpotent group which is not abelian.
Wouter Van Limbeek suggested to the authors the following simple construction of a self-embedding of $G_0$, which yields a group chain that is not normal. Define $\phi \colon G_0 \to G_0$ given by $\phi(a,b,c) = (2a, 2b, 4c)$, which yields a group monomorphism. Let $$G_{\ell} = \phi^{\ell}(G_0) = \{ (2^{\ell}a, 2^{\ell}b, 4^{\ell} c)\mid a,b,c \in {\mathbb Z}\}$$ It is then an exercise to check that the normal core of $G_{\ell}$ is the subgroup $$C_{\ell} = \{ (4^{\ell}a, 4^{\ell}b, 4^{\ell} c)\mid a,b,c \in {\mathbb Z}\} \subset G_{\ell} .$$ Note that $G_{2\ell} \subset C_{\ell}$ so the group chain $\{G_{\ell} \mid \ell \geq 0\}$ is equivalent to a normal chain, hence the solenoid defined by self-coverings of $M_0 = {\mathcal H}/G_0$ will be homogeneous by \cite{FO2002}, and $M$-like by construction.
On the other hand, Jessica Dyer gave in her thesis \cite{Dyer2015} examples of group chains in $G_0 \subset {\mathbb H}$ for which the discriminant of the Cantor action constructed from the chain is a Cantor group. We recall Example~8.5 from \cite{DHL2016a}. Choose distinct primes $p > 1$ and $q>1$, and define $$H_{\ell} = \{ (p^{\ell} a, q^{\ell} b , p^{\ell} c)\mid a,b,c \in {\mathbb Z}\} ~ .$$ Note that $H_1$ is the image of the set map $\phi(a,b,c) = (pa, qb, pc)$, but this $\phi$ is not a homomorphism for the group law on ${\mathbb H}$. On the other hand, it is shown in \cite{DHL2016a} that the solenoid defined by the associated coverings $M_{\ell} = {\mathcal H}/H_{\ell}$ of $M_0 = {\mathcal H}/{\mathbb H}$ is not homogeneous.
The following is an interesting open problem:
\begin{prob}\label{prob-heisenberg}
Let $G_0 \subset {\mathbb H}$ be as above. Does there exists a subgroup $G_1 \subset G_0$ of finite index, such that for $M = {\mathcal H}/G_1$, there exists an $M$-like weak solenoid which is not homogeneous?
In terms of group chains, does there exists $G_1 \subset G_0$ of finite index, and a proper self-embedding $\phi \colon G_1 \to G_1$, such that the associated group chain ${\mathcal G}(\phi) \equiv \{G_{\ell} = \phi^{\ell}(G_1) \mid \ell \geq 1\}$ has trivial kernel, and the associated weak solenoid is not homogeneous?
\end{prob}
\subsection{Nil-manifolds}\label{subsec-nilmanifolds}
There are many generalizations of the above constructions of weak solenoids obtained from lattices in the Heisenberg group. The Heisenberg group ${\mathcal H}$ is replaced with any connected nilpotent Lie group ${\mathcal N}$, and ${\mathbb H}$ is replaced with a cocompact lattice subgroup $\Gamma \subset {\mathcal N}$. For example, Belegradek considered in \cite{Belegradek2003} when such a lattice must be co-Hopfian, and in particular when they are not. Non co-Hopfian subgroups of nilpotent Lie groups were also studied by Dekimpe, Lee and Potyagailo in \cite{DL2003a,DL2003b,DD2016}, and by Cornulier in \cite{Cornulier2016}. Here is a general version of Problem~\ref{prob-heisenberg}:
\begin{prob}\label{prob-nilpotent}
Let ${\mathcal N}$ be a connected Lie group. Does there exists a lattice subgroup $\Gamma \subset {\mathcal N}$ which admits a proper self-embedding $\phi \colon \Gamma \to \Gamma$, such that the group chain ${\mathcal G}(\phi) \equiv \{G_{\ell} = \phi^{\ell}(\Gamma) \mid \ell \geq 0\}$ has trivial kernel, and the associated weak solenoid is not homogeneous.
\end{prob}
This problem is closely related to the works of Reid \cite{Reid2014}, Willis \cite{Willis2015} and Van Limbeek in \cite{vanLimbeek2018,vanLimbeek2017} on the properties of profinite groups derived from a self-embedding of a non co-Hopfian group, in terms of contracting maps for profinite groups.
\subsection{Non co-Hopfian 3-manifolds}\label{subsec-3manifolds} The question of which compact $3$-manifolds admit proper self-coverings has been studied in detail by Gonz{\'a}lez-Acu{\~n}a, Litherland and Whitten in the works \cite{GLiW1994} and \cite{GW1994}, and see also the work \cite{WangWu1994} by Wang and Wu.
\subsection{Higher dimensional constructions}\label{subsec-higher}
Note that the product of any finite collection of non co-Hopfian manifolds is again non co-Hopfian. Thus, we can form the products of an arbitrary collection of examples as given above to obtain a non co-Hopfian manifold of arbitrarily large dimension.
A more general approach is to consider the problem of which finitely-presented groups $G_0$ are non co-Hopfian? One must then also ask, when such a group can be realized as the fundamental group of a non co-Hopfian manifold? Such constructions usually result in a base manifold of dimension $m \geq 4$. We recall some results in the literature on when a group is non co-Hopfian.
Endimioni and Robinson give in \cite{ER2005} some sufficient conditions for a group to be non co-Hopfian.
Delgado and Timm consider in \cite{DT2003} the co-Hopfian condition for the fundamental groups of connected finite complexes, which is certainly a requirement for the construction of non co-Hopfian manifolds.
Ohshika and Potyagailo gave example in \cite{OP1998} of a freely indecomposable geometrically finite torsion-free non-elementary Kleinian group which is non co-Hopfian. Delzant and Potyagailo study in \cite{DP2003} non-elementary geometrically finite Kleinian groups are non co-Hopfian, and Kapovich and Weiss considered the co-Hopf property for word hyperbolic groups in \cite{KW2001}.
On the other hand, the following problem appears to be open:
\begin{prob}\label{prob-realize} Let $G_0$ be a finitely-generated non co-Hopfian group. Find conditions on $G_0$ so that it is realized as the fundamental group of a non co-Hopfian closed manifold $M_0$.
\end{prob}
\subsection{Scale-invariant groups}\label{subsec-scale}
A finitely generated infinite group $G$ is called \emph{scale-invariant} by Nekrashevych and Pete \cite{NekkyPete2011}, if there is a group chain ${\mathcal G} = \{G_{\ell} \mid \ell \geq 0\}$ that each $G_{\ell}$ is isomorphic to $G$, and whose kernel $K({\mathcal G}) = \cap_{\ell \geq 0} \ G_{\ell}$ is a finite group. The assumption that the kernel is finite is a fundamental aspect of this concept.
Remark~\ref{rmk-kernels} identifies $K({\mathcal G})$ with the fundamental group of the leaf $L_x$ containing the basepoint $x \in {\mathfrak{M}}$ used to define the group chain. Thus, if ${\mathfrak{M}}$ is an $M$-like matchbox manifold and ${\mathcal F}_{{\mathfrak{M}}}$ admits a leaf with finite fundamental group, Theorem~\ref{thm-main1} implies that the fundamental group $G_0 = \pi_1(M, b)$ is scale-invariant.
A non co-Hopfian group $G_0$ with proper embedding $\phi \colon G_0 \to G_0$ is scale-invariant if the intersection $\cap_{\ell \geq 0} \ \phi^{\ell}(G_0)$ is a finite group. Nekrashevych and Pete addressed in \cite{NekkyPete2011} the problem whether there exists scale invariant groups which are non co-Hopfian. In particular, they gave examples of scale invariant but non co-Hopfian groups, obtained from a cross-product of groups, some of which are nilpotent, and some are not. We briefly recall a special case of this construction.
Fix $n \geq 1$ and let $G \subset {\rm GL}(n, {\mathbb Z})$. Define
$$H = \{(\vec{n}, g) \mid \vec{n} \in {\mathbb Z}^n ~ , ~ g \in G\}$$
where the product is defined by
$(\vec{n}_1, g_1) \star (\vec{n}_2, g_2) = (\vec{n}_1 + g_1 \cdot \vec{n}_2 , g_1 g_2)$.
Then $H$ is nilpotent if and only if the subgroup $G$ is nilpotent. Choose $p > 1$ and define the monomorphism
$\phi(\vec{n}_1, g_1) = (p \cdot \vec{n}_1, g_1)$. The Rogers and Tollefson Example~\ref{subsec-RT} is the simplest example of this construction.
Suppose that $G = \pi_1(B, b)$ for a closed manifold $B$ which satisfies the strong Borel conjecture. Then let $M$ be the closed manifold which fibers over $B$, with fiber ${\mathbb T}^n$ twisted by the action of $G$. Then $M$ is again aspherical, and satisfies the strong Borel Conjecture. For example, we can take $G$ to be a group which acts freely on ${\mathbb R}^m$ by isometries and let $B = {\mathbb R}^m/G$ be the quotient. Then $M$ is a non co-Hopfian manifold, and the weak solenoid determined by the map $\phi$ is $M$-like. There are many variations on this construction.
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